Lambert, Interval Cycles, Spectrum
-
Upload
jeremie-michael -
Category
Documents
-
view
216 -
download
0
Transcript of Lambert, Interval Cycles, Spectrum
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 1/41
Society for Music Theory
Interval Cycles as Compositional Resources in the Music of Charles IvesAuthor(s): J. Philip LambertSource: Music Theory Spectrum, Vol. 12, No. 1 (Spring, 1990), pp. 43-82Published by: {oupl} on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/746146Accessed: 23-09-2015 16:49 UTC
EFEREN ES
Linked references are available on JSTOR for this article:http://www.jstor.org/stable/746146?seq=1&cid=pdf-reference#references_tab_contents
You may need to log in to JSTOR to access the linked references.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of contentin a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship.For more information about JSTOR, please contact [email protected].
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 2/41
I n t e r v a l
y c l e s
s
ompositional
esources
in
t h e
u s i c
o f
h a r l e s
v e s
J.
Philip
Lambert
I n t e r v a l
y c l e s
s
ompositional
esources
in
t h e
u s i c
o f
h a r l e s
v e s
J.
Philip
Lambert
I n t e r v a l
y c l e s
s
ompositional
esources
in
t h e
u s i c
o f
h a r l e s
v e s
J.
Philip
Lambert
Students
of
the music
of
CharlesIves
over the
past
20
years
have made
convincing
arguments
or
elevating
his musical
quo-
tations to
a
level of structure ar above that
of
the musicalsur-
face. Dennis
Marshall
provided
a seminal definitionof the
is-
sue,
asking
whether borrowed materials are
part
of the
surface-level "manner"or are more
integrated
nto the "sub-
stance"
of a
composition,1
and
analysts
have
since affirmed hat
Ives's
quotations
are more than tidbitsof musical Americana
added to enhance the nationalistic lavor.2Recently, J. Peter
Burkholderhas identified
and defined
specific echniques,
such
as
"modeling"
and
"paraphrasing,"
hat Ives
employs
n incor-
porating
borrowed
material,
supporting
the assertion
that
'Dennis
Marshall,
"Charles Ives's
Quotations:
Manner or
Substance?"
Perspectives f
New Music
6/2
(1968),
45-56;
repr.
n
Perspectives
n American
Composers,
ed.
Benjamin
Boretz and EdwardT. Cone
(New
York:
Norton,
1971),
13-24.
Henry
and
Sidney
Cowell address
he
same ssues n Charles ves
and His
Music
(1955;
reissued
with additional
material,
1969;
repr.
New
York:
OxfordUniversityPress, 1975), 147.
2Some
representative
studies are Gordon
Cyr,
"Intervallic
Structural
Ele-
ments
n Ives's Fourth
Symphony,"
Perspectives f
New Music
9/2-10/1
(1971),
291-303;
Christopher
Ballantine,
"Charles ves and
the
Meaning
of
Quotation
in
Music,"
Musical
Quarterly
5
(1979),
167-184;
Stuart
Feder,
"Decoration
Day:
A
Boyhood Memory
of
Charles
Ives,"
Musical
Quarterly
6
(1980),
234-
261.
For a
typicalopposing
view,
see Kurt
Stone,
"Ives'sFourth
Symphony:
A
Review,"
Musical
Quarterly
2
(1966),
1-16.
Students
of
the music
of
CharlesIves
over the
past
20
years
have made
convincing
arguments
or
elevating
his musical
quo-
tations to
a
level of structure ar above that
of
the musicalsur-
face. Dennis
Marshall
provided
a seminal definitionof the
is-
sue,
asking
whether borrowed materials are
part
of the
surface-level "manner"or are more
integrated
nto the "sub-
stance"
of a
composition,1
and
analysts
have
since affirmed hat
Ives's
quotations
are more than tidbitsof musical Americana
added to enhance the nationalistic lavor.2Recently, J. Peter
Burkholderhas identified
and defined
specific echniques,
such
as
"modeling"
and
"paraphrasing,"
hat Ives
employs
n incor-
porating
borrowed
material,
supporting
the assertion
that
'Dennis
Marshall,
"Charles Ives's
Quotations:
Manner or
Substance?"
Perspectives f
New Music
6/2
(1968),
45-56;
repr.
n
Perspectives
n American
Composers,
ed.
Benjamin
Boretz and EdwardT. Cone
(New
York:
Norton,
1971),
13-24.
Henry
and
Sidney
Cowell address
he
same ssues n Charles ves
and His
Music
(1955;
reissued
with additional
material,
1969;
repr.
New
York:
OxfordUniversityPress, 1975), 147.
2Some
representative
studies are Gordon
Cyr,
"Intervallic
Structural
Ele-
ments
n Ives's Fourth
Symphony,"
Perspectives f
New Music
9/2-10/1
(1971),
291-303;
Christopher
Ballantine,
"Charles ves and
the
Meaning
of
Quotation
in
Music,"
Musical
Quarterly
5
(1979),
167-184;
Stuart
Feder,
"Decoration
Day:
A
Boyhood Memory
of
Charles
Ives,"
Musical
Quarterly
6
(1980),
234-
261.
For a
typicalopposing
view,
see Kurt
Stone,
"Ives'sFourth
Symphony:
A
Review,"
Musical
Quarterly
2
(1966),
1-16.
Students
of
the music
of
CharlesIves
over the
past
20
years
have made
convincing
arguments
or
elevating
his musical
quo-
tations to
a
level of structure ar above that
of
the musicalsur-
face. Dennis
Marshall
provided
a seminal definitionof the
is-
sue,
asking
whether borrowed materials are
part
of the
surface-level "manner"or are more
integrated
nto the "sub-
stance"
of a
composition,1
and
analysts
have
since affirmed hat
Ives's
quotations
are more than tidbitsof musical Americana
added to enhance the nationalistic lavor.2Recently, J. Peter
Burkholderhas identified
and defined
specific echniques,
such
as
"modeling"
and
"paraphrasing,"
hat Ives
employs
n incor-
porating
borrowed
material,
supporting
the assertion
that
'Dennis
Marshall,
"Charles Ives's
Quotations:
Manner or
Substance?"
Perspectives f
New Music
6/2
(1968),
45-56;
repr.
n
Perspectives
n American
Composers,
ed.
Benjamin
Boretz and EdwardT. Cone
(New
York:
Norton,
1971),
13-24.
Henry
and
Sidney
Cowell address
he
same ssues n Charles ves
and His
Music
(1955;
reissued
with additional
material,
1969;
repr.
New
York:
OxfordUniversityPress, 1975), 147.
2Some
representative
studies are Gordon
Cyr,
"Intervallic
Structural
Ele-
ments
n Ives's Fourth
Symphony,"
Perspectives f
New Music
9/2-10/1
(1971),
291-303;
Christopher
Ballantine,
"Charles ves and
the
Meaning
of
Quotation
in
Music,"
Musical
Quarterly
5
(1979),
167-184;
Stuart
Feder,
"Decoration
Day:
A
Boyhood Memory
of
Charles
Ives,"
Musical
Quarterly
6
(1980),
234-
261.
For a
typicalopposing
view,
see Kurt
Stone,
"Ives'sFourth
Symphony:
A
Review,"
Musical
Quarterly
2
(1966),
1-16.
"Ives's
reworking
of
existing
music
is
the
single
most central
technique
in his
process
of
creation."3From
this
perspective,
much of Ives's music
responds favorably
to close
analytical
scrutiny,
and
the
incorporation
and
integration
of
quotations
s
revealed to be a vital
organizing
orce in
a
musical
language
breakingaway
from
tonality.4
While Ives
may
have been
a
supremepractitioner
f musical
borrowing,
however,
he was also a
composer
of
complex
nter-
ests who refused to be confined to a single compositionalpos-
ture.
His methods of
achieving
musical
unity
without
tonality
may
have relied
heavily,
and
successfully,
on
"modeling"pro-
cedures concentratedon
existing
deas,
but
they
also embraced
attempts
at
developing
a more abstract
anguage,
ndependent
of
any
structural ramework hat borrowedmaterial
mightpro-
vide.
Works without
musical
quotations
may
receive some uni-
fying
contributions rom
textual or other extramusical
actors,
but
theymight
also
pursue
a
structural
purity
born of relation-
3J.
Peter
Burkholder,
"
'Quotation'
and
Emulation:
Charles
ves's
Uses
of
His
Models,"
Musical
Quarterly
1
(1985),
20.
See also
Burkholder,
"
'Quota-
tion'
and
Paraphrase
n
Ives's
Second
Symphony,"Nineteenth-Century
Music
11
(1987),
3-25.
4Robert
Morgan
discusses
Ives's
quotations
n this
ight
in
"Rewriting
Mu-
sic
History:
Second
Thoughts
on Ives
and
Varese,"
Musical Newsletter3/1
(January1973),
8-12.
"Ives's
reworking
of
existing
music
is
the
single
most central
technique
in his
process
of
creation."3From
this
perspective,
much of Ives's music
responds favorably
to close
analytical
scrutiny,
and
the
incorporation
and
integration
of
quotations
s
revealed to be a vital
organizing
orce in
a
musical
language
breakingaway
from
tonality.4
While Ives
may
have been
a
supremepractitioner
f musical
borrowing,
however,
he was also a
composer
of
complex
nter-
ests who refused to be confined to a single compositionalpos-
ture.
His methods of
achieving
musical
unity
without
tonality
may
have relied
heavily,
and
successfully,
on
"modeling"pro-
cedures concentratedon
existing
deas,
but
they
also embraced
attempts
at
developing
a more abstract
anguage,
ndependent
of
any
structural ramework hat borrowedmaterial
mightpro-
vide.
Works without
musical
quotations
may
receive some uni-
fying
contributions rom
textual or other extramusical
actors,
but
theymight
also
pursue
a
structural
purity
born of relation-
3J.
Peter
Burkholder,
"
'Quotation'
and
Emulation:
Charles
ves's
Uses
of
His
Models,"
Musical
Quarterly
1
(1985),
20.
See also
Burkholder,
"
'Quota-
tion'
and
Paraphrase
n
Ives's
Second
Symphony,"Nineteenth-Century
Music
11
(1987),
3-25.
4Robert
Morgan
discusses
Ives's
quotations
n this
ight
in
"Rewriting
Mu-
sic
History:
Second
Thoughts
on Ives
and
Varese,"
Musical Newsletter3/1
(January1973),
8-12.
"Ives's
reworking
of
existing
music
is
the
single
most central
technique
in his
process
of
creation."3From
this
perspective,
much of Ives's music
responds favorably
to close
analytical
scrutiny,
and
the
incorporation
and
integration
of
quotations
s
revealed to be a vital
organizing
orce in
a
musical
language
breakingaway
from
tonality.4
While Ives
may
have been
a
supremepractitioner
f musical
borrowing,
however,
he was also a
composer
of
complex
nter-
ests who refused to be confined to a single compositionalpos-
ture.
His methods of
achieving
musical
unity
without
tonality
may
have relied
heavily,
and
successfully,
on
"modeling"pro-
cedures concentratedon
existing
deas,
but
they
also embraced
attempts
at
developing
a more abstract
anguage,
ndependent
of
any
structural ramework hat borrowedmaterial
mightpro-
vide.
Works without
musical
quotations
may
receive some uni-
fying
contributions rom
textual or other extramusical
actors,
but
theymight
also
pursue
a
structural
purity
born of relation-
3J.
Peter
Burkholder,
"
'Quotation'
and
Emulation:
Charles
ves's
Uses
of
His
Models,"
Musical
Quarterly
1
(1985),
20.
See also
Burkholder,
"
'Quota-
tion'
and
Paraphrase
n
Ives's
Second
Symphony,"Nineteenth-Century
Music
11
(1987),
3-25.
4Robert
Morgan
discusses
Ives's
quotations
n this
ight
in
"Rewriting
Mu-
sic
History:
Second
Thoughts
on Ives
and
Varese,"
Musical Newsletter3/1
(January1973),
8-12.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 3/41
44
Music
Theory Spectrum
4
Music
Theory Spectrum
4
Music
Theory Spectrum
ships
defined
contextually
within
the
available resources.
While a single broadlydefined process may characterize he
creative
evolution of an
Ives
composition,
the
organizational
problems posed
by
the
individual
approaches
and solutions n
variousworks
are
numerousand
diverse.5
Indeed,
it is in
the
works without
quotations
where
Ives's
compositional
objectives
are most
overt,
his search or
organi-
zationalalternativesmost
consistently
apparent.
These works
generallyseparate
hemselvesfrom
the main
body
of his
music,
distinguished by
their
searches
for
components
of
abstract
structureand by their attention to technical detail. Ives gives
many
of them
titles such
as
"Study"
or
"Exercise,"
making
clear that his
compositional purpose
is to work
out technical
problems.6
He also refers
to works of
this
type
as "memos in
notes,"
a
phrase
that
alludes to their
unrefined tate as
well
as
the
very private
nature of
his
perspective
on them.7 These
pieces
are,
in
fact,
"experiments"
n
musical
organization-
musical
ncarnationsof
ideas about
structure hat
may
receive
theirformulation
only
in
the courseof
composing
he work. Al-
togetherIves'sexperimentalworks form a distinctive ubset of
his
music,
serving
as a
forum for the isolation
and
exploration
of
specific
echnical ssues.8
5Burkholder eels that
Ives's
"working
out
of technical
problems
and his
creationof musical
analogues
o
texts and to
programmatic
onceptions
ollow
the same
pattern
as his
elaborationof
borrowed
material,"
and that
the
process
common to these activities would be
the basis of a
"unifiedview
of
Ives's
ap-
proach
o
composition"
"
'Quotation'
and
Emulation,"20,
n.
36).
Testing
or
this
hyphothesis
will come
as each
compositional
area s
subjected
o
thorough
investigation.
6For
example,
the
piano
pieces
that fall in this
category
are entitled"Stud-
ies."
Similarly,
the
song Soliloquy
is
entitled
"A
Study
in 7ths and Other
Things."
Also
typical
s the
subtitle to
Chromdtimelodtune:
Ear-Study
aural
& mental
exercise )."
7Charles
E.
Ives, Memos,
edited and with
appendicesby
John
Kirkpatrick
(New
York:
Norton,
1972),
64.
8Ives's
memoirs describe an
active influenceof his
father on these
experi-
mental
attitudes,
although
the
reliability
of
Ives's accounthas been
questioned
ships
defined
contextually
within
the
available resources.
While a single broadlydefined process may characterize he
creative
evolution of an
Ives
composition,
the
organizational
problems posed
by
the
individual
approaches
and solutions n
variousworks
are
numerousand
diverse.5
Indeed,
it is in
the
works without
quotations
where
Ives's
compositional
objectives
are most
overt,
his search or
organi-
zationalalternativesmost
consistently
apparent.
These works
generallyseparate
hemselvesfrom
the main
body
of his
music,
distinguished by
their
searches
for
components
of
abstract
structureand by their attention to technical detail. Ives gives
many
of them
titles such
as
"Study"
or
"Exercise,"
making
clear that his
compositional purpose
is to work
out technical
problems.6
He also refers
to works of
this
type
as "memos in
notes,"
a
phrase
that
alludes to their
unrefined tate as
well
as
the
very private
nature of
his
perspective
on them.7 These
pieces
are,
in
fact,
"experiments"
n
musical
organization-
musical
ncarnationsof
ideas about
structure hat
may
receive
theirformulation
only
in
the courseof
composing
he work. Al-
togetherIves'sexperimentalworks form a distinctive ubset of
his
music,
serving
as a
forum for the isolation
and
exploration
of
specific
echnical ssues.8
5Burkholder eels that
Ives's
"working
out
of technical
problems
and his
creationof musical
analogues
o
texts and to
programmatic
onceptions
ollow
the same
pattern
as his
elaborationof
borrowed
material,"
and that
the
process
common to these activities would be
the basis of a
"unifiedview
of
Ives's
ap-
proach
o
composition"
"
'Quotation'
and
Emulation,"20,
n.
36).
Testing
or
this
hyphothesis
will come
as each
compositional
area s
subjected
o
thorough
investigation.
6For
example,
the
piano
pieces
that fall in this
category
are entitled"Stud-
ies."
Similarly,
the
song Soliloquy
is
entitled
"A
Study
in 7ths and Other
Things."
Also
typical
s the
subtitle to
Chromdtimelodtune:
Ear-Study
aural
& mental
exercise )."
7Charles
E.
Ives, Memos,
edited and with
appendicesby
John
Kirkpatrick
(New
York:
Norton,
1972),
64.
8Ives's
memoirs describe an
active influenceof his
father on these
experi-
mental
attitudes,
although
the
reliability
of
Ives's accounthas been
questioned
ships
defined
contextually
within
the
available resources.
While a single broadlydefined process may characterize he
creative
evolution of an
Ives
composition,
the
organizational
problems posed
by
the
individual
approaches
and solutions n
variousworks
are
numerousand
diverse.5
Indeed,
it is in
the
works without
quotations
where
Ives's
compositional
objectives
are most
overt,
his search or
organi-
zationalalternativesmost
consistently
apparent.
These works
generallyseparate
hemselvesfrom
the main
body
of his
music,
distinguished by
their
searches
for
components
of
abstract
structureand by their attention to technical detail. Ives gives
many
of them
titles such
as
"Study"
or
"Exercise,"
making
clear that his
compositional purpose
is to work
out technical
problems.6
He also refers
to works of
this
type
as "memos in
notes,"
a
phrase
that
alludes to their
unrefined tate as
well
as
the
very private
nature of
his
perspective
on them.7 These
pieces
are,
in
fact,
"experiments"
n
musical
organization-
musical
ncarnationsof
ideas about
structure hat
may
receive
theirformulation
only
in
the courseof
composing
he work. Al-
togetherIves'sexperimentalworks form a distinctive ubset of
his
music,
serving
as a
forum for the isolation
and
exploration
of
specific
echnical ssues.8
5Burkholder eels that
Ives's
"working
out
of technical
problems
and his
creationof musical
analogues
o
texts and to
programmatic
onceptions
ollow
the same
pattern
as his
elaborationof
borrowed
material,"
and that
the
process
common to these activities would be
the basis of a
"unifiedview
of
Ives's
ap-
proach
o
composition"
"
'Quotation'
and
Emulation,"20,
n.
36).
Testing
or
this
hyphothesis
will come
as each
compositional
area s
subjected
o
thorough
investigation.
6For
example,
the
piano
pieces
that fall in this
category
are entitled"Stud-
ies."
Similarly,
the
song Soliloquy
is
entitled
"A
Study
in 7ths and Other
Things."
Also
typical
s the
subtitle to
Chromdtimelodtune:
Ear-Study
aural
& mental
exercise )."
7Charles
E.
Ives, Memos,
edited and with
appendicesby
John
Kirkpatrick
(New
York:
Norton,
1972),
64.
8Ives's
memoirs describe an
active influenceof his
father on these
experi-
mental
attitudes,
although
the
reliability
of
Ives's accounthas been
questioned
Central
to the issues under
investigation
n Ives's
experi-
mentation is his focus on intervallicconstructionsas primary
structural
components.
Even in his earliest
compositional
ef-
forts,
the interval
provided
he means
for
constructing
northo-
dox but
orderly
melodic
and harmonic
structures,
often based
on standard
types
of formations
employing
atypical
units of
combination.
Ives
recalls,
for
example,
his father's
suggestion
that
"If one can use
chords
of
3rds
and make
them mean
some-
thing, why
not chordsof 4ths?"9 ves
took these
and similar
suggestions
o
heart,
leading
to his well-known
experiments
n
"quartal"harmony, such as the piano accompaniment o the
song
"The
Cage,"
and
other instances
of intervallic satura-
tion.10
Analysts
have also
noted more
complex
approaches
o
intervallic
tructure,
such
as a
system
of
permutations
f a cer-
tain intervallic
profilel
and
a
process
of
shifting
ocusfromone
interval
or
group
of intervalsto
another over the course of
a
by Maynard
Solomon,
"Charles ves:
Some
Questions
of
Veracity,"
Journal
of
the American
MusicologicalSociety
40
(1987),
443-470.
Solomon
speculates
that Ives
exaggerated
his father's nfluence
as
part
of a
mourningprocess
that
included
an idealizationof his father'smusical nnovationsand a realization
of
some of his father's musical
aspirations.
Solomon
discusses
mportant
ssues
about
the Ives
biography,
but his views remainas
questions
that are left
unan-
swered;
any
substantiationof his thesis will come
only
after extensive
non-
speculativescrutiny
of the availableevidence.
The conventionalview of Ives's debt to his father's
experimental
attitudes
is
described
by
J. Peter
Burkholder,
Charles
ves: The Ideas
behind he Music
(New
Haven
and
London:
Yale
UniversityPress, 1985),
45-50. The author
makesa useful distinctionbetween
"experimental"
nd
"concert"
music,
em-
phasizing
the
private
nature of the former as
opposed
to an attitude
toward
concert
works that
encourages
revision
and
refinement,
ostensibly eading
o a
public presentation.
9Ives,Memos,
140.
10Soliloquy,
he
"Study
n
7ths,"
is another
example.
"Cyr,
"Intervallic
Structural
Elements
in Ives's Fourth
Symphony."
Cyr's
observations
establish
a
link between
the intervallic
constructions
and
the
structure
of
many
of the tunes
quoted
in
the
symphony.
Central
to the issues under
investigation
n Ives's
experi-
mentation is his focus on intervallicconstructionsas primary
structural
components.
Even in his earliest
compositional
ef-
forts,
the interval
provided
he means
for
constructing
northo-
dox but
orderly
melodic
and harmonic
structures,
often based
on standard
types
of formations
employing
atypical
units of
combination.
Ives
recalls,
for
example,
his father's
suggestion
that
"If one can use
chords
of
3rds
and make
them mean
some-
thing, why
not chordsof 4ths?"9 ves
took these
and similar
suggestions
o
heart,
leading
to his well-known
experiments
n
"quartal"harmony, such as the piano accompaniment o the
song
"The
Cage,"
and
other instances
of intervallic satura-
tion.10
Analysts
have also
noted more
complex
approaches
o
intervallic
tructure,
such
as a
system
of
permutations
f a cer-
tain intervallic
profilel
and
a
process
of
shifting
ocusfromone
interval
or
group
of intervalsto
another over the course of
a
by Maynard
Solomon,
"Charles ves:
Some
Questions
of
Veracity,"
Journal
of
the American
MusicologicalSociety
40
(1987),
443-470.
Solomon
speculates
that Ives
exaggerated
his father's nfluence
as
part
of a
mourningprocess
that
included
an idealizationof his father'smusical nnovationsand a realization
of
some of his father's musical
aspirations.
Solomon
discusses
mportant
ssues
about
the Ives
biography,
but his views remainas
questions
that are left
unan-
swered;
any
substantiationof his thesis will come
only
after extensive
non-
speculativescrutiny
of the availableevidence.
The conventionalview of Ives's debt to his father's
experimental
attitudes
is
described
by
J. Peter
Burkholder,
Charles
ves: The Ideas
behind he Music
(New
Haven
and
London:
Yale
UniversityPress, 1985),
45-50. The author
makesa useful distinctionbetween
"experimental"
nd
"concert"
music,
em-
phasizing
the
private
nature of the former as
opposed
to an attitude
toward
concert
works that
encourages
revision
and
refinement,
ostensibly eading
o a
public presentation.
9Ives,Memos,
140.
10Soliloquy,
he
"Study
n
7ths,"
is another
example.
"Cyr,
"Intervallic
Structural
Elements
in Ives's Fourth
Symphony."
Cyr's
observations
establish
a
link between
the intervallic
constructions
and
the
structure
of
many
of the tunes
quoted
in
the
symphony.
Central
to the issues under
investigation
n Ives's
experi-
mentation is his focus on intervallicconstructionsas primary
structural
components.
Even in his earliest
compositional
ef-
forts,
the interval
provided
he means
for
constructing
northo-
dox but
orderly
melodic
and harmonic
structures,
often based
on standard
types
of formations
employing
atypical
units of
combination.
Ives
recalls,
for
example,
his father's
suggestion
that
"If one can use
chords
of
3rds
and make
them mean
some-
thing, why
not chordsof 4ths?"9 ves
took these
and similar
suggestions
o
heart,
leading
to his well-known
experiments
n
"quartal"harmony, such as the piano accompaniment o the
song
"The
Cage,"
and
other instances
of intervallic satura-
tion.10
Analysts
have also
noted more
complex
approaches
o
intervallic
tructure,
such
as a
system
of
permutations
f a cer-
tain intervallic
profilel
and
a
process
of
shifting
ocusfromone
interval
or
group
of intervalsto
another over the course of
a
by Maynard
Solomon,
"Charles ves:
Some
Questions
of
Veracity,"
Journal
of
the American
MusicologicalSociety
40
(1987),
443-470.
Solomon
speculates
that Ives
exaggerated
his father's nfluence
as
part
of a
mourningprocess
that
included
an idealizationof his father'smusical nnovationsand a realization
of
some of his father's musical
aspirations.
Solomon
discusses
mportant
ssues
about
the Ives
biography,
but his views remainas
questions
that are left
unan-
swered;
any
substantiationof his thesis will come
only
after extensive
non-
speculativescrutiny
of the availableevidence.
The conventionalview of Ives's debt to his father's
experimental
attitudes
is
described
by
J. Peter
Burkholder,
Charles
ves: The Ideas
behind he Music
(New
Haven
and
London:
Yale
UniversityPress, 1985),
45-50. The author
makesa useful distinctionbetween
"experimental"
nd
"concert"
music,
em-
phasizing
the
private
nature of the former as
opposed
to an attitude
toward
concert
works that
encourages
revision
and
refinement,
ostensibly eading
o a
public presentation.
9Ives,Memos,
140.
10Soliloquy,
he
"Study
n
7ths,"
is another
example.
"Cyr,
"Intervallic
Structural
Elements
in Ives's Fourth
Symphony."
Cyr's
observations
establish
a
link between
the intervallic
constructions
and
the
structure
of
many
of the tunes
quoted
in
the
symphony.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 4/41
Interval
ycles
as
Compositional
esources
45nterval
ycles
as
Compositional
esources
45nterval
ycles
as
Compositional
esources
45
work.12n
maintaining
an active
nterest n alternatives
o struc-
tural conventions, Ives took the interval as his primaryre-
source,
and the evolution
of his
experimentation
centers
in
large part
on this
element
of
musical
structure.
This
orientation,
combined
with an
ever-present
oncern
or
maximizingpitch-class
variety,
develops finally
nto
an aware-
ness
of
the size
and characterof
repetitive
ntervallic
tructures,
viewed
as
cycles,
and their
potential
for
compositional
xploita-
tion.
Many
of his
experiments,
including
those
exhibiting
higher degrees
of
technical
sophistication,
are centered on
in-
tervalcyclesas contributors o acontextuallydefinedharmonic
language
and a unified
catalogue
of
developmental
devices.
Ives thus allies
himself,
spiritually,
with his
contemporaries:
Berg's
"master
array"
of
interval
cycles,
as described
by
George
Perle,
and the role of
cyclic
structures
in
early
Stravinsky
as
explained
by
Elliott
Antokoletz,
demonstrate
similar
approaches
o intervallicconstruction.13 s an alterna-
tive
to traditional
compositional
resources,
the interval
cycles
12Nors
S.
Josephson,
"Charles Ives: Intervallische Permutationen
im
Spatwerk,"
Zeitschrift ur
Musiktheorie /2
(1978),
27-33.
Josephson's
study
encompasses
works of various
types
from all
periods
of Ives's
compositional
career.
13George
Perle,
"Berg's
Master
Array
of the Interval
Cycles,"
Musical
Quarterly
3
(1977),
1-30;
Elliott
Antokoletz,
"Interval
Cycles
n
Stravinsky's
Early
Ballets,"
Journal
of
the
American
MusicologicalSociety
34
(1986),
578-
614;
Marianne
Kielian-Gilbert,
"Relationships
of
Symmetrical
Pitch-Class
Sets and
Stravinsky's
Metaphor
of
Polarity," Perspectivesof
New Music 21
(1982-83), 209-240; MenachemZur, "TonalAmbiguitiesas a Constructive
Force
in
the
Language
of
Stravinsky,"
Musical
Quarterly
8
(1982),
516-526.
Cyclic
ntervallic
repetitions
are also central
to
the studies of Bart6k
by
Erno
Lendvai,
The
Workshop of
Bart6k and
Kodaly
(Budapest:
Editio
Musica,
1983)
and
by
Elliott
Antokoletz,
TheMusic
of
Bela Bart6k:A
Studyof
Tonality
and
Progression
n
Twentieth-Century
Music
(Berkeley: University
of
Califor-
nia
Press,
1984).
Another
composer
from this
period
who workedwith nterval
cycles
is Karol
Szymanowski
1882-1937);
see Ann K.
McNamee,
"Bitonality,
Mode,
and Interval in the Music of Karol
Szymanowski,"
Journal
of
Music
Theory
29
(1985),
61-84.
work.12n
maintaining
an active
nterest n alternatives
o struc-
tural conventions, Ives took the interval as his primaryre-
source,
and the evolution
of his
experimentation
centers
in
large part
on this
element
of
musical
structure.
This
orientation,
combined
with an
ever-present
oncern
or
maximizingpitch-class
variety,
develops finally
nto
an aware-
ness
of
the size
and characterof
repetitive
ntervallic
tructures,
viewed
as
cycles,
and their
potential
for
compositional
xploita-
tion.
Many
of his
experiments,
including
those
exhibiting
higher degrees
of
technical
sophistication,
are centered on
in-
tervalcyclesas contributors o acontextuallydefinedharmonic
language
and a unified
catalogue
of
developmental
devices.
Ives thus allies
himself,
spiritually,
with his
contemporaries:
Berg's
"master
array"
of
interval
cycles,
as described
by
George
Perle,
and the role of
cyclic
structures
in
early
Stravinsky
as
explained
by
Elliott
Antokoletz,
demonstrate
similar
approaches
o intervallicconstruction.13 s an alterna-
tive
to traditional
compositional
resources,
the interval
cycles
12Nors
S.
Josephson,
"Charles Ives: Intervallische Permutationen
im
Spatwerk,"
Zeitschrift ur
Musiktheorie /2
(1978),
27-33.
Josephson's
study
encompasses
works of various
types
from all
periods
of Ives's
compositional
career.
13George
Perle,
"Berg's
Master
Array
of the Interval
Cycles,"
Musical
Quarterly
3
(1977),
1-30;
Elliott
Antokoletz,
"Interval
Cycles
n
Stravinsky's
Early
Ballets,"
Journal
of
the
American
MusicologicalSociety
34
(1986),
578-
614;
Marianne
Kielian-Gilbert,
"Relationships
of
Symmetrical
Pitch-Class
Sets and
Stravinsky's
Metaphor
of
Polarity," Perspectivesof
New Music 21
(1982-83), 209-240; MenachemZur, "TonalAmbiguitiesas a Constructive
Force
in
the
Language
of
Stravinsky,"
Musical
Quarterly
8
(1982),
516-526.
Cyclic
ntervallic
repetitions
are also central
to
the studies of Bart6k
by
Erno
Lendvai,
The
Workshop of
Bart6k and
Kodaly
(Budapest:
Editio
Musica,
1983)
and
by
Elliott
Antokoletz,
TheMusic
of
Bela Bart6k:A
Studyof
Tonality
and
Progression
n
Twentieth-Century
Music
(Berkeley: University
of
Califor-
nia
Press,
1984).
Another
composer
from this
period
who workedwith nterval
cycles
is Karol
Szymanowski
1882-1937);
see Ann K.
McNamee,
"Bitonality,
Mode,
and Interval in the Music of Karol
Szymanowski,"
Journal
of
Music
Theory
29
(1985),
61-84.
work.12n
maintaining
an active
nterest n alternatives
o struc-
tural conventions, Ives took the interval as his primaryre-
source,
and the evolution
of his
experimentation
centers
in
large part
on this
element
of
musical
structure.
This
orientation,
combined
with an
ever-present
oncern
or
maximizingpitch-class
variety,
develops finally
nto
an aware-
ness
of
the size
and characterof
repetitive
ntervallic
tructures,
viewed
as
cycles,
and their
potential
for
compositional
xploita-
tion.
Many
of his
experiments,
including
those
exhibiting
higher degrees
of
technical
sophistication,
are centered on
in-
tervalcyclesas contributors o acontextuallydefinedharmonic
language
and a unified
catalogue
of
developmental
devices.
Ives thus allies
himself,
spiritually,
with his
contemporaries:
Berg's
"master
array"
of
interval
cycles,
as described
by
George
Perle,
and the role of
cyclic
structures
in
early
Stravinsky
as
explained
by
Elliott
Antokoletz,
demonstrate
similar
approaches
o intervallicconstruction.13 s an alterna-
tive
to traditional
compositional
resources,
the interval
cycles
12Nors
S.
Josephson,
"Charles Ives: Intervallische Permutationen
im
Spatwerk,"
Zeitschrift ur
Musiktheorie /2
(1978),
27-33.
Josephson's
study
encompasses
works of various
types
from all
periods
of Ives's
compositional
career.
13George
Perle,
"Berg's
Master
Array
of the Interval
Cycles,"
Musical
Quarterly
3
(1977),
1-30;
Elliott
Antokoletz,
"Interval
Cycles
n
Stravinsky's
Early
Ballets,"
Journal
of
the
American
MusicologicalSociety
34
(1986),
578-
614;
Marianne
Kielian-Gilbert,
"Relationships
of
Symmetrical
Pitch-Class
Sets and
Stravinsky's
Metaphor
of
Polarity," Perspectivesof
New Music 21
(1982-83), 209-240; MenachemZur, "TonalAmbiguitiesas a Constructive
Force
in
the
Language
of
Stravinsky,"
Musical
Quarterly
8
(1982),
516-526.
Cyclic
ntervallic
repetitions
are also central
to
the studies of Bart6k
by
Erno
Lendvai,
The
Workshop of
Bart6k and
Kodaly
(Budapest:
Editio
Musica,
1983)
and
by
Elliott
Antokoletz,
TheMusic
of
Bela Bart6k:A
Studyof
Tonality
and
Progression
n
Twentieth-Century
Music
(Berkeley: University
of
Califor-
nia
Press,
1984).
Another
composer
from this
period
who workedwith nterval
cycles
is Karol
Szymanowski
1882-1937);
see Ann K.
McNamee,
"Bitonality,
Mode,
and Interval in the Music of Karol
Szymanowski,"
Journal
of
Music
Theory
29
(1985),
61-84.
form a viable
system
of
internally
defined
pitch
relationships
and suggestfertile means for musicaldevelopmentand trans-
formation.
The
description
of
Ives's
incorporation
of interval
cycles
in
the
following
pages
citesevidencefrom musicwritten
at various
stages
in his
composing period, primarily
ncluding
works that
are of the
experimental
ype, though
not exclusive
to this cate-
gory.
The discussion
centers first on
cyclic repetitions
of
single
intervals,
and
then
on
alternatingrepetitions
of
two
different
intervals,
or combination
ycles.
Each
topic
includes
definitions
of terminologyand establishmentof analyticalmethodology.A
finalareaof discussion llustrates
particular
ompositional
ap-
plications
of
cycles, generally
n
larger
musicalcontexts.
SINGLE-INTERVAL
YCLES. linear
pitch-class
(pc) presenta-
tion translates
to a
segment
notated
as
integers(C
=
0) sepa-
rated
by
commas within
angled
brackets.14 he
adjacent
nter-
vals,
or ordered
pitch-class
intervals,
form the INT
of a
segment,
notated as
integers (1-11) separated
by
dashes and
enclosedinangledbrackets.15 xample1illustratesheapplica-
tion
of
these notations to
a
violin line from Ives's
Largo
Riso-
luto No.
1
(1906).16
An INT
containing
exclusive
repetitions
of
14The erm
segment
s used here as defined
n Robert D.
Morris,
Composi-
tion
with
Pitch
Classes:
A
Theoryof Compositional
Design (New
Haven
and
London:
Yale
University
Press,
1987),
37,
64.
Morris defines
segments
of
pitches
(pseg)
and
pitch
classes
(pcseg).
A
segment
s ordered
by
definition.
15Anordered
pc
interval,
or directed nterval n Milton
Babbitt's erminol-
ogy, is calculatedby subtracting mod 12) the firstpc from the second. See
Morris,
62
and John
Rahn,
Basic Atonal
Theory New
York:
Longman,
1980),
25-27.
The term INT
is defined n
Morris,
107.
'6Dates
of
composition
are those
given
by
John
Kirkpatrick
n The
New
Grove
Dictionaryof
Music and
Musicians,
6th
ed.,
s.v.
"Ives,
Charles
E."
These
dates are
primarily
based
on
evidence
from
Ives's scoresand
memoirs,
although
the
reliability
of these sources
has been
questioned
by
Solomon
("Questions
of
Veracity").
Until
Solomon's
questions
and
speculations
are
subjected
to further
analysis,
Kirkpatrick's
ates
represent
he most
accurate
informationavailable.
form a viable
system
of
internally
defined
pitch
relationships
and suggestfertile means for musicaldevelopmentand trans-
formation.
The
description
of
Ives's
incorporation
of interval
cycles
in
the
following
pages
citesevidencefrom musicwritten
at various
stages
in his
composing period, primarily
ncluding
works that
are of the
experimental
ype, though
not exclusive
to this cate-
gory.
The discussion
centers first on
cyclic repetitions
of
single
intervals,
and
then
on
alternatingrepetitions
of
two
different
intervals,
or combination
ycles.
Each
topic
includes
definitions
of terminologyand establishmentof analyticalmethodology.A
finalareaof discussion llustrates
particular
ompositional
ap-
plications
of
cycles, generally
n
larger
musicalcontexts.
SINGLE-INTERVAL
YCLES. linear
pitch-class
(pc) presenta-
tion translates
to a
segment
notated
as
integers(C
=
0) sepa-
rated
by
commas within
angled
brackets.14 he
adjacent
nter-
vals,
or ordered
pitch-class
intervals,
form the INT
of a
segment,
notated as
integers (1-11) separated
by
dashes and
enclosedinangledbrackets.15 xample1illustratesheapplica-
tion
of
these notations to
a
violin line from Ives's
Largo
Riso-
luto No.
1
(1906).16
An INT
containing
exclusive
repetitions
of
14The erm
segment
s used here as defined
n Robert D.
Morris,
Composi-
tion
with
Pitch
Classes:
A
Theoryof Compositional
Design (New
Haven
and
London:
Yale
University
Press,
1987),
37,
64.
Morris defines
segments
of
pitches
(pseg)
and
pitch
classes
(pcseg).
A
segment
s ordered
by
definition.
15Anordered
pc
interval,
or directed nterval n Milton
Babbitt's erminol-
ogy, is calculatedby subtracting mod 12) the firstpc from the second. See
Morris,
62
and John
Rahn,
Basic Atonal
Theory New
York:
Longman,
1980),
25-27.
The term INT
is defined n
Morris,
107.
'6Dates
of
composition
are those
given
by
John
Kirkpatrick
n The
New
Grove
Dictionaryof
Music and
Musicians,
6th
ed.,
s.v.
"Ives,
Charles
E."
These
dates are
primarily
based
on
evidence
from
Ives's scoresand
memoirs,
although
the
reliability
of these sources
has been
questioned
by
Solomon
("Questions
of
Veracity").
Until
Solomon's
questions
and
speculations
are
subjected
to further
analysis,
Kirkpatrick's
ates
represent
he most
accurate
informationavailable.
form a viable
system
of
internally
defined
pitch
relationships
and suggestfertile means for musicaldevelopmentand trans-
formation.
The
description
of
Ives's
incorporation
of interval
cycles
in
the
following
pages
citesevidencefrom musicwritten
at various
stages
in his
composing period, primarily
ncluding
works that
are of the
experimental
ype, though
not exclusive
to this cate-
gory.
The discussion
centers first on
cyclic repetitions
of
single
intervals,
and
then
on
alternatingrepetitions
of
two
different
intervals,
or combination
ycles.
Each
topic
includes
definitions
of terminologyand establishmentof analyticalmethodology.A
finalareaof discussion llustrates
particular
ompositional
ap-
plications
of
cycles, generally
n
larger
musicalcontexts.
SINGLE-INTERVAL
YCLES. linear
pitch-class
(pc) presenta-
tion translates
to a
segment
notated
as
integers(C
=
0) sepa-
rated
by
commas within
angled
brackets.14 he
adjacent
nter-
vals,
or ordered
pitch-class
intervals,
form the INT
of a
segment,
notated as
integers (1-11) separated
by
dashes and
enclosedinangledbrackets.15 xample1illustratesheapplica-
tion
of
these notations to
a
violin line from Ives's
Largo
Riso-
luto No.
1
(1906).16
An INT
containing
exclusive
repetitions
of
14The erm
segment
s used here as defined
n Robert D.
Morris,
Composi-
tion
with
Pitch
Classes:
A
Theoryof Compositional
Design (New
Haven
and
London:
Yale
University
Press,
1987),
37,
64.
Morris defines
segments
of
pitches
(pseg)
and
pitch
classes
(pcseg).
A
segment
s ordered
by
definition.
15Anordered
pc
interval,
or directed nterval n Milton
Babbitt's erminol-
ogy, is calculatedby subtracting mod 12) the firstpc from the second. See
Morris,
62
and John
Rahn,
Basic Atonal
Theory New
York:
Longman,
1980),
25-27.
The term INT
is defined n
Morris,
107.
'6Dates
of
composition
are those
given
by
John
Kirkpatrick
n The
New
Grove
Dictionaryof
Music and
Musicians,
6th
ed.,
s.v.
"Ives,
Charles
E."
These
dates are
primarily
based
on
evidence
from
Ives's scoresand
memoirs,
although
the
reliability
of these sources
has been
questioned
by
Solomon
("Questions
of
Veracity").
Until
Solomon's
questions
and
speculations
are
subjected
to further
analysis,
Kirkpatrick's
ates
represent
he most
accurate
informationavailable.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 5/41
46 Music
Theory
Spectrum
6 Music
Theory
Spectrum
6 Music
Theory
Spectrum
Example
1.
Largo
Risoluto No.
1,
mm.
24-25,
first violin.
xample
1.
Largo
Risoluto No.
1,
mm.
24-25,
first violin.
xample
1.
Largo
Risoluto No.
1,
mm.
24-25,
first violin.
3
A I4
-I-
>I
,>
3
A I4
-I-
>I
,>
3
A I4
-I-
>I
,>
I
J
r
^
)-
^
6.-
I
,^J
r
^
)-
^
6.-
I
,^J
r
^
)-
^
6.-
I
,^
2555
Y
L
I i
3
3I
pc <4,
9, 2,
7,
0, 5, 10,
3, 8, 1,
6,
11>
<4,...
INT
<5-5-5-5-5-5-5-5-5-5-5>
a
single
interval
generates
a
segment
that is
cyclic
n that
it re-
turnsto its
point
of
origin;
the
repetition
of interval5
in Exam-
ple
1is aninterval-5
ycle
that
generates
he returnof
pc
4 atthe
downbeat
of m.
25.17
The numberof
pcs generatedby
the
cyclic
repetition
of a
given
interval
s the
cardinality,
r
CARD
of
the
cycle,
which totals
12 in
Example
1,
a
completion
of the
aggre-
gate.
The familiar
cyclicpitch-class
tructures
have CARDs of
12
(interval-1,
-5,
-7,
or -11
cycles),
6
(interval-2
or
-10),
4
(interval-3
or
-9),
3
(interval-4
or
-8),
or 2
(interval-6).
The or-
derof
presentation
of
the
pcs generated
by
a
cycle
may
be indi-
cated
by
italicized
integers representing
order
position
(op),
extending
from0 for the initial
pc
to CARD-1forthe last.
Ives's interest in
cyclic
structures
originates
with the
in-
stancesof chromaticand whole-tone scales
appearing
n
music
from various
stages
of his life.
In his
Memos,
he describes
an
early,
father-influenced xercise
in
which he
played
the chro-
matic
scale
with octave
displacements,
creating
"wide
umps
n
the
counterpoint
and lines."18 ves'schromatic cale
presenta-
Notes in brackets in musical excerpts are correctionsof misprints n the
published
score,
confirmed
through comparison
with
the
original
manuscript
(housed
in the
Ives Collection
at
Yale).
17Morris efines a
cycle (pcyc
and
pccyc)
more
generally
to
apply
to
any
reiterative
segment,
regardless
of whether an intervallic
repetition
occurs
in
the
INT
(pp.
37,
65).
The
resulting
ntervallic
uccession s the
cyclic
INT,
or
CINT(pp.
40,
107).
18Memos,
44
(and
musical
example). Kirkpatrick
ecalls that
"Ives told
George
Roberts
[one
of
his
copyists]
that
his father
had him do
chromatic
scales
with each
interval
a
minor 9th"
(Memos,
44,
n.
5).
Y
L
I i
3
3I
pc <4,
9, 2,
7,
0, 5, 10,
3, 8, 1,
6,
11>
<4,...
INT
<5-5-5-5-5-5-5-5-5-5-5>
a
single
interval
generates
a
segment
that is
cyclic
n that
it re-
turnsto its
point
of
origin;
the
repetition
of interval5
in Exam-
ple
1is aninterval-5
ycle
that
generates
he returnof
pc
4 atthe
downbeat
of m.
25.17
The numberof
pcs generatedby
the
cyclic
repetition
of a
given
interval
s the
cardinality,
r
CARD
of
the
cycle,
which totals
12 in
Example
1,
a
completion
of the
aggre-
gate.
The familiar
cyclicpitch-class
tructures
have CARDs of
12
(interval-1,
-5,
-7,
or -11
cycles),
6
(interval-2
or
-10),
4
(interval-3
or
-9),
3
(interval-4
or
-8),
or 2
(interval-6).
The or-
derof
presentation
of
the
pcs generated
by
a
cycle
may
be indi-
cated
by
italicized
integers representing
order
position
(op),
extending
from0 for the initial
pc
to CARD-1forthe last.
Ives's interest in
cyclic
structures
originates
with the
in-
stancesof chromaticand whole-tone scales
appearing
n
music
from various
stages
of his life.
In his
Memos,
he describes
an
early,
father-influenced xercise
in
which he
played
the chro-
matic
scale
with octave
displacements,
creating
"wide
umps
n
the
counterpoint
and lines."18 ves'schromatic cale
presenta-
Notes in brackets in musical excerpts are correctionsof misprints n the
published
score,
confirmed
through comparison
with
the
original
manuscript
(housed
in the
Ives Collection
at
Yale).
17Morris efines a
cycle (pcyc
and
pccyc)
more
generally
to
apply
to
any
reiterative
segment,
regardless
of whether an intervallic
repetition
occurs
in
the
INT
(pp.
37,
65).
The
resulting
ntervallic
uccession s the
cyclic
INT,
or
CINT(pp.
40,
107).
18Memos,
44
(and
musical
example). Kirkpatrick
ecalls that
"Ives told
George
Roberts
[one
of
his
copyists]
that
his father
had him do
chromatic
scales
with each
interval
a
minor 9th"
(Memos,
44,
n.
5).
Y
L
I i
3
3I
pc <4,
9, 2,
7,
0, 5, 10,
3, 8, 1,
6,
11>
<4,...
INT
<5-5-5-5-5-5-5-5-5-5-5>
a
single
interval
generates
a
segment
that is
cyclic
n that
it re-
turnsto its
point
of
origin;
the
repetition
of interval5
in Exam-
ple
1is aninterval-5
ycle
that
generates
he returnof
pc
4 atthe
downbeat
of m.
25.17
The numberof
pcs generatedby
the
cyclic
repetition
of a
given
interval
s the
cardinality,
r
CARD
of
the
cycle,
which totals
12 in
Example
1,
a
completion
of the
aggre-
gate.
The familiar
cyclicpitch-class
tructures
have CARDs of
12
(interval-1,
-5,
-7,
or -11
cycles),
6
(interval-2
or
-10),
4
(interval-3
or
-9),
3
(interval-4
or
-8),
or 2
(interval-6).
The or-
derof
presentation
of
the
pcs generated
by
a
cycle
may
be indi-
cated
by
italicized
integers representing
order
position
(op),
extending
from0 for the initial
pc
to CARD-1forthe last.
Ives's interest in
cyclic
structures
originates
with the
in-
stancesof chromaticand whole-tone scales
appearing
n
music
from various
stages
of his life.
In his
Memos,
he describes
an
early,
father-influenced xercise
in
which he
played
the chro-
matic
scale
with octave
displacements,
creating
"wide
umps
n
the
counterpoint
and lines."18 ves'schromatic cale
presenta-
Notes in brackets in musical excerpts are correctionsof misprints n the
published
score,
confirmed
through comparison
with
the
original
manuscript
(housed
in the
Ives Collection
at
Yale).
17Morris efines a
cycle (pcyc
and
pccyc)
more
generally
to
apply
to
any
reiterative
segment,
regardless
of whether an intervallic
repetition
occurs
in
the
INT
(pp.
37,
65).
The
resulting
ntervallic
uccession s the
cyclic
INT,
or
CINT(pp.
40,
107).
18Memos,
44
(and
musical
example). Kirkpatrick
ecalls that
"Ives told
George
Roberts
[one
of
his
copyists]
that
his father
had him do
chromatic
scales
with each
interval
a
minor 9th"
(Memos,
44,
n.
5).
tions often take this
form,
bringing
the
scale,
and
thus the
interval-1 or -11 cycle, to its completion in a melodic setting
that either reverses directions
frequently,
producing
linear
an-
gularity,
or
moves
in the
same direction to
cover a
large regis-
tral
span.
The
early
sketch shown
in
Example
2a
employs
the
"wide
jumps" technique
in a melodic line with direction
changes, establishing
a
displacement
pattern
in the first mea-
sure that is
duplicated
a
fourth
higher
in the second.19
Example
2b
gives
a
portion
of
the
piano part
to the
song
Soliloquy
that
employs
the unidirectional
approach,
stating
scale
segments
first with interval 1 (pitch interval 13), and then with interval
11.
The second
arpeggio
(mm. 4-5)
presents
six
pitch-classes
that
are not stated in the
arpeggio
of the
previous
two
mea-
sures,
leaving
only
pc
0
absent from the
upper
voice
of the
passage.20
Ives's
best-known
incorporations
of the
whole-tone
scale
are in the Finales
to the
Second
String
Quartet
(1907-13)
and
Fourth
Symphony
(1909-16).
Both works conclude
with
thick,
layered
textures
of
repeated figures
above reiterated
whole-
tone scales, producing an arrival point of stability that repre-
sents
a kind of resolution of the
many
musical
and extramusical
conflicts
that
have
previously
been
prevalent.21
The
beginning
19The ketch
appears
n Charles Ives's hand
in
George
Ives's
Copybook.
See
John
Kirkpatrick,
A
TemporaryMimeographedCatalogue
of
the Music
Manuscripts
and Related Materials
of
CharlesEdward
Ives 1874-1954
(New
Haven:
Library
of the Yale
University
School of
Music,
1960),
214,
Cat.
No.
7A2.
The
pagination
of the
Copybook
s
Kirkpatrick's.
The
catalogue p.
214)
liststhe probabledatesfor Ives'ssketchings n the Copybook
as 1890-93.
This
scale
setting
reappears
on
p. [71]
of the
Copybook
in a
short
organ
work
Kirkpatrick
alls
"Burlesque
Postlude"
Kirkpatrick,
Catalogue,
19,
Cat.
No.
7C6).
Ives uses the scale
in
canon,
preceded
by
chromatic
ines
in
contrary
mo-
tion.
20Similar
echniques appear
n Over he Pavements
1906-13),
mm.
81-92,
piano,
and
in four works named
by Kirkpatrick
n
Memos,
44,
n.
5.
21Ives's
itle for the finale of the
Quartet
s "The
Callof the
Mountains,"
indicating
that the
"4 men"
personified
by
the
instruments
have
concluded
their "Discussions"of the firstmovement
and
"Arguments"
f the second
and
tions often take this
form,
bringing
the
scale,
and
thus the
interval-1 or -11 cycle, to its completion in a melodic setting
that either reverses directions
frequently,
producing
linear
an-
gularity,
or
moves
in the
same direction to
cover a
large regis-
tral
span.
The
early
sketch shown
in
Example
2a
employs
the
"wide
jumps" technique
in a melodic line with direction
changes, establishing
a
displacement
pattern
in the first mea-
sure that is
duplicated
a
fourth
higher
in the second.19
Example
2b
gives
a
portion
of
the
piano part
to the
song
Soliloquy
that
employs
the unidirectional
approach,
stating
scale
segments
first with interval 1 (pitch interval 13), and then with interval
11.
The second
arpeggio
(mm. 4-5)
presents
six
pitch-classes
that
are not stated in the
arpeggio
of the
previous
two
mea-
sures,
leaving
only
pc
0
absent from the
upper
voice
of the
passage.20
Ives's
best-known
incorporations
of the
whole-tone
scale
are in the Finales
to the
Second
String
Quartet
(1907-13)
and
Fourth
Symphony
(1909-16).
Both works conclude
with
thick,
layered
textures
of
repeated figures
above reiterated
whole-
tone scales, producing an arrival point of stability that repre-
sents
a kind of resolution of the
many
musical
and extramusical
conflicts
that
have
previously
been
prevalent.21
The
beginning
19The ketch
appears
n Charles Ives's hand
in
George
Ives's
Copybook.
See
John
Kirkpatrick,
A
TemporaryMimeographedCatalogue
of
the Music
Manuscripts
and Related Materials
of
CharlesEdward
Ives 1874-1954
(New
Haven:
Library
of the Yale
University
School of
Music,
1960),
214,
Cat.
No.
7A2.
The
pagination
of the
Copybook
s
Kirkpatrick's.
The
catalogue p.
214)
liststhe probabledatesfor Ives'ssketchings n the Copybook
as 1890-93.
This
scale
setting
reappears
on
p. [71]
of the
Copybook
in a
short
organ
work
Kirkpatrick
alls
"Burlesque
Postlude"
Kirkpatrick,
Catalogue,
19,
Cat.
No.
7C6).
Ives uses the scale
in
canon,
preceded
by
chromatic
ines
in
contrary
mo-
tion.
20Similar
echniques appear
n Over he Pavements
1906-13),
mm.
81-92,
piano,
and
in four works named
by Kirkpatrick
n
Memos,
44,
n.
5.
21Ives's
itle for the finale of the
Quartet
s "The
Callof the
Mountains,"
indicating
that the
"4 men"
personified
by
the
instruments
have
concluded
their "Discussions"of the firstmovement
and
"Arguments"
f the second
and
tions often take this
form,
bringing
the
scale,
and
thus the
interval-1 or -11 cycle, to its completion in a melodic setting
that either reverses directions
frequently,
producing
linear
an-
gularity,
or
moves
in the
same direction to
cover a
large regis-
tral
span.
The
early
sketch shown
in
Example
2a
employs
the
"wide
jumps" technique
in a melodic line with direction
changes, establishing
a
displacement
pattern
in the first mea-
sure that is
duplicated
a
fourth
higher
in the second.19
Example
2b
gives
a
portion
of
the
piano part
to the
song
Soliloquy
that
employs
the unidirectional
approach,
stating
scale
segments
first with interval 1 (pitch interval 13), and then with interval
11.
The second
arpeggio
(mm. 4-5)
presents
six
pitch-classes
that
are not stated in the
arpeggio
of the
previous
two
mea-
sures,
leaving
only
pc
0
absent from the
upper
voice
of the
passage.20
Ives's
best-known
incorporations
of the
whole-tone
scale
are in the Finales
to the
Second
String
Quartet
(1907-13)
and
Fourth
Symphony
(1909-16).
Both works conclude
with
thick,
layered
textures
of
repeated figures
above reiterated
whole-
tone scales, producing an arrival point of stability that repre-
sents
a kind of resolution of the
many
musical
and extramusical
conflicts
that
have
previously
been
prevalent.21
The
beginning
19The ketch
appears
n Charles Ives's hand
in
George
Ives's
Copybook.
See
John
Kirkpatrick,
A
TemporaryMimeographedCatalogue
of
the Music
Manuscripts
and Related Materials
of
CharlesEdward
Ives 1874-1954
(New
Haven:
Library
of the Yale
University
School of
Music,
1960),
214,
Cat.
No.
7A2.
The
pagination
of the
Copybook
s
Kirkpatrick's.
The
catalogue p.
214)
liststhe probabledatesfor Ives'ssketchings n the Copybook
as 1890-93.
This
scale
setting
reappears
on
p. [71]
of the
Copybook
in a
short
organ
work
Kirkpatrick
alls
"Burlesque
Postlude"
Kirkpatrick,
Catalogue,
19,
Cat.
No.
7C6).
Ives uses the scale
in
canon,
preceded
by
chromatic
ines
in
contrary
mo-
tion.
20Similar
echniques appear
n Over he Pavements
1906-13),
mm.
81-92,
piano,
and
in four works named
by Kirkpatrick
n
Memos,
44,
n.
5.
21Ives's
itle for the finale of the
Quartet
s "The
Callof the
Mountains,"
indicating
that the
"4 men"
personified
by
the
instruments
have
concluded
their "Discussions"of the firstmovement
and
"Arguments"
f the second
and
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 6/41
Interval
ycles
as
Compositional
esources 47nterval
ycles
as
Compositional
esources 47nterval
ycles
as
Compositional
esources 47
Example
2. "Wide
jumps"
reatmentof the chromatic
cale.
a. GeorgeIves'scopybook, p. [68].
pc<0,
1,
2, 3, 4, 5, 6,
7, 8,
9
10, 11,
1,
0>
b.
Soliloquy,
mm.
2-5,
piano.
A
A
4
A1
18
18
18
18
"'
1>
IL
'
i\if
^^
^
Example
2. "Wide
jumps"
reatmentof the chromatic
cale.
a. GeorgeIves'scopybook, p. [68].
pc<0,
1,
2, 3, 4, 5, 6,
7, 8,
9
10, 11,
1,
0>
b.
Soliloquy,
mm.
2-5,
piano.
A
A
4
A1
18
18
18
18
"'
1>
IL
'
i\if
^^
^
Example
2. "Wide
jumps"
reatmentof the chromatic
cale.
a. GeorgeIves'scopybook, p. [68].
pc<0,
1,
2, 3, 4, 5, 6,
7, 8,
9
10, 11,
1,
0>
b.
Soliloquy,
mm.
2-5,
piano.
A
A
4
A1
18
18
18
18
"'
1>
IL
'
i\if
^^
^
upper-
voice
v
pc:
<1, 2,
3,
4,
5>
INT: < 1--
1-
1
-
1
>
upper-
voice
v
pc:
<1, 2,
3,
4,
5>
INT: < 1--
1-
1
-
1
>
upper-
voice
v
pc:
<1, 2,
3,
4,
5>
INT: < 1--
1-
1
-
1
>
va-
- J
loco
8va
<
11,
10,
9,
8,
7,
6>
< 11-11-11-11 -
11>
va-
- J
loco
8va
<
11,
10,
9,
8,
7,
6>
< 11-11-11-11 -
11>
va-
- J
loco
8va
<
11,
10,
9,
8,
7,
6>
< 11-11-11-11 -
11>
of
this
section of the
Quartet,
shown n
Example
3,
is anchored
on
the
repetition
of the
scale in
the
cello,
forming
a
seven-beat
ostinatothat is
noncoincidentalwith
the
one-measure
pattern
in
the
viola,
the ten-beat
ostinato in
the second
violin,
and the
fragment
of the tune
"Bethany"
n
the first
violin,
all
of
which
support
a tonal
center of D
(see
brackets n
Ex.
3).
The absence
of
rhythmic
variation in
the
whole-tone
presentation
in
the
cello, the lack of intervallicvariety nthe cello'srepeatedmate-
rial,
and
the avoidance
of metric
correlation with
the other
have now walked
"up
the
mountainside o view
the firmament "
Kirkpatrick,
Catalogue,
60).
Ives describes
he
Finale to the
Symphony
as "an
apotheosis
of
the
preceding
content,
in terms
that
have
something
to do
with the
reality
of
existence and its
religious
experience."
See CharlesE.
Ives,
"TheFourth
Sym-
phony
for
Large
Orchestra,"
New
Music
Quarterly
/2
(January
1929): [ii].
of
this
section of the
Quartet,
shown n
Example
3,
is anchored
on
the
repetition
of the
scale in
the
cello,
forming
a
seven-beat
ostinatothat is
noncoincidentalwith
the
one-measure
pattern
in
the
viola,
the ten-beat
ostinato in
the second
violin,
and the
fragment
of the tune
"Bethany"
n
the first
violin,
all
of
which
support
a tonal
center of D
(see
brackets n
Ex.
3).
The absence
of
rhythmic
variation in
the
whole-tone
presentation
in
the
cello, the lack of intervallicvariety nthe cello'srepeatedmate-
rial,
and
the avoidance
of metric
correlation with
the other
have now walked
"up
the
mountainside o view
the firmament "
Kirkpatrick,
Catalogue,
60).
Ives describes
he
Finale to the
Symphony
as "an
apotheosis
of
the
preceding
content,
in terms
that
have
something
to do
with the
reality
of
existence and its
religious
experience."
See CharlesE.
Ives,
"TheFourth
Sym-
phony
for
Large
Orchestra,"
New
Music
Quarterly
/2
(January
1929): [ii].
of
this
section of the
Quartet,
shown n
Example
3,
is anchored
on
the
repetition
of the
scale in
the
cello,
forming
a
seven-beat
ostinatothat is
noncoincidentalwith
the
one-measure
pattern
in
the
viola,
the ten-beat
ostinato in
the second
violin,
and the
fragment
of the tune
"Bethany"
n
the first
violin,
all
of
which
support
a tonal
center of D
(see
brackets n
Ex.
3).
The absence
of
rhythmic
variation in
the
whole-tone
presentation
in
the
cello, the lack of intervallicvariety nthe cello'srepeatedmate-
rial,
and
the avoidance
of metric
correlation with
the other
have now walked
"up
the
mountainside o view
the firmament "
Kirkpatrick,
Catalogue,
60).
Ives describes
he
Finale to the
Symphony
as "an
apotheosis
of
the
preceding
content,
in terms
that
have
something
to do
with the
reality
of
existence and its
religious
experience."
See CharlesE.
Ives,
"TheFourth
Sym-
phony
for
Large
Orchestra,"
New
Music
Quarterly
/2
(January
1929): [ii].
parts
contribute
to a
minimizationof
temporal
qualities, sup-
portingthe projectionof an ethereal, spiritually ranscendent
quality
or the
conclusion
to the work.22
Two basic
features of
the
chromaticand
whole-tone scales
are
relevantto Ives's
pitch
structures n
general.
First,
thecom-
plete uniformity
of interval
sizes in
the scales
s a
desirablechar-
acteristic hat
Ives
exploits
to
highlight
heeffects of
repetition,
as at the
end of the
Second
Quartet,
and to
create a
consistency
of
pitch
structure rom
intervallic
aturation.
Applied
to other
intervals,
repetition
may generate
pitch
materials hat are
less
attractiveowing to their tonal connotations and low cardinal-
ity;
this
may
be
true of
the
repetitions
of
intervals3 and 4 or
their
nverses,
and of
interval
6.
However,
intervals
5
and 7
pro-
vide
rich
resources for
harmonic
saturation
and
have been
widely
used to
ensure
nontraditional
constructionaluniform-
ity.23
ves
views his
song
"The
Cage,"
for
example,
as "a
study
of
how chordsof
4thsand
5ths
may
throw
melodies
away
rom a
set
tonality,"
noting
that "To
make music in
no
particular
key
has a
nice name
nowadays-'atonality.'
"24
The secondinfluential eature of the scales,as foranyof the
cycles,
is
their
insurance of
pitch-class
variety,
or
non-
repetition.
Ives's concern
for avoidance of
pc repetition
allies
22Theseactors
demonstrate he
"spatial"
qualities
of Ives'smusic
observed
by
Robert
Morgan,
"Spatial
Form
n
Ives,"
in An Ives
Celebration:
apers
and
Panels
of
theIves Centennial
Festival-Conference,
d. H.
Wiley
Hitchcockand
Vivian
Perlis
(Urbana:
University
of Illinois
Press,
1977),
145-158.
Morgan
cites, amongother factors, "fragmentation" nd "thesimultaneouscombina-
tion of two
or more
independent,
though
related
musicalcontinuities"
as
con-
tributors o
spatial
effects,
both of
which would
apply
to this
passage
in the
Quartet.
23This
s the
intent of
Schoenberg,
for
example,
in the
chapter
entitled
"Chords Constructed
in
Fourths,"
in
Theory
of Harmony,
trans.
Roy
E.
Carter
(Berkeley:
University
of California
Press,
1978),
399-410. See
also
"Chords
by
Fourths,"
in Vincent
Persichetti,
Twentieth-Century
armony
(New
York:
Norton,
1961),
93-108.
24Ives,Memos,
56.
parts
contribute
to a
minimizationof
temporal
qualities, sup-
portingthe projectionof an ethereal, spiritually ranscendent
quality
or the
conclusion
to the work.22
Two basic
features of
the
chromaticand
whole-tone scales
are
relevantto Ives's
pitch
structures n
general.
First,
thecom-
plete uniformity
of interval
sizes in
the scales
s a
desirablechar-
acteristic hat
Ives
exploits
to
highlight
heeffects of
repetition,
as at the
end of the
Second
Quartet,
and to
create a
consistency
of
pitch
structure rom
intervallic
aturation.
Applied
to other
intervals,
repetition
may generate
pitch
materials hat are
less
attractiveowing to their tonal connotations and low cardinal-
ity;
this
may
be
true of
the
repetitions
of
intervals3 and 4 or
their
nverses,
and of
interval
6.
However,
intervals
5
and 7
pro-
vide
rich
resources for
harmonic
saturation
and
have been
widely
used to
ensure
nontraditional
constructionaluniform-
ity.23
ves
views his
song
"The
Cage,"
for
example,
as "a
study
of
how chordsof
4thsand
5ths
may
throw
melodies
away
rom a
set
tonality,"
noting
that "To
make music in
no
particular
key
has a
nice name
nowadays-'atonality.'
"24
The secondinfluential eature of the scales,as foranyof the
cycles,
is
their
insurance of
pitch-class
variety,
or
non-
repetition.
Ives's concern
for avoidance of
pc repetition
allies
22Theseactors
demonstrate he
"spatial"
qualities
of Ives'smusic
observed
by
Robert
Morgan,
"Spatial
Form
n
Ives,"
in An Ives
Celebration:
apers
and
Panels
of
theIves Centennial
Festival-Conference,
d. H.
Wiley
Hitchcockand
Vivian
Perlis
(Urbana:
University
of Illinois
Press,
1977),
145-158.
Morgan
cites, amongother factors, "fragmentation" nd "thesimultaneouscombina-
tion of two
or more
independent,
though
related
musicalcontinuities"
as
con-
tributors o
spatial
effects,
both of
which would
apply
to this
passage
in the
Quartet.
23This
s the
intent of
Schoenberg,
for
example,
in the
chapter
entitled
"Chords Constructed
in
Fourths,"
in
Theory
of Harmony,
trans.
Roy
E.
Carter
(Berkeley:
University
of California
Press,
1978),
399-410. See
also
"Chords
by
Fourths,"
in Vincent
Persichetti,
Twentieth-Century
armony
(New
York:
Norton,
1961),
93-108.
24Ives,Memos,
56.
parts
contribute
to a
minimizationof
temporal
qualities, sup-
portingthe projectionof an ethereal, spiritually ranscendent
quality
or the
conclusion
to the work.22
Two basic
features of
the
chromaticand
whole-tone scales
are
relevantto Ives's
pitch
structures n
general.
First,
thecom-
plete uniformity
of interval
sizes in
the scales
s a
desirablechar-
acteristic hat
Ives
exploits
to
highlight
heeffects of
repetition,
as at the
end of the
Second
Quartet,
and to
create a
consistency
of
pitch
structure rom
intervallic
aturation.
Applied
to other
intervals,
repetition
may generate
pitch
materials hat are
less
attractiveowing to their tonal connotations and low cardinal-
ity;
this
may
be
true of
the
repetitions
of
intervals3 and 4 or
their
nverses,
and of
interval
6.
However,
intervals
5
and 7
pro-
vide
rich
resources for
harmonic
saturation
and
have been
widely
used to
ensure
nontraditional
constructionaluniform-
ity.23
ves
views his
song
"The
Cage,"
for
example,
as "a
study
of
how chordsof
4thsand
5ths
may
throw
melodies
away
rom a
set
tonality,"
noting
that "To
make music in
no
particular
key
has a
nice name
nowadays-'atonality.'
"24
The secondinfluential eature of the scales,as foranyof the
cycles,
is
their
insurance of
pitch-class
variety,
or
non-
repetition.
Ives's concern
for avoidance of
pc repetition
allies
22Theseactors
demonstrate he
"spatial"
qualities
of Ives'smusic
observed
by
Robert
Morgan,
"Spatial
Form
n
Ives,"
in An Ives
Celebration:
apers
and
Panels
of
theIves Centennial
Festival-Conference,
d. H.
Wiley
Hitchcockand
Vivian
Perlis
(Urbana:
University
of Illinois
Press,
1977),
145-158.
Morgan
cites, amongother factors, "fragmentation" nd "thesimultaneouscombina-
tion of two
or more
independent,
though
related
musicalcontinuities"
as
con-
tributors o
spatial
effects,
both of
which would
apply
to this
passage
in the
Quartet.
23This
s the
intent of
Schoenberg,
for
example,
in the
chapter
entitled
"Chords Constructed
in
Fourths,"
in
Theory
of Harmony,
trans.
Roy
E.
Carter
(Berkeley:
University
of California
Press,
1978),
399-410. See
also
"Chords
by
Fourths,"
in Vincent
Persichetti,
Twentieth-Century
armony
(New
York:
Norton,
1961),
93-108.
24Ives,Memos,
56.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 7/41
48
Music
Theory
Spectrum
8
Music
Theory
Spectrum
8
Music
Theory
Spectrum
Example
3.
Second
String
Quartet,
third
movement,
mm.
123-126.
xample
3.
Second
String
Quartet,
third
movement,
mm.
123-126.
xample
3.
Second
String
Quartet,
third
movement,
mm.
123-126.
Adagio
maestoso
125
8va
..
.-----------
--
- 3 3
,
3
,
Otlt
-a
It9Rt
-e
iff: -&
Adagio
maestoso
125
8va
..
.-----------
--
- 3 3
,
3
,
Otlt
-a
It9Rt
-e
iff: -&
Adagio
maestoso
125
8va
..
.-----------
--
- 3 3
,
3
,
Otlt
-a
It9Rt
-e
iff: -&
v
i
I I
I I
I
I
I
ff
ff
a n_
IXtt
3
-
3
,--_ ---3
---
-
--
3--
--
3 -
ir
i
r
IT
i
ff
I I
F
l-
dJu a I
v
i
I I
I I
I
I
I
ff
ff
a n_
IXtt
3
-
3
,--_ ---3
---
-
--
3--
--
3 -
ir
i
r
IT
i
ff
I I
F
l-
dJu a I
v
i
I I
I I
I
I
I
ff
ff
a n_
IXtt
3
-
3
,--_ ---3
---
-
--
3--
--
3 -
ir
i
r
IT
i
ff
I I
F
l-
dJu a I
'
t?-
F
t
--4
I
I.
r
f4w
-E-
-.
'
t?-
F
t
--4
I
I.
r
f4w
-E-
-.
'
t?-
F
t
--4
I
I.
r
f4w
-E-
-.
him with his friend Carl
Ruggles,
who is said to have advocated
stating
at least "sevenor
eight
differentnotes
[pitchclasses]
na
melody"
before
introducing
a
repetition.25
he
composer
s as-
sured of maximal
pitch-class variety,
without
repetition,
in
chromatic
completions
and in structuresbased
on
intervals5
or
7,
such as the melodic line illustrated n
Example
1.
Methodical
generation
of the
aggregate
s also
possible
in
a
complementa-
tion of the other
cycles,
such
as,
for
example,
a combination
of
pitch
classes
from "odd" and
"even"
whole-tone scales.26
2sHenry
Cowell,
New Musical Resources
(1930;repr.
New York:
Knopf,
1950),
41-42. See
Steven
E.
Gilbert,
"The 'Twelve-Tone
System'
of
Carl
Rug-
gles:
A
Study
of the Evocations or
Piano,"
Journal
of
Music
Theory
14
(1970),
68-91.
26DaveHeadlam defines this
type
of
complementation
s an "extension"
of
an
interval
cycle
in "The Derivation of Rows in
Lulu,"
Perspectives
of
New
Music 24/1
(1985),
203.
him with his friend Carl
Ruggles,
who is said to have advocated
stating
at least "sevenor
eight
differentnotes
[pitchclasses]
na
melody"
before
introducing
a
repetition.25
he
composer
s as-
sured of maximal
pitch-class variety,
without
repetition,
in
chromatic
completions
and in structuresbased
on
intervals5
or
7,
such as the melodic line illustrated n
Example
1.
Methodical
generation
of the
aggregate
s also
possible
in
a
complementa-
tion of the other
cycles,
such
as,
for
example,
a combination
of
pitch
classes
from "odd" and
"even"
whole-tone scales.26
2sHenry
Cowell,
New Musical Resources
(1930;repr.
New York:
Knopf,
1950),
41-42. See
Steven
E.
Gilbert,
"The 'Twelve-Tone
System'
of
Carl
Rug-
gles:
A
Study
of the Evocations or
Piano,"
Journal
of
Music
Theory
14
(1970),
68-91.
26DaveHeadlam defines this
type
of
complementation
s an "extension"
of
an
interval
cycle
in "The Derivation of Rows in
Lulu,"
Perspectives
of
New
Music 24/1
(1985),
203.
him with his friend Carl
Ruggles,
who is said to have advocated
stating
at least "sevenor
eight
differentnotes
[pitchclasses]
na
melody"
before
introducing
a
repetition.25
he
composer
s as-
sured of maximal
pitch-class variety,
without
repetition,
in
chromatic
completions
and in structuresbased
on
intervals5
or
7,
such as the melodic line illustrated n
Example
1.
Methodical
generation
of the
aggregate
s also
possible
in
a
complementa-
tion of the other
cycles,
such
as,
for
example,
a combination
of
pitch
classes
from "odd" and
"even"
whole-tone scales.26
2sHenry
Cowell,
New Musical Resources
(1930;repr.
New York:
Knopf,
1950),
41-42. See
Steven
E.
Gilbert,
"The 'Twelve-Tone
System'
of
Carl
Rug-
gles:
A
Study
of the Evocations or
Piano,"
Journal
of
Music
Theory
14
(1970),
68-91.
26DaveHeadlam defines this
type
of
complementation
s an "extension"
of
an
interval
cycle
in "The Derivation of Rows in
Lulu,"
Perspectives
of
New
Music 24/1
(1985),
203.
The union
of intervallic
uniformity
and
pitch-classvariety
may
yield
a
plan
of
organization
or
melodic
or
harmonic truc-
tures.
In a
brief sketch that Ives
gives
the title
Song
in
5's,
for
example,
chords of stackedfifths are related
through
associa-
tion
with a
generating
interval-5
cycle.27
The
sketch,
tran-
scribed
n
Example
4,
consists
of
four sonoritiesconnected
by
an
upper
melodic
line that
is itself
a
whole-tone
pentachord,
a
combination reminiscent of the
juxtaposition
of
chords in
fourths
with a
whole-tone
melody
in "The
Cage."
The
pitch
classes
in the lowest voices of chords 1 and 2 connect
to
the
up-
per
voices
of
the
ensuing
chords as continuations
of the
interval-5
cycle
indicated
below the score.
Aggregate
comple-
27Kirkpatrick, atalogue,
226,
No.
7E38.
The
compiler
does
not
suggest
a
date for the sketch. It
appears
within materials
relating
to
the "Thoreau"
movement of the
Concord
Sonata,
which was
composed
around
1910-15.
The union
of intervallic
uniformity
and
pitch-classvariety
may
yield
a
plan
of
organization
or
melodic
or
harmonic truc-
tures.
In a
brief sketch that Ives
gives
the title
Song
in
5's,
for
example,
chords of stackedfifths are related
through
associa-
tion
with a
generating
interval-5
cycle.27
The
sketch,
tran-
scribed
n
Example
4,
consists
of
four sonoritiesconnected
by
an
upper
melodic
line that
is itself
a
whole-tone
pentachord,
a
combination reminiscent of the
juxtaposition
of
chords in
fourths
with a
whole-tone
melody
in "The
Cage."
The
pitch
classes
in the lowest voices of chords 1 and 2 connect
to
the
up-
per
voices
of
the
ensuing
chords as continuations
of the
interval-5
cycle
indicated
below the score.
Aggregate
comple-
27Kirkpatrick, atalogue,
226,
No.
7E38.
The
compiler
does
not
suggest
a
date for the sketch. It
appears
within materials
relating
to
the "Thoreau"
movement of the
Concord
Sonata,
which was
composed
around
1910-15.
The union
of intervallic
uniformity
and
pitch-classvariety
may
yield
a
plan
of
organization
or
melodic
or
harmonic truc-
tures.
In a
brief sketch that Ives
gives
the title
Song
in
5's,
for
example,
chords of stackedfifths are related
through
associa-
tion
with a
generating
interval-5
cycle.27
The
sketch,
tran-
scribed
n
Example
4,
consists
of
four sonoritiesconnected
by
an
upper
melodic
line that
is itself
a
whole-tone
pentachord,
a
combination reminiscent of the
juxtaposition
of
chords in
fourths
with a
whole-tone
melody
in "The
Cage."
The
pitch
classes
in the lowest voices of chords 1 and 2 connect
to
the
up-
per
voices
of
the
ensuing
chords as continuations
of the
interval-5
cycle
indicated
below the score.
Aggregate
comple-
27Kirkpatrick, atalogue,
226,
No.
7E38.
The
compiler
does
not
suggest
a
date for the sketch. It
appears
within materials
relating
to
the "Thoreau"
movement of the
Concord
Sonata,
which was
composed
around
1910-15.
I
ff
I
ff
I
ff
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 8/41
Interval
ycles
as
Compositional
esources
49
nterval
ycles
as
Compositional
esources
49
nterval
ycles
as
Compositional
esources
49
Example
4.
Song
in
5's.
chord
1 2
Example
4.
Song
in
5's.
chord
1 2
Example
4.
Song
in
5's.
chord
1 2 3
444
n 7
d
3
'
[7]
b
I:
6
n 7
d
3
'
[7]
b
I:
6
n 7
d
3
'
[7]
b
I:
6
chord: 1 2 3
pc<9,
2,
7,
0,
5
5,
10,
3,
8,
1
1,6, 11,
4>
INT
<5
-
5
-5
-5 . . .
4
tion is
achieved
in
chord
3,
followed
in the final
sonority
by
a
repetition
of
pitch
classes from the latter
part
of the
cycle.
In
the
early
choral work
Psalm
54
(1894?),
a bidimensional
projection
of
repetitive
ntervallic tructures
xploits
the
circu-
larity nherent in the materials,achievinga moreextensive ex-
position
of
cyclic
deas than n similar
passages
rom the
Finales
of
the Second
String
Quartet
and Fourth
Symphony.
Example
5
is
a reduction in
short score
of the
first
of
the
seven
verses,
which exhibits
a
pattern
of
half-note chords n the
lower voices
embellished
by
added notes in the
upper
voices.
This same
ar-
rangement
is used in the
setting
of verse
2
(mm. 7-13)
and
again
n
the
setting
of
the
final
verse
(mm.
44-57)
with the
roles
exchanged
(half-note
chords in the
upper
voices,
and so
forth).28
The
upperembellishments circled
n Ex.
5)
decorate
a
lower-rangeprojection
of
augmented
riadsrooted
on notes
of
the
descending
whole-tone
scale,
completing
an
octave
de-
scent at the downbeat of m. 4. The
projection
of the triads hen
reverses direction and shifts to the
complementary
whole-tone
28Verses
through
6
are set in a
contrasting
contrapuntal tyle
employing
double canon.
chord: 1 2 3
pc<9,
2,
7,
0,
5
5,
10,
3,
8,
1
1,6, 11,
4>
INT
<5
-
5
-5
-5 . . .
4
tion is
achieved
in
chord
3,
followed
in the final
sonority
by
a
repetition
of
pitch
classes from the latter
part
of the
cycle.
In
the
early
choral work
Psalm
54
(1894?),
a bidimensional
projection
of
repetitive
ntervallic tructures
xploits
the
circu-
larity nherent in the materials,achievinga moreextensive ex-
position
of
cyclic
deas than n similar
passages
rom the
Finales
of
the Second
String
Quartet
and Fourth
Symphony.
Example
5
is
a reduction in
short score
of the
first
of
the
seven
verses,
which exhibits
a
pattern
of
half-note chords n the
lower voices
embellished
by
added notes in the
upper
voices.
This same
ar-
rangement
is used in the
setting
of verse
2
(mm. 7-13)
and
again
n
the
setting
of
the
final
verse
(mm.
44-57)
with the
roles
exchanged
(half-note
chords in the
upper
voices,
and so
forth).28
The
upperembellishments circled
n Ex.
5)
decorate
a
lower-rangeprojection
of
augmented
riadsrooted
on notes
of
the
descending
whole-tone
scale,
completing
an
octave
de-
scent at the downbeat of m. 4. The
projection
of the triads hen
reverses direction and shifts to the
complementary
whole-tone
28Verses
through
6
are set in a
contrasting
contrapuntal tyle
employing
double canon.
chord: 1 2 3
pc<9,
2,
7,
0,
5
5,
10,
3,
8,
1
1,6, 11,
4>
INT
<5
-
5
-5
-5 . . .
4
tion is
achieved
in
chord
3,
followed
in the final
sonority
by
a
repetition
of
pitch
classes from the latter
part
of the
cycle.
In
the
early
choral work
Psalm
54
(1894?),
a bidimensional
projection
of
repetitive
ntervallic tructures
xploits
the
circu-
larity nherent in the materials,achievinga moreextensive ex-
position
of
cyclic
deas than n similar
passages
rom the
Finales
of
the Second
String
Quartet
and Fourth
Symphony.
Example
5
is
a reduction in
short score
of the
first
of
the
seven
verses,
which exhibits
a
pattern
of
half-note chords n the
lower voices
embellished
by
added notes in the
upper
voices.
This same
ar-
rangement
is used in the
setting
of verse
2
(mm. 7-13)
and
again
n
the
setting
of
the
final
verse
(mm.
44-57)
with the
roles
exchanged
(half-note
chords in the
upper
voices,
and so
forth).28
The
upperembellishments circled
n Ex.
5)
decorate
a
lower-rangeprojection
of
augmented
riadsrooted
on notes
of
the
descending
whole-tone
scale,
completing
an
octave
de-
scent at the downbeat of m. 4. The
projection
of the triads hen
reverses direction and shifts to the
complementary
whole-tone
28Verses
through
6
are set in a
contrasting
contrapuntal tyle
employing
double canon.
scale,
eventually
stating
an
augmented
triad on
each
of the 12
possibleroots.29
Because each
pc representation
of an
interval-4
ycle
is
sym-
metrically
ituated within one of the whole-tone
scales,
the lin-
ear whole-tone
presentationmay
be
viewed as
a
gradual
"un-
folding"
of
alternate members
of
the
interval-4
cycles.
Correspondingly,
he
augmented
triads built above
each scale
step
alternate
appearances
as notated below the score
n
Exam-
ple
5:
cycles
I and III
alternate
n
the initial descent
(mm. 1-4)
and
cycles
II and IV
alternate n the
subsequent
ascent. Each
cycleexhibitsacompleterotationof its verticalarrangemento
exhaust
the
possible
augmented-triad
oots and thus the
aggre-
gate. Cycle
I,
for
instance,
moves from
pc
<0,4,8>
in m. 1 to
<8,4,0>
in
m.
2
and to
<4,8,0>
in
m.
3. The
passage
thus
be-
comes saturated with
cyclic
formations from the horizontal
whole-tone
cycles (interval-2
or
-10)
and their
support
of verti-
cal interval-4
cycles
undergoing
regularpatterns
of rotation.
The
excerpt
from Psalm54 in
Example
5
is
typical
n
that the
pure
whole-tone
saturation n the lower
voices
provides
a
cyclic
framework or a processof embellishment n the uppervoices
that introduces
pitch
classes from outside
the
prevailing
har-
monies. The
cycle
thus
serves
a
role that
might
be fulfilled
by
a
diatonic
scale
or
scale
segment
in a tonal
context.
This
ap-
proach
is also evident in
Ives's music as
an
embellishment
of
repetitive
ntervallic tructures hat do not
reach
cycliccomple-
tion,
includingpassages
based on
seconds,
fourths,
or
fifths as
controllersof
linear motion.30
29The
assage
is described
similarly
n H.
Wiley
Hitchcock,
Ives: A
Survey
of
the Music
(London:
Oxford
University
Press, 1977;
repr.
New York: Insti-
tute for
Studies in American
Music,
1983),
29-31. The chords could also be
viewed as
resulting
from
three whole-tone scales
moving
in
parallel
major
thirds.
30See,
or
example,
the
embellishmentof an
interval-5
equence
n the
first
movement
of
the Second
String
Quartet,
mm.
28-33,
firstviolin.
scale,
eventually
stating
an
augmented
triad on
each
of the 12
possibleroots.29
Because each
pc representation
of an
interval-4
ycle
is
sym-
metrically
ituated within one of the whole-tone
scales,
the lin-
ear whole-tone
presentationmay
be
viewed as
a
gradual
"un-
folding"
of
alternate members
of
the
interval-4
cycles.
Correspondingly,
he
augmented
triads built above
each scale
step
alternate
appearances
as notated below the score
n
Exam-
ple
5:
cycles
I and III
alternate
n
the initial descent
(mm. 1-4)
and
cycles
II and IV
alternate n the
subsequent
ascent. Each
cycleexhibitsacompleterotationof its verticalarrangemento
exhaust
the
possible
augmented-triad
oots and thus the
aggre-
gate. Cycle
I,
for
instance,
moves from
pc
<0,4,8>
in m. 1 to
<8,4,0>
in
m.
2
and to
<4,8,0>
in
m.
3. The
passage
thus
be-
comes saturated with
cyclic
formations from the horizontal
whole-tone
cycles (interval-2
or
-10)
and their
support
of verti-
cal interval-4
cycles
undergoing
regularpatterns
of rotation.
The
excerpt
from Psalm54 in
Example
5
is
typical
n
that the
pure
whole-tone
saturation n the lower
voices
provides
a
cyclic
framework or a processof embellishment n the uppervoices
that introduces
pitch
classes from outside
the
prevailing
har-
monies. The
cycle
thus
serves
a
role that
might
be fulfilled
by
a
diatonic
scale
or
scale
segment
in a tonal
context.
This
ap-
proach
is also evident in
Ives's music as
an
embellishment
of
repetitive
ntervallic tructures hat do not
reach
cycliccomple-
tion,
includingpassages
based on
seconds,
fourths,
or
fifths as
controllersof
linear motion.30
29The
assage
is described
similarly
n H.
Wiley
Hitchcock,
Ives: A
Survey
of
the Music
(London:
Oxford
University
Press, 1977;
repr.
New York: Insti-
tute for
Studies in American
Music,
1983),
29-31. The chords could also be
viewed as
resulting
from
three whole-tone scales
moving
in
parallel
major
thirds.
30See,
or
example,
the
embellishmentof an
interval-5
equence
n the
first
movement
of
the Second
String
Quartet,
mm.
28-33,
firstviolin.
scale,
eventually
stating
an
augmented
triad on
each
of the 12
possibleroots.29
Because each
pc representation
of an
interval-4
ycle
is
sym-
metrically
ituated within one of the whole-tone
scales,
the lin-
ear whole-tone
presentationmay
be
viewed as
a
gradual
"un-
folding"
of
alternate members
of
the
interval-4
cycles.
Correspondingly,
he
augmented
triads built above
each scale
step
alternate
appearances
as notated below the score
n
Exam-
ple
5:
cycles
I and III
alternate
n
the initial descent
(mm. 1-4)
and
cycles
II and IV
alternate n the
subsequent
ascent. Each
cycleexhibitsacompleterotationof its verticalarrangemento
exhaust
the
possible
augmented-triad
oots and thus the
aggre-
gate. Cycle
I,
for
instance,
moves from
pc
<0,4,8>
in m. 1 to
<8,4,0>
in
m.
2
and to
<4,8,0>
in
m.
3. The
passage
thus
be-
comes saturated with
cyclic
formations from the horizontal
whole-tone
cycles (interval-2
or
-10)
and their
support
of verti-
cal interval-4
cycles
undergoing
regularpatterns
of rotation.
The
excerpt
from Psalm54 in
Example
5
is
typical
n
that the
pure
whole-tone
saturation n the lower
voices
provides
a
cyclic
framework or a processof embellishment n the uppervoices
that introduces
pitch
classes from outside
the
prevailing
har-
monies. The
cycle
thus
serves
a
role that
might
be fulfilled
by
a
diatonic
scale
or
scale
segment
in a tonal
context.
This
ap-
proach
is also evident in
Ives's music as
an
embellishment
of
repetitive
ntervallic tructures hat do not
reach
cycliccomple-
tion,
includingpassages
based on
seconds,
fourths,
or
fifths as
controllersof
linear motion.30
29The
assage
is described
similarly
n H.
Wiley
Hitchcock,
Ives: A
Survey
of
the Music
(London:
Oxford
University
Press, 1977;
repr.
New York: Insti-
tute for
Studies in American
Music,
1983),
29-31. The chords could also be
viewed as
resulting
from
three whole-tone scales
moving
in
parallel
major
thirds.
30See,
or
example,
the
embellishmentof an
interval-5
equence
n the
first
movement
of
the Second
String
Quartet,
mm.
28-33,
firstviolin.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 9/41
50
Music
Theory
Spectrum
0
Music
Theory
Spectrum
0
Music
Theory
Spectrum
Example
5. Psalm
54,
mm. 1-6.
xample
5. Psalm
54,
mm. 1-6.
xample
5. Psalm
54,
mm. 1-6.
Save
me,
[Largo
maestoso]
Save
me,
[Largo
maestoso]
Save
me,
[Largo
maestoso]
O
God,God,God,
I
III
I
III
I
III
name,
and
judge
me
by
I
III
I
III
I
III
name,
and
judge
me
by
I
III
I
III
I
III
name,
and
judge
me
by
thy
strength,
hy
strength,
hy
strength,
I
II
IV
II
IV
II
IV
II
IV
II
IV
II
IV
II
IV
II
IV
II
IV
I.
<0,
4,
8>
.
<0,
4,
8>
.
<0,
4,
8>
IV.
<3,
7,
11>
V.
<3,
7,
11>
V.
<3,
7,
11>
Embellishment
techniques
also
help
interpret passages
where,
unlike
the Psalm
54
excerpt,
the
underlying
ycle
is less
explicitly
stated
within
the
texture.
An
elaboration
of an
interval-11
cycle,
realized
as a
descending
chromatic
cale,
oc-
curs
in the
passage
from
Ives's
Study
No.
20
(1908?)
given
in
Example 6a, spanningeight barsand reachingcompletionin
the
bass
register
four
octaves
below
its initial
pitch.
The
chro-
matic descent
begins
with the
C#6
in beat
1 of
m.
6,
stated
as the
top
note
in an instance
of 3-5
[0,1,6].31
Thistrichord
appears
31Set-class
abels
throughout
his
article
are those
of Allen Forte.
See
The
Structure
of
Atonal
Music
(New
Haven
and London:
Yale
University
Press,
1973), Appendix
1(179-181).
Embellishment
techniques
also
help
interpret passages
where,
unlike
the Psalm
54
excerpt,
the
underlying
ycle
is less
explicitly
stated
within
the
texture.
An
elaboration
of an
interval-11
cycle,
realized
as a
descending
chromatic
cale,
oc-
curs
in the
passage
from
Ives's
Study
No.
20
(1908?)
given
in
Example 6a, spanningeight barsand reachingcompletionin
the
bass
register
four
octaves
below
its initial
pitch.
The
chro-
matic descent
begins
with the
C#6
in beat
1 of
m.
6,
stated
as the
top
note
in an instance
of 3-5
[0,1,6].31
Thistrichord
appears
31Set-class
abels
throughout
his
article
are those
of Allen Forte.
See
The
Structure
of
Atonal
Music
(New
Haven
and London:
Yale
University
Press,
1973), Appendix
1(179-181).
Embellishment
techniques
also
help
interpret passages
where,
unlike
the Psalm
54
excerpt,
the
underlying
ycle
is less
explicitly
stated
within
the
texture.
An
elaboration
of an
interval-11
cycle,
realized
as a
descending
chromatic
cale,
oc-
curs
in the
passage
from
Ives's
Study
No.
20
(1908?)
given
in
Example 6a, spanningeight barsand reachingcompletionin
the
bass
register
four
octaves
below
its initial
pitch.
The
chro-
matic descent
begins
with the
C#6
in beat
1 of
m.
6,
stated
as the
top
note
in an instance
of 3-5
[0,1,6].31
Thistrichord
appears
31Set-class
abels
throughout
his
article
are those
of Allen Forte.
See
The
Structure
of
Atonal
Music
(New
Haven
and London:
Yale
University
Press,
1973), Appendix
1(179-181).
beneath
several
of
the scale
steps,
combining
he
intervals
of a
fourth
or fifth
plus
a
tritone
to
accompany
he
participants
n
the
chromatic
descent.
The
circled
notes
in
Example
6a
follow
the
descent
of the
scale
down
a
fifth
in the
right
hand
to
F#5
in m. 8.
Singularly
absent from the scalarunfoldingis pc 9, whichoccursas the
bass
grace
note
in
mm.
6-8
and
as the
pedal
starting
n m.
1 of
the
Study.
Example
6bsummarizes
he
descent
plus
the
accom-
panying
3-5s,
incorporating
he
missingpc
9
in
its
appropriate
position
and
actual
register.
When
the
line
transfers
o
the
left
hand
in m.
9,
two
scale
tones
arestated
simultaneously
s
part
of a 3-5
and are
subsequently
reiterated
n
the
bass
of m.
10
(the
reiteration
s enclosed
in brackets
n Ex.
6b).
The
remain-
beneath
several
of
the scale
steps,
combining
he
intervals
of a
fourth
or fifth
plus
a
tritone
to
accompany
he
participants
n
the
chromatic
descent.
The
circled
notes
in
Example
6a
follow
the
descent
of the
scale
down
a
fifth
in the
right
hand
to
F#5
in m. 8.
Singularly
absent from the scalarunfoldingis pc 9, whichoccursas the
bass
grace
note
in
mm.
6-8
and
as the
pedal
starting
n m.
1 of
the
Study.
Example
6bsummarizes
he
descent
plus
the
accom-
panying
3-5s,
incorporating
he
missingpc
9
in
its
appropriate
position
and
actual
register.
When
the
line
transfers
o
the
left
hand
in m.
9,
two
scale
tones
arestated
simultaneously
s
part
of a 3-5
and are
subsequently
reiterated
n
the
bass
of m.
10
(the
reiteration
s enclosed
in brackets
n Ex.
6b).
The
remain-
beneath
several
of
the scale
steps,
combining
he
intervals
of a
fourth
or fifth
plus
a
tritone
to
accompany
he
participants
n
the
chromatic
descent.
The
circled
notes
in
Example
6a
follow
the
descent
of the
scale
down
a
fifth
in the
right
hand
to
F#5
in m. 8.
Singularly
absent from the scalarunfoldingis pc 9, whichoccursas the
bass
grace
note
in
mm.
6-8
and
as the
pedal
starting
n m.
1 of
the
Study.
Example
6bsummarizes
he
descent
plus
the
accom-
panying
3-5s,
incorporating
he
missingpc
9
in
its
appropriate
position
and
actual
register.
When
the
line
transfers
o
the
left
hand
in m.
9,
two
scale
tones
arestated
simultaneously
s
part
of a 3-5
and are
subsequently
reiterated
n
the
bass
of m.
10
(the
reiteration
s enclosed
in brackets
n Ex.
6b).
The
remain-
byyy
thyhyhy
II.
<1,
5,
9>
III.
<2,
6, 10>I.
<1,
5,
9>
III.
<2,
6, 10>I.
<1,
5,
9>
III.
<2,
6, 10>
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 10/41
Interval
ycles
as
Compositional
esources
51
nterval
ycles
as
Compositional
esources
51
nterval
ycles
as
Compositional
esources
51
Example
6.
Study
No.
20,
mm. 6-14.
a.
tem~
.
tmpo
pnmo
06
1
2 13
4i
J
-'
-TI ,J ,~
^^
n
i
I I
i
,
l^^r
-
,^1
"
dlJ '^n^
rK
^n
Example
6.
Study
No.
20,
mm. 6-14.
a.
tem~
.
tmpo
pnmo
06
1
2 13
4i
J
-'
-TI ,J ,~
^^
n
i
I I
i
,
l^^r
-
,^1
"
dlJ '^n^
rK
^n
Example
6.
Study
No.
20,
mm. 6-14.
a.
tem~
.
tmpo
pnmo
06
1
2 13
4i
J
-'
-TI ,J ,~
^^
n
i
I I
i
,
l^^r
-
,^1
"
dlJ '^n^
rK
^n
derof the descent isaccomplishedby successive ranspositions
of
a
three-beat
pattern
n the left
handthat
ncludes
his
reitera-
tion
and
comes to rest
on the
3-5;
the
firststatementof the
pat-
tern is
m. 10
through
the first
beat
of m.
11,
bracketed
below
the
score
in
Example
6a. Since each three-beat
unit
is
trans-
posed
down a half
step,
the descent continueswith each new
transposition
evel,
as
highlightedby
the
stemmed
notes
in
Ex-
ample
6b. After
arriving
at
C#2
on the third beat
of m.
13,
the
pattern
returns
to its
original
pitch
level,
with mm.
14-17
re-
peatingmm. 10-13 exactly.
The
adjacent
intervals
forming
the 3-5s beneath the scale
tones,
as notatedin
Example
6b,
reverse their
positions
n the
course
of
the
presentation
rom
7
above
6
at the firsttwo scale
tones
to
6 above
7
in
m. 7
and in mm.
9-13,
portraying
he final
left-hand trichordsas mirrorsof
the first wo
in
the
right
hand.
These
trichords,
which also include chromatic intervals be-
tween the
outer
notes,
provide
a consistent character or the
derof the descent isaccomplishedby successive ranspositions
of
a
three-beat
pattern
n the left
handthat
ncludes
his
reitera-
tion
and
comes to rest
on the
3-5;
the
firststatementof the
pat-
tern is
m. 10
through
the first
beat
of m.
11,
bracketed
below
the
score
in
Example
6a. Since each three-beat
unit
is
trans-
posed
down a half
step,
the descent continueswith each new
transposition
evel,
as
highlightedby
the
stemmed
notes
in
Ex-
ample
6b. After
arriving
at
C#2
on the third beat
of m.
13,
the
pattern
returns
to its
original
pitch
level,
with mm.
14-17
re-
peatingmm. 10-13 exactly.
The
adjacent
intervals
forming
the 3-5s beneath the scale
tones,
as notatedin
Example
6b,
reverse their
positions
n the
course
of
the
presentation
rom
7
above
6
at the firsttwo scale
tones
to
6 above
7
in
m. 7
and in mm.
9-13,
portraying
he final
left-hand trichordsas mirrorsof
the first wo
in
the
right
hand.
These
trichords,
which also include chromatic intervals be-
tween the
outer
notes,
provide
a consistent character or the
derof the descent isaccomplishedby successive ranspositions
of
a
three-beat
pattern
n the left
handthat
ncludes
his
reitera-
tion
and
comes to rest
on the
3-5;
the
firststatementof the
pat-
tern is
m. 10
through
the first
beat
of m.
11,
bracketed
below
the
score
in
Example
6a. Since each three-beat
unit
is
trans-
posed
down a half
step,
the descent continueswith each new
transposition
evel,
as
highlightedby
the
stemmed
notes
in
Ex-
ample
6b. After
arriving
at
C#2
on the third beat
of m.
13,
the
pattern
returns
to its
original
pitch
level,
with mm.
14-17
re-
peatingmm. 10-13 exactly.
The
adjacent
intervals
forming
the 3-5s beneath the scale
tones,
as notatedin
Example
6b,
reverse their
positions
n the
course
of
the
presentation
rom
7
above
6
at the firsttwo scale
tones
to
6 above
7
in
m. 7
and in mm.
9-13,
portraying
he final
left-hand trichordsas mirrorsof
the first wo
in
the
right
hand.
These
trichords,
which also include chromatic intervals be-
tween the
outer
notes,
provide
a consistent character or the
steps in the chromatic ine, contributing o the associationof
pitches
that are
registrally eparated
as
part
of a
unified state-
ment of the
cyclic
pitch
source.
Other
embellishment
procedures
alter a source not
through
addition
of notes
surrounding
he
cyclic unfolding
but
through
rearrangement
f the
pitch
classes
of the source alone.
A
cycli-
cally generated
pc
set
is
thus used
as an
unordered
collection
that
provides
material or a melodic
or
harmonic
etting.
With
cycles
of
cardinality
12,
the
source
provides
a convenient
point
of departure or aggregateconstructions; ves typicallyretains
features
of the source so that the
origins
of the
material
are
im-
mediately
apparent.
A
principal
heme in
the Robert
Browning
Overture
1908-12) employs
a
displacement
of a
single pitch
class
within the
generation
of an interval-7
cycle,
producing
an
aggregate
ordering
and an INT
of
<7-7-2-7-7-7-7-7-7-11-8>.
One
transposition
of
this
theme,
for
example,
is an alteration
of
<3,10,5,0,7,2,9,4,11,6,1,8>,
a
strict
repetition
of
interval
7,
to
steps in the chromatic ine, contributing o the associationof
pitches
that are
registrally eparated
as
part
of a
unified state-
ment of the
cyclic
pitch
source.
Other
embellishment
procedures
alter a source not
through
addition
of notes
surrounding
he
cyclic unfolding
but
through
rearrangement
f the
pitch
classes
of the source alone.
A
cycli-
cally generated
pc
set
is
thus used
as an
unordered
collection
that
provides
material or a melodic
or
harmonic
etting.
With
cycles
of
cardinality
12,
the
source
provides
a convenient
point
of departure or aggregateconstructions; ves typicallyretains
features
of the source so that the
origins
of the
material
are
im-
mediately
apparent.
A
principal
heme in
the Robert
Browning
Overture
1908-12) employs
a
displacement
of a
single pitch
class
within the
generation
of an interval-7
cycle,
producing
an
aggregate
ordering
and an INT
of
<7-7-2-7-7-7-7-7-7-11-8>.
One
transposition
of
this
theme,
for
example,
is an alteration
of
<3,10,5,0,7,2,9,4,11,6,1,8>,
a
strict
repetition
of
interval
7,
to
steps in the chromatic ine, contributing o the associationof
pitches
that are
registrally eparated
as
part
of a
unified state-
ment of the
cyclic
pitch
source.
Other
embellishment
procedures
alter a source not
through
addition
of notes
surrounding
he
cyclic unfolding
but
through
rearrangement
f the
pitch
classes
of the source alone.
A
cycli-
cally generated
pc
set
is
thus used
as an
unordered
collection
that
provides
material or a melodic
or
harmonic
etting.
With
cycles
of
cardinality
12,
the
source
provides
a convenient
point
of departure or aggregateconstructions; ves typicallyretains
features
of the source so that the
origins
of the
material
are
im-
mediately
apparent.
A
principal
heme in
the Robert
Browning
Overture
1908-12) employs
a
displacement
of a
single pitch
class
within the
generation
of an interval-7
cycle,
producing
an
aggregate
ordering
and an INT
of
<7-7-2-7-7-7-7-7-7-11-8>.
One
transposition
of
this
theme,
for
example,
is an alteration
of
<3,10,5,0,7,2,9,4,11,6,1,8>,
a
strict
repetition
of
interval
7,
to
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 11/41
52 Music
Theory
Spectrum
2 Music
Theory
Spectrum
2 Music
Theory
Spectrum
Example
7.
Three-Page
Sonata,
derivation
of left
hand,
mm.77-79.
a.
source
interval-5
op
0
1
2 3
4 5 6 7
8
9
10 11
cycle
p
--
cycl:0
?
b"o
to
t0
f
lo
o
pc
9 2
7
0 5 10 3
8 1
6
11
4
b.
left
pc
9
2
7
0
5
6
1
4
10 8 3 8 11
hand,
mm 77 3 r
3-
78 ---3 r-- 79 ----- 3-m. - 7 3
.-
79
77-79
-
I
K
I
,
.
r
INT
<7-5--5
-
5--1-7-3-
--
10-7-5-3>
produce
<3,10,5,7,2,9,4,11,6,1,0,8>,
accomplished
by
mov-
ing
pc
0 from
op
3 to
op
10.32
A
pitch-class
succession
n Ives's
Three-Page
Sonata
(1905)
employsa moreextensivereorderingn anaggregateconstruc-
tion
while still
retaining
a
display
of elements
of its intervallic
source.
Example
7a
displays
he interval-5
ycle
that serves as
a
point
of
departure
or the
aggregate
of
Example
7b
(the
final
pc
2 of m. 79
begins
a
repetition
of the
pitch-class
material).33
f-
ter
five
pitch
classes
drawn
without variation rom
the
source,
the line
presents
pc
<6,1>,
or a reversal
of source
ops
8 and
9,
followed
by pc
4,
which s the final
note in the
source,
and then
pc
<10,8,3,8>,
a
rearrangement
of
ops
5, 6,
and
7. The INT
notatedbelowExample7billustrates hepresenceof interval5
and
ts inverse
7
within
a
sequence
that also
includes ntervals
1,
3, 6,
and
10. Within
this intervallic
variety,
ntervals
5 and
7 re-
32This ccurs
n the
cello,
mm. 50-52. The
first
presentation
of this
theme
(bassoon
and
trombone,
mm.
46-49)
has some
pitch
naccuracies
hat
prevent
aggregatecompletion,
probably
he results
of
copying
or calculation
rrors.
33In
he
original,
this line is doubled
an octave
lower.
Example
7.
Three-Page
Sonata,
derivation
of left
hand,
mm.77-79.
a.
source
interval-5
op
0
1
2 3
4 5 6 7
8
9
10 11
cycle
p
--
cycl:0
?
b"o
to
t0
f
lo
o
pc
9 2
7
0 5 10 3
8 1
6
11
4
b.
left
pc
9
2
7
0
5
6
1
4
10 8 3 8 11
hand,
mm 77 3 r
3-
78 ---3 r-- 79 ----- 3-m. - 7 3
.-
79
77-79
-
I
K
I
,
.
r
INT
<7-5--5
-
5--1-7-3-
--
10-7-5-3>
produce
<3,10,5,7,2,9,4,11,6,1,0,8>,
accomplished
by
mov-
ing
pc
0 from
op
3 to
op
10.32
A
pitch-class
succession
n Ives's
Three-Page
Sonata
(1905)
employsa moreextensivereorderingn anaggregateconstruc-
tion
while still
retaining
a
display
of elements
of its intervallic
source.
Example
7a
displays
he interval-5
ycle
that serves as
a
point
of
departure
or the
aggregate
of
Example
7b
(the
final
pc
2 of m. 79
begins
a
repetition
of the
pitch-class
material).33
f-
ter
five
pitch
classes
drawn
without variation rom
the
source,
the line
presents
pc
<6,1>,
or a reversal
of source
ops
8 and
9,
followed
by pc
4,
which s the final
note in the
source,
and then
pc
<10,8,3,8>,
a
rearrangement
of
ops
5, 6,
and
7. The INT
notatedbelowExample7billustrates hepresenceof interval5
and
ts inverse
7
within
a
sequence
that also
includes ntervals
1,
3, 6,
and
10. Within
this intervallic
variety,
ntervals
5 and
7 re-
32This ccurs
n the
cello,
mm. 50-52. The
first
presentation
of this
theme
(bassoon
and
trombone,
mm.
46-49)
has some
pitch
naccuracies
hat
prevent
aggregatecompletion,
probably
he results
of
copying
or calculation
rrors.
33In
he
original,
this line is doubled
an octave
lower.
Example
7.
Three-Page
Sonata,
derivation
of left
hand,
mm.77-79.
a.
source
interval-5
op
0
1
2 3
4 5 6 7
8
9
10 11
cycle
p
--
cycl:0
?
b"o
to
t0
f
lo
o
pc
9 2
7
0 5 10 3
8 1
6
11
4
b.
left
pc
9
2
7
0
5
6
1
4
10 8 3 8 11
hand,
mm 77 3 r
3-
78 ---3 r-- 79 ----- 3-m. - 7 3
.-
79
77-79
-
I
K
I
,
.
r
INT
<7-5--5
-
5--1-7-3-
--
10-7-5-3>
produce
<3,10,5,7,2,9,4,11,6,1,0,8>,
accomplished
by
mov-
ing
pc
0 from
op
3 to
op
10.32
A
pitch-class
succession
n Ives's
Three-Page
Sonata
(1905)
employsa moreextensivereorderingn anaggregateconstruc-
tion
while still
retaining
a
display
of elements
of its intervallic
source.
Example
7a
displays
he interval-5
ycle
that serves as
a
point
of
departure
or the
aggregate
of
Example
7b
(the
final
pc
2 of m. 79
begins
a
repetition
of the
pitch-class
material).33
f-
ter
five
pitch
classes
drawn
without variation rom
the
source,
the line
presents
pc
<6,1>,
or a reversal
of source
ops
8 and
9,
followed
by pc
4,
which s the final
note in the
source,
and then
pc
<10,8,3,8>,
a
rearrangement
of
ops
5, 6,
and
7. The INT
notatedbelowExample7billustrates hepresenceof interval5
and
ts inverse
7
within
a
sequence
that also
includes ntervals
1,
3, 6,
and
10. Within
this intervallic
variety,
ntervals
5 and
7 re-
32This ccurs
n the
cello,
mm. 50-52. The
first
presentation
of this
theme
(bassoon
and
trombone,
mm.
46-49)
has some
pitch
naccuracies
hat
prevent
aggregatecompletion,
probably
he results
of
copying
or calculation
rrors.
33In
he
original,
this line is doubled
an octave
lower.
ceive
emphasis
at
the
beginning
of m. 79
from the
repetition
of
pc 8, matching he repetitionof pc 2 in m. 77; this reinforcesa
less
direct connection
with
the source
cycle
in
the latter
part
of
the
pattern.
Ives's most extensive
and most
systematically
omprehen-
sive
employment
of
single-interval
cycles
occurs
in the
early
choral
work
Psalm
24
(1894?).34
The
setting
of each
of
the
ten
versesis based
on a
mirroring
f
inversionally omplementary
cycles
that makes
possible
a
gradual egistral
xpansion
or con-
traction,
simulating
a
registral"wedge"
betweenlines
moving
in contrarymotion. Typically,a mirroring f a givencyclewill
provide
the
pitch-class
source
material,
or
"model,"
for the
verse,
while the
actual
pitch
realization
may employ
octave
dis-
placements
o obscure
a literal
registral
nactment
of the
wedge
shape.
In the
pitch-class
model,
the
gradual
inear
changespro-
duce
gradations
of vertical nterval
sizes
and a consistent
sym-
metrical
axis,
while
octave variations
on the
pattern
n the mu-
sical
settings disrupt
the vertical intervallic
regularity
and
mobilize
the axisof
symmetry.
Verse 1 of the Psalm,forexample,is basedon themirroring
of
intervals
1 and 11 outlined
n
Figure
1. This chromatic
wedge
expands
the vertical
distances
between
voices in
increments
of
two from
0 to 24.
Ives's realization
of the
model,
however,
em-
ploys
the "wide
jumps" echnique
(see Example
2),
so that
the
cyclic
model
determines
only
the
pitch-class
content
of the
voices
involved,
not
a
pitch-specific
egistral
pattern.
Example
8
is
a two-stave
reduction
of verse
1,
illustrating
he octave-
displaced
cycles
of intervals
1 and 11
in the outer
voices,
sup-
ported by chromatic ines in the innervoices. In the firsttwo
measures,
the octave
displacements
are
themselves
mirrored:
both
soprano
and
bass shiftto outer octaves
at the
firstchord
of
m.
2,
emphasizing
the
word "Lord's."
Because
both voices
34As s the
case
with Psalm
54,
Ives
probably
worked on
Psalm 24
with his
father
around
1893-94,
though
no
specific
recollection
of a collaboration
ap-
pears
in Memos or elsewhere.
(See
Memos,
47.)
ceive
emphasis
at
the
beginning
of m. 79
from the
repetition
of
pc 8, matching he repetitionof pc 2 in m. 77; this reinforcesa
less
direct connection
with
the source
cycle
in
the latter
part
of
the
pattern.
Ives's most extensive
and most
systematically
omprehen-
sive
employment
of
single-interval
cycles
occurs
in the
early
choral
work
Psalm
24
(1894?).34
The
setting
of each
of
the
ten
versesis based
on a
mirroring
f
inversionally omplementary
cycles
that makes
possible
a
gradual egistral
xpansion
or con-
traction,
simulating
a
registral"wedge"
betweenlines
moving
in contrarymotion. Typically,a mirroring f a givencyclewill
provide
the
pitch-class
source
material,
or
"model,"
for the
verse,
while the
actual
pitch
realization
may employ
octave
dis-
placements
o obscure
a literal
registral
nactment
of the
wedge
shape.
In the
pitch-class
model,
the
gradual
inear
changespro-
duce
gradations
of vertical nterval
sizes
and a consistent
sym-
metrical
axis,
while
octave variations
on the
pattern
n the mu-
sical
settings disrupt
the vertical intervallic
regularity
and
mobilize
the axisof
symmetry.
Verse 1 of the Psalm,forexample,is basedon themirroring
of
intervals
1 and 11 outlined
n
Figure
1. This chromatic
wedge
expands
the vertical
distances
between
voices in
increments
of
two from
0 to 24.
Ives's realization
of the
model,
however,
em-
ploys
the "wide
jumps" echnique
(see Example
2),
so that
the
cyclic
model
determines
only
the
pitch-class
content
of the
voices
involved,
not
a
pitch-specific
egistral
pattern.
Example
8
is
a two-stave
reduction
of verse
1,
illustrating
he octave-
displaced
cycles
of intervals
1 and 11
in the outer
voices,
sup-
ported by chromatic ines in the innervoices. In the firsttwo
measures,
the octave
displacements
are
themselves
mirrored:
both
soprano
and
bass shiftto outer octaves
at the
firstchord
of
m.
2,
emphasizing
the
word "Lord's."
Because
both voices
34As s the
case
with Psalm
54,
Ives
probably
worked on
Psalm 24
with his
father
around
1893-94,
though
no
specific
recollection
of a collaboration
ap-
pears
in Memos or elsewhere.
(See
Memos,
47.)
ceive
emphasis
at
the
beginning
of m. 79
from the
repetition
of
pc 8, matching he repetitionof pc 2 in m. 77; this reinforcesa
less
direct connection
with
the source
cycle
in
the latter
part
of
the
pattern.
Ives's most extensive
and most
systematically
omprehen-
sive
employment
of
single-interval
cycles
occurs
in the
early
choral
work
Psalm
24
(1894?).34
The
setting
of each
of
the
ten
versesis based
on a
mirroring
f
inversionally omplementary
cycles
that makes
possible
a
gradual egistral
xpansion
or con-
traction,
simulating
a
registral"wedge"
betweenlines
moving
in contrarymotion. Typically,a mirroring f a givencyclewill
provide
the
pitch-class
source
material,
or
"model,"
for the
verse,
while the
actual
pitch
realization
may employ
octave
dis-
placements
o obscure
a literal
registral
nactment
of the
wedge
shape.
In the
pitch-class
model,
the
gradual
inear
changespro-
duce
gradations
of vertical nterval
sizes
and a consistent
sym-
metrical
axis,
while
octave variations
on the
pattern
n the mu-
sical
settings disrupt
the vertical intervallic
regularity
and
mobilize
the axisof
symmetry.
Verse 1 of the Psalm,forexample,is basedon themirroring
of
intervals
1 and 11 outlined
n
Figure
1. This chromatic
wedge
expands
the vertical
distances
between
voices in
increments
of
two from
0 to 24.
Ives's realization
of the
model,
however,
em-
ploys
the "wide
jumps" echnique
(see Example
2),
so that
the
cyclic
model
determines
only
the
pitch-class
content
of the
voices
involved,
not
a
pitch-specific
egistral
pattern.
Example
8
is
a two-stave
reduction
of verse
1,
illustrating
he octave-
displaced
cycles
of intervals
1 and 11
in the outer
voices,
sup-
ported by chromatic ines in the innervoices. In the firsttwo
measures,
the octave
displacements
are
themselves
mirrored:
both
soprano
and
bass shiftto outer octaves
at the
firstchord
of
m.
2,
emphasizing
the
word "Lord's."
Because
both voices
34As s the
case
with Psalm
54,
Ives
probably
worked on
Psalm 24
with his
father
around
1893-94,
though
no
specific
recollection
of a collaboration
ap-
pears
in Memos or elsewhere.
(See
Memos,
47.)
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 12/41
Interval
ycles
as
Compositional
esources
53
nterval
ycles
as
Compositional
esources
53
nterval
ycles
as
Compositional
esources
53
Example
8.
Psalm
24,
verse
1,
two-stavereduction.
The
earth is the Lord's
and
the
ful- ness
there-of,
the world
and
they
that dwell there
-
in.
1i
2 3
4
6
^
^
..
-
F^^r
Example
8.
Psalm
24,
verse
1,
two-stavereduction.
The
earth is the Lord's
and
the
ful- ness
there-of,
the world
and
they
that dwell there
-
in.
1i
2 3
4
6
^
^
..
-
F^^r
Example
8.
Psalm
24,
verse
1,
two-stavereduction.
The
earth is the Lord's
and
the
ful- ness
there-of,
the world
and
they
that dwell there
-
in.
1i
2 3
4
6
^
^
..
-
F^^r
Figure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).igure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).igure 1. Pitch-class model of Psalm 24, verse 1 (mm. 1-6).
12
3 45
6 789
2
3 45
6 789
2
3 45
6 789
0
11
10 9
1
10 9
1
10 9
8
7
6
5
4
3
7
6
5
4
3
7
6
5
4
3
0 2 4
6 8 10
12
14 16 182 4
6 8 10
12
14 16 182 4
6 8 10
12
14 16 18
10
11
2
1
20
22
10
11
2
1
20
22
10
11
2
1
20
22
0
0
0
0
0
0
4444
shift,
the
symmetrical
axis remains stable
(at C4).
Subse-
quently,
in
m.
3,
the
soprano
employs
the octave
leap
while the
bass continues its
half-step
descent,
avoiding
pitches
in the
lower
part
of the bass
register;
this shiftsthe axis
upward
(to
F#4)
for the first two
beats of
m. 3.
The unmirrored
hifts con-
tinue in
the remainder
of
the
verse,
continuallymoving
the axis
andunpredictably arying he sizes of the vertical ntervals.
Similar
proceduresgovern
a
mirroring
f
intervals
2 and 10
in
verse
2,
initiating
a
verse-by-verse
pattern
of
progressive
n-
creases
in
the interval sizes of the
cyclic
model that culminates
with a
cycle
of interval 7
mirrored
by
5
beginning
in
verse
7
(mm.
35-44).
This is followed
by
a
rapid
reduction
n interval
sizes in the final two
verses,
with several
cycles
used
in
each
verse,
concluding
with an interval-1
source
in the final
phrase
shift,
the
symmetrical
axis remains stable
(at C4).
Subse-
quently,
in
m.
3,
the
soprano
employs
the octave
leap
while the
bass continues its
half-step
descent,
avoiding
pitches
in the
lower
part
of the bass
register;
this shiftsthe axis
upward
(to
F#4)
for the first two
beats of
m. 3.
The unmirrored
hifts con-
tinue in
the remainder
of
the
verse,
continuallymoving
the axis
andunpredictably arying he sizes of the vertical ntervals.
Similar
proceduresgovern
a
mirroring
f
intervals
2 and 10
in
verse
2,
initiating
a
verse-by-verse
pattern
of
progressive
n-
creases
in
the interval sizes of the
cyclic
model that culminates
with a
cycle
of interval 7
mirrored
by
5
beginning
in
verse
7
(mm.
35-44).
This is followed
by
a
rapid
reduction
n interval
sizes in the final two
verses,
with several
cycles
used
in
each
verse,
concluding
with an interval-1
source
in the final
phrase
shift,
the
symmetrical
axis remains stable
(at C4).
Subse-
quently,
in
m.
3,
the
soprano
employs
the octave
leap
while the
bass continues its
half-step
descent,
avoiding
pitches
in the
lower
part
of the bass
register;
this shiftsthe axis
upward
(to
F#4)
for the first two
beats of
m. 3.
The unmirrored
hifts con-
tinue in
the remainder
of
the
verse,
continuallymoving
the axis
andunpredictably arying he sizes of the vertical ntervals.
Similar
proceduresgovern
a
mirroring
f
intervals
2 and 10
in
verse
2,
initiating
a
verse-by-verse
pattern
of
progressive
n-
creases
in
the interval sizes of the
cyclic
model that culminates
with a
cycle
of interval 7
mirrored
by
5
beginning
in
verse
7
(mm.
35-44).
This is followed
by
a
rapid
reduction
n interval
sizes in the final two
verses,
with several
cycles
used
in
each
verse,
concluding
with an interval-1
source
in the final
phrase
(mm.
53-57,
not mirrored n the bass
voice).
Thus the overall
structure
of Psalm
24
is based on
a
wedge-like
expansion
and
contraction
of
interval sizes that
is reflected
n
the
cyclic
pitch-
class
model of
each
verse.35
Ives's interest in
maximizingpitch-classvariety
s
naturally
relevant
to a
systematicexploration
of
cycles
such
as Psalm
24.
Intervals1and 2 are sourcesfor verses1 and2, intervals5 and7
control
verses 5 and
7,
respectively,
and interval
3,
despite
a
low
cardinality (4),
is used
(with
pc
repetitions)
in verse
3.
Rather than
employing
nterval
4
(CARD
3)
in verse
4 and in-
(mm.
53-57,
not mirrored n the bass
voice).
Thus the overall
structure
of Psalm
24
is based on
a
wedge-like
expansion
and
contraction
of
interval sizes that
is reflected
n
the
cyclic
pitch-
class
model of
each
verse.35
Ives's interest in
maximizingpitch-classvariety
s
naturally
relevant
to a
systematicexploration
of
cycles
such
as Psalm
24.
Intervals1and 2 are sourcesfor verses1 and2, intervals5 and7
control
verses 5 and
7,
respectively,
and interval
3,
despite
a
low
cardinality (4),
is used
(with
pc
repetitions)
in verse
3.
Rather than
employing
nterval
4
(CARD
3)
in verse
4 and in-
(mm.
53-57,
not mirrored n the bass
voice).
Thus the overall
structure
of Psalm
24
is based on
a
wedge-like
expansion
and
contraction
of
interval sizes that
is reflected
n
the
cyclic
pitch-
class
model of
each
verse.35
Ives's interest in
maximizingpitch-classvariety
s
naturally
relevant
to a
systematicexploration
of
cycles
such
as Psalm
24.
Intervals1and 2 are sourcesfor verses1 and2, intervals5 and7
control
verses 5 and
7,
respectively,
and interval
3,
despite
a
low
cardinality (4),
is used
(with
pc
repetitions)
in verse
3.
Rather than
employing
nterval
4
(CARD
3)
in verse
4 and in-
35Hitchcock escribes
Psalm24
similarly
n Ives:A
Surveyof
the
Music,
31-
32.
35Hitchcock escribes
Psalm24
similarly
n Ives:A
Surveyof
the
Music,
31-
32.
35Hitchcock escribes
Psalm24
similarly
n Ives:A
Surveyof
the
Music,
31-
32.
int.
1
cycle:
int.
11
cycle:
vertical
distance:
int.
1
cycle:
int.
11
cycle:
vertical
distance:
int.
1
cycle:
int.
11
cycle:
vertical
distance:
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 13/41
54 Music
Theory
Spectrum
4 Music
Theory
Spectrum
4 Music
Theory
Spectrum
terval 6
(CARD
2)
in verse
6,
however,
Ives introduces
varia-
tions in the scheme that promote greater pitch-classvariety.
The
pitch-class
models for verses
4 and
6
are based
on alterna-
tions
of intervals:verse
4
alternates
ntervals3
and
4
(mirrored
by
intervals 9 and
8),
and verse 6 alternates
ntervals5
and 6
(mirroredby
7 and
6).
Cyclic
repetitions
of this
type,
essentially
combinations
of
two
single-interval
ycles, represent
a fruitful
area
for further
nquiry
nto the
more
sophisticated
xperimen-
tation of Ives's later
years.
COMBINATIONYCLES. he idea of a "combination cycle," or
a
cyclic
alternation
of two
intervals,
presents
itself
early
in
Ives's works
as,
for
example,
a
combination
of
major
and mi-
nor thirds
(intervals
4 and
3).
In addition
to their
role in the
pitch-classwedge
material
of verse
4 of Psalm
24,
these
inter-
vals combine
to form chords
of
"stacked
hirds"n
organ
works
from the
1890s,
ncluding
an interlude
or a church
ervice36
nd
organ
parts
rom the cantata
The
Celestial
Country
1898-99).37
These
structures
realize a
suggestion
from Ives's
father,
re-
calledby Ives in his Memos: "If two majoror minor3rds can
make
up
a
chord,
why
not more?"38While
Ives's chords
exhibit
different distributions
of the
two
intervals,
some,
such
as the
sonorities
n the Introduction
o No. 7 in The Celestial
Country,
employ
a strict alternation
between the two
intervals
o
gener-
ate a
large, regularly
tructured hord
without
pitch-class
dupli-
cations.39 ves's
exploration
of other intervallic
alternations
raises
questions
about the
gamut
of
possibilities
or these
types
36Transcribed
n
Henry
and
Sidney
Cowell,
Charles
ves and
His
Music,
35.
Kirkpatrick
Catalogue,
107) gives
the
probable
datefor the
interlude
as 1892.
37See,
or
example,
the Prelude
before No.
2 and the Introduction
o
No.
7.
38Ives,
Memos,
47.
See
also
related
recollections,
33 and 120.
39Another nstance
occurs
in Psalm
90,
m. 2. The
stacked
thirds
of
m. 2
recur
hroughout
he
work to serve
as
part
of a "harmonic
eitmotif"
system,
as
described
by
Donald
Grantham,
"A Harmonic
'Leitmotif'
System
in Ives's
Psalm
90,"
In
TheoryOnly
5/2
(1979),
3-14.
terval 6
(CARD
2)
in verse
6,
however,
Ives introduces
varia-
tions in the scheme that promote greater pitch-classvariety.
The
pitch-class
models for verses
4 and
6
are based
on alterna-
tions
of intervals:verse
4
alternates
ntervals3
and
4
(mirrored
by
intervals 9 and
8),
and verse 6 alternates
ntervals5
and 6
(mirroredby
7 and
6).
Cyclic
repetitions
of this
type,
essentially
combinations
of
two
single-interval
ycles, represent
a fruitful
area
for further
nquiry
nto the
more
sophisticated
xperimen-
tation of Ives's later
years.
COMBINATIONYCLES. he idea of a "combination cycle," or
a
cyclic
alternation
of two
intervals,
presents
itself
early
in
Ives's works
as,
for
example,
a
combination
of
major
and mi-
nor thirds
(intervals
4 and
3).
In addition
to their
role in the
pitch-classwedge
material
of verse
4 of Psalm
24,
these
inter-
vals combine
to form chords
of
"stacked
hirds"n
organ
works
from the
1890s,
ncluding
an interlude
or a church
ervice36
nd
organ
parts
rom the cantata
The
Celestial
Country
1898-99).37
These
structures
realize a
suggestion
from Ives's
father,
re-
calledby Ives in his Memos: "If two majoror minor3rds can
make
up
a
chord,
why
not more?"38While
Ives's chords
exhibit
different distributions
of the
two
intervals,
some,
such
as the
sonorities
n the Introduction
o No. 7 in The Celestial
Country,
employ
a strict alternation
between the two
intervals
o
gener-
ate a
large, regularly
tructured hord
without
pitch-class
dupli-
cations.39 ves's
exploration
of other intervallic
alternations
raises
questions
about the
gamut
of
possibilities
or these
types
36Transcribed
n
Henry
and
Sidney
Cowell,
Charles
ves and
His
Music,
35.
Kirkpatrick
Catalogue,
107) gives
the
probable
datefor the
interlude
as 1892.
37See,
or
example,
the Prelude
before No.
2 and the Introduction
o
No.
7.
38Ives,
Memos,
47.
See
also
related
recollections,
33 and 120.
39Another nstance
occurs
in Psalm
90,
m. 2. The
stacked
thirds
of
m. 2
recur
hroughout
he
work to serve
as
part
of a "harmonic
eitmotif"
system,
as
described
by
Donald
Grantham,
"A Harmonic
'Leitmotif'
System
in Ives's
Psalm
90,"
In
TheoryOnly
5/2
(1979),
3-14.
terval 6
(CARD
2)
in verse
6,
however,
Ives introduces
varia-
tions in the scheme that promote greater pitch-classvariety.
The
pitch-class
models for verses
4 and
6
are based
on alterna-
tions
of intervals:verse
4
alternates
ntervals3
and
4
(mirrored
by
intervals 9 and
8),
and verse 6 alternates
ntervals5
and 6
(mirroredby
7 and
6).
Cyclic
repetitions
of this
type,
essentially
combinations
of
two
single-interval
ycles, represent
a fruitful
area
for further
nquiry
nto the
more
sophisticated
xperimen-
tation of Ives's later
years.
COMBINATIONYCLES. he idea of a "combination cycle," or
a
cyclic
alternation
of two
intervals,
presents
itself
early
in
Ives's works
as,
for
example,
a
combination
of
major
and mi-
nor thirds
(intervals
4 and
3).
In addition
to their
role in the
pitch-classwedge
material
of verse
4 of Psalm
24,
these
inter-
vals combine
to form chords
of
"stacked
hirds"n
organ
works
from the
1890s,
ncluding
an interlude
or a church
ervice36
nd
organ
parts
rom the cantata
The
Celestial
Country
1898-99).37
These
structures
realize a
suggestion
from Ives's
father,
re-
calledby Ives in his Memos: "If two majoror minor3rds can
make
up
a
chord,
why
not more?"38While
Ives's chords
exhibit
different distributions
of the
two
intervals,
some,
such
as the
sonorities
n the Introduction
o No. 7 in The Celestial
Country,
employ
a strict alternation
between the two
intervals
o
gener-
ate a
large, regularly
tructured hord
without
pitch-class
dupli-
cations.39 ves's
exploration
of other intervallic
alternations
raises
questions
about the
gamut
of
possibilities
or these
types
36Transcribed
n
Henry
and
Sidney
Cowell,
Charles
ves and
His
Music,
35.
Kirkpatrick
Catalogue,
107) gives
the
probable
datefor the
interlude
as 1892.
37See,
or
example,
the Prelude
before No.
2 and the Introduction
o
No.
7.
38Ives,
Memos,
47.
See
also
related
recollections,
33 and 120.
39Another nstance
occurs
in Psalm
90,
m. 2. The
stacked
thirds
of
m. 2
recur
hroughout
he
work to serve
as
part
of a "harmonic
eitmotif"
system,
as
described
by
Donald
Grantham,
"A Harmonic
'Leitmotif'
System
in Ives's
Psalm
90,"
In
TheoryOnly
5/2
(1979),
3-14.
of
structures,
their
ability
to
generate pitch-class
variety,
and
theircontributionso Ives'scompositional anguage.
The
alternation
of
intervals
4
and 3
in
Figure
2,
a
"4/3"
com-
bination
cycle,
can be viewed
as a combination
of
an "A set"
of
pitch
classes
in
the
even-numbered
order
positions
with a
"B
set"
in the
odd-numbered
positions.
Both the
A
and B
sets
in
this
case
are
generated
by
cycles
of interval
7,
which s the
sum
(mod
12)
of the two
alternating
ntervals.
Conceptually,
he
A
and
B sets
constitute
an
"overlay"
of a
cycle
with itself
ata
par-
ticular
nterval:
he 4/3
cycle
overlays
wo interval-7
ycles
at
a
distanceof interval4.40ViewingIves's ntervallic lternations s
combination
cycles
assumes
a
transpositional
quivalence
be-
tween
the combined
cycles,
and
encompasses
any possible
wo-
interval
alternation.41
n
general
terms,
x +
y
=
n
(mod
12),
where
x and
y
are
any
two
alternating
ntervals
overlaying
ycles
of interval
n.
The
possible
values
of x and
y
for
a
given
n are
defined
by
the
operator
cycles
TnI
(see
Table
1).
For
example,
where
n
=
7,
as in
Figure
2,
the
possible
values
of
x and
y
are
indicated
by
T7I
cycles:
x/y
=
0/7, 7/0,
1/6, 6/1,2/5, 5/2,
3/4, 4/3, 8/11,
11/8,
9/10,
or
10/9.
A
simple
exchange
of the
x/y
values,
as in
converting
9/10
to
10/9,
has
only
a subtle
effect
on the
presentation
of the
two
al-
4This
approach
bears
similarities
o that
of Elliott
Antokoletz,
in The
Mu-
sic
ofBela
Bart6k,
in which
nterval-1
cycles
are
combined
n order
to
explain
symmetrical
pitch
constructions,
hough
his
study
does not
extend
the
concept
to
encompass
combinations
of other
cycles.
Along
these
same
lines,
the
com-
positionalsystemof GeorgePerle outlined n Twelve-ToneTonalityBerkeley:
University
of California
Press,
1977)
combines
cycles
of
every
interval,
but is
primarily
concerned
with
inverse-related
cycles
and their
compositional
applications.
41The
oncept
may
be
naturally
extended
to
encompass
more
thantwo
in-
tervals
n
alternation,
thus
possibly
nvolving
a combination
of several
cycles.
Ives
rarely
uses
such a
sequence,
and the
present
discussion
s
limited
to
two
alternating
ntervals.
Morris's
oncept
of
a
cyclic
INT
would
apply
o
cyclicrep-
etitions
of
any type (Composition
WithPitch
Classes,40,
107).
of
structures,
their
ability
to
generate pitch-class
variety,
and
theircontributionso Ives'scompositional anguage.
The
alternation
of
intervals
4
and 3
in
Figure
2,
a
"4/3"
com-
bination
cycle,
can be viewed
as a combination
of
an "A set"
of
pitch
classes
in
the
even-numbered
order
positions
with a
"B
set"
in the
odd-numbered
positions.
Both the
A
and B
sets
in
this
case
are
generated
by
cycles
of interval
7,
which s the
sum
(mod
12)
of the two
alternating
ntervals.
Conceptually,
he
A
and
B sets
constitute
an
"overlay"
of a
cycle
with itself
ata
par-
ticular
nterval:
he 4/3
cycle
overlays
wo interval-7
ycles
at
a
distanceof interval4.40ViewingIves's ntervallic lternations s
combination
cycles
assumes
a
transpositional
quivalence
be-
tween
the combined
cycles,
and
encompasses
any possible
wo-
interval
alternation.41
n
general
terms,
x +
y
=
n
(mod
12),
where
x and
y
are
any
two
alternating
ntervals
overlaying
ycles
of interval
n.
The
possible
values
of x and
y
for
a
given
n are
defined
by
the
operator
cycles
TnI
(see
Table
1).
For
example,
where
n
=
7,
as in
Figure
2,
the
possible
values
of
x and
y
are
indicated
by
T7I
cycles:
x/y
=
0/7, 7/0,
1/6, 6/1,2/5, 5/2,
3/4, 4/3, 8/11,
11/8,
9/10,
or
10/9.
A
simple
exchange
of the
x/y
values,
as in
converting
9/10
to
10/9,
has
only
a subtle
effect
on the
presentation
of the
two
al-
4This
approach
bears
similarities
o that
of Elliott
Antokoletz,
in The
Mu-
sic
ofBela
Bart6k,
in which
nterval-1
cycles
are
combined
n order
to
explain
symmetrical
pitch
constructions,
hough
his
study
does not
extend
the
concept
to
encompass
combinations
of other
cycles.
Along
these
same
lines,
the
com-
positionalsystemof GeorgePerle outlined n Twelve-ToneTonalityBerkeley:
University
of California
Press,
1977)
combines
cycles
of
every
interval,
but is
primarily
concerned
with
inverse-related
cycles
and their
compositional
applications.
41The
oncept
may
be
naturally
extended
to
encompass
more
thantwo
in-
tervals
n
alternation,
thus
possibly
nvolving
a combination
of several
cycles.
Ives
rarely
uses
such a
sequence,
and the
present
discussion
s
limited
to
two
alternating
ntervals.
Morris's
oncept
of
a
cyclic
INT
would
apply
o
cyclicrep-
etitions
of
any type (Composition
WithPitch
Classes,40,
107).
of
structures,
their
ability
to
generate pitch-class
variety,
and
theircontributionso Ives'scompositional anguage.
The
alternation
of
intervals
4
and 3
in
Figure
2,
a
"4/3"
com-
bination
cycle,
can be viewed
as a combination
of
an "A set"
of
pitch
classes
in
the
even-numbered
order
positions
with a
"B
set"
in the
odd-numbered
positions.
Both the
A
and B
sets
in
this
case
are
generated
by
cycles
of interval
7,
which s the
sum
(mod
12)
of the two
alternating
ntervals.
Conceptually,
he
A
and
B sets
constitute
an
"overlay"
of a
cycle
with itself
ata
par-
ticular
nterval:
he 4/3
cycle
overlays
wo interval-7
ycles
at
a
distanceof interval4.40ViewingIves's ntervallic lternations s
combination
cycles
assumes
a
transpositional
quivalence
be-
tween
the combined
cycles,
and
encompasses
any possible
wo-
interval
alternation.41
n
general
terms,
x +
y
=
n
(mod
12),
where
x and
y
are
any
two
alternating
ntervals
overlaying
ycles
of interval
n.
The
possible
values
of x and
y
for
a
given
n are
defined
by
the
operator
cycles
TnI
(see
Table
1).
For
example,
where
n
=
7,
as in
Figure
2,
the
possible
values
of
x and
y
are
indicated
by
T7I
cycles:
x/y
=
0/7, 7/0,
1/6, 6/1,2/5, 5/2,
3/4, 4/3, 8/11,
11/8,
9/10,
or
10/9.
A
simple
exchange
of the
x/y
values,
as in
converting
9/10
to
10/9,
has
only
a subtle
effect
on the
presentation
of the
two
al-
4This
approach
bears
similarities
o that
of Elliott
Antokoletz,
in The
Mu-
sic
ofBela
Bart6k,
in which
nterval-1
cycles
are
combined
n order
to
explain
symmetrical
pitch
constructions,
hough
his
study
does not
extend
the
concept
to
encompass
combinations
of other
cycles.
Along
these
same
lines,
the
com-
positionalsystemof GeorgePerle outlined n Twelve-ToneTonalityBerkeley:
University
of California
Press,
1977)
combines
cycles
of
every
interval,
but is
primarily
concerned
with
inverse-related
cycles
and their
compositional
applications.
41The
oncept
may
be
naturally
extended
to
encompass
more
thantwo
in-
tervals
n
alternation,
thus
possibly
nvolving
a combination
of several
cycles.
Ives
rarely
uses
such a
sequence,
and the
present
discussion
s
limited
to
two
alternating
ntervals.
Morris's
oncept
of
a
cyclic
INT
would
apply
o
cyclicrep-
etitions
of
any type (Composition
WithPitch
Classes,40,
107).
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 14/41
Interval
ycles
as
Compositional
esources
55
nterval
ycles
as
Compositional
esources
55
nterval
ycles
as
Compositional
esources
55
Figure
2. 4/3 combination
cycle.
igure
2. 4/3 combination
cycle.
igure
2. 4/3 combination
cycle.
op:
0
B set:
(pc)
A set:
0
(pc)
I
op:
0
B set:
(pc)
A set:
0
(pc)
I
op:
0
B set:
(pc)
A set:
0
(pc)
I
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19
20
21 22
23
4 11 6
1
8 3 10
5 0 7
2
9
7 2 9 4 11
6
1
8 3 10
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19
20
21 22
23
4 11 6
1
8 3 10
5 0 7
2
9
7 2 9 4 11
6
1
8 3 10
5
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15 16 17 18 19
20
21 22
23
4 11 6
1
8 3 10
5 0 7
2
9
7 2 9 4 11
6
1
8 3 10
5
PCL
=
8CL
=
8CL
=
8
first
repetition
irst
repetition
irst
repetition
ternating
intervals,
but
may
more
significantly
nfluence as-
pects
of
the
pitch-class
succession.
Where n is
even,
the
presence
of a
singleton
cycle
indicates
that
x
=
y,
expressing
a
single-interval ycle
as an alternation
of two identical
ntervals.
If
n
=
10,
for
example, x/y
can be 5/5
or 11/11 n addition o
0/10,
1/9,
andso
on,
because 5 and11are
the
singletons
of
the
T1oI
cycles.
A subdivision
of
this
type
within
an
interval-5
cycle
(x/y
=
5/5)
is
apparent
n
Example
1,
where Ives highlights he underlyingnterval-10cycle through
accentuationand
registral
association.
Though
a
combination
nvolving
a CARD-12
cycle
will con-
tinue
to
op
23,
Ives
is
typically
most concernedwith the number
of
pcs prior
to
repetition,
the
"pitch-class
ength"
(PCL)
of a
combination
cycle.
The PCL of the 4/3
cycle
illustrated
n
Fig-
ure 2
is
8,
because
eight pcs (op 0-7)
are stated
prior
o the first
repetition:pc
4 at
op
8
is
a
repetition
of
the
pc
at
op
1.
Thus,
in a
numbering
of order
positions
that
begins
with
0,
the
op
of
the
firstrepeated pc will correspond o the value of the PCL.
The value of the PCLis a
function
of
the CARD
of
the over-
laid
single-interval ycle
and of
the content
of
the
A
and
B
sets.
For
combinations
of
cycles
of CARD
12,
the
pitch
classes
of A
and B will
always
be
the
same,
allowing
a
variety
of
possible
PCL
values. Overlaid
cycles
with
cardinalities maller
han
12,
however,
exhibit
either
complete equivalence
of
pitch-class
content
between the A and B
sets or
complete
nonequivalence.
ternating
intervals,
but
may
more
significantly
nfluence as-
pects
of
the
pitch-class
succession.
Where n is
even,
the
presence
of a
singleton
cycle
indicates
that
x
=
y,
expressing
a
single-interval ycle
as an alternation
of two identical
ntervals.
If
n
=
10,
for
example, x/y
can be 5/5
or 11/11 n addition o
0/10,
1/9,
andso
on,
because 5 and11are
the
singletons
of
the
T1oI
cycles.
A subdivision
of
this
type
within
an
interval-5
cycle
(x/y
=
5/5)
is
apparent
n
Example
1,
where Ives highlights he underlyingnterval-10cycle through
accentuationand
registral
association.
Though
a
combination
nvolving
a CARD-12
cycle
will con-
tinue
to
op
23,
Ives
is
typically
most concernedwith the number
of
pcs prior
to
repetition,
the
"pitch-class
ength"
(PCL)
of a
combination
cycle.
The PCL of the 4/3
cycle
illustrated
n
Fig-
ure 2
is
8,
because
eight pcs (op 0-7)
are stated
prior
o the first
repetition:pc
4 at
op
8
is
a
repetition
of
the
pc
at
op
1.
Thus,
in a
numbering
of order
positions
that
begins
with
0,
the
op
of
the
firstrepeated pc will correspond o the value of the PCL.
The value of the PCLis a
function
of
the CARD
of
the over-
laid
single-interval ycle
and of
the content
of
the
A
and
B
sets.
For
combinations
of
cycles
of CARD
12,
the
pitch
classes
of A
and B will
always
be
the
same,
allowing
a
variety
of
possible
PCL
values. Overlaid
cycles
with
cardinalities maller
han
12,
however,
exhibit
either
complete equivalence
of
pitch-class
content
between the A and B
sets or
complete
nonequivalence.
ternating
intervals,
but
may
more
significantly
nfluence as-
pects
of
the
pitch-class
succession.
Where n is
even,
the
presence
of a
singleton
cycle
indicates
that
x
=
y,
expressing
a
single-interval ycle
as an alternation
of two identical
ntervals.
If
n
=
10,
for
example, x/y
can be 5/5
or 11/11 n addition o
0/10,
1/9,
andso
on,
because 5 and11are
the
singletons
of
the
T1oI
cycles.
A subdivision
of
this
type
within
an
interval-5
cycle
(x/y
=
5/5)
is
apparent
n
Example
1,
where Ives highlights he underlyingnterval-10cycle through
accentuationand
registral
association.
Though
a
combination
nvolving
a CARD-12
cycle
will con-
tinue
to
op
23,
Ives
is
typically
most concernedwith the number
of
pcs prior
to
repetition,
the
"pitch-class
ength"
(PCL)
of a
combination
cycle.
The PCL of the 4/3
cycle
illustrated
n
Fig-
ure 2
is
8,
because
eight pcs (op 0-7)
are stated
prior
o the first
repetition:pc
4 at
op
8
is
a
repetition
of
the
pc
at
op
1.
Thus,
in a
numbering
of order
positions
that
begins
with
0,
the
op
of
the
firstrepeated pc will correspond o the value of the PCL.
The value of the PCLis a
function
of
the CARD
of
the over-
laid
single-interval ycle
and of
the content
of
the
A
and
B
sets.
For
combinations
of
cycles
of CARD
12,
the
pitch
classes
of A
and B will
always
be
the
same,
allowing
a
variety
of
possible
PCL
values. Overlaid
cycles
with
cardinalities maller
han
12,
however,
exhibit
either
complete equivalence
of
pitch-class
content
between the A and B
sets or
complete
nonequivalence.
The combinationwill be
termed "reiterative"where the
pitch-
class content of A and
B
is
the
same,
or
"nonreiterative"where
the
pitch-class
content of
the A and B sets
is different. Mem-
bership
of
any
combination none of these two
categories
s dis-
played
by
the
x/y
values: if x and
y
are
multiples
mod
12)
of
n,
thecombination s
reiterative,
and
f x
and
y
are not
multiples
of
n the combination s
nonreiterative.
The four
CARD-12
cycles
(n
=
1, 5, 7,
11)
generate only
reiterative
combinations
be-
cause allpossible x/yvalues aremultiples mod 12)of the n val-
ues,
while the other
cycles may
or
may
not
generate
reiterative
combinations.
For
example,
where
n
=
2,
reiterative
ombina-
tions result
when
x/y
=
(multiples
of
2)
0/2, 2/0,
4/10,
10/4,
6/8,
and
8/6,
but
nonreiterative
combinations
result when
x/y
=
(non-multiples)
1/1, 3/11,
11/3,
5/9,
9/5,
and 7/7. In the
caseof interval-2
ycles
(and
nterval-10),
a nonreiterative om-
bination
produces
aggregate completion
from
combining
the
odd
and even whole-tone collections.
Prediction of the PCL in nonreiterativecombinationsis
achieved
by doubling
the
CARD
of the
n-cycle:
PCL
=
CARD(n)
x
2. That
is,
the numberof
pitch
classes in both
A
and
B
together
totals the number
of
unique
pitch
classes n the
segment.
PCL
prediction
or
reiterativecombinations
ntails a
more extensive calculation
procedure.Everypossible
combina-
tion of a
cycle
with
itself,
and
thus
every
PCL
value,
is
displayed
in a
comparison
of a
constant
A
set
with
each rotation
of
The combinationwill be
termed "reiterative"where the
pitch-
class content of A and
B
is
the
same,
or
"nonreiterative"where
the
pitch-class
content of
the A and B sets
is different. Mem-
bership
of
any
combination none of these two
categories
s dis-
played
by
the
x/y
values: if x and
y
are
multiples
mod
12)
of
n,
thecombination s
reiterative,
and
f x
and
y
are not
multiples
of
n the combination s
nonreiterative.
The four
CARD-12
cycles
(n
=
1, 5, 7,
11)
generate only
reiterative
combinations
be-
cause allpossible x/yvalues aremultiples mod 12)of the n val-
ues,
while the other
cycles may
or
may
not
generate
reiterative
combinations.
For
example,
where
n
=
2,
reiterative
ombina-
tions result
when
x/y
=
(multiples
of
2)
0/2, 2/0,
4/10,
10/4,
6/8,
and
8/6,
but
nonreiterative
combinations
result when
x/y
=
(non-multiples)
1/1, 3/11,
11/3,
5/9,
9/5,
and 7/7. In the
caseof interval-2
ycles
(and
nterval-10),
a nonreiterative om-
bination
produces
aggregate completion
from
combining
the
odd
and even whole-tone collections.
Prediction of the PCL in nonreiterativecombinationsis
achieved
by doubling
the
CARD
of the
n-cycle:
PCL
=
CARD(n)
x
2. That
is,
the numberof
pitch
classes in both
A
and
B
together
totals the number
of
unique
pitch
classes n the
segment.
PCL
prediction
or
reiterativecombinations
ntails a
more extensive calculation
procedure.Everypossible
combina-
tion of a
cycle
with
itself,
and
thus
every
PCL
value,
is
displayed
in a
comparison
of a
constant
A
set
with
each rotation
of
The combinationwill be
termed "reiterative"where the
pitch-
class content of A and
B
is
the
same,
or
"nonreiterative"where
the
pitch-class
content of
the A and B sets
is different. Mem-
bership
of
any
combination none of these two
categories
s dis-
played
by
the
x/y
values: if x and
y
are
multiples
mod
12)
of
n,
thecombination s
reiterative,
and
f x
and
y
are not
multiples
of
n the combination s
nonreiterative.
The four
CARD-12
cycles
(n
=
1, 5, 7,
11)
generate only
reiterative
combinations
be-
cause allpossible x/yvalues aremultiples mod 12)of the n val-
ues,
while the other
cycles may
or
may
not
generate
reiterative
combinations.
For
example,
where
n
=
2,
reiterative
ombina-
tions result
when
x/y
=
(multiples
of
2)
0/2, 2/0,
4/10,
10/4,
6/8,
and
8/6,
but
nonreiterative
combinations
result when
x/y
=
(non-multiples)
1/1, 3/11,
11/3,
5/9,
9/5,
and 7/7. In the
caseof interval-2
ycles
(and
nterval-10),
a nonreiterative om-
bination
produces
aggregate completion
from
combining
the
odd
and even whole-tone collections.
Prediction of the PCL in nonreiterativecombinationsis
achieved
by doubling
the
CARD
of the
n-cycle:
PCL
=
CARD(n)
x
2. That
is,
the numberof
pitch
classes in both
A
and
B
together
totals the number
of
unique
pitch
classes n the
segment.
PCL
prediction
or
reiterativecombinations
ntails a
more extensive calculation
procedure.Everypossible
combina-
tion of a
cycle
with
itself,
and
thus
every
PCL
value,
is
displayed
in a
comparison
of a
constant
A
set
with
each rotation
of
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 15/41
56
Music
Theory
Spectrum
6
Music
Theory
Spectrum
6
Music
Theory
Spectrum
Table 1.
Values for
x/y
and
PCL
n combination
ycles.
[x/y,PCL] [etc.]
n R=0
R= 1 R=2 R=3 R=4 R=
1
0/1,
1
1/0,
2
2/11,4 3/10,6 4/9,8
5/E
2
0/2,
1
2/0,
2
4/10,4 6/8,
6
8/6,
5 10/,
*1/1,12 3/11,12 5/9,12 7/7,12
9/5,12
11/
3
0/3,
1
3/0,
2
6/9,
4
9/6,
3
*1/2,
8
4/11,8 7/8,
8
10/5,8
*2/1,
8
5/10,8
8/7,
8
11/4,8
4
0/4,
1
4/0,
2
8/8,
3
*1/3,
6
5/11,6 9/7,
6
*2/2,
6
6/10,6 10/6,6
*3/1,
6
7/9,
6
11/5,6
5
0/5,
1
5/0,
2
10/7,4 3/2,
6
8/9,
8 1/'
6
0/6,
1
6/0,
2
*1/5,
4
7/11,4
*2/4,
4
8/10,4
*3/3,
4
9/9,
4
*4/2,
4
10/8,
4
*5/1,
4
11/7,
4
7
0/7,
1
7/0,
2
2/5,
4
9/10,
6
4/3,
8 11/
8
0/8,
1
8/0,
2
4/4,
3
*1/7,
6
9/11,6 5/3,
6
*2/6,
6
10/10,6 6/2,
6
*3/5,
6
11/9,
6
7/1,
6
9
0/9,
1
9/0,
2
6/3,
4
3/6,
3
*1/8,
8
10/11,8 7/2,
8
4/5,
8
*2/7,
8
11/10,8 8/1,
8
5/4,
8
10
0/10,1 10/0,
2
8/2,
4
6/4,
6
4/6,
5
2/
*1/9,12
11/11,12
9/1,12 7/3,12
5/5,12
3/
11
0/11,1
11/0,
2
10/1,
4
9/2,
6
8/3,
8
7/
Table 1.
Values for
x/y
and
PCL
n combination
ycles.
[x/y,PCL] [etc.]
n R=0
R= 1 R=2 R=3 R=4 R=
1
0/1,
1
1/0,
2
2/11,4 3/10,6 4/9,8
5/E
2
0/2,
1
2/0,
2
4/10,4 6/8,
6
8/6,
5 10/,
*1/1,12 3/11,12 5/9,12 7/7,12
9/5,12
11/
3
0/3,
1
3/0,
2
6/9,
4
9/6,
3
*1/2,
8
4/11,8 7/8,
8
10/5,8
*2/1,
8
5/10,8
8/7,
8
11/4,8
4
0/4,
1
4/0,
2
8/8,
3
*1/3,
6
5/11,6 9/7,
6
*2/2,
6
6/10,6 10/6,6
*3/1,
6
7/9,
6
11/5,6
5
0/5,
1
5/0,
2
10/7,4 3/2,
6
8/9,
8 1/'
6
0/6,
1
6/0,
2
*1/5,
4
7/11,4
*2/4,
4
8/10,4
*3/3,
4
9/9,
4
*4/2,
4
10/8,
4
*5/1,
4
11/7,
4
7
0/7,
1
7/0,
2
2/5,
4
9/10,
6
4/3,
8 11/
8
0/8,
1
8/0,
2
4/4,
3
*1/7,
6
9/11,6 5/3,
6
*2/6,
6
10/10,6 6/2,
6
*3/5,
6
11/9,
6
7/1,
6
9
0/9,
1
9/0,
2
6/3,
4
3/6,
3
*1/8,
8
10/11,8 7/2,
8
4/5,
8
*2/7,
8
11/10,8 8/1,
8
5/4,
8
10
0/10,1 10/0,
2
8/2,
4
6/4,
6
4/6,
5
2/
*1/9,12
11/11,12
9/1,12 7/3,12
5/5,12
3/
11
0/11,1
11/0,
2
10/1,
4
9/2,
6
8/3,
8
7/
Table 1.
Values for
x/y
and
PCL
n combination
ycles.
[x/y,PCL] [etc.]
n R=0
R= 1 R=2 R=3 R=4 R=
1
0/1,
1
1/0,
2
2/11,4 3/10,6 4/9,8
5/E
2
0/2,
1
2/0,
2
4/10,4 6/8,
6
8/6,
5 10/,
*1/1,12 3/11,12 5/9,12 7/7,12
9/5,12
11/
3
0/3,
1
3/0,
2
6/9,
4
9/6,
3
*1/2,
8
4/11,8 7/8,
8
10/5,8
*2/1,
8
5/10,8
8/7,
8
11/4,8
4
0/4,
1
4/0,
2
8/8,
3
*1/3,
6
5/11,6 9/7,
6
*2/2,
6
6/10,6 10/6,6
*3/1,
6
7/9,
6
11/5,6
5
0/5,
1
5/0,
2
10/7,4 3/2,
6
8/9,
8 1/'
6
0/6,
1
6/0,
2
*1/5,
4
7/11,4
*2/4,
4
8/10,4
*3/3,
4
9/9,
4
*4/2,
4
10/8,
4
*5/1,
4
11/7,
4
7
0/7,
1
7/0,
2
2/5,
4
9/10,
6
4/3,
8 11/
8
0/8,
1
8/0,
2
4/4,
3
*1/7,
6
9/11,6 5/3,
6
*2/6,
6
10/10,6 6/2,
6
*3/5,
6
11/9,
6
7/1,
6
9
0/9,
1
9/0,
2
6/3,
4
3/6,
3
*1/8,
8
10/11,8 7/2,
8
4/5,
8
*2/7,
8
11/10,8 8/1,
8
5/4,
8
10
0/10,1 10/0,
2
8/2,
4
6/4,
6
4/6,
5
2/
*1/9,12
11/11,12
9/1,12 7/3,12
5/5,12
3/
11
0/11,1
11/0,
2
10/1,
4
9/2,
6
8/3,
8
7/
= 5
3,10
4,3
3,12
= 5
3,10
4,3
3,12
= 5
3,10
4,3
3,12
R=6
6/7,12
R=6
6/7,12
R=6
6/7,12
R=7
7/6,11
R=7
7/6,11
R=7
7/6,11
4,10
6/11,12 11/6,11
,10
6/11,12 11/6,11
,10
6/11,12 11/6,11
8,10 6/1,12
8,
3
7,12
4,10
6/5,12
8,10 6/1,12
8,
3
7,12
4,10
6/5,12
8,10 6/1,12
8,
3
7,12
4,10
6/5,12
R=8
8/5,9
R=8
8/5,9
R=8
8/5,9
R=9
9/4,7
R=9
9/4,7
R=9
9/4,7
R
=
10 R
=
11
10/3,5 11/2,3
R
=
10 R
=
11
10/3,5 11/2,3
R
=
10 R
=
11
10/3,5 11/2,3
4/1,
9
9/8,
7
2/3,
5
7/10,3/1,
9
9/8,
7
2/3,
5
7/10,3/1,
9
9/8,
7
2/3,
5
7/10,3
1/6,11
8/11,9
/6,11
8/11,9
/6,11
8/11,9 3/4,7 10/9,5 5/2,3/4,7 10/9,5 5/2,3/4,7 10/9,5 5/2,3
5/6,11 4/7,
9
3/8,7 2/9,5
1/10,3
*rows
preceded
by
asterisks ist nonreiterativecombinations
5/6,11 4/7,
9
3/8,7 2/9,5
1/10,3
*rows
preceded
by
asterisks ist nonreiterativecombinations
5/6,11 4/7,
9
3/8,7 2/9,5
1/10,3
*rows
preceded
by
asterisks ist nonreiterativecombinations
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 16/41
Interval
ycles
as
Compositional
esources 57
nterval
ycles
as
Compositional
esources 57
nterval
ycles
as
Compositional
esources 57
Table
1
(cont.)
able
1
(cont.)
able
1
(cont.)
TI
Cycles
I
Cycles
I
Cycles
T1I
(0-1)
T2I
(0-2)
T31
(0-3)
T41
(0-4)
T5I
(0-5)
T61
(0-6)
T7I
(0-7)
T8I
(0-8)
TgI
(0-9)
TloI (0-10)
TllI (0-11)
T1I
(0-1)
T2I
(0-2)
T31
(0-3)
T41
(0-4)
T5I
(0-5)
T61
(0-6)
T7I
(0-7)
T8I
(0-8)
TgI
(0-9)
TloI (0-10)
TllI (0-11)
T1I
(0-1)
T2I
(0-2)
T31
(0-3)
T41
(0-4)
T5I
(0-5)
T61
(0-6)
T7I
(0-7)
T8I
(0-8)
TgI
(0-9)
TloI (0-10)
TllI (0-11)
(2-11)
(1)
(1-2)
(1-3)
(1-4)
(1-5)
(1-6)
(1-7)
(1-8)
(1-9)
(1-10)
(2-11)
(1)
(1-2)
(1-3)
(1-4)
(1-5)
(1-6)
(1-7)
(1-8)
(1-9)
(1-10)
(2-11)
(1)
(1-2)
(1-3)
(1-4)
(1-5)
(1-6)
(1-7)
(1-8)
(1-9)
(1-10)
(3-10)
(3-11)
(4-11)
(2)
(2-3)
(2-4)
(2-5)
(2-6)
(2-7)
(2-8)
(2-9)
(3-10)
(3-11)
(4-11)
(2)
(2-3)
(2-4)
(2-5)
(2-6)
(2-7)
(2-8)
(2-9)
(3-10)
(3-11)
(4-11)
(2)
(2-3)
(2-4)
(2-5)
(2-6)
(2-7)
(2-8)
(2-9)
(4-9)
(4-10)
(5-10)
(5-11)
(6-11)
(3)
(3-4)
(3-5)
(3-6)
(3-7)
(3-8)
(4-9)
(4-10)
(5-10)
(5-11)
(6-11)
(3)
(3-4)
(3-5)
(3-6)
(3-7)
(3-8)
(4-9)
(4-10)
(5-10)
(5-11)
(6-11)
(3)
(3-4)
(3-5)
(3-6)
(3-7)
(3-8)
(5-8)
(5-9)
(6-9)
(6-10)
(7-10)
(7-11)
(8-11)
(4)
(4-5)
(4-6)
(4-7)
(5-8)
(5-9)
(6-9)
(6-10)
(7-10)
(7-11)
(8-11)
(4)
(4-5)
(4-6)
(4-7)
(5-8)
(5-9)
(6-9)
(6-10)
(7-10)
(7-11)
(8-11)
(4)
(4-5)
(4-6)
(4-7)
(6-7)
(6-8)
(7-8)
(7-9)
(8-9)
(8-10)
(9-10)
(9-11)
(10-11)
(5)
(5-6)
(6-7)
(6-8)
(7-8)
(7-9)
(8-9)
(8-10)
(9-10)
(9-11)
(10-11)
(5)
(5-6)
(6-7)
(6-8)
(7-8)
(7-9)
(8-9)
(8-10)
(9-10)
(9-11)
(10-11)
(5)
(5-6)
the
(pitch-class
equivalent)
B
set;
CARD
different rotations
are
possible.
The PCL of a
specific
combination, herefore,
s a
result of the
number of
rotations of B with
respect
to A.
The
calculation
procedure
for
the
PCL
begins
with the establish-
ment of
order
position
indicators or
the
A
set, or,
in Morris's
notation,
the order
mapping
of
segment A, OMAA.42
The
OMAA
s
numbered from 0
through
CARD-
1,
as shown be-
low
the A set of
the 3/2
combination
cycle
in
Figure
3a
(n
=
5),
for
comparison
with
OMAB,
the order
mappings
of B with re-
spect
to A.
A
four-step
procedure
calculates he PCL:
1.
Determine
R
=
OMABo.
In other
words,
R is the A set or-
der
position
of
the first
pitch
class of the B
set,
showing
the
numberof times B has
been rotated with
respect
to
A.
2.
Find
R',
defined
as the
mod(CARD)
complement
of R.
R + R' = 0 (mod(CARD)).
3.
Compare
R and
R'.
If R
(and
R')
O
andR
<
R'
(Case
),
the
first
pc repetition
occurs n
the A
set,
repeating
he ini-
tial
pc
of
the B
set.
If R
(and
R')
=
0 or
R
>
R'
(Case
2),
42Morris,
15
(DEF 3.13).
The
OMAA
s a
mapping
based on
the
segment
A such that
OMAAs-os.
the
(pitch-class
equivalent)
B
set;
CARD
different rotations
are
possible.
The PCL of a
specific
combination, herefore,
s a
result of the
number of
rotations of B with
respect
to A.
The
calculation
procedure
for
the
PCL
begins
with the establish-
ment of
order
position
indicators or
the
A
set, or,
in Morris's
notation,
the order
mapping
of
segment A, OMAA.42
The
OMAA
s
numbered from 0
through
CARD-
1,
as shown be-
low
the A set of
the 3/2
combination
cycle
in
Figure
3a
(n
=
5),
for
comparison
with
OMAB,
the order
mappings
of B with re-
spect
to A.
A
four-step
procedure
calculates he PCL:
1.
Determine
R
=
OMABo.
In other
words,
R is the A set or-
der
position
of
the first
pitch
class of the B
set,
showing
the
numberof times B has
been rotated with
respect
to
A.
2.
Find
R',
defined
as the
mod(CARD)
complement
of R.
R + R' = 0 (mod(CARD)).
3.
Compare
R and
R'.
If R
(and
R')
O
andR
<
R'
(Case
),
the
first
pc repetition
occurs n
the A
set,
repeating
he ini-
tial
pc
of
the B
set.
If R
(and
R')
=
0 or
R
>
R'
(Case
2),
42Morris,
15
(DEF 3.13).
The
OMAA
s a
mapping
based on
the
segment
A such that
OMAAs-os.
the
(pitch-class
equivalent)
B
set;
CARD
different rotations
are
possible.
The PCL of a
specific
combination, herefore,
s a
result of the
number of
rotations of B with
respect
to A.
The
calculation
procedure
for
the
PCL
begins
with the establish-
ment of
order
position
indicators or
the
A
set, or,
in Morris's
notation,
the order
mapping
of
segment A, OMAA.42
The
OMAA
s
numbered from 0
through
CARD-
1,
as shown be-
low
the A set of
the 3/2
combination
cycle
in
Figure
3a
(n
=
5),
for
comparison
with
OMAB,
the order
mappings
of B with re-
spect
to A.
A
four-step
procedure
calculates he PCL:
1.
Determine
R
=
OMABo.
In other
words,
R is the A set or-
der
position
of
the first
pitch
class of the B
set,
showing
the
numberof times B has
been rotated with
respect
to
A.
2.
Find
R',
defined
as the
mod(CARD)
complement
of R.
R + R' = 0 (mod(CARD)).
3.
Compare
R and
R'.
If R
(and
R')
O
andR
<
R'
(Case
),
the
first
pc repetition
occurs n
the A
set,
repeating
he ini-
tial
pc
of
the B
set.
If R
(and
R')
=
0 or
R
>
R'
(Case
2),
42Morris,
15
(DEF 3.13).
The
OMAA
s a
mapping
based on
the
segment
A such that
OMAAs-os.
the first
pc repetition
occurs
in
the B
set,
repeating
the ini-
tial
pc
of
the A set.
4.
Casel: PCL
=
R
x
2. Case2: PCL
=
(R'
x
2)
+
1.
Figure
3a
executes these four
steps
in
calculating
he PCL
of
the 3/2 combination.At
step
1,
R =
3,
because
OMABo
s
3:
pc
9,
the
first
pc
of
B,
occursat
op
3 inA. R'
is
9
for
step
2,
because
3 +
9 = 0
(mod
12).
Step
3 determines hat
case 1
applies,
and
step
4 calculates
he PCL as 6. The first
pc repetition
occurs
ust
after the sixth
pc
in the
segment,
at
pc
9
in
the
A
set,
which
first
occurs at the beginningof the B set.
Figure
3b
gives
another rotation
of the
same
pitch-class
ma-
terials with the
four-step
PCL calculation.
The value
of
R,
cal-
culated
at
step
1,
is 9
and its
complement,
R' in
step
2,
is 3.
In
step
3,
case
2
applies,
so the
PCL is calculated o be 7 in
step
4.
The first
pc
repetition
occurs
ust
after
he seventh
pc
in the
seg-
ment,
at
pc
6 in the B
set,
which
firstoccursat the
beginning
of
the
A set.
WhereR =
0,
meaning
he
ordering
of A and B is
identical,
the initialpc repetitionoccurs at its earliestpossible position,
the
beginning
of
the
B
set. The
PCL, therefore,
follows thecase
2
conditions,
where the
first
pitch
class
of A
recurs
in B
to
define the
PCL. At
step
4,
either R or R'
can be inserted
n
the
formula,
because both
values
are
0.
Where n =
0,
meaning
x and
y
are
mod
12
complements,
the calculation
procedure
does not
apply,
since,
technically,
this
indicates an
overlay
of
cycles
of
interval
0. For
example,
x/y
= 5/7
defines the
segment
<0,5>.
The PCL
is 2
where
n = 0, except for x/y = 0/0, where the PCL is 1.
Figure
4
calculatesPCL
values for two
reiterativecombina-
tions of
interval-3
cycles.
TheCARD of this
cycle
s
4,
requiring
referenceto
the mod 4
complement
pairs
isted on line a. In
Fig-
ure
4b,
the
x/y
valuesare
3/0,
producing
an
earlypc duplication
that can
easily
be
observed
by inspection,
or can
be derived
through
the
four-step
calculation
procedure,
as listed to the
right
of
the
segment.
The
value
of
R is
1,
which s smaller han
the first
pc repetition
occurs
in
the B
set,
repeating
the ini-
tial
pc
of
the A set.
4.
Casel: PCL
=
R
x
2. Case2: PCL
=
(R'
x
2)
+
1.
Figure
3a
executes these four
steps
in
calculating
he PCL
of
the 3/2 combination.At
step
1,
R =
3,
because
OMABo
s
3:
pc
9,
the
first
pc
of
B,
occursat
op
3 inA. R'
is
9
for
step
2,
because
3 +
9 = 0
(mod
12).
Step
3 determines hat
case 1
applies,
and
step
4 calculates
he PCL as 6. The first
pc repetition
occurs
ust
after the sixth
pc
in the
segment,
at
pc
9
in
the
A
set,
which
first
occurs at the beginningof the B set.
Figure
3b
gives
another rotation
of the
same
pitch-class
ma-
terials with the
four-step
PCL calculation.
The value
of
R,
cal-
culated
at
step
1,
is 9
and its
complement,
R' in
step
2,
is 3.
In
step
3,
case
2
applies,
so the
PCL is calculated o be 7 in
step
4.
The first
pc
repetition
occurs
ust
after
he seventh
pc
in the
seg-
ment,
at
pc
6 in the B
set,
which
firstoccursat the
beginning
of
the
A set.
WhereR =
0,
meaning
he
ordering
of A and B is
identical,
the initialpc repetitionoccurs at its earliestpossible position,
the
beginning
of
the
B
set. The
PCL, therefore,
follows thecase
2
conditions,
where the
first
pitch
class
of A
recurs
in B
to
define the
PCL. At
step
4,
either R or R'
can be inserted
n
the
formula,
because both
values
are
0.
Where n =
0,
meaning
x and
y
are
mod
12
complements,
the calculation
procedure
does not
apply,
since,
technically,
this
indicates an
overlay
of
cycles
of
interval
0. For
example,
x/y
= 5/7
defines the
segment
<0,5>.
The PCL
is 2
where
n = 0, except for x/y = 0/0, where the PCL is 1.
Figure
4
calculatesPCL
values for two
reiterativecombina-
tions of
interval-3
cycles.
TheCARD of this
cycle
s
4,
requiring
referenceto
the mod 4
complement
pairs
isted on line a. In
Fig-
ure
4b,
the
x/y
valuesare
3/0,
producing
an
earlypc duplication
that can
easily
be
observed
by inspection,
or can
be derived
through
the
four-step
calculation
procedure,
as listed to the
right
of
the
segment.
The
value
of
R is
1,
which s smaller han
the first
pc repetition
occurs
in
the B
set,
repeating
the ini-
tial
pc
of
the A set.
4.
Casel: PCL
=
R
x
2. Case2: PCL
=
(R'
x
2)
+
1.
Figure
3a
executes these four
steps
in
calculating
he PCL
of
the 3/2 combination.At
step
1,
R =
3,
because
OMABo
s
3:
pc
9,
the
first
pc
of
B,
occursat
op
3 inA. R'
is
9
for
step
2,
because
3 +
9 = 0
(mod
12).
Step
3 determines hat
case 1
applies,
and
step
4 calculates
he PCL as 6. The first
pc repetition
occurs
ust
after the sixth
pc
in the
segment,
at
pc
9
in
the
A
set,
which
first
occurs at the beginningof the B set.
Figure
3b
gives
another rotation
of the
same
pitch-class
ma-
terials with the
four-step
PCL calculation.
The value
of
R,
cal-
culated
at
step
1,
is 9
and its
complement,
R' in
step
2,
is 3.
In
step
3,
case
2
applies,
so the
PCL is calculated o be 7 in
step
4.
The first
pc
repetition
occurs
ust
after
he seventh
pc
in the
seg-
ment,
at
pc
6 in the B
set,
which
firstoccursat the
beginning
of
the
A set.
WhereR =
0,
meaning
he
ordering
of A and B is
identical,
the initialpc repetitionoccurs at its earliestpossible position,
the
beginning
of
the
B
set. The
PCL, therefore,
follows thecase
2
conditions,
where the
first
pitch
class
of A
recurs
in B
to
define the
PCL. At
step
4,
either R or R'
can be inserted
n
the
formula,
because both
values
are
0.
Where n =
0,
meaning
x and
y
are
mod
12
complements,
the calculation
procedure
does not
apply,
since,
technically,
this
indicates an
overlay
of
cycles
of
interval
0. For
example,
x/y
= 5/7
defines the
segment
<0,5>.
The PCL
is 2
where
n = 0, except for x/y = 0/0, where the PCL is 1.
Figure
4
calculatesPCL
values for two
reiterativecombina-
tions of
interval-3
cycles.
TheCARD of this
cycle
s
4,
requiring
referenceto
the mod 4
complement
pairs
isted on line a. In
Fig-
ure
4b,
the
x/y
valuesare
3/0,
producing
an
earlypc duplication
that can
easily
be
observed
by inspection,
or can
be derived
through
the
four-step
calculation
procedure,
as listed to the
right
of
the
segment.
The
value
of
R is
1,
which s smaller han
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 17/41
58
Music
TheorySpectrum
8
Music
TheorySpectrum
8
Music
TheorySpectrum
Figure
3.
PCL
calculations.
a. 3/2 combination
cycle.
Figure
3.
PCL
calculations.
a. 3/2 combination
cycle.
Figure
3.
PCL
calculations.
a. 3/2 combination
cycle.
B set:
A set:
OMAA:
B set:
A set:
OMAA:
B set:
A set:
OMAA:
9 2
7 0 5
10 3
8 1
6
11
4
2
7 0 5
10 3
8 1
6
11
4
2
7 0 5
10 3
8 1
6
11
4
6 11
0 1
6 11
0 1
6 11
0 1
4
9 2 7
0 5
10 3
8
1
2 3 4
5 6 7
8 9
10
11
4
9 2 7
0 5
10 3
8
1
2 3 4
5 6 7
8 9
10
11
4
9 2 7
0 5
10 3
8
1
2 3 4
5 6 7
8 9
10
11
1.
R
2. R'
3. R
4.
PCL
PCL
PCL
1.
R
2. R'
3. R
4.
PCL
PCL
PCL
1.
R
2. R'
3. R
4.
PCL
PCL
PCL
=
OMABo
=
3
=
9
0 and R
<
R' (case 1)
=
Rx2
=
3x2
=
6
=
OMABo
=
3
=
9
0 and R
<
R' (case 1)
=
Rx2
=
3x2
=
6
=
OMABo
=
3
=
9
0 and R
<
R' (case 1)
=
Rx2
=
3x2
=
6
b.
9/8
combination
cycle.
.
9/8
combination
cycle.
.
9/8
combination
cycle.
3 8 1
6 11 4 9
2 7
0
5
10
8 1
6 11 4 9
2 7
0
5
10
8 1
6 11 4 9
2 7
0
5
10
6 11
0
1
6 11
0
1
6 11
0
1
4 9 2 7 0 5 10 3 8 1
2
3
4
5 6 7
8
9
10
11
4 9 2 7 0 5 10 3 8 1
2
3
4
5 6 7
8
9
10
11
4 9 2 7 0 5 10 3 8 1
2
3
4
5 6 7
8
9
10
11
1. R
=
OMABo
=9
2.
R'
=
3
3. R >
R'(case2)
4. PCL =
(R'
x
2)
+
1
PCL
=
(3
x
2)+
1
PCL
=
6+1
PCL = 7
1. R
=
OMABo
=9
2.
R'
=
3
3. R >
R'(case2)
4. PCL =
(R'
x
2)
+
1
PCL
=
(3
x
2)+
1
PCL
=
6+1
PCL = 7
1. R
=
OMABo
=9
2.
R'
=
3
3. R >
R'(case2)
4. PCL =
(R'
x
2)
+
1
PCL
=
(3
x
2)+
1
PCL
=
6+1
PCL = 7
its mod
4
complement,
3,
requiring
he
case-1 PCL calculation
shown at
step
4.
The PCL
is 2:
both the A
and B sets state
only
one
pitch
class
before a
duplication
occurs.In
Figure
4c,
differ-
ent
dispositions
of the
same
A
and
B sets
demonstrate
a
case-2
calculationand a
PCL of 3.
its mod
4
complement,
3,
requiring
he
case-1 PCL calculation
shown at
step
4.
The PCL
is 2:
both the A
and B sets state
only
one
pitch
class
before a
duplication
occurs.In
Figure
4c,
differ-
ent
dispositions
of the
same
A
and
B sets
demonstrate
a
case-2
calculationand a
PCL of 3.
its mod
4
complement,
3,
requiring
he
case-1 PCL calculation
shown at
step
4.
The PCL
is 2:
both the A
and B sets state
only
one
pitch
class
before a
duplication
occurs.In
Figure
4c,
differ-
ent
dispositions
of the
same
A
and
B sets
demonstrate
a
case-2
calculationand a
PCL of 3.
Once the
pitch-class
succession
of an
entire
segment
has
been
established,
rotational
operations applied
to the
segment-that
is,
to the combinationof A and
B
sets,
not
just
to
the
B
set alone-can
result in
exchanges
of
x/y
values and
variable
PCLs. The values
of
x and
y
for
any complete segment
Once the
pitch-class
succession
of an
entire
segment
has
been
established,
rotational
operations applied
to the
segment-that
is,
to the combinationof A and
B
sets,
not
just
to
the
B
set alone-can
result in
exchanges
of
x/y
values and
variable
PCLs. The values
of
x and
y
for
any complete segment
Once the
pitch-class
succession
of an
entire
segment
has
been
established,
rotational
operations applied
to the
segment-that
is,
to the combinationof A and
B
sets,
not
just
to
the
B
set alone-can
result in
exchanges
of
x/y
values and
variable
PCLs. The values
of
x and
y
for
any complete segment
B
set:
A set:
OMAA:
B
set:
A set:
OMAA:
B
set:
A set:
OMAA:
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 18/41
Interval
ycles
as
Compositional
esources 59
nterval
ycles
as
Compositional
esources 59
nterval
ycles
as
Compositional
esources 59
Figure
4.
PCL
calculations for smaller
cardinalities.
Figure
4.
PCL
calculations for smaller
cardinalities.
Figure
4.
PCL
calculations for smaller
cardinalities.
a. mod
4
complements:
. mod
4
complements:
. mod
4
complements:
0 1
2
0 3 2
0 1
2
0 3 2
0 1
2
0 3 2
3
1
3
1
3
1
b.
B set:
A set: 5
OMAA:
0
b.
B set:
A set: 5
OMAA:
0
b.
B set:
A set: 5
OMAA:
0
c.
B
set:
A set: 5
OMAA:
0
c.
B
set:
A set: 5
OMAA:
0
c.
B
set:
A set: 5
OMAA:
0
8
11 2 5
8
11 2
1
2
3
8
11 2 5
8
11 2
1
2
3
8
11 2 5
8
11 2
1
2
3
2
5
8
8
11
2
1 2
3
2
5
8
8
11
2
1 2
3
2
5
8
8
11
2
1 2
3
1. R
2.
R'
3. R
4. PCL
PCL
PCL
1. R
2.
R'
3. R
4. PCL
PCL
PCL
1. R
2.
R'
3. R
4. PCL
PCL
PCL
1111
1. R
2.
R'
3. R
4.
PCL
PCL
PCL
PCL
1. R
2.
R'
3. R
4.
PCL
PCL
PCL
PCL
1. R
2.
R'
3. R
4.
PCL
PCL
PCL
PCL
OMABo
=
1
3
0 and R
<
R'
(case
1)
Rx2
1x2
2
OMABo
=
1
3
0 and R
<
R'
(case
1)
Rx2
1x2
2
OMABo
=
1
3
0 and R
<
R'
(case
1)
Rx2
1x2
2
OMABo
=
3
1
R'
(case 2)
(R'
x
2)
+ 1
(1
x
2)
+
1
2 + 1
3
OMABo
=
3
1
R'
(case 2)
(R'
x
2)
+ 1
(1
x
2)
+
1
2 + 1
3
OMABo
=
3
1
R'
(case 2)
(R'
x
2)
+ 1
(1
x
2)
+
1
2 + 1
3
beginning
on a memberof the A set
exchange positions
for a
segment
starting
with a B set member. In
Figure
4c,
for exam-
ple, x/y
=
9/6
for the
original segment
<5,2,8,5,11,8,2,11>
and for rotations that
begin
with other membersof the
A
set,
but
x/y
=
6/9
for
a
single
rotation
to
<2,8,5,11,8,2,11,5>
or to
a segment beginningon any other memberof the B set. This
distinction
between
x/y exchanges
in reiterativecombinations
is reflected
by
the PCL.
Obviously,
the PCL is not variable
n
rotations and
x/y exchanges
within
nonreiterative
combina-
tions,
where the PCL is
always
wice
the CARD.
However,
ro-
tations of
reiterativecombinationsexhibit
one
of two
PCL
val-
ues,
with
a
differenceof
1,
depending
on whether
hey
begin
on
a
member of the A set or a
member
of
the
B
set.
The
original
beginning
on a memberof the A set
exchange positions
for a
segment
starting
with a B set member. In
Figure
4c,
for exam-
ple, x/y
=
9/6
for the
original segment
<5,2,8,5,11,8,2,11>
and for rotations that
begin
with other membersof the
A
set,
but
x/y
=
6/9
for
a
single
rotation
to
<2,8,5,11,8,2,11,5>
or to
a segment beginningon any other memberof the B set. This
distinction
between
x/y exchanges
in reiterativecombinations
is reflected
by
the PCL.
Obviously,
the PCL is not variable
n
rotations and
x/y exchanges
within
nonreiterative
combina-
tions,
where the PCL is
always
wice
the CARD.
However,
ro-
tations of
reiterativecombinationsexhibit
one
of two
PCL
val-
ues,
with
a
differenceof
1,
depending
on whether
hey
begin
on
a
member of the A set or a
member
of
the
B
set.
The
original
beginning
on a memberof the A set
exchange positions
for a
segment
starting
with a B set member. In
Figure
4c,
for exam-
ple, x/y
=
9/6
for the
original segment
<5,2,8,5,11,8,2,11>
and for rotations that
begin
with other membersof the
A
set,
but
x/y
=
6/9
for
a
single
rotation
to
<2,8,5,11,8,2,11,5>
or to
a segment beginningon any other memberof the B set. This
distinction
between
x/y exchanges
in reiterativecombinations
is reflected
by
the PCL.
Obviously,
the PCL is not variable
n
rotations and
x/y exchanges
within
nonreiterative
combina-
tions,
where the PCL is
always
wice
the CARD.
However,
ro-
tations of
reiterativecombinationsexhibit
one
of two
PCL
val-
ues,
with
a
differenceof
1,
depending
on whether
hey
begin
on
a
member of the A set or a
member
of
the
B
set.
The
original
segment
in
Figure
4c and
any
rotations to
A
set members ex-
hibit PCLsof
3,
while all rotations o B
set members how PCLs
of 4. In
essence,
an
exchange
of
x/y
values in a
reiterativecom-
bination
cycle
increasesor
decreasesthe PCL
by
one.
Table 1
summarizes
x/y
values
arising
rom combinationsof
interval-ncycles and the resultingPCLs. The possiblevalues
for n are
given by
the
cycles
of
TnI
shown
separately
n
the
ta-
ble. Rows in thechart ist differentcombinations f the same n-
cycles
in both
reiterative
and
nonreiterative ombinations
the
latter ndicated
by
asterisks).
Columns
represent
differentrota-
tions of the B set with
respect
to the
A
set. For reiterativecom-
binations,
R
=
0 in column
1,
indicating
hat the orderof theA
and B sets is
identical,
R
=
1 in column
2,
meaning
B is one
segment
in
Figure
4c and
any
rotations to
A
set members ex-
hibit PCLsof
3,
while all rotations o B
set members how PCLs
of 4. In
essence,
an
exchange
of
x/y
values in a
reiterativecom-
bination
cycle
increasesor
decreasesthe PCL
by
one.
Table 1
summarizes
x/y
values
arising
rom combinationsof
interval-ncycles and the resultingPCLs. The possiblevalues
for n are
given by
the
cycles
of
TnI
shown
separately
n
the
ta-
ble. Rows in thechart ist differentcombinations f the same n-
cycles
in both
reiterative
and
nonreiterative ombinations
the
latter ndicated
by
asterisks).
Columns
represent
differentrota-
tions of the B set with
respect
to the
A
set. For reiterativecom-
binations,
R
=
0 in column
1,
indicating
hat the orderof theA
and B sets is
identical,
R
=
1 in column
2,
meaning
B is one
segment
in
Figure
4c and
any
rotations to
A
set members ex-
hibit PCLsof
3,
while all rotations o B
set members how PCLs
of 4. In
essence,
an
exchange
of
x/y
values in a
reiterativecom-
bination
cycle
increasesor
decreasesthe PCL
by
one.
Table 1
summarizes
x/y
values
arising
rom combinationsof
interval-ncycles and the resultingPCLs. The possiblevalues
for n are
given by
the
cycles
of
TnI
shown
separately
n
the
ta-
ble. Rows in thechart ist differentcombinations f the same n-
cycles
in both
reiterative
and
nonreiterative ombinations
the
latter ndicated
by
asterisks).
Columns
represent
differentrota-
tions of the B set with
respect
to the
A
set. For reiterativecom-
binations,
R
=
0 in column
1,
indicating
hat the orderof theA
and B sets is
identical,
R
=
1 in column
2,
meaning
B is one
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 19/41
60 Music
TheorySpectrum
0 Music
TheorySpectrum
0 Music
TheorySpectrum
rotation
of
A,
and
so forth. For
nonreiterative
combinations,
the firstcolumnliststhe combinationswiththe smallestvalues
of
x and their
subsequent readings
after rotations
of B. In
the
rotations of interval-3
cycle
combinations,
for
example,
the
first
nonreiterative
x/y
combination s
1/2,
listed
at the
begin-
ning
of the row that
includesrotationsof B
to
produce
4/11,
7/8,
and 10/5.
A
thirdrow then
completes
the
listings
or
n
=
3,
be-
ginning
with
x/y
=
2/1,
the
next smallest availablevalue
of
x,
and
including
B set rotations to
generate
5/10, 8/7,
and
11/4.
The
chart omits values for n
=
0,
where
x/y
are mod-12 com-
plement pairswith PCLsof 1 (x/y = 0/0)or2 (x/y = 1/11,2/10,
3/9,
4/8, 5/7,
6/6).
With
regard
to
pitch-class
content,
the rows in the chartas-
sociated with each value of
n
represent
he different
pitch-class
collections
that are
possible
from the
indicated
combinations.
The
number of
rows
for
each value of
n
corresponds
o the
number
of
differentcollectionsthat can be
generated:
wo rows
are listed
for
n
=
2 or 10 because there are two whole-tone
pitch-class
collections,
three rows are listed for n
=
3 or 9 be-
cause there are three interval-3or -9 collections (diminished
seventh
chords),
and so forth. In
every
case,
combinations
n
the
first
(reiterative)
row can
generate
CARD(n)
unique
pitch
classes,
because
A
and
B
are
pitch-class quivalent,
while com-
binations
n
any subsequent nonreiterative)
ows can
generate
(CARD(n)
x
2)
unique pitch
classes,
because
A
and
B
are not
equivalent.
Assuming,
for
purposes
of
illustration,
that the
pitch
classes
of the A
set are the same
for
each
row,
the
pitch
classes
of
the
B set are increased
by
one in successive rows.
Where n = 3, forexample,a pitch-class epresentationof row
1 could read as follows:
rotation
of
A,
and
so forth. For
nonreiterative
combinations,
the firstcolumnliststhe combinationswiththe smallestvalues
of
x and their
subsequent readings
after rotations
of B. In
the
rotations of interval-3
cycle
combinations,
for
example,
the
first
nonreiterative
x/y
combination s
1/2,
listed
at the
begin-
ning
of the row that
includesrotationsof B
to
produce
4/11,
7/8,
and 10/5.
A
thirdrow then
completes
the
listings
or
n
=
3,
be-
ginning
with
x/y
=
2/1,
the
next smallest availablevalue
of
x,
and
including
B set rotations to
generate
5/10, 8/7,
and
11/4.
The
chart omits values for n
=
0,
where
x/y
are mod-12 com-
plement pairswith PCLsof 1 (x/y = 0/0)or2 (x/y = 1/11,2/10,
3/9,
4/8, 5/7,
6/6).
With
regard
to
pitch-class
content,
the rows in the chartas-
sociated with each value of
n
represent
he different
pitch-class
collections
that are
possible
from the
indicated
combinations.
The
number of
rows
for
each value of
n
corresponds
o the
number
of
differentcollectionsthat can be
generated:
wo rows
are listed
for
n
=
2 or 10 because there are two whole-tone
pitch-class
collections,
three rows are listed for n
=
3 or 9 be-
cause there are three interval-3or -9 collections (diminished
seventh
chords),
and so forth. In
every
case,
combinations
n
the
first
(reiterative)
row can
generate
CARD(n)
unique
pitch
classes,
because
A
and
B
are
pitch-class quivalent,
while com-
binations
n
any subsequent nonreiterative)
ows can
generate
(CARD(n)
x
2)
unique pitch
classes,
because
A
and
B
are not
equivalent.
Assuming,
for
purposes
of
illustration,
that the
pitch
classes
of the A
set are the same
for
each
row,
the
pitch
classes
of
the
B set are increased
by
one in successive rows.
Where n = 3, forexample,a pitch-class epresentationof row
1 could read as follows:
rotation
of
A,
and
so forth. For
nonreiterative
combinations,
the firstcolumnliststhe combinationswiththe smallestvalues
of
x and their
subsequent readings
after rotations
of B. In
the
rotations of interval-3
cycle
combinations,
for
example,
the
first
nonreiterative
x/y
combination s
1/2,
listed
at the
begin-
ning
of the row that
includesrotationsof B
to
produce
4/11,
7/8,
and 10/5.
A
thirdrow then
completes
the
listings
or
n
=
3,
be-
ginning
with
x/y
=
2/1,
the
next smallest availablevalue
of
x,
and
including
B set rotations to
generate
5/10, 8/7,
and
11/4.
The
chart omits values for n
=
0,
where
x/y
are mod-12 com-
plement pairswith PCLsof 1 (x/y = 0/0)or2 (x/y = 1/11,2/10,
3/9,
4/8, 5/7,
6/6).
With
regard
to
pitch-class
content,
the rows in the chartas-
sociated with each value of
n
represent
he different
pitch-class
collections
that are
possible
from the
indicated
combinations.
The
number of
rows
for
each value of
n
corresponds
o the
number
of
differentcollectionsthat can be
generated:
wo rows
are listed
for
n
=
2 or 10 because there are two whole-tone
pitch-class
collections,
three rows are listed for n
=
3 or 9 be-
cause there are three interval-3or -9 collections (diminished
seventh
chords),
and so forth. In
every
case,
combinations
n
the
first
(reiterative)
row can
generate
CARD(n)
unique
pitch
classes,
because
A
and
B
are
pitch-class quivalent,
while com-
binations
n
any subsequent nonreiterative)
ows can
generate
(CARD(n)
x
2)
unique pitch
classes,
because
A
and
B
are not
equivalent.
Assuming,
for
purposes
of
illustration,
that the
pitch
classes
of the A
set are the same
for
each
row,
the
pitch
classes
of
the
B set are increased
by
one in successive rows.
Where n = 3, forexample,a pitch-class epresentationof row
1 could read as follows:
B
set:
A
set:
x/y
=
B
set:
A
set:
x/y
=
B
set:
A
set:
x/y
=
R=0
0369
0369
0/3
R=0
0369
0369
0/3
R=0
0369
0369
0/3
R=1
R=2
3690
6903
0369 0369
3/0
6/9
R=1
R=2
3690
6903
0369 0369
3/0
6/9
R=1
R=2
3690
6903
0369 0369
3/0
6/9
R=3
9036
0369
9/6
R=3
9036
0369
9/6
R=3
9036
0369
9/6
Row 2
could
then
appear
as
a
combination
of
the same
A set
witha B set of pc values that are one greater han the previous
row. The
B
set at R
=
0
would
be
<1,4,7,10>,
rotated as
fol-
lows:
Row 2
could
then
appear
as
a
combination
of
the same
A set
witha B set of pc values that are one greater han the previous
row. The
B
set at R
=
0
would
be
<1,4,7,10>,
rotated as
fol-
lows:
Row 2
could
then
appear
as
a
combination
of
the same
A set
witha B set of pc values that are one greater han the previous
row. The
B
set at R
=
0
would
be
<1,4,7,10>,
rotated as
fol-
lows:
B
set:
A set:
x/
=
B
set:
A set:
x/
=
B
set:
A set:
x/
=
R=0
1 4 7
10
0369
1/2
R=0
1 4 7
10
0369
1/2
R=0
1 4 7
10
0369
1/2
R=1
4 7
10
1
0369
4/11
R=1
4 7
10
1
0369
4/11
R=1
4 7
10
1
0369
4/11
R=2
7
10 1
4
03
69
7/8
R=2
7
10 1
4
03
69
7/8
R=2
7
10 1
4
03
69
7/8
R=3
10 1
4 7
0
369
10/5
R=3
10 1
4 7
0
369
10/5
R=3
10 1
4 7
0
369
10/5
Similarly,row 3 couldbeginwith the B set <2,5,8,11>, or val-
ues one
greater
than
<1,4,7,10>,
rotated to
generate
the re-
maining
x/y
combinations.
Complementary
n values contain nversevalues
of
x/y:
every
x/y
in
Table
1
corresponds
o another
x/y
that is its mod 12
in-
verse.
These inverse
pairs
exhibit he same
R
values
and do not
differ
in PCL.
For
example,
n values of 5 and
7
generate
the
inverse
x/y
values
of
1/4 and
11/8,
both
generated
from
R
=
5
and
exhibiting
a
PCL of
10.
Because
interval 6 is
its own in-
verse, all inversepairsfor n = 6 arecontained n the same six
rows of the chart.
Certainly,
if
pitch-class
variety
is
a
primarycompositional
aim,
many
of the combination
cycles
in Table
1
may
be
of lim-
ited
value. Combinations
of
intervals
4, 6,
or
8,
for
example,
cannot
generate
a PCL
greater
than 6.
This does
not
mean,
however,
that
the smallerPCLs
are without
compositional
ap-
plicability.
Indeed,
the
set-class
ypes
of
many
of the combina-
tion
cycles,
regardless
of the
sizeof their
PCLs,
are some
of
the
most common pitch-classstructures n Ives's music, as in the
music of
other
composers
of his era. In the
interval-6
ombina-
tions,
for
example,
the
nonreiterative
istings
on the second
and
sixth
lines
of Table
1
produce
4-9
[0,1,6,7],
a basic
element
of
Ives's
music
at
every point
in his
development.43
Other
familiar
43See
or
example,
the
combinations
of
half-step
related tritones
in
Ives's
Second
String
Quartet
(first
movement,
mm.
27,
37
[vln.
1,
via.],
38
[vln.
2,
Similarly,row 3 couldbeginwith the B set <2,5,8,11>, or val-
ues one
greater
than
<1,4,7,10>,
rotated to
generate
the re-
maining
x/y
combinations.
Complementary
n values contain nversevalues
of
x/y:
every
x/y
in
Table
1
corresponds
o another
x/y
that is its mod 12
in-
verse.
These inverse
pairs
exhibit he same
R
values
and do not
differ
in PCL.
For
example,
n values of 5 and
7
generate
the
inverse
x/y
values
of
1/4 and
11/8,
both
generated
from
R
=
5
and
exhibiting
a
PCL of
10.
Because
interval 6 is
its own in-
verse, all inversepairsfor n = 6 arecontained n the same six
rows of the chart.
Certainly,
if
pitch-class
variety
is
a
primarycompositional
aim,
many
of the combination
cycles
in Table
1
may
be
of lim-
ited
value. Combinations
of
intervals
4, 6,
or
8,
for
example,
cannot
generate
a PCL
greater
than 6.
This does
not
mean,
however,
that
the smallerPCLs
are without
compositional
ap-
plicability.
Indeed,
the
set-class
ypes
of
many
of the combina-
tion
cycles,
regardless
of the
sizeof their
PCLs,
are some
of
the
most common pitch-classstructures n Ives's music, as in the
music of
other
composers
of his era. In the
interval-6
ombina-
tions,
for
example,
the
nonreiterative
istings
on the second
and
sixth
lines
of Table
1
produce
4-9
[0,1,6,7],
a basic
element
of
Ives's
music
at
every point
in his
development.43
Other
familiar
43See
or
example,
the
combinations
of
half-step
related tritones
in
Ives's
Second
String
Quartet
(first
movement,
mm.
27,
37
[vln.
1,
via.],
38
[vln.
2,
Similarly,row 3 couldbeginwith the B set <2,5,8,11>, or val-
ues one
greater
than
<1,4,7,10>,
rotated to
generate
the re-
maining
x/y
combinations.
Complementary
n values contain nversevalues
of
x/y:
every
x/y
in
Table
1
corresponds
o another
x/y
that is its mod 12
in-
verse.
These inverse
pairs
exhibit he same
R
values
and do not
differ
in PCL.
For
example,
n values of 5 and
7
generate
the
inverse
x/y
values
of
1/4 and
11/8,
both
generated
from
R
=
5
and
exhibiting
a
PCL of
10.
Because
interval 6 is
its own in-
verse, all inversepairsfor n = 6 arecontained n the same six
rows of the chart.
Certainly,
if
pitch-class
variety
is
a
primarycompositional
aim,
many
of the combination
cycles
in Table
1
may
be
of lim-
ited
value. Combinations
of
intervals
4, 6,
or
8,
for
example,
cannot
generate
a PCL
greater
than 6.
This does
not
mean,
however,
that
the smallerPCLs
are without
compositional
ap-
plicability.
Indeed,
the
set-class
ypes
of
many
of the combina-
tion
cycles,
regardless
of the
sizeof their
PCLs,
are some
of
the
most common pitch-classstructures n Ives's music, as in the
music of
other
composers
of his era. In the
interval-6
ombina-
tions,
for
example,
the
nonreiterative
istings
on the second
and
sixth
lines
of Table
1
produce
4-9
[0,1,6,7],
a basic
element
of
Ives's
music
at
every point
in his
development.43
Other
familiar
43See
or
example,
the
combinations
of
half-step
related tritones
in
Ives's
Second
String
Quartet
(first
movement,
mm.
27,
37
[vln.
1,
via.],
38
[vln.
2,
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 20/41
Interval
ycles
as
Compositional
esources 61nterval
ycles
as
Compositional
esources 61nterval
ycles
as
Compositional
esources 61
Example
9.
Musical
applications
of combination
ycles.
a. SecondStringQuartet,secondmovement,mm. 17-18, firstviolin.
Example
9.
Musical
applications
of combination
ycles.
a. SecondStringQuartet,secondmovement,mm. 17-18, firstviolin.
Example
9.
Musical
applications
of combination
ycles.
a. SecondStringQuartet,secondmovement,mm. 17-18, firstviolin.
A
17
_
L^-tt- 4,
17
_
L^-tt- 4,
17
_
L^-tt- 4,
W
r[
I
-
I
fr
I
I
PI
m
-
. LI;
I
f*
Le
F- I
rll
r[
I
-
I
fr
I
I
PI
m
-
. LI;
I
f*
Le
F- I
rll
r[
I
-
I
fr
I
I
PI
m
-
. LI;
I
f*
Le
F- I
rll
PCL=
12
CL=
12
CL=
12
I
-
-
I=I
'='l
-
-
I=I
'='l
-
-
I=I
'='l
Bset: 6 11 4 9 2
1 4 9
2 7
A
set: 0
5 10 3 8
1
0
5 10 3
8 1
(n
=
5)
x/y
=
6/11
b. Over the
Pavements,
mm.
18-22,
clarinet
concertpitch).
Bset: 6 11 4 9 2
1 4 9
2 7
A
set: 0
5 10 3 8
1
0
5 10 3
8 1
(n
=
5)
x/y
=
6/11
b. Over the
Pavements,
mm.
18-22,
clarinet
concertpitch).
Bset: 6 11 4 9 2
1 4 9
2 7
A
set: 0
5 10 3 8
1
0
5 10 3
8 1
(n
=
5)
x/y
=
6/11
b. Over the
Pavements,
mm.
18-22,
clarinet
concertpitch).
21
.
22
1
.
22
1
.
22
,
d
L
J
I -
V
I
LJ
-
IL
I
i
i
i'i
I
,
d
L
J
I -
V
I
LJ
-
IL
I
i
i
i'i
I
,
d
L
J
I -
V
I
LJ
-
IL
I
i
i
i'i
I
x/y
=
x/y
=
x/y
=
I I 1
4
/
7
I I 1
4
/
7
I I 1
4
/
7
(n=
11)
n=
11)
n=
11)
Bset:
3 2
1
0
11 10
A set:
11
10
9
8 7
5
[
7I
6
PCL
=9
\
1st
rep.
Bset:
3 2
1
0
11 10
A set:
11
10
9
8 7
5
[
7I
6
PCL
=9
\
1st
rep.
Bset:
3 2
1
0
11 10
A set:
11
10
9
8 7
5
[
7I
6
PCL
=9
\
1st
rep.
structures
in
the table
include the whole-tone subset 4-25
[0,2,6,8]
on the third and
fifth rows of the
interval-6combina-
tions,
and 6-20
[0,1,4,5,8,9],
one of the all-combinatorial
exa-
chords,
on
the second and fourth
rows
of
the interval-4or
-8
combinations.
When he
uses these combinations o
project cyclic
nterval-
lic
repetitions,
however,
Ives
typically
favors the
larger
PCL
cello]).
In TheStructure
f
Atonal
Music,
Forte cites six
examples
of this tetra-
chord n music
of
Webern,
Scriabin,
Stravinsky,
and
Berg, including
he well-
known
extensive
usage
in
Webern's Five
Movements
or String
Quartet,
Op.
5
No. 4
(p.
27).
This set class s
George
Perle's
"y"
cell in his
analysis
of
Op.
5 No.
4 in
Serial
Composition
and
Atonality,
5th ed.
(Berkeley:
University
of
Califor-
nia
Press,
1981),
16-18. Lendvai's
study
of Bart6k
also focuses on this tetra-
chord,
identifying
t as one of
the
repetitive
nterval
"models"common n Bar-
t6k's
pitch
language.
structures
in
the table
include the whole-tone subset 4-25
[0,2,6,8]
on the third and
fifth rows of the
interval-6combina-
tions,
and 6-20
[0,1,4,5,8,9],
one of the all-combinatorial
exa-
chords,
on
the second and fourth
rows
of
the interval-4or
-8
combinations.
When he
uses these combinations o
project cyclic
nterval-
lic
repetitions,
however,
Ives
typically
favors the
larger
PCL
cello]).
In TheStructure
f
Atonal
Music,
Forte cites six
examples
of this tetra-
chord n music
of
Webern,
Scriabin,
Stravinsky,
and
Berg, including
he well-
known
extensive
usage
in
Webern's Five
Movements
or String
Quartet,
Op.
5
No. 4
(p.
27).
This set class s
George
Perle's
"y"
cell in his
analysis
of
Op.
5 No.
4 in
Serial
Composition
and
Atonality,
5th ed.
(Berkeley:
University
of
Califor-
nia
Press,
1981),
16-18. Lendvai's
study
of Bart6k
also focuses on this tetra-
chord,
identifying
t as one of
the
repetitive
nterval
"models"common n Bar-
t6k's
pitch
language.
structures
in
the table
include the whole-tone subset 4-25
[0,2,6,8]
on the third and
fifth rows of the
interval-6combina-
tions,
and 6-20
[0,1,4,5,8,9],
one of the all-combinatorial
exa-
chords,
on
the second and fourth
rows
of
the interval-4or
-8
combinations.
When he
uses these combinations o
project cyclic
nterval-
lic
repetitions,
however,
Ives
typically
favors the
larger
PCL
cello]).
In TheStructure
f
Atonal
Music,
Forte cites six
examples
of this tetra-
chord n music
of
Webern,
Scriabin,
Stravinsky,
and
Berg, including
he well-
known
extensive
usage
in
Webern's Five
Movements
or String
Quartet,
Op.
5
No. 4
(p.
27).
This set class s
George
Perle's
"y"
cell in his
analysis
of
Op.
5 No.
4 in
Serial
Composition
and
Atonality,
5th ed.
(Berkeley:
University
of
Califor-
nia
Press,
1981),
16-18. Lendvai's
study
of Bart6k
also focuses on this tetra-
chord,
identifying
t as one of
the
repetitive
nterval
"models"common n Bar-
t6k's
pitch
language.
values in reiterative andnonreiterativecombinations.
Among
the
more common
larger
structures s the octatonic
collection,
8-28
[0,1,3,4,6,7,9,10],
derived n nonreiterative
ombinations
of
intervals 3
or
9. For
combinations
of the
CARD-12
cycles,
every possible
PCL value
(1-12)
is available
(see
Tab.
1),
pro-
viding
material or
some
of
Ives's most
frequent
combinations.
In the
violin
line from his
Second
String
Quartet
given
n
Exam-
ple 9a, for example,Ives employs the 6/11 interval-5combina-
tion to
complete
the
aggregate
hroughprojection
of
the maxi-
mal PCL.
The
segment
s
repeatedbeginning
on the fourthbeat
of m.
17,
following
the
arrivalof
the twelfth
pitch
class,
so that
the
cycles
in the A
and B
sets
individually
do not continue
past
their
midpoints.
However,
since A
and B
are
literal
comple-
ments,
the reiteration
might
be viewed as a continuationof the
individual
cycles
with the
relative order
positions exchanged:
values in reiterative andnonreiterativecombinations.
Among
the
more common
larger
structures s the octatonic
collection,
8-28
[0,1,3,4,6,7,9,10],
derived n nonreiterative
ombinations
of
intervals 3
or
9. For
combinations
of the
CARD-12
cycles,
every possible
PCL value
(1-12)
is available
(see
Tab.
1),
pro-
viding
material or
some
of
Ives's most
frequent
combinations.
In the
violin
line from his
Second
String
Quartet
given
n
Exam-
ple 9a, for example,Ives employs the 6/11 interval-5combina-
tion to
complete
the
aggregate
hroughprojection
of
the maxi-
mal PCL.
The
segment
s
repeatedbeginning
on the fourthbeat
of m.
17,
following
the
arrivalof
the twelfth
pitch
class,
so that
the
cycles
in the A
and B
sets
individually
do not continue
past
their
midpoints.
However,
since A
and B
are
literal
comple-
ments,
the reiteration
might
be viewed as a continuationof the
individual
cycles
with the
relative order
positions exchanged:
values in reiterative andnonreiterativecombinations.
Among
the
more common
larger
structures s the octatonic
collection,
8-28
[0,1,3,4,6,7,9,10],
derived n nonreiterative
ombinations
of
intervals 3
or
9. For
combinations
of the
CARD-12
cycles,
every possible
PCL value
(1-12)
is available
(see
Tab.
1),
pro-
viding
material or
some
of
Ives's most
frequent
combinations.
In the
violin
line from his
Second
String
Quartet
given
n
Exam-
ple 9a, for example,Ives employs the 6/11 interval-5combina-
tion to
complete
the
aggregate
hroughprojection
of
the maxi-
mal PCL.
The
segment
s
repeatedbeginning
on the fourthbeat
of m.
17,
following
the
arrivalof
the twelfth
pitch
class,
so that
the
cycles
in the A
and B
sets
individually
do not continue
past
their
midpoints.
However,
since A
and B
are
literal
comple-
ments,
the reiteration
might
be viewed as a continuationof the
individual
cycles
with the
relative order
positions exchanged:
In L_
-f
n L_
-f
n L_
-f
l8
8
,
..
20
_p..
8
8
,
..
20
_p..
8
8
,
..
20
_p..
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 21/41
62 Music
TheorySpectrum
2 Music
TheorySpectrum
2 Music
TheorySpectrum
the
pitch
classes
of the initialA
set
complete
an interval-5
ycle
with the pitchclasses of the reiterationof the B set, and vice
versa.
The
clarinet ine
from Ives's
Over
the
Pavements
1906-13)
given
in
Example
9b
projects
a PCL of 9 from
a
combination
of
interval-11
cycles
in
a
4/7 alternation.Similar
o the extraction
of
interval-10
ycles
from the 5/5
cycle
in
Example
1,
the
gener-
ating
interval-11
cycles
in
Example
9b are
registrally
associ-
ated,
in the manner
of
a
"compoundmelody."
In
mm.
18-21,
each note in the combination
cycle
is of three sixteenths'
dura-
tion, and this durationalconsistency s abandonedwiththe ar-
rival of
the
final
element
of
the
PCL,
pc
7 in m. 22. After this
point,
the
durations
are
shortened
and
the
A and B sets
do
not
continue
to
their individual
cyclic
completions.
As with
single-interval ycles,
combination
cycles
may pro-
vide skeletons
for
patterns
of
embellishment
over
larger
musi-
cal
spans.
This can
generate
an
unsystematic type
of
embellishment-as
is the case for the interval-11
cycle
of Ex-
ample
6-or
a framework or
sequential
repetition.
In Ives's
pi-
ano workRough and Ready (1906-07), for example,the ten-
note
sequential pattern
bracketed
above
the
excerpt
in
Example
10
occurs
in six
transpositions
elated
by descending
whole-steps.44
Accents and slurs
highlight
the first and
sixth
sonorities
n
each
sequential
unit,
projecting
an
alternating
de-
scent
of
even
and
odd whole-tone scales
among
the
upper
ac-
cented
notes.
Since
the
overlay
distance
s
3,
the result s the 3/7
combination
cycle,
a nonreiterative
whole-tone combination.
In the sketchesfor his Universe
Symphony
(1911-28),
Ives
begins to catalogue the possibilitiesfor combinationcycles,
44Kirkpatrick,
atalogue,
96
(Cat.
No.
3B16ii).
This
work,
the full
title of
which s
"Rough
and
Ready
et al. and/or the
JumpingFrog,"
is the second
of
the
Five
Take-Offs
or
piano,
as named
by Kirkpatrick.
t is
transcribed
n Al-
bert
Lotto,
"ExperimentalAspects
of the
Completed
Short
Piano
Pieces
of
CharlesIves"
(D.M.A.
thesis, Juilliard,
1978),
112-118.
the
pitch
classes
of the initialA
set
complete
an interval-5
ycle
with the pitchclasses of the reiterationof the B set, and vice
versa.
The
clarinet ine
from Ives's
Over
the
Pavements
1906-13)
given
in
Example
9b
projects
a PCL of 9 from
a
combination
of
interval-11
cycles
in
a
4/7 alternation.Similar
o the extraction
of
interval-10
ycles
from the 5/5
cycle
in
Example
1,
the
gener-
ating
interval-11
cycles
in
Example
9b are
registrally
associ-
ated,
in the manner
of
a
"compoundmelody."
In
mm.
18-21,
each note in the combination
cycle
is of three sixteenths'
dura-
tion, and this durationalconsistency s abandonedwiththe ar-
rival of
the
final
element
of
the
PCL,
pc
7 in m. 22. After this
point,
the
durations
are
shortened
and
the
A and B sets
do
not
continue
to
their individual
cyclic
completions.
As with
single-interval ycles,
combination
cycles
may pro-
vide skeletons
for
patterns
of
embellishment
over
larger
musi-
cal
spans.
This can
generate
an
unsystematic type
of
embellishment-as
is the case for the interval-11
cycle
of Ex-
ample
6-or
a framework or
sequential
repetition.
In Ives's
pi-
ano workRough and Ready (1906-07), for example,the ten-
note
sequential pattern
bracketed
above
the
excerpt
in
Example
10
occurs
in six
transpositions
elated
by descending
whole-steps.44
Accents and slurs
highlight
the first and
sixth
sonorities
n
each
sequential
unit,
projecting
an
alternating
de-
scent
of
even
and
odd whole-tone scales
among
the
upper
ac-
cented
notes.
Since
the
overlay
distance
s
3,
the result s the 3/7
combination
cycle,
a nonreiterative
whole-tone combination.
In the sketchesfor his Universe
Symphony
(1911-28),
Ives
begins to catalogue the possibilitiesfor combinationcycles,
44Kirkpatrick,
atalogue,
96
(Cat.
No.
3B16ii).
This
work,
the full
title of
which s
"Rough
and
Ready
et al. and/or the
JumpingFrog,"
is the second
of
the
Five
Take-Offs
or
piano,
as named
by Kirkpatrick.
t is
transcribed
n Al-
bert
Lotto,
"ExperimentalAspects
of the
Completed
Short
Piano
Pieces
of
CharlesIves"
(D.M.A.
thesis, Juilliard,
1978),
112-118.
the
pitch
classes
of the initialA
set
complete
an interval-5
ycle
with the pitchclasses of the reiterationof the B set, and vice
versa.
The
clarinet ine
from Ives's
Over
the
Pavements
1906-13)
given
in
Example
9b
projects
a PCL of 9 from
a
combination
of
interval-11
cycles
in
a
4/7 alternation.Similar
o the extraction
of
interval-10
ycles
from the 5/5
cycle
in
Example
1,
the
gener-
ating
interval-11
cycles
in
Example
9b are
registrally
associ-
ated,
in the manner
of
a
"compoundmelody."
In
mm.
18-21,
each note in the combination
cycle
is of three sixteenths'
dura-
tion, and this durationalconsistency s abandonedwiththe ar-
rival of
the
final
element
of
the
PCL,
pc
7 in m. 22. After this
point,
the
durations
are
shortened
and
the
A and B sets
do
not
continue
to
their individual
cyclic
completions.
As with
single-interval ycles,
combination
cycles
may pro-
vide skeletons
for
patterns
of
embellishment
over
larger
musi-
cal
spans.
This can
generate
an
unsystematic type
of
embellishment-as
is the case for the interval-11
cycle
of Ex-
ample
6-or
a framework or
sequential
repetition.
In Ives's
pi-
ano workRough and Ready (1906-07), for example,the ten-
note
sequential pattern
bracketed
above
the
excerpt
in
Example
10
occurs
in six
transpositions
elated
by descending
whole-steps.44
Accents and slurs
highlight
the first and
sixth
sonorities
n
each
sequential
unit,
projecting
an
alternating
de-
scent
of
even
and
odd whole-tone scales
among
the
upper
ac-
cented
notes.
Since
the
overlay
distance
s
3,
the result s the 3/7
combination
cycle,
a nonreiterative
whole-tone combination.
In the sketchesfor his Universe
Symphony
(1911-28),
Ives
begins to catalogue the possibilitiesfor combinationcycles,
44Kirkpatrick,
atalogue,
96
(Cat.
No.
3B16ii).
This
work,
the full
title of
which s
"Rough
and
Ready
et al. and/or the
JumpingFrog,"
is the second
of
the
Five
Take-Offs
or
piano,
as named
by Kirkpatrick.
t is
transcribed
n Al-
bert
Lotto,
"ExperimentalAspects
of the
Completed
Short
Piano
Pieces
of
CharlesIves"
(D.M.A.
thesis, Juilliard,
1978),
112-118.
amassing pitch
materials
for his
"Universe
...
in
tones."45
Sketchpage 3038, as numberedby Kirkpatrick,istscyclesof
varioussizes
and combinational
distances,
apparently
n
prepa-
ration for the musical
settings
of
some
of these structures
on
sketch
page
3036.46
Example
11
literally
transcribes
3038,
in-
cluding
the
composer's marginal
notations
indicating
nterval
sizes and instrumental
pecifications,
while
excludingonly
ex-
traneous
markings-some
of which
apparently
cross
out
material-and erasures
or otherwise ndistinctnotations.
In the
transcription,only
bracketed
material,
ncluding
clef
signs
and
the numberingof the stavesin the left margin, s not original.
Along
the
left side
of
the
page,
Ives
writes letters
"a," "b,"
"C,"
and "D" to subdivide he texture
nto four
parts,
although
the musical
notations within
the
parts
of
the
page
correspond-
ing
to each letter
do not
obviously
align
as a score:
the
majority
of the
notations are
clearly
the abstract
pitch
resources
on
whicheachof the
correspondingparts
are
to be based.
Two
groups
of ideas on 3038 are
not abstract
cyclic
struc-
tures. In staves
4
through
8,
roughly
the
right
half
of the
page
containsa melodiclinewithsustainedaccompaniment inbass
clef,
with bar lines
interspersed)
that
apparently
continues
from the
end of staves 5 and 6 to the
middle
of
staves
7
and 8.
A
second
area of
compositionalsetting appears
on staves
11-13.
All other materials
on
the
page
are
cyclicpitch
repetitions,
usu-
ally
notated
in whole
notes,
often
circled,
and often
accompa-
nied
by
an indication
of
the interval
or intervals
hat constitute
the
cyclic repetition.
In a few
cases,
Ives also
makesnote
of the
total number
of
unique pitch
classes that
is
generated.
Includedamong
the
cyclic
notations
on
page
3038
are
single
cycles
of intervals
10 and
11
at the
beginning
of staves7
and
8.
Ives labels
these,
respectively,
"all MIN
7" and
"all
MAJ
7,"
45Memos, 106;
Kirkpatrick,
Catalogue,
27
(Cat.
No.
1A9).
A
"Facsimile/Transcription"
f
the
sketches
s in
preparation
by
Peer-Southern.
46These
numbers are the
photostat
negative
numbers
given
to the
right
of
each
entry
in
Kirkpatrick'sCatalogue.
amassing pitch
materials
for his
"Universe
...
in
tones."45
Sketchpage 3038, as numberedby Kirkpatrick,istscyclesof
varioussizes
and combinational
distances,
apparently
n
prepa-
ration for the musical
settings
of
some
of these structures
on
sketch
page
3036.46
Example
11
literally
transcribes
3038,
in-
cluding
the
composer's marginal
notations
indicating
nterval
sizes and instrumental
pecifications,
while
excludingonly
ex-
traneous
markings-some
of which
apparently
cross
out
material-and erasures
or otherwise ndistinctnotations.
In the
transcription,only
bracketed
material,
ncluding
clef
signs
and
the numberingof the stavesin the left margin, s not original.
Along
the
left side
of
the
page,
Ives
writes letters
"a," "b,"
"C,"
and "D" to subdivide he texture
nto four
parts,
although
the musical
notations within
the
parts
of
the
page
correspond-
ing
to each letter
do not
obviously
align
as a score:
the
majority
of the
notations are
clearly
the abstract
pitch
resources
on
whicheachof the
correspondingparts
are
to be based.
Two
groups
of ideas on 3038 are
not abstract
cyclic
struc-
tures. In staves
4
through
8,
roughly
the
right
half
of the
page
containsa melodiclinewithsustainedaccompaniment inbass
clef,
with bar lines
interspersed)
that
apparently
continues
from the
end of staves 5 and 6 to the
middle
of
staves
7
and 8.
A
second
area of
compositionalsetting appears
on staves
11-13.
All other materials
on
the
page
are
cyclicpitch
repetitions,
usu-
ally
notated
in whole
notes,
often
circled,
and often
accompa-
nied
by
an indication
of
the interval
or intervals
hat constitute
the
cyclic repetition.
In a few
cases,
Ives also
makesnote
of the
total number
of
unique pitch
classes that
is
generated.
Includedamong
the
cyclic
notations
on
page
3038
are
single
cycles
of intervals
10 and
11
at the
beginning
of staves7
and
8.
Ives labels
these,
respectively,
"all MIN
7" and
"all
MAJ
7,"
45Memos, 106;
Kirkpatrick,
Catalogue,
27
(Cat.
No.
1A9).
A
"Facsimile/Transcription"
f
the
sketches
s in
preparation
by
Peer-Southern.
46These
numbers are the
photostat
negative
numbers
given
to the
right
of
each
entry
in
Kirkpatrick'sCatalogue.
amassing pitch
materials
for his
"Universe
...
in
tones."45
Sketchpage 3038, as numberedby Kirkpatrick,istscyclesof
varioussizes
and combinational
distances,
apparently
n
prepa-
ration for the musical
settings
of
some
of these structures
on
sketch
page
3036.46
Example
11
literally
transcribes
3038,
in-
cluding
the
composer's marginal
notations
indicating
nterval
sizes and instrumental
pecifications,
while
excludingonly
ex-
traneous
markings-some
of which
apparently
cross
out
material-and erasures
or otherwise ndistinctnotations.
In the
transcription,only
bracketed
material,
ncluding
clef
signs
and
the numberingof the stavesin the left margin, s not original.
Along
the
left side
of
the
page,
Ives
writes letters
"a," "b,"
"C,"
and "D" to subdivide he texture
nto four
parts,
although
the musical
notations within
the
parts
of
the
page
correspond-
ing
to each letter
do not
obviously
align
as a score:
the
majority
of the
notations are
clearly
the abstract
pitch
resources
on
whicheachof the
correspondingparts
are
to be based.
Two
groups
of ideas on 3038 are
not abstract
cyclic
struc-
tures. In staves
4
through
8,
roughly
the
right
half
of the
page
containsa melodiclinewithsustainedaccompaniment inbass
clef,
with bar lines
interspersed)
that
apparently
continues
from the
end of staves 5 and 6 to the
middle
of
staves
7
and 8.
A
second
area of
compositionalsetting appears
on staves
11-13.
All other materials
on
the
page
are
cyclicpitch
repetitions,
usu-
ally
notated
in whole
notes,
often
circled,
and often
accompa-
nied
by
an indication
of
the interval
or intervals
hat constitute
the
cyclic repetition.
In a few
cases,
Ives also
makesnote
of the
total number
of
unique pitch
classes that
is
generated.
Includedamong
the
cyclic
notations
on
page
3038
are
single
cycles
of intervals
10 and
11
at the
beginning
of staves7
and
8.
Ives labels
these,
respectively,
"all MIN
7" and
"all
MAJ
7,"
45Memos, 106;
Kirkpatrick,
Catalogue,
27
(Cat.
No.
1A9).
A
"Facsimile/Transcription"
f
the
sketches
s in
preparation
by
Peer-Southern.
46These
numbers are the
photostat
negative
numbers
given
to the
right
of
each
entry
in
Kirkpatrick'sCatalogue.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 22/41
Interval
ycles
as
Compositional
esources 63
nterval
ycles
as
Compositional
esources 63
nterval
ycles
as
Compositional
esources 63
Example
10.
Rough
and
Ready,
mm.
8-15,
right
hand.
sequential
unit:
1
2
3
4
5
6
A8
Et.Br^JrTr
I
^ T
1
r
rrT^rrl Tr^
r
1^2
A
A^^
15
8
A A
A
13A
A
14
A
J"
'"-"
-K
L
Ti
PIMal
6"~~~~~~~~~~~~~~~~~~~~~~~~~~~
ta
Example
10.
Rough
and
Ready,
mm.
8-15,
right
hand.
sequential
unit:
1
2
3
4
5
6
A8
Et.Br^JrTr
I
^ T
1
r
rrT^rrl Tr^
r
1^2
A
A^^
15
8
A A
A
13A
A
14
A
J"
'"-"
-K
L
Ti
PIMal
6"~~~~~~~~~~~~~~~~~~~~~~~~~~~
ta
Example
10.
Rough
and
Ready,
mm.
8-15,
right
hand.
sequential
unit:
1
2
3
4
5
6
A8
Et.Br^JrTr
I
^ T
1
r
rrT^rrl Tr^
r
1^2
A
A^^
15
8
A A
A
13A
A
14
A
J"
'"-"
-K
L
Ti
PIMal
6"~~~~~~~~~~~~~~~~~~~~~~~~~~~
ta
__-i
@
-
I
|=I
'lEWl
W
iU;;
I
I
II;
_-i
@
-
I
|=I
'lEWl
W
iU;;
I
I
II;
_-i
@
-
I
|=I
'lEWl
W
iU;;
I
I
II;
x/y= 3 / 7/y= 3 / 7/y= 3 / 7
1111
9
7
6
PCL = 12CL = 12CL = 12
and allows octave
transpositions
n the
latter
out of
necessity
(see
Ex.
11).
On
staff
8,
immediately following
the
major-
seventh
sequence,
he
begins
a
sequence
of
decreasing
nterval
sizes that states
only
intervals
7, 6,
and
5.47
The combination
cycles
on the
page
display
many
of the
possibilities
for
higher
PCLvalues fromcombiningcyclesof intervals1,2, 3, 9, and11.
Figure
5 summarizes he structureof
all
the
cyclic
ormations
n
Example
11,
giving
their
x/y,
n,
and PCL values. Parts a and b
of the
figure
illustrate combination
cycles
found in
the
upper
half of
the
page,
parts
e
and f show
structures
on staves 11 and
12,
near
the
right
margin,
and
parts g through correspond
o
materials
on
the bottom two
staves,
in order from
left
to
right.
The
cycles
of
intervals 10 and 11 on
staves
7
and 8 are listed as
10/10 and 11/11 n
parts
c and d of
Figure
5,
viewed
as
overlays
of interval-8andinterval-10cycles, respectively.
47Ives uses
sequences
of
descending
intervals,
similar to Fritz Henrich
Klein's
"Pyramidenakkord,"
on
several
occasions. An
early example
appears
in his father's
Copybook, p. [165] (Kirkpatrick
Catalogue
No.
7E77).
See
also
Tone Roads No.
1,
downbeat of
m.
12.
See
Fritz Heinrich
Klein,
"Die Grenze
der
Halbtonwelt,"
Die Musik 17/4
(January 1925),
284.
and allows octave
transpositions
n the
latter
out of
necessity
(see
Ex.
11).
On
staff
8,
immediately following
the
major-
seventh
sequence,
he
begins
a
sequence
of
decreasing
nterval
sizes that states
only
intervals
7, 6,
and
5.47
The combination
cycles
on the
page
display
many
of the
possibilities
for
higher
PCLvalues fromcombiningcyclesof intervals1,2, 3, 9, and11.
Figure
5 summarizes he structureof
all
the
cyclic
ormations
n
Example
11,
giving
their
x/y,
n,
and PCL values. Parts a and b
of the
figure
illustrate combination
cycles
found in
the
upper
half of
the
page,
parts
e
and f show
structures
on staves 11 and
12,
near
the
right
margin,
and
parts g through correspond
o
materials
on
the bottom two
staves,
in order from
left
to
right.
The
cycles
of
intervals 10 and 11 on
staves
7
and 8 are listed as
10/10 and 11/11 n
parts
c and d of
Figure
5,
viewed
as
overlays
of interval-8andinterval-10cycles, respectively.
47Ives uses
sequences
of
descending
intervals,
similar to Fritz Henrich
Klein's
"Pyramidenakkord,"
on
several
occasions. An
early example
appears
in his father's
Copybook, p. [165] (Kirkpatrick
Catalogue
No.
7E77).
See
also
Tone Roads No.
1,
downbeat of
m.
12.
See
Fritz Heinrich
Klein,
"Die Grenze
der
Halbtonwelt,"
Die Musik 17/4
(January 1925),
284.
and allows octave
transpositions
n the
latter
out of
necessity
(see
Ex.
11).
On
staff
8,
immediately following
the
major-
seventh
sequence,
he
begins
a
sequence
of
decreasing
nterval
sizes that states
only
intervals
7, 6,
and
5.47
The combination
cycles
on the
page
display
many
of the
possibilities
for
higher
PCLvalues fromcombiningcyclesof intervals1,2, 3, 9, and11.
Figure
5 summarizes he structureof
all
the
cyclic
ormations
n
Example
11,
giving
their
x/y,
n,
and PCL values. Parts a and b
of the
figure
illustrate combination
cycles
found in
the
upper
half of
the
page,
parts
e
and f show
structures
on staves 11 and
12,
near
the
right
margin,
and
parts g through correspond
o
materials
on
the bottom two
staves,
in order from
left
to
right.
The
cycles
of
intervals 10 and 11 on
staves
7
and 8 are listed as
10/10 and 11/11 n
parts
c and d of
Figure
5,
viewed
as
overlays
of interval-8andinterval-10cycles, respectively.
47Ives uses
sequences
of
descending
intervals,
similar to Fritz Henrich
Klein's
"Pyramidenakkord,"
on
several
occasions. An
early example
appears
in his father's
Copybook, p. [165] (Kirkpatrick
Catalogue
No.
7E77).
See
also
Tone Roads No.
1,
downbeat of
m.
12.
See
Fritz Heinrich
Klein,
"Die Grenze
der
Halbtonwelt,"
Die Musik 17/4
(January 1925),
284.
The
9/4
cycle
on staves 1
and 2
(Fig. 5a),
the
5/8 combination
labeled
by
Ives "5E"on staves 11
and 12
(Fig. 5e),
and
the 6/7
structureon staves 15 and
16,
near the
center
(Fig. 5i)
all
com-
bine
cycles
of interval 1 to
project
different PCL
values.
Ives
indicates,
with the
number
"12,"
that
the latter
generates
a
completeaggregate roman alternationof a "MIN5 [sic]"and
"Perfect5th."48
The
other two
interval-1
ombinations
annot,
of
course,
complete
the
aggregate
without
pc repetitions,
and
Ives continues
the
intervallic
sequences only
to
the final
notes
in
the PCLs.
On staves 1
and
2,
the
9/4
combination
tops
after
the seventh
element
(see
Fig. 5a),
avoiding
he
continuation o
pc
0,
which would
reiterate
the
pitch
class on which
the se-
quence begins.
The
5/8 combination
on
staves 11 and 12
(with
registral shift)
stops
after the
tenth element
(see
Fig.
5e)
to
avoidreiteratingpc 5, firstpresentedas the secondelement.
Interval-11
combinationson
the
page,
displayingx/y
values
that are
complementary
o
those for
interval
1,
present
two ad-
ditional PCL
possibilities.
The 4/7
combination at
the
right
margin
of
staves
11 and 12
(Fig. 5f)
continues
past
the end of
The
9/4
cycle
on staves 1
and 2
(Fig. 5a),
the
5/8 combination
labeled
by
Ives "5E"on staves 11
and 12
(Fig. 5e),
and
the 6/7
structureon staves 15 and
16,
near the
center
(Fig. 5i)
all
com-
bine
cycles
of interval 1 to
project
different PCL
values.
Ives
indicates,
with the
number
"12,"
that
the latter
generates
a
completeaggregate roman alternationof a "MIN5 [sic]"and
"Perfect5th."48
The
other two
interval-1
ombinations
annot,
of
course,
complete
the
aggregate
without
pc repetitions,
and
Ives continues
the
intervallic
sequences only
to
the final
notes
in
the PCLs.
On staves 1
and
2,
the
9/4
combination
tops
after
the seventh
element
(see
Fig. 5a),
avoiding
he
continuation o
pc
0,
which would
reiterate
the
pitch
class on which
the se-
quence begins.
The
5/8 combination
on
staves 11 and 12
(with
registral shift)
stops
after the
tenth element
(see
Fig.
5e)
to
avoidreiteratingpc 5, firstpresentedas the secondelement.
Interval-11
combinationson
the
page,
displayingx/y
values
that are
complementary
o
those for
interval
1,
present
two ad-
ditional PCL
possibilities.
The 4/7
combination at
the
right
margin
of
staves
11 and 12
(Fig. 5f)
continues
past
the end of
The
9/4
cycle
on staves 1
and 2
(Fig. 5a),
the
5/8 combination
labeled
by
Ives "5E"on staves 11
and 12
(Fig. 5e),
and
the 6/7
structureon staves 15 and
16,
near the
center
(Fig. 5i)
all
com-
bine
cycles
of interval 1 to
project
different PCL
values.
Ives
indicates,
with the
number
"12,"
that
the latter
generates
a
completeaggregate roman alternationof a "MIN5 [sic]"and
"Perfect5th."48
The
other two
interval-1
ombinations
annot,
of
course,
complete
the
aggregate
without
pc repetitions,
and
Ives continues
the
intervallic
sequences only
to
the final
notes
in
the PCLs.
On staves 1
and
2,
the
9/4
combination
tops
after
the seventh
element
(see
Fig. 5a),
avoiding
he
continuation o
pc
0,
which would
reiterate
the
pitch
class on which
the se-
quence begins.
The
5/8 combination
on
staves 11 and 12
(with
registral shift)
stops
after the
tenth element
(see
Fig.
5e)
to
avoidreiteratingpc 5, firstpresentedas the secondelement.
Interval-11
combinationson
the
page,
displayingx/y
values
that are
complementary
o
those for
interval
1,
present
two ad-
ditional PCL
possibilities.
The 4/7
combination at
the
right
margin
of
staves
11 and 12
(Fig. 5f)
continues
past
the end of
48A
perusal
of
the
marginalia
in
Example
11
will reveal
that Ives occasion-
ally
mislabels the
quality
of
intervals.
48A
perusal
of
the
marginalia
in
Example
11
will reveal
that Ives occasion-
ally
mislabels the
quality
of
intervals.
48A
perusal
of
the
marginalia
in
Example
11
will reveal
that Ives occasion-
ally
mislabels the
quality
of
intervals.
B set:
A set: 0
B set:
A set: 0
B set:
A set: 0
3
1000
8
(n = 10)n = 10)n = 10)
IMM"WMM"WMM"W
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 23/41
64
Music
TheorySpectrum
Example
11.
Universe
Symphony,
sketch
page
q3038,
iteral
ranscription.
64
Music
TheorySpectrum
Example
11.
Universe
Symphony,
sketch
page
q3038,
iteral
ranscription.
64
Music
TheorySpectrum
Example
11.
Universe
Symphony,
sketch
page
q3038,
iteral
ranscription.
Maj6 bassoon
@r0n -
|
f^ /ft^^ u
6Maj
^^
.
, i
--*?-1
Mva
<^
r"
-
^f
-3 -
--4P
M:L
/5^
j/J
J
.
-J
0
r.W
Maj6 bassoon
@r0n -
|
f^ /ft^^ u
6Maj
^^
.
, i
--*?-1
Mva
<^
r"
-
^f
-3 -
--4P
M:L
/5^
j/J
J
.
-J
0
r.W
Maj6 bassoon
@r0n -
|
f^ /ft^^ u
6Maj
^^
.
, i
--*?-1
Mva
<^
r"
-
^f
-3 -
--4P
M:L
/5^
j/J
J
.
-J
0
r.W
at]
ti4N
alMAJ7
t]
ti4N
alMAJ7
t]
ti4N
alMAJ7
S--5E--5E--5E
Tuba
3 horn
-nn
43
6
W
M?
/
?
TJ, <
^ ^it
5$3 ,
^
1
y
u
/;
f0
<?
/
trombone
j
3
1
v ...
(J
tn
t
"
~;
'
..../M
^-^
'
.^^^'
L~~~~~~~~~~~~~~~~~~~~~~~broken
y
\
p_4 3rd
6
2 Ma 2
v61
?
o
"?
?
"')
"
t~
r^*-
.r
.
_
~~b
ab"~MIN
- ;Min2
b_ horn 2 - ..o_
0
- -o'-
[J?1
(o'0"^
^?*'c
^
/^
12
Tuba
3 horn
-nn
43
6
W
M?
/
?
TJ, <
^ ^it
5$3 ,
^
1
y
u
/;
f0
<?
/
trombone
j
3
1
v ...
(J
tn
t
"
~;
'
..../M
^-^
'
.^^^'
L~~~~~~~~~~~~~~~~~~~~~~~broken
y
\
p_4 3rd
6
2 Ma 2
v61
?
o
"?
?
"')
"
t~
r^*-
.r
.
_
~~b
ab"~MIN
- ;Min2
b_ horn 2 - ..o_
0
- -o'-
[J?1
(o'0"^
^?*'c
^
/^
12
Tuba
3 horn
-nn
43
6
W
M?
/
?
TJ, <
^ ^it
5$3 ,
^
1
y
u
/;
f0
<?
/
trombone
j
3
1
v ...
(J
tn
t
"
~;
'
..../M
^-^
'
.^^^'
L~~~~~~~~~~~~~~~~~~~~~~~broken
y
\
p_4 3rd
6
2 Ma 2
v61
?
o
"?
?
"')
"
t~
r^*-
.r
.
_
~~b
ab"~MIN
- ;Min2
b_ horn 2 - ..o_
0
- -o'-
[J?1
(o'0"^
^?*'c
^
/^
12
1222
wm2 Mtaj
"
-m
A
m2
Maj
2
.
m3
wm2 Mtaj
"
-m
A
m2
Maj
2
.
m3
wm2 Mtaj
"
-m
A
m2
Maj
2
.
m3
9r
/
Ma]3
9:1 itev F
'
\
9:~~~~~~~~~~~t,
9"-
.'-
1,1
.i--~ .0
-tO-
""
9r
/
Ma]3
9:1 itev F
'
\
9:~~~~~~~~~~~t,
9"-
.'-
1,1
.i--~ .0
-tO-
""
9r
/
Ma]3
9:1 itev F
'
\
9:~~~~~~~~~~~t,
9"-
.'-
1,1
.i--~ .0
-tO-
""
Maj3
"
4
3
4
Maj3
"
4
3
4
Maj3
"
4
3
4
Tuba
-
MIN
uba
'\ MIN5
Cor
Perfect
5
Ba
Tuba
-
MIN
uba
'\ MIN5
Cor
Perfect
5
Ba
Tuba
-
MIN
uba
'\ MIN5
Cor
Perfect
5
Ba
Maj
2
min 2
etc.
Maj
2
min 2
etc.
Maj
2
min 2
etc.
maj
2
min 2
mi
3
maj
2
min 2
mi
3
maj
2
min 2
mi
3
[21
[2]
[4]
[5]
[6]
[7]
[8]
[21
[2]
[4]
[5]
[6]
[7]
[8]
[21
[2]
[4]
[5]
[6]
[7]
[8]
[9]
7
9]
7
9]
7
[10]
[
11
I
[
12]
[ 13]
[ 14]
[
15
1
[10]
[
11
I
[
12]
[ 13]
[ 14]
[
15
1
[10]
[
11
I
[
12]
[ 13]
[ 14]
[
15
1
[
161161161
LV
J
r I
LV
J
r I
LV
J
r I
tF
/
4th mm
5
4th
min 5
tF
/
4th mm
5
4th
min 5
tF
/
4th mm
5
4th
min 5
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 24/41
Interval
ycles
as
Compositional
esources
65nterval
ycles
as
Compositional
esources
65nterval
ycles
as
Compositional
esources
65
Figure
5.
Summary
of
cycles
in
Example
11.
igure
5.
Summary
of
cycles
in
Example
11.
igure
5.
Summary
of
cycles
in
Example
11.
a. Lines 1-2.
pc
0 9 1 10 2
11
3
x/y
=
9/4 n= 1 PCL=7
b. Lines 5-6.
pc
0
5 2 7 4 9
6
11
8
1
10 3
x/y=
5/9 n
=
2 PCL
=
12
c. Lines 7-8.
pc
0 10 8 6
4 2 0
x/y
=
10/10
n
=
8
PCL
=
6
d. Lines 7-8.
pc
0 11 10 9 8 7
6
5 4 3 2 1
x/y=
11/11 n
=
10 PCL
=
12
e. Lines 11-12.
pc
0 5 1 6 2 7 3 8 4 9
x/y=
5
/8
n
=
1 PCL
=
10
the
PCL
at the ninth element to return
o
pc
0,
thus
forming
a
sonority
with the same
top
and bottom note. In the
lower left-
hand
corer
of
the
page,
a
5/6 combination
Fig.
5g) presents
a
PCL of
11,
followed
by
a
repetition
of
pc
6 and the
aggregate-
completing pc 0. Ives draws a line to separatethe 11 members
of the PCL from the final
two
notes,
highlighting
he
nonrepeti-
tive
portion
of the
sequence,
while he also notes
that 12
pitch
classes are
present
in
the entire
formation,
deemphasizing
he
single pitch-classrepetition.
The other
combination
cycles
notated
in
Example
11 and
n-
cluded in
Figure
5
are nonreiterativecombinations
of
other
cy-
cles.
Nonintersecting
nterval-2
cycles
are
overlaidon
staves
5-
a. Lines 1-2.
pc
0 9 1 10 2
11
3
x/y
=
9/4 n= 1 PCL=7
b. Lines 5-6.
pc
0
5 2 7 4 9
6
11
8
1
10 3
x/y=
5/9 n
=
2 PCL
=
12
c. Lines 7-8.
pc
0 10 8 6
4 2 0
x/y
=
10/10
n
=
8
PCL
=
6
d. Lines 7-8.
pc
0 11 10 9 8 7
6
5 4 3 2 1
x/y=
11/11 n
=
10 PCL
=
12
e. Lines 11-12.
pc
0 5 1 6 2 7 3 8 4 9
x/y=
5
/8
n
=
1 PCL
=
10
the
PCL
at the ninth element to return
o
pc
0,
thus
forming
a
sonority
with the same
top
and bottom note. In the
lower left-
hand
corer
of
the
page,
a
5/6 combination
Fig.
5g) presents
a
PCL of
11,
followed
by
a
repetition
of
pc
6 and the
aggregate-
completing pc 0. Ives draws a line to separatethe 11 members
of the PCL from the final
two
notes,
highlighting
he
nonrepeti-
tive
portion
of the
sequence,
while he also notes
that 12
pitch
classes are
present
in
the entire
formation,
deemphasizing
he
single pitch-classrepetition.
The other
combination
cycles
notated
in
Example
11 and
n-
cluded in
Figure
5
are nonreiterativecombinations
of
other
cy-
cles.
Nonintersecting
nterval-2
cycles
are
overlaidon
staves
5-
a. Lines 1-2.
pc
0 9 1 10 2
11
3
x/y
=
9/4 n= 1 PCL=7
b. Lines 5-6.
pc
0
5 2 7 4 9
6
11
8
1
10 3
x/y=
5/9 n
=
2 PCL
=
12
c. Lines 7-8.
pc
0 10 8 6
4 2 0
x/y
=
10/10
n
=
8
PCL
=
6
d. Lines 7-8.
pc
0 11 10 9 8 7
6
5 4 3 2 1
x/y=
11/11 n
=
10 PCL
=
12
e. Lines 11-12.
pc
0 5 1 6 2 7 3 8 4 9
x/y=
5
/8
n
=
1 PCL
=
10
the
PCL
at the ninth element to return
o
pc
0,
thus
forming
a
sonority
with the same
top
and bottom note. In the
lower left-
hand
corer
of
the
page,
a
5/6 combination
Fig.
5g) presents
a
PCL of
11,
followed
by
a
repetition
of
pc
6 and the
aggregate-
completing pc 0. Ives draws a line to separatethe 11 members
of the PCL from the final
two
notes,
highlighting
he
nonrepeti-
tive
portion
of the
sequence,
while he also notes
that 12
pitch
classes are
present
in
the entire
formation,
deemphasizing
he
single pitch-classrepetition.
The other
combination
cycles
notated
in
Example
11 and
n-
cluded in
Figure
5
are nonreiterativecombinations
of
other
cy-
cles.
Nonintersecting
nterval-2
cycles
are
overlaidon
staves
5-
f.
Lines
11-12.
pc
0
4
11 3 10
2
9
1
8
x/y=
4/7 n=11 PCL =9
g.
Lines 15-16.
pc
6
11
5 10
4
9
3
8
2 7
1
6
0
x/y
=
5/6
n
=
11 PCL = 11
h. Lines 15-16.
pc
0 4 9 1
6
10 3 7 0
x/y
=
4/ 5 n
=
9 PCL
=
8
i.
Lines 15-16.
pc
0
6
1
7 2 8 3
9
4 10 5
11
x/y
=
6/7 n
=
1 PCL
=
12
j.
Lines 15-16.
pc
0 2 3
5 6 8 9
11
x/y
=
2 /1
n
=
3 PCL
=
8
6
(Fig. 5b)
in a 5/9
alternationthat extends to its PCL
of 12.
Octatonic collections result from a 4/5
combination
overlaying
interval-9
cycles
inthe center of lines
15-16
(Fig. 5h),
and from
a 2/1 alternationon staff
15
in
the lower
right-hand
orner
of the
page(Fig. 5j) overlaying nterval-3cycles.These combinations
complete
a
comprehensive
accumulation f
larger
PCLs,
rang-
ing
from
a PCL
of 6 for the interval-10
cycle
to
completion
of
the
aggregate
by
three
others,
and
includingevery
value
be-
tween. Table
2 charts
this
accumulation,
including
the two
single-interval
ycles
(Fig.
5c and
5d)
notated as combinations.
Noticeably
absent from the n values
are
cycles
of
intervals
5 and
7;
combinations
of
these
cycles,
of
course,
display
a
pattern
of
f.
Lines
11-12.
pc
0
4
11 3 10
2
9
1
8
x/y=
4/7 n=11 PCL =9
g.
Lines 15-16.
pc
6
11
5 10
4
9
3
8
2 7
1
6
0
x/y
=
5/6
n
=
11 PCL = 11
h. Lines 15-16.
pc
0 4 9 1
6
10 3 7 0
x/y
=
4/ 5 n
=
9 PCL
=
8
i.
Lines 15-16.
pc
0
6
1
7 2 8 3
9
4 10 5
11
x/y
=
6/7 n
=
1 PCL
=
12
j.
Lines 15-16.
pc
0 2 3
5 6 8 9
11
x/y
=
2 /1
n
=
3 PCL
=
8
6
(Fig. 5b)
in a 5/9
alternationthat extends to its PCL
of 12.
Octatonic collections result from a 4/5
combination
overlaying
interval-9
cycles
inthe center of lines
15-16
(Fig. 5h),
and from
a 2/1 alternationon staff
15
in
the lower
right-hand
orner
of the
page(Fig. 5j) overlaying nterval-3cycles.These combinations
complete
a
comprehensive
accumulation f
larger
PCLs,
rang-
ing
from
a PCL
of 6 for the interval-10
cycle
to
completion
of
the
aggregate
by
three
others,
and
includingevery
value
be-
tween. Table
2 charts
this
accumulation,
including
the two
single-interval
ycles
(Fig.
5c and
5d)
notated as combinations.
Noticeably
absent from the n values
are
cycles
of
intervals
5 and
7;
combinations
of
these
cycles,
of
course,
display
a
pattern
of
f.
Lines
11-12.
pc
0
4
11 3 10
2
9
1
8
x/y=
4/7 n=11 PCL =9
g.
Lines 15-16.
pc
6
11
5 10
4
9
3
8
2 7
1
6
0
x/y
=
5/6
n
=
11 PCL = 11
h. Lines 15-16.
pc
0 4 9 1
6
10 3 7 0
x/y
=
4/ 5 n
=
9 PCL
=
8
i.
Lines 15-16.
pc
0
6
1
7 2 8 3
9
4 10 5
11
x/y
=
6/7 n
=
1 PCL
=
12
j.
Lines 15-16.
pc
0 2 3
5 6 8 9
11
x/y
=
2 /1
n
=
3 PCL
=
8
6
(Fig. 5b)
in a 5/9
alternationthat extends to its PCL
of 12.
Octatonic collections result from a 4/5
combination
overlaying
interval-9
cycles
inthe center of lines
15-16
(Fig. 5h),
and from
a 2/1 alternationon staff
15
in
the lower
right-hand
orner
of the
page(Fig. 5j) overlaying nterval-3cycles.These combinations
complete
a
comprehensive
accumulation f
larger
PCLs,
rang-
ing
from
a PCL
of 6 for the interval-10
cycle
to
completion
of
the
aggregate
by
three
others,
and
includingevery
value
be-
tween. Table
2 charts
this
accumulation,
including
the two
single-interval
ycles
(Fig.
5c and
5d)
notated as combinations.
Noticeably
absent from the n values
are
cycles
of
intervals
5 and
7;
combinations
of
these
cycles,
of
course,
display
a
pattern
of
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 25/41
66
Music
TheorySpectrum
6
Music
TheorySpectrum
6
Music
TheorySpectrum
Table 2.
Summary
of
Figure
5
organized by
PCL.
able 2.
Summary
of
Figure
5
organized by
PCL.
able 2.
Summary
of
Figure
5
organized by
PCL.
PCL
6
PCL
6
PCL
6
x/y
10/10
x/y
10/10
x/y
10/10
7
9/4
1
9/4
1
9/4
1
n
Ex.
11,
lines:
Fig.
5,
part:
8 4-5 c
n
Ex.
11,
lines:
Fig.
5,
part:
8 4-5 c
n
Ex.
11,
lines:
Fig.
5,
part:
8 4-5 c
1-2-2-2 a
sions of the combination
cycles
that
precede
them,
forming
me-
lodic
skeletons
from
the
cyclic
sources. These
sketchings
dem-
onstratea
compositionalapproach
hat carries
hrough
o other
pages
of the
Symphony,
as the combination
cycles
provide
sources
of
pitch-class
material or
subsequent
development
and
transformation.
sions of the combination
cycles
that
precede
them,
forming
me-
lodic
skeletons
from
the
cyclic
sources. These
sketchings
dem-
onstratea
compositionalapproach
hat carries
hrough
o other
pages
of the
Symphony,
as the combination
cycles
provide
sources
of
pitch-class
material or
subsequent
development
and
transformation.
sions of the combination
cycles
that
precede
them,
forming
me-
lodic
skeletons
from
the
cyclic
sources. These
sketchings
dem-
onstratea
compositionalapproach
hat carries
hrough
o other
pages
of the
Symphony,
as the combination
cycles
provide
sources
of
pitch-class
material or
subsequent
development
and
transformation.
8 4/5 9
15-16
2/1
3
15-16
9 4/7 11 11-12
10 5/8 1 11-12
11 5/6 11
15-16
8 4/5 9
15-16
2/1
3
15-16
9 4/7 11 11-12
10 5/8 1 11-12
11 5/6 11
15-16
8 4/5 9
15-16
2/1
3
15-16
9 4/7 11 11-12
10 5/8 1 11-12
11 5/6 11
15-16
12
5/9
11/11
6/7
12
5/9
11/11
6/7
12
5/9
11/11
6/7
2
2
1
2
2
1
2
2
1
5-6
7-8
15-16
5-6
7-8
15-16
5-6
7-8
15-16
h
J
f
e
g
b
d
i
h
J
f
e
g
b
d
i
h
J
f
e
g
b
d
i
PCLdistributionanalogousto that for intervals1 and 11, and
thus offer no
unique
contributions
o
the PCL
summary,
al-
though
they
do
display contrasting possibilities
for
x/y
combinations.
The bottom three staves of
Example
11
contain additional
notations
correlating
to
Ives's
cyclic conception
of the
pitch
structures n
the Universe
Symphony.
Severaloftheseare
cyclic
repetitions
of
different
types, including
an interval-3
ycle plus
two added
notes
(right
margin
of staff
14)
andtwo instances
of a
three-intervalalternation: epetitionsof intervals1-2-3and2-1-
3 form octatonic
subsets at the
right margin
of staff 16.49The
other notes on
these bottom
three
staves,
mostly
notated
with
darkened
note
heads,
are linearizedand
slightly
reorderedver-
49Below he
1-2-3
sequence,
Ives indicates the
alternating
ntervallic
pat-
tern
"Maj
2
/
min
2
/
etc,"
referring
o the
sequence
directly
above
this but on
staff15.
PCLdistributionanalogousto that for intervals1 and 11, and
thus offer no
unique
contributions
o
the PCL
summary,
al-
though
they
do
display contrasting possibilities
for
x/y
combinations.
The bottom three staves of
Example
11
contain additional
notations
correlating
to
Ives's
cyclic conception
of the
pitch
structures n
the Universe
Symphony.
Severaloftheseare
cyclic
repetitions
of
different
types, including
an interval-3
ycle plus
two added
notes
(right
margin
of staff
14)
andtwo instances
of a
three-intervalalternation: epetitionsof intervals1-2-3and2-1-
3 form octatonic
subsets at the
right margin
of staff 16.49The
other notes on
these bottom
three
staves,
mostly
notated
with
darkened
note
heads,
are linearizedand
slightly
reorderedver-
49Below he
1-2-3
sequence,
Ives indicates the
alternating
ntervallic
pat-
tern
"Maj
2
/
min
2
/
etc,"
referring
o the
sequence
directly
above
this but on
staff15.
PCLdistributionanalogousto that for intervals1 and 11, and
thus offer no
unique
contributions
o
the PCL
summary,
al-
though
they
do
display contrasting possibilities
for
x/y
combinations.
The bottom three staves of
Example
11
contain additional
notations
correlating
to
Ives's
cyclic conception
of the
pitch
structures n
the Universe
Symphony.
Severaloftheseare
cyclic
repetitions
of
different
types, including
an interval-3
ycle plus
two added
notes
(right
margin
of staff
14)
andtwo instances
of a
three-intervalalternation: epetitionsof intervals1-2-3and2-1-
3 form octatonic
subsets at the
right margin
of staff 16.49The
other notes on
these bottom
three
staves,
mostly
notated
with
darkened
note
heads,
are linearizedand
slightly
reorderedver-
49Below he
1-2-3
sequence,
Ives indicates the
alternating
ntervallic
pat-
tern
"Maj
2
/
min
2
/
etc,"
referring
o the
sequence
directly
above
this but on
staff15.
CYCLES AS COMPOSITIONAL
OURCES.The
linearizationson
the bottom staves of the
Universe
Symphonypage
(Ex.
11) rep-
resent
a first
step
toward
a
musicalrealizationof the
cyclic
pitch
structures.The
5/6
cycle
notated in
whole notes at the left mar-
gin
(staves
15 and
16),
for
example,
is
immediately
ollowed
by
a series
of
darkenednote
heads,
withand then without
stems,
that
presents
the
notes
of the
cycle
in
order,
except
that the or-
der
of
the
thirdand
fourthnotes
(pcs
5 and
10)
is
reversed,
and
the
penultimatepc
6,
which s a
duplication
of
the
initialnote of
the
cycle,
is omitted. In
essence, then,
the
whole-note notation
is an abstract
expression
of
the
pitch-class
ource
material hat
is
subsequently
realized,
without
rhythmic
values,
in a
specific
register. Presumably,
each
cycle
notated
on
the
page
is
to
pro-
vide source
material of this
nature
in
the
composition
of
the
Symphony,
in
order that
many
musical
deas within the
depic-
tion of a "Universe in tones"
originate
with
a
predetermined,
systematically
conceived
pitch-class
structure. The
cyclic
sources
determine
a
particular itch-class
uccession,
subject
o
slight
order
variations,
as well as
a
uniform ntervallic ucces-
sion and
a
specificpitch-class ength.50
Ives's methods
of
realizing compositional
sources are
not,
however,
limitedto
the
simple
inearizations hown
here
and n
Examples
1 and
9.
In more
complex
incorporations,
cyclic
structuresserve as sources
for
intermingled
vertical
and
hori-
50Similarly,
Michael
J. Babcock
explores
the
circle of fifths
as a
composi-
tional source for the "Thoreau"
movement
of Ives's ConcordSonata
n "Ives's
'Thoreau':
A
Point of
Order,"
Proceedings f
the American
Society
of
Univer-
sity Composers
9 and 10
(1976),
89-102.
CYCLES AS COMPOSITIONAL
OURCES.The
linearizationson
the bottom staves of the
Universe
Symphonypage
(Ex.
11) rep-
resent
a first
step
toward
a
musicalrealizationof the
cyclic
pitch
structures.The
5/6
cycle
notated in
whole notes at the left mar-
gin
(staves
15 and
16),
for
example,
is
immediately
ollowed
by
a series
of
darkenednote
heads,
withand then without
stems,
that
presents
the
notes
of the
cycle
in
order,
except
that the or-
der
of
the
thirdand
fourthnotes
(pcs
5 and
10)
is
reversed,
and
the
penultimatepc
6,
which s a
duplication
of
the
initialnote of
the
cycle,
is omitted. In
essence, then,
the
whole-note notation
is an abstract
expression
of
the
pitch-class
ource
material hat
is
subsequently
realized,
without
rhythmic
values,
in a
specific
register. Presumably,
each
cycle
notated
on
the
page
is
to
pro-
vide source
material of this
nature
in
the
composition
of
the
Symphony,
in
order that
many
musical
deas within the
depic-
tion of a "Universe in tones"
originate
with
a
predetermined,
systematically
conceived
pitch-class
structure. The
cyclic
sources
determine
a
particular itch-class
uccession,
subject
o
slight
order
variations,
as well as
a
uniform ntervallic ucces-
sion and
a
specificpitch-class ength.50
Ives's methods
of
realizing compositional
sources are
not,
however,
limitedto
the
simple
inearizations hown
here
and n
Examples
1 and
9.
In more
complex
incorporations,
cyclic
structuresserve as sources
for
intermingled
vertical
and
hori-
50Similarly,
Michael
J. Babcock
explores
the
circle of fifths
as a
composi-
tional source for the "Thoreau"
movement
of Ives's ConcordSonata
n "Ives's
'Thoreau':
A
Point of
Order,"
Proceedings f
the American
Society
of
Univer-
sity Composers
9 and 10
(1976),
89-102.
CYCLES AS COMPOSITIONAL
OURCES.The
linearizationson
the bottom staves of the
Universe
Symphonypage
(Ex.
11) rep-
resent
a first
step
toward
a
musicalrealizationof the
cyclic
pitch
structures.The
5/6
cycle
notated in
whole notes at the left mar-
gin
(staves
15 and
16),
for
example,
is
immediately
ollowed
by
a series
of
darkenednote
heads,
withand then without
stems,
that
presents
the
notes
of the
cycle
in
order,
except
that the or-
der
of
the
thirdand
fourthnotes
(pcs
5 and
10)
is
reversed,
and
the
penultimatepc
6,
which s a
duplication
of
the
initialnote of
the
cycle,
is omitted. In
essence, then,
the
whole-note notation
is an abstract
expression
of
the
pitch-class
ource
material hat
is
subsequently
realized,
without
rhythmic
values,
in a
specific
register. Presumably,
each
cycle
notated
on
the
page
is
to
pro-
vide source
material of this
nature
in
the
composition
of
the
Symphony,
in
order that
many
musical
deas within the
depic-
tion of a "Universe in tones"
originate
with
a
predetermined,
systematically
conceived
pitch-class
structure. The
cyclic
sources
determine
a
particular itch-class
uccession,
subject
o
slight
order
variations,
as well as
a
uniform ntervallic ucces-
sion and
a
specificpitch-class ength.50
Ives's methods
of
realizing compositional
sources are
not,
however,
limitedto
the
simple
inearizations hown
here
and n
Examples
1 and
9.
In more
complex
incorporations,
cyclic
structuresserve as sources
for
intermingled
vertical
and
hori-
50Similarly,
Michael
J. Babcock
explores
the
circle of fifths
as a
composi-
tional source for the "Thoreau"
movement
of Ives's ConcordSonata
n "Ives's
'Thoreau':
A
Point of
Order,"
Proceedings f
the American
Society
of
Univer-
sity Composers
9 and 10
(1976),
89-102.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 26/41
Interval
ycles
as
Compositional
esources 67
nterval
ycles
as
Compositional
esources 67
nterval
ycles
as
Compositional
esources 67
zontal
ideas,
potentially employing
more extensive distortions
of originalordering.The consistent set-class substructure hat
naturally
arises from an
intervallic
equence
may
be
preserved
and
highlighted,
while the
underlying
ntervallic
epetitionmay
be
to some extent
suppressed.
A "sourceset" of
interval
cycles
must exert
unequivocal
control over
pitch
constructions,
ince
substantial order variations
may
obscure
original
structural
properties,
and,
indeed,
largercycles
are
distinguishable
rom
each other
only
by
their
intervallic
tructure,
not
by
their
pitch-
class content.
Still,
the
boundaries et forth
by
comprehensibil-
ity and fidelity to a source allow for ample compositional
freedom.
The
overall
unityprovidedby
this
type
of structural etermi-
nant
appears
n works
composed
as
early
as
the
choral
Psalms,
including
versesofPsalm24 that follow the
excerptgiven
as Ex-
ample
8. While the
chromaticism
f
verse
1
of this Psalm is dis-
played
in the outer voices
only, cyclic
sources in other verses
influence he
complete
texture,
so that an
entire
passage
s satu-
rated with a
distinctive constructional
principle.
In
verse
2 of
Psalm 24 (mm. 7-11), forexample,the notes of the outer-voice
whole-tone
scale also
determine
the
pitch-class
content of the
inner
voices,
with the
result that the
pervasive
pitch
resource s
indeed
a
"scale,"
in the
traditional
ense,
that defines the har-
monic
language
of
the
passage.
With a small
cardinality,
his
source
is
easily
characterized
by
content,
and
thus
requires
no
restrictionsof
ordering
o
retain ts
integrity;
notes
for
the
alto
and
tenor can be
selected from
any
portion
of
the outer-voice
scale,
without
regard
for their
scalar
ordering.
The
resulting
verticalstructuresare whole-tone subsets formed by various
combinations of even-numberedintervals.
Intervals 3 and 9
similarly
prescribe
the harmonic
anguage
of
verse 3
(mm.
12-
16).
For
cyclic
structures hat
generate
higher
numbersof
pitch
classes,
this
type
of
intervallic
aturation annot
be achieved
by
simple
distributionof
pitch
classes from
the
source
throughout
the
texture.
Because
the
source is
distinguished
primarily
by
zontal
ideas,
potentially employing
more extensive distortions
of originalordering.The consistent set-class substructure hat
naturally
arises from an
intervallic
equence
may
be
preserved
and
highlighted,
while the
underlying
ntervallic
epetitionmay
be
to some extent
suppressed.
A "sourceset" of
interval
cycles
must exert
unequivocal
control over
pitch
constructions,
ince
substantial order variations
may
obscure
original
structural
properties,
and,
indeed,
largercycles
are
distinguishable
rom
each other
only
by
their
intervallic
tructure,
not
by
their
pitch-
class content.
Still,
the
boundaries et forth
by
comprehensibil-
ity and fidelity to a source allow for ample compositional
freedom.
The
overall
unityprovidedby
this
type
of structural etermi-
nant
appears
n works
composed
as
early
as
the
choral
Psalms,
including
versesofPsalm24 that follow the
excerptgiven
as Ex-
ample
8. While the
chromaticism
f
verse
1
of this Psalm is dis-
played
in the outer voices
only, cyclic
sources in other verses
influence he
complete
texture,
so that an
entire
passage
s satu-
rated with a
distinctive constructional
principle.
In
verse
2 of
Psalm 24 (mm. 7-11), forexample,the notes of the outer-voice
whole-tone
scale also
determine
the
pitch-class
content of the
inner
voices,
with the
result that the
pervasive
pitch
resource s
indeed
a
"scale,"
in the
traditional
ense,
that defines the har-
monic
language
of
the
passage.
With a small
cardinality,
his
source
is
easily
characterized
by
content,
and
thus
requires
no
restrictionsof
ordering
o
retain ts
integrity;
notes
for
the
alto
and
tenor can be
selected from
any
portion
of
the outer-voice
scale,
without
regard
for their
scalar
ordering.
The
resulting
verticalstructuresare whole-tone subsets formed by various
combinations of even-numberedintervals.
Intervals 3 and 9
similarly
prescribe
the harmonic
anguage
of
verse 3
(mm.
12-
16).
For
cyclic
structures hat
generate
higher
numbersof
pitch
classes,
this
type
of
intervallic
aturation annot
be achieved
by
simple
distributionof
pitch
classes from
the
source
throughout
the
texture.
Because
the
source is
distinguished
primarily
by
zontal
ideas,
potentially employing
more extensive distortions
of originalordering.The consistent set-class substructure hat
naturally
arises from an
intervallic
equence
may
be
preserved
and
highlighted,
while the
underlying
ntervallic
epetitionmay
be
to some extent
suppressed.
A "sourceset" of
interval
cycles
must exert
unequivocal
control over
pitch
constructions,
ince
substantial order variations
may
obscure
original
structural
properties,
and,
indeed,
largercycles
are
distinguishable
rom
each other
only
by
their
intervallic
tructure,
not
by
their
pitch-
class content.
Still,
the
boundaries et forth
by
comprehensibil-
ity and fidelity to a source allow for ample compositional
freedom.
The
overall
unityprovidedby
this
type
of structural etermi-
nant
appears
n works
composed
as
early
as
the
choral
Psalms,
including
versesofPsalm24 that follow the
excerptgiven
as Ex-
ample
8. While the
chromaticism
f
verse
1
of this Psalm is dis-
played
in the outer voices
only, cyclic
sources in other verses
influence he
complete
texture,
so that an
entire
passage
s satu-
rated with a
distinctive constructional
principle.
In
verse
2 of
Psalm 24 (mm. 7-11), forexample,the notes of the outer-voice
whole-tone
scale also
determine
the
pitch-class
content of the
inner
voices,
with the
result that the
pervasive
pitch
resource s
indeed
a
"scale,"
in the
traditional
ense,
that defines the har-
monic
language
of
the
passage.
With a small
cardinality,
his
source
is
easily
characterized
by
content,
and
thus
requires
no
restrictionsof
ordering
o
retain ts
integrity;
notes
for
the
alto
and
tenor can be
selected from
any
portion
of
the outer-voice
scale,
without
regard
for their
scalar
ordering.
The
resulting
verticalstructuresare whole-tone subsets formed by various
combinations of even-numberedintervals.
Intervals 3 and 9
similarly
prescribe
the harmonic
anguage
of
verse 3
(mm.
12-
16).
For
cyclic
structures hat
generate
higher
numbersof
pitch
classes,
this
type
of
intervallic
aturation annot
be achieved
by
simple
distributionof
pitch
classes from
the
source
throughout
the
texture.
Because
the
source is
distinguished
primarily
by
order-solely by
order in the
case of
aggregatecompletion-
the process changesfrom a retention of pitchclasses from the
source to a
perpetuation
of the
intervallic
adjacencies
of the
source.
Thus,
in verse 5 of Psalm 24
(mm. 22-27),
outer-voice
cycles
of
ascending
and
descending perfect
fourths are
sup-
portedexclusivelyby quartal
verticalities.Even when a
source
may
be less
audibly
amiliar,
as
in
some
combination
ycles,
for
example,
a certain
type
of intervallic structure will be
pre-
scribed,
and this
can be
projected
despite
selected
reorderings
and other distortionsof the source.
The structureof Ives's orchestral"tonepoem" TheFourth
of
July
(1911-13)
is based both on
quotations
of familiar
tunes-as is thatof
many
of his
longer
orchestralworks-and
types
of
compositional
calculations associated more with
the
shorter
experimentalpieces.
In
recalling
his
composition
of
the
work,
Ives writes
of
"a
feeling
of freedom as a
boy
has ... who
wants
to do
anything
he wants to do" while at the same time
working
out
"combinations f tones and
rhythms ery
carefully
by
kind
of
prescriptions,
in
the
way
a
chemical
compound
which makes explosions would be made."51The network of
quotations
s indeed
diverse,
resembling
he sort of
"free
asso-
ciation" Ives seems to
describe,
yet
the choices
of
tunes are
hardly
made at
random,
and,
as
Dennis
Marshallhas demon-
strated,
the tune
"Red,
White,
and Blue"
(RWB)
standsat the
structuralcore of the
movement.52Both the tune
quotations
and the
pitch-rhythm
calculations contribute
to a
program-
matic
depiction
of a civic
celebration,
establishing
a series of
musical
interrelationships
and extramusicalassociations that
helpto portray he multiplicityof the experience.53
51Ives,Memos,
104.
52Marshall,
"Charles Ives's
Quotations,"
54-55.
The
following
analysis
supports
Marshall'sobservation
hat
RWB is
used as
"both a
melodic
and har-
monic source" n the
opening
of
the
work.
53See Mark D.
Nelson,
"Beyond
Mimesis: Transcendentalism nd
Pro-
cesses of
Analogy
in
Charles Ives's The Fourth
of July,"
Perspectives
f
New
Music 22/1-2
(1983-84),
353-384.
order-solely by
order in the
case of
aggregatecompletion-
the process changesfrom a retention of pitchclasses from the
source to a
perpetuation
of the
intervallic
adjacencies
of the
source.
Thus,
in verse 5 of Psalm 24
(mm. 22-27),
outer-voice
cycles
of
ascending
and
descending perfect
fourths are
sup-
portedexclusivelyby quartal
verticalities.Even when a
source
may
be less
audibly
amiliar,
as
in
some
combination
ycles,
for
example,
a certain
type
of intervallic structure will be
pre-
scribed,
and this
can be
projected
despite
selected
reorderings
and other distortionsof the source.
The structureof Ives's orchestral"tonepoem" TheFourth
of
July
(1911-13)
is based both on
quotations
of familiar
tunes-as is thatof
many
of his
longer
orchestralworks-and
types
of
compositional
calculations associated more with
the
shorter
experimentalpieces.
In
recalling
his
composition
of
the
work,
Ives writes
of
"a
feeling
of freedom as a
boy
has ... who
wants
to do
anything
he wants to do" while at the same time
working
out
"combinations f tones and
rhythms ery
carefully
by
kind
of
prescriptions,
in
the
way
a
chemical
compound
which makes explosions would be made."51The network of
quotations
s indeed
diverse,
resembling
he sort of
"free
asso-
ciation" Ives seems to
describe,
yet
the choices
of
tunes are
hardly
made at
random,
and,
as
Dennis
Marshallhas demon-
strated,
the tune
"Red,
White,
and Blue"
(RWB)
standsat the
structuralcore of the
movement.52Both the tune
quotations
and the
pitch-rhythm
calculations contribute
to a
program-
matic
depiction
of a civic
celebration,
establishing
a series of
musical
interrelationships
and extramusicalassociations that
helpto portray he multiplicityof the experience.53
51Ives,Memos,
104.
52Marshall,
"Charles Ives's
Quotations,"
54-55.
The
following
analysis
supports
Marshall'sobservation
hat
RWB is
used as
"both a
melodic
and har-
monic source" n the
opening
of
the
work.
53See Mark D.
Nelson,
"Beyond
Mimesis: Transcendentalism nd
Pro-
cesses of
Analogy
in
Charles Ives's The Fourth
of July,"
Perspectives
f
New
Music 22/1-2
(1983-84),
353-384.
order-solely by
order in the
case of
aggregatecompletion-
the process changesfrom a retention of pitchclasses from the
source to a
perpetuation
of the
intervallic
adjacencies
of the
source.
Thus,
in verse 5 of Psalm 24
(mm. 22-27),
outer-voice
cycles
of
ascending
and
descending perfect
fourths are
sup-
portedexclusivelyby quartal
verticalities.Even when a
source
may
be less
audibly
amiliar,
as
in
some
combination
ycles,
for
example,
a certain
type
of intervallic structure will be
pre-
scribed,
and this
can be
projected
despite
selected
reorderings
and other distortionsof the source.
The structureof Ives's orchestral"tonepoem" TheFourth
of
July
(1911-13)
is based both on
quotations
of familiar
tunes-as is thatof
many
of his
longer
orchestralworks-and
types
of
compositional
calculations associated more with
the
shorter
experimentalpieces.
In
recalling
his
composition
of
the
work,
Ives writes
of
"a
feeling
of freedom as a
boy
has ... who
wants
to do
anything
he wants to do" while at the same time
working
out
"combinations f tones and
rhythms ery
carefully
by
kind
of
prescriptions,
in
the
way
a
chemical
compound
which makes explosions would be made."51The network of
quotations
s indeed
diverse,
resembling
he sort of
"free
asso-
ciation" Ives seems to
describe,
yet
the choices
of
tunes are
hardly
made at
random,
and,
as
Dennis
Marshallhas demon-
strated,
the tune
"Red,
White,
and Blue"
(RWB)
standsat the
structuralcore of the
movement.52Both the tune
quotations
and the
pitch-rhythm
calculations contribute
to a
program-
matic
depiction
of a civic
celebration,
establishing
a series of
musical
interrelationships
and extramusicalassociations that
helpto portray he multiplicityof the experience.53
51Ives,Memos,
104.
52Marshall,
"Charles Ives's
Quotations,"
54-55.
The
following
analysis
supports
Marshall'sobservation
hat
RWB is
used as
"both a
melodic
and har-
monic source" n the
opening
of
the
work.
53See Mark D.
Nelson,
"Beyond
Mimesis: Transcendentalism nd
Pro-
cesses of
Analogy
in
Charles Ives's The Fourth
of July,"
Perspectives
f
New
Music 22/1-2
(1983-84),
353-384.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 27/41
68 Music
TheorySpectrum
8 Music
TheorySpectrum
8 Music
TheorySpectrum
Evidence of
cyclic compositional
ources s
prominent
rom
the
openingsection of the work. Ivesnotes, inthemarginof an
early
score-sketch,
that
some "chords n
'[The] Cage'
"
repre-
sent the
origins
of the firstsection of TheFourth
of July,
appar-
ently referring
o
the series of
sonoritiesconstructed
of
fourths
and fifths in the
strings
of mm. 8-13.54
The
sustained
notes in
the condensed score of this
passage
given
in
Example
12a form
a
cyclic unity
of
intervals
5 and
7
(chords
a and
c,
respectively,
in the
example) alternating
withwhole-tone sonorities
(chord
b)
in mm.
8-11.55
Then in
mm. 12-13 chords a and c are re-
peatedwithout the interruptionof chordb. The lower melodic
voice
is
mostly
independent
of
the
chords.
The notationsbelow
the score
in the
example
trace chord
a,
beginning
with
the
pc
0
of
the
lower
melodic
line,
through
the first seven
elements
of
the interval-5
cycle, connecting
o
chord
c for
the
cycliccomple-
tion.56Chord
c
is constructed
of
fifths and thus "inverts"
hord
a;
in
effect,
the
cycle
progressesupward hrough
a,
connects
n
the
upper
register
of
both
chords,
and then
progresses
down-
ward
hrough
c. The
intervals
n
chord
b
are
also
inverted
from
2 to 10), but the pitchclassesdo not change:both occurrences
of
b state the five-tone whole-tone
subset,
as illustratedbelow
the
score.
The
lower
melodic voice in
this
passagepresents
he firstes-
sentiallycomplete
statement of the first
phrase
of RWB.
Com-
54Kirkpatrick,
Catalogue,
11. Ives
indicates elsewhere the influence
of
other works on The Fourth
of July,
including
March
and
Overture
1776
(Memos,
83)
and The GeneralSlocum
(Memos, 105).
55Ives's ssociationof quartaland whole-tone structures allsto mindparts
of
Schoenberg's
Kammersymphonie,Op.
9
(for
instance,
mm.
1-3).
Schoen-
berg's usages display
the
voice-leading
connections discussed n
his
Theory
of
Harmony,
406.
56ArthurMaisel describes
these
cyclic
completions
as
"mutually
xclusive
collections"
(that
is,
literal
complements)
in "The Fourth
of July
by
Charles
Ives: Mixed Harmonic Criteria n
a
Twentieth-Century
Classic,"
Theory
and
Practice
6/1
(1981),
3-32. These
collections
assume
considerable
ignificance
n
Maisel's
analysis
of the work.
Evidence of
cyclic compositional
ources s
prominent
rom
the
openingsection of the work. Ivesnotes, inthemarginof an
early
score-sketch,
that
some "chords n
'[The] Cage'
"
repre-
sent the
origins
of the firstsection of TheFourth
of July,
appar-
ently referring
o
the series of
sonoritiesconstructed
of
fourths
and fifths in the
strings
of mm. 8-13.54
The
sustained
notes in
the condensed score of this
passage
given
in
Example
12a form
a
cyclic unity
of
intervals
5 and
7
(chords
a and
c,
respectively,
in the
example) alternating
withwhole-tone sonorities
(chord
b)
in mm.
8-11.55
Then in
mm. 12-13 chords a and c are re-
peatedwithout the interruptionof chordb. The lower melodic
voice
is
mostly
independent
of
the
chords.
The notationsbelow
the score
in the
example
trace chord
a,
beginning
with
the
pc
0
of
the
lower
melodic
line,
through
the first seven
elements
of
the interval-5
cycle, connecting
o
chord
c for
the
cycliccomple-
tion.56Chord
c
is constructed
of
fifths and thus "inverts"
hord
a;
in
effect,
the
cycle
progressesupward hrough
a,
connects
n
the
upper
register
of
both
chords,
and then
progresses
down-
ward
hrough
c. The
intervals
n
chord
b
are
also
inverted
from
2 to 10), but the pitchclassesdo not change:both occurrences
of
b state the five-tone whole-tone
subset,
as illustratedbelow
the
score.
The
lower
melodic voice in
this
passagepresents
he firstes-
sentiallycomplete
statement of the first
phrase
of RWB.
Com-
54Kirkpatrick,
Catalogue,
11. Ives
indicates elsewhere the influence
of
other works on The Fourth
of July,
including
March
and
Overture
1776
(Memos,
83)
and The GeneralSlocum
(Memos, 105).
55Ives's ssociationof quartaland whole-tone structures allsto mindparts
of
Schoenberg's
Kammersymphonie,Op.
9
(for
instance,
mm.
1-3).
Schoen-
berg's usages display
the
voice-leading
connections discussed n
his
Theory
of
Harmony,
406.
56ArthurMaisel describes
these
cyclic
completions
as
"mutually
xclusive
collections"
(that
is,
literal
complements)
in "The Fourth
of July
by
Charles
Ives: Mixed Harmonic Criteria n
a
Twentieth-Century
Classic,"
Theory
and
Practice
6/1
(1981),
3-32. These
collections
assume
considerable
ignificance
n
Maisel's
analysis
of the work.
Evidence of
cyclic compositional
ources s
prominent
rom
the
openingsection of the work. Ivesnotes, inthemarginof an
early
score-sketch,
that
some "chords n
'[The] Cage'
"
repre-
sent the
origins
of the firstsection of TheFourth
of July,
appar-
ently referring
o
the series of
sonoritiesconstructed
of
fourths
and fifths in the
strings
of mm. 8-13.54
The
sustained
notes in
the condensed score of this
passage
given
in
Example
12a form
a
cyclic unity
of
intervals
5 and
7
(chords
a and
c,
respectively,
in the
example) alternating
withwhole-tone sonorities
(chord
b)
in mm.
8-11.55
Then in
mm. 12-13 chords a and c are re-
peatedwithout the interruptionof chordb. The lower melodic
voice
is
mostly
independent
of
the
chords.
The notationsbelow
the score
in the
example
trace chord
a,
beginning
with
the
pc
0
of
the
lower
melodic
line,
through
the first seven
elements
of
the interval-5
cycle, connecting
o
chord
c for
the
cycliccomple-
tion.56Chord
c
is constructed
of
fifths and thus "inverts"
hord
a;
in
effect,
the
cycle
progressesupward hrough
a,
connects
n
the
upper
register
of
both
chords,
and then
progresses
down-
ward
hrough
c. The
intervals
n
chord
b
are
also
inverted
from
2 to 10), but the pitchclassesdo not change:both occurrences
of
b state the five-tone whole-tone
subset,
as illustratedbelow
the
score.
The
lower
melodic voice in
this
passagepresents
he firstes-
sentiallycomplete
statement of the first
phrase
of RWB.
Com-
54Kirkpatrick,
Catalogue,
11. Ives
indicates elsewhere the influence
of
other works on The Fourth
of July,
including
March
and
Overture
1776
(Memos,
83)
and The GeneralSlocum
(Memos, 105).
55Ives's ssociationof quartaland whole-tone structures allsto mindparts
of
Schoenberg's
Kammersymphonie,Op.
9
(for
instance,
mm.
1-3).
Schoen-
berg's usages display
the
voice-leading
connections discussed n
his
Theory
of
Harmony,
406.
56ArthurMaisel describes
these
cyclic
completions
as
"mutually
xclusive
collections"
(that
is,
literal
complements)
in "The Fourth
of July
by
Charles
Ives: Mixed Harmonic Criteria n
a
Twentieth-Century
Classic,"
Theory
and
Practice
6/1
(1981),
3-32. These
collections
assume
considerable
ignificance
n
Maisel's
analysis
of the work.
Example
12.
The Fourth
of
July,
mm. 8-13.
a. Thirdand fourthviolins, cello, bass.
Example
12.
The Fourth
of
July,
mm. 8-13.
a. Thirdand fourthviolins, cello, bass.
Example
12.
The Fourth
of
July,
mm. 8-13.
a. Thirdand fourthviolins, cello, bass.
9
I
I
a
I
I
a
I
I
a
\
^
7
i
-^
*
-
d -
i
r
o
r
i
i
-a
d
t
r
i
X
F
-
p
"
F
I
F
6^A
t
__
X
0
X
J
,
J
TL3--i
r
\
^
7
i
-^
*
-
d -
i
r
o
r
i
i
-a
d
t
r
i
X
F
-
p
"
F
I
F
6^A
t
__
X
0
X
J
,
J
TL3--i
r
\
^
7
i
-^
*
-
d -
i
r
o
r
i
i
-a
d
t
r
i
X
F
-
p
"
F
I
F
6^A
t
__
X
0
X
J
,
J
TL3--i
r
I I I I II
1
I
i
b
c
b a c
I I I I II
1
I
i
b
c
b a c
I I I I II
1
I
i
b
c
b a c
int.5:
pc
0 5
10 3
8
1
6 11
4 9 2
7 int.2:
pc
0
2
4
6
8 10
I
I
I
I
I
b
c b
b.
"Red, White,
and
Blue,"
first
phrase.
INT <5
-
2-5>
parison
of the lowest voice from
Example
12a with the
more
familiarversion of the first
phrase
notated
n
Example
12b
con-
firms that the former is
rhythmically
aried,
and
only
the
first
note,
the G
anacrusis,
s
missing.
In
the bars
preceding
m. 8
a
motivic
interplay nvolvingprimarily
he first our
pitch
classes
of
the
tune
(including
the
anacrusis)
orecasts the later
more
complete
versions.
Example
13a
gives
a
condensed
score
of
these first
seven
bars, omitting only
a chromatic
neighboring
figure
to
C#
that occurs n
the second violins
of mm.
4-7. The
third violin
opens
the workwith a statement
of
the
motive in
the
key
of
Ct: G#-C#t-C#-D#-Gt
n m.
1,
extending
to
the
downbeat
of
m.
2,
displays
he
interval uccession
5-2-5
that s
the characteristic
beginning
of
RWB
(see
Ex.
12b). Example
13b isolates each
motivic occurrence.In m. 2 the fourth
violin
answers
with a variant n which
the central
nterval s
replaced
int.5:
pc
0 5
10 3
8
1
6 11
4 9 2
7 int.2:
pc
0
2
4
6
8 10
I
I
I
I
I
b
c b
b.
"Red, White,
and
Blue,"
first
phrase.
INT <5
-
2-5>
parison
of the lowest voice from
Example
12a with the
more
familiarversion of the first
phrase
notated
n
Example
12b
con-
firms that the former is
rhythmically
aried,
and
only
the
first
note,
the G
anacrusis,
s
missing.
In
the bars
preceding
m. 8
a
motivic
interplay nvolvingprimarily
he first our
pitch
classes
of
the
tune
(including
the
anacrusis)
orecasts the later
more
complete
versions.
Example
13a
gives
a
condensed
score
of
these first
seven
bars, omitting only
a chromatic
neighboring
figure
to
C#
that occurs n
the second violins
of mm.
4-7. The
third violin
opens
the workwith a statement
of
the
motive in
the
key
of
Ct: G#-C#t-C#-D#-Gt
n m.
1,
extending
to
the
downbeat
of
m.
2,
displays
he
interval uccession
5-2-5
that s
the characteristic
beginning
of
RWB
(see
Ex.
12b). Example
13b isolates each
motivic occurrence.In m. 2 the fourth
violin
answers
with a variant n which
the central
nterval s
replaced
int.5:
pc
0 5
10 3
8
1
6 11
4 9 2
7 int.2:
pc
0
2
4
6
8 10
I
I
I
I
I
b
c b
b.
"Red, White,
and
Blue,"
first
phrase.
INT <5
-
2-5>
parison
of the lowest voice from
Example
12a with the
more
familiarversion of the first
phrase
notated
n
Example
12b
con-
firms that the former is
rhythmically
aried,
and
only
the
first
note,
the G
anacrusis,
s
missing.
In
the bars
preceding
m. 8
a
motivic
interplay nvolvingprimarily
he first our
pitch
classes
of
the
tune
(including
the
anacrusis)
orecasts the later
more
complete
versions.
Example
13a
gives
a
condensed
score
of
these first
seven
bars, omitting only
a chromatic
neighboring
figure
to
C#
that occurs n
the second violins
of mm.
4-7. The
third violin
opens
the workwith a statement
of
the
motive in
the
key
of
Ct: G#-C#t-C#-D#-Gt
n m.
1,
extending
to
the
downbeat
of
m.
2,
displays
he
interval uccession
5-2-5
that s
the characteristic
beginning
of
RWB
(see
Ex.
12b). Example
13b isolates each
motivic occurrence.In m. 2 the fourth
violin
answers
with a variant n which
the central
nterval s
replaced
l10o
11
.
21
,b, J13),
J
10o
11
.
21
,b, J13),
J
10o
11
.
21
,b, J13),
J
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 28/41
Interval
ycles
as
Compositional
esources 69
nterval
ycles
as
Compositional
esources 69
nterval
ycles
as
Compositional
esources 69
Example
13.
TheFourth
of
July,
mm.
1-7,
strings.
a. condensed
score
Example
13.
TheFourth
of
July,
mm.
1-7,
strings.
a. condensed
score
Example
13.
TheFourth
of
July,
mm.
1-7,
strings.
a. condensed
score
vln.
2
ln.
2
ln.
2
+vln.
1
vln.
1
vln.
1
f2 Hr-3---
3
--345
6
7
yin
l
3
2
cb.
b. motivic
structure
ni
I
IJ
t
lT
l
?.J7
J7
f2 Hr-3---
3
--345
6
7
yin
l
3
2
cb.
b. motivic
structure
ni
I
IJ
t
lT
l
?.J7
J7
f2 Hr-3---
3
--345
6
7
yin
l
3
2
cb.
b. motivic
structure
ni
I
IJ
t
lT
l
?.J7
J7
INT:
<5
- 2 -
5>
<5
-
10
-
5>
NT:
<5
- 2 -
5>
<5
-
10
-
5>
NT:
<5
- 2 -
5>
<5
-
10
-
5>
<5
-
2
-
5>
5
-
2
-
5>
5
-
2
-
5>
with its
inverse,
and in mm.
4-5
an exact
transposition
of the
motive in the bass
implies
the
key
of B.
In addition to its
relationship
o
the
primary
quoted
tune
of
the
work,
the 5-2-5 motive exhibits a
repetitive
intervallic
structure
hat
may
be tied
to
more abstract
pitch
resources.
The
first our
pitch
classes
of
RWB
are situatedwithinthe 5/2 com-
bination
cycle,
an
overlay
of
interval-7
cycles
at a distance
of
interval
5.
Taking
he
most
complete
statement
of
RWB,
the
C-
major
version in mm.
8-12
(plus
the
missing anacrusis),
as a
point
of
departure,
a
full
expression
of the
intervallicalterna-
tion
would
read:
with its
inverse,
and in mm.
4-5
an exact
transposition
of the
motive in the bass
implies
the
key
of B.
In addition to its
relationship
o
the
primary
quoted
tune
of
the
work,
the 5-2-5 motive exhibits a
repetitive
intervallic
structure
hat
may
be tied
to
more abstract
pitch
resources.
The
first our
pitch
classes
of
RWB
are situatedwithinthe 5/2 com-
bination
cycle,
an
overlay
of
interval-7
cycles
at a distance
of
interval
5.
Taking
he
most
complete
statement
of
RWB,
the
C-
major
version in mm.
8-12
(plus
the
missing anacrusis),
as a
point
of
departure,
a
full
expression
of the
intervallicalterna-
tion
would
read:
with its
inverse,
and in mm.
4-5
an exact
transposition
of the
motive in the bass
implies
the
key
of B.
In addition to its
relationship
o
the
primary
quoted
tune
of
the
work,
the 5-2-5 motive exhibits a
repetitive
intervallic
structure
hat
may
be tied
to
more abstract
pitch
resources.
The
first our
pitch
classes
of
RWB
are situatedwithinthe 5/2 com-
bination
cycle,
an
overlay
of
interval-7
cycles
at a distance
of
interval
5.
Taking
he
most
complete
statement
of
RWB,
the
C-
major
version in mm.
8-12
(plus
the
missing anacrusis),
as a
point
of
departure,
a
full
expression
of the
intervallicalterna-
tion
would
read:
B
set:
A set:
[x/y
=
5/2]
B
set:
A set:
[x/y
=
5/2]
B
set:
A set:
[x/y
=
5/2]
0
7
2
9
4
11 6 1
7 2
9
4
11
6 1 8
n=7
0
7
2
9
4
11 6 1
7 2
9
4
11
6 1 8
n=7
0
7
2
9
4
11 6 1
7 2
9
4
11
6 1 8
n=7
8 3 10 5
3 10 5 0
8 3 10 5
3 10 5 0
8 3 10 5
3 10 5 0
This
version
of
the 5/2 combination
begins
with the four
pitch
classes of the
motive
as it
appears
n
C
major,
pc
<7,0,2,7>;
any
four-element
segment beginning
on an
A set
member s a
version of the
5-2-5 motive.
The
small PCL
(3)
is,
of
course,
apparent
as a
pitch-classrepetition
between
the
first
and
fourth
pitch
classes in the motive.
This
version
of
the 5/2 combination
begins
with the four
pitch
classes of the
motive
as it
appears
n
C
major,
pc
<7,0,2,7>;
any
four-element
segment beginning
on an
A set
member s a
version of the
5-2-5 motive.
The
small PCL
(3)
is,
of
course,
apparent
as a
pitch-classrepetition
between
the
first
and
fourth
pitch
classes in the motive.
This
version
of
the 5/2 combination
begins
with the four
pitch
classes of the
motive
as it
appears
n
C
major,
pc
<7,0,2,7>;
any
four-element
segment beginning
on an
A set
member s a
version of the
5-2-5 motive.
The
small PCL
(3)
is,
of
course,
apparent
as a
pitch-classrepetition
between
the
first
and
fourth
pitch
classes in the motive.
va.
vc.
cb.
va.
vc.
cb.
va.
vc.
cb.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 29/41
70
Music
Theory
Spectrum
0
Music
Theory
Spectrum
0
Music
Theory
Spectrum
While
the 5/2
combination
cycle
cannot
be viewed
as
a
tone-
row-like
source
set
for
this
or
any portion
of The
Fourth
of
July,
it can
provide
a
logical
backdrop
or the
pitch
structure
andthe
integration
of the
quoted
tune.
Consistent
reference
to the
characteristic
ntervals
n the
combination
highlights
he
impor-
tance
of the
repetitive
intervallic
structures
n
the
pitch
lan-
guage
of the
composition.
Indeed,
the
two
complete
motives
from
mm.
1-7
(Ex.
13)
arelinked
as
adjacencies
n
the
source,
in reverse
order:
pc
<6,11,1,6>
(mm.
4-5)
immediately
pre-
cedes
pc
<8,1,3,8>
(mm.
1-2)
in
the 5/2combination:
Bset: 0 7 2 9 4 11 6 1 8 3 10 5
Aset:
7 2
9
4
11
6
1
83
10
50
(mm.
4-5Xmm.
1-2)
Further,
the two
alternating
ntervals
ndividually
are
primary
structural
omponents.
This
is
most
obviously apparent
n
the
passage
immediately
ollowing
the
motivic
nterplay,
shown
in
Example
12a,
as chord
structures
built
from
cycles
of intervals
5
and
2
(and
their
inverses).
In the
firstseven
measures,
each
in-
terval
and
ts
inverse
are
prominent
even
apart
rom
exact
repe-
titions
of
the 5-2-5
motive: the
continuation,
or
example,
of
the
initial
motivic
statement
n the
third
violin
of
m. 2 consists
of
intervals
10
and
7,
and
the
violinssustain
nterval
2
through-
out
mm.
4-7
(two
muted
violins
continue
sustaining
his
inter-
val
through
m.
91).
Whole-tone
relationships
are
also
pro-
jected
by
the
pitch
levels
of the
motivic
statements
in the
opening
measures,
as thesecond
statement
vln.
4,
m.
2)
begins
a whole
stephigher
than
the
initial
motive
(vln.
3,
m.
1),
while
the
version
beginning
n m.
4 is
a whole
step
lowerthanthatof
m. 1.
Cyclicpitch
derivations
also
play
an
integral
role
in In
re con
moto
etal
(1913),
a
work
for
pianoquintet
n
which
experimen-
tationwith
complex
methods
of
organizing
pitch
and
rhythm
reaches
a
sort
of saturation
point.
Amid
a
diversity
of
pitch
structures
derived
through
cyclic
and
other
means,
this
work
is
While
the 5/2
combination
cycle
cannot
be viewed
as
a
tone-
row-like
source
set
for
this
or
any portion
of The
Fourth
of
July,
it can
provide
a
logical
backdrop
or the
pitch
structure
andthe
integration
of the
quoted
tune.
Consistent
reference
to the
characteristic
ntervals
n the
combination
highlights
he
impor-
tance
of the
repetitive
intervallic
structures
n
the
pitch
lan-
guage
of the
composition.
Indeed,
the
two
complete
motives
from
mm.
1-7
(Ex.
13)
arelinked
as
adjacencies
n
the
source,
in reverse
order:
pc
<6,11,1,6>
(mm.
4-5)
immediately
pre-
cedes
pc
<8,1,3,8>
(mm.
1-2)
in
the 5/2combination:
Bset: 0 7 2 9 4 11 6 1 8 3 10 5
Aset:
7 2
9
4
11
6
1
83
10
50
(mm.
4-5Xmm.
1-2)
Further,
the two
alternating
ntervals
ndividually
are
primary
structural
omponents.
This
is
most
obviously apparent
n
the
passage
immediately
ollowing
the
motivic
nterplay,
shown
in
Example
12a,
as chord
structures
built
from
cycles
of intervals
5
and
2
(and
their
inverses).
In the
firstseven
measures,
each
in-
terval
and
ts
inverse
are
prominent
even
apart
rom
exact
repe-
titions
of
the 5-2-5
motive: the
continuation,
or
example,
of
the
initial
motivic
statement
n the
third
violin
of
m. 2 consists
of
intervals
10
and
7,
and
the
violinssustain
nterval
2
through-
out
mm.
4-7
(two
muted
violins
continue
sustaining
his
inter-
val
through
m.
91).
Whole-tone
relationships
are
also
pro-
jected
by
the
pitch
levels
of the
motivic
statements
in the
opening
measures,
as thesecond
statement
vln.
4,
m.
2)
begins
a whole
stephigher
than
the
initial
motive
(vln.
3,
m.
1),
while
the
version
beginning
n m.
4 is
a whole
step
lowerthanthatof
m. 1.
Cyclicpitch
derivations
also
play
an
integral
role
in In
re con
moto
etal
(1913),
a
work
for
pianoquintet
n
which
experimen-
tationwith
complex
methods
of
organizing
pitch
and
rhythm
reaches
a
sort
of saturation
point.
Amid
a
diversity
of
pitch
structures
derived
through
cyclic
and
other
means,
this
work
is
While
the 5/2
combination
cycle
cannot
be viewed
as
a
tone-
row-like
source
set
for
this
or
any portion
of The
Fourth
of
July,
it can
provide
a
logical
backdrop
or the
pitch
structure
andthe
integration
of the
quoted
tune.
Consistent
reference
to the
characteristic
ntervals
n the
combination
highlights
he
impor-
tance
of the
repetitive
intervallic
structures
n
the
pitch
lan-
guage
of the
composition.
Indeed,
the
two
complete
motives
from
mm.
1-7
(Ex.
13)
arelinked
as
adjacencies
n
the
source,
in reverse
order:
pc
<6,11,1,6>
(mm.
4-5)
immediately
pre-
cedes
pc
<8,1,3,8>
(mm.
1-2)
in
the 5/2combination:
Bset: 0 7 2 9 4 11 6 1 8 3 10 5
Aset:
7 2
9
4
11
6
1
83
10
50
(mm.
4-5Xmm.
1-2)
Further,
the two
alternating
ntervals
ndividually
are
primary
structural
omponents.
This
is
most
obviously apparent
n
the
passage
immediately
ollowing
the
motivic
nterplay,
shown
in
Example
12a,
as chord
structures
built
from
cycles
of intervals
5
and
2
(and
their
inverses).
In the
firstseven
measures,
each
in-
terval
and
ts
inverse
are
prominent
even
apart
rom
exact
repe-
titions
of
the 5-2-5
motive: the
continuation,
or
example,
of
the
initial
motivic
statement
n the
third
violin
of
m. 2 consists
of
intervals
10
and
7,
and
the
violinssustain
nterval
2
through-
out
mm.
4-7
(two
muted
violins
continue
sustaining
his
inter-
val
through
m.
91).
Whole-tone
relationships
are
also
pro-
jected
by
the
pitch
levels
of the
motivic
statements
in the
opening
measures,
as thesecond
statement
vln.
4,
m.
2)
begins
a whole
stephigher
than
the
initial
motive
(vln.
3,
m.
1),
while
the
version
beginning
n m.
4 is
a whole
step
lowerthanthatof
m. 1.
Cyclicpitch
derivations
also
play
an
integral
role
in In
re con
moto
etal
(1913),
a
work
for
pianoquintet
n
which
experimen-
tationwith
complex
methods
of
organizing
pitch
and
rhythm
reaches
a
sort
of saturation
point.
Amid
a
diversity
of
pitch
structures
derived
through
cyclic
and
other
means,
this
work
is
structured
ccording
o
repeated
projections
of a number
eries
that
determines
meter
changes,
phrase
lengths,
or
rhythmic
groupings.
In
Memos,
Ives
gives
the numberseriesas "2-3-5-
7-11-7-5-3-2,"
a
symmetrical
arrangement
f the
five
prime
numbers
greater
than
1;
the
various
presentations
of
the
"Prime Series"
(PS)
are
sometimes
altered
by
stopping
after
the
midpoint
or
by
inverting
he
order
to
begin
and
end
on
11,
placing
2 in the
center.
Each
occurrence
of the
PS
in
the
work
can be
viewed
as a
"variation,"
projecting
an
overall
form
of:
introduction,
nitial
presentation
of the
PS,
and
the
variations.
In
commenting
on
the structure
of the
work,
Ives
implies
that
only
the PS
represents
an element of
constancy
n the work,
while the
methods
of
projection
are
constantly
hanging;
he
re-
fers
to
repetitions
of the PS
as
"cycles"
that
"grow,
expand,
ebb,
but
never
literally
repeat."57
Prior
to the
first
presentation
of the
PS,
a
one-measure
n-
troduction
n the
strings
displays
he
cyclic
origins
hat
will
simi-
larly
characterize
pitch
structures
n
other
portions
of the
work.
Constructed
according
o
a
proportional
cheme
of
4:3:2:1
be-
tween
the
note
valuesof
the
four
voices
(top
to
bottom),
this
measuresubdividesinto the three
aggregates
hat are boxed
and numbered
n
Example
14. Each
aggregate
s
derived
roma
source
interval-5
cycle,
following
the
op
labels
placed
beside
each
note
in
the score.
The
initial
aggregate
unfolds
the
pitch
classes
in direct
temporal
succession,
starting
with
ops
0-3
stated
simultaneously,
followed
by
op
4
on
the
next
pitch
change
(vln.
1),
leading
to
ops
5 and
6 stated
together
on
beat
2,
and
continuing
n this
manner
to
op
11
on
beat
3
in
the cello.
The
two
other
aggregates
contain
slight
disorderings
of the
57Memos,
01.
Thisview
of
the form
generally
corresponds
o
that
of John
McLain
Rinehart,
"Ives's
Compositional
dioms:
An
Investigation
f Selected
Short
Compositions
as
Microcosms
of
His Musical
Language"
Ph.D.
disserta-
tion,
Ohio State
University,
1970),
48-61.
See
also Ulrich
Maske,
Charles
ves
in seiner
Kammermusik
ur
drei
bis sechs
Instrumente,
Kolner
Beitrage
zur
Mu-
sikforschung,
vol. 64
(Regensburg:
G.
Bosse,
1971),
121-123.
structured
ccording
o
repeated
projections
of a number
eries
that
determines
meter
changes,
phrase
lengths,
or
rhythmic
groupings.
In
Memos,
Ives
gives
the numberseriesas "2-3-5-
7-11-7-5-3-2,"
a
symmetrical
arrangement
f the
five
prime
numbers
greater
than
1;
the
various
presentations
of
the
"Prime Series"
(PS)
are
sometimes
altered
by
stopping
after
the
midpoint
or
by
inverting
he
order
to
begin
and
end
on
11,
placing
2 in the
center.
Each
occurrence
of the
PS
in
the
work
can be
viewed
as a
"variation,"
projecting
an
overall
form
of:
introduction,
nitial
presentation
of the
PS,
and
the
variations.
In
commenting
on
the structure
of the
work,
Ives
implies
that
only
the PS
represents
an element of
constancy
n the work,
while the
methods
of
projection
are
constantly
hanging;
he
re-
fers
to
repetitions
of the PS
as
"cycles"
that
"grow,
expand,
ebb,
but
never
literally
repeat."57
Prior
to the
first
presentation
of the
PS,
a
one-measure
n-
troduction
n the
strings
displays
he
cyclic
origins
hat
will
simi-
larly
characterize
pitch
structures
n
other
portions
of the
work.
Constructed
according
o
a
proportional
cheme
of
4:3:2:1
be-
tween
the
note
valuesof
the
four
voices
(top
to
bottom),
this
measuresubdividesinto the three
aggregates
hat are boxed
and numbered
n
Example
14. Each
aggregate
s
derived
roma
source
interval-5
cycle,
following
the
op
labels
placed
beside
each
note
in
the score.
The
initial
aggregate
unfolds
the
pitch
classes
in direct
temporal
succession,
starting
with
ops
0-3
stated
simultaneously,
followed
by
op
4
on
the
next
pitch
change
(vln.
1),
leading
to
ops
5 and
6 stated
together
on
beat
2,
and
continuing
n this
manner
to
op
11
on
beat
3
in
the cello.
The
two
other
aggregates
contain
slight
disorderings
of the
57Memos,
01.
Thisview
of
the form
generally
corresponds
o
that
of John
McLain
Rinehart,
"Ives's
Compositional
dioms:
An
Investigation
f Selected
Short
Compositions
as
Microcosms
of
His Musical
Language"
Ph.D.
disserta-
tion,
Ohio State
University,
1970),
48-61.
See
also Ulrich
Maske,
Charles
ves
in seiner
Kammermusik
ur
drei
bis sechs
Instrumente,
Kolner
Beitrage
zur
Mu-
sikforschung,
vol. 64
(Regensburg:
G.
Bosse,
1971),
121-123.
structured
ccording
o
repeated
projections
of a number
eries
that
determines
meter
changes,
phrase
lengths,
or
rhythmic
groupings.
In
Memos,
Ives
gives
the numberseriesas "2-3-5-
7-11-7-5-3-2,"
a
symmetrical
arrangement
f the
five
prime
numbers
greater
than
1;
the
various
presentations
of
the
"Prime Series"
(PS)
are
sometimes
altered
by
stopping
after
the
midpoint
or
by
inverting
he
order
to
begin
and
end
on
11,
placing
2 in the
center.
Each
occurrence
of the
PS
in
the
work
can be
viewed
as a
"variation,"
projecting
an
overall
form
of:
introduction,
nitial
presentation
of the
PS,
and
the
variations.
In
commenting
on
the structure
of the
work,
Ives
implies
that
only
the PS
represents
an element of
constancy
n the work,
while the
methods
of
projection
are
constantly
hanging;
he
re-
fers
to
repetitions
of the PS
as
"cycles"
that
"grow,
expand,
ebb,
but
never
literally
repeat."57
Prior
to the
first
presentation
of the
PS,
a
one-measure
n-
troduction
n the
strings
displays
he
cyclic
origins
hat
will
simi-
larly
characterize
pitch
structures
n
other
portions
of the
work.
Constructed
according
o
a
proportional
cheme
of
4:3:2:1
be-
tween
the
note
valuesof
the
four
voices
(top
to
bottom),
this
measuresubdividesinto the three
aggregates
hat are boxed
and numbered
n
Example
14. Each
aggregate
s
derived
roma
source
interval-5
cycle,
following
the
op
labels
placed
beside
each
note
in
the score.
The
initial
aggregate
unfolds
the
pitch
classes
in direct
temporal
succession,
starting
with
ops
0-3
stated
simultaneously,
followed
by
op
4
on
the
next
pitch
change
(vln.
1),
leading
to
ops
5 and
6 stated
together
on
beat
2,
and
continuing
n this
manner
to
op
11
on
beat
3
in
the cello.
The
two
other
aggregates
contain
slight
disorderings
of the
57Memos,
01.
Thisview
of
the form
generally
corresponds
o
that
of John
McLain
Rinehart,
"Ives's
Compositional
dioms:
An
Investigation
f Selected
Short
Compositions
as
Microcosms
of
His Musical
Language"
Ph.D.
disserta-
tion,
Ohio State
University,
1970),
48-61.
See
also Ulrich
Maske,
Charles
ves
in seiner
Kammermusik
ur
drei
bis sechs
Instrumente,
Kolner
Beitrage
zur
Mu-
sikforschung,
vol. 64
(Regensburg:
G.
Bosse,
1971),
121-123.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 30/41
Interval
ycles
as
Compositional
esources 71nterval
ycles
as
Compositional
esources 71nterval
ycles
as
Compositional
esources 71
Example
14.
In re con moto et
al,
m. 1.
xample
14.
In re con moto et
al,
m. 1.
xample
14.
In re con moto et
al,
m. 1.
r-., 5 7
.10
-., 5 7
.10
-., 5 7
.10
vln.
3
4
'"'1
2 3
4
5
7
10
3
,,f
A
A
^
A
A
va.
4I
1J
.
J
mf
8
6
3
vln.
3
4
'"'1
2 3
4
5
7
10
3
,,f
A
A
^
A
A
va.
4I
1J
.
J
mf
8
6
3
vln.
3
4
'"'1
2 3
4
5
7
10
3
,,f
A
A
^
A
A
va.
4I
1J
.
J
mf
8
6
3
)
aggregates:ggregates:ggregates:
11
1
1
1
1
1
1...
7
8
2...
3...
op
0 1
2 3 4
5
6 7
8 9 10
11
int. 5
cycle:
pc
1
6
11 4
9
2
7 0 5
10
3 8
op
0 1
2 3 4
5
6 7
8 9 10
11
int. 5
cycle:
pc
1
6
11 4
9
2
7 0 5
10
3 8
op
0 1
2 3 4
5
6 7
8 9 10
11
int. 5
cycle:
pc
1
6
11 4
9
2
7 0 5
10
3 8
source,
so
that,
for
example,
op
7
occurs before
op
6 in
aggre-
gate
2
(cello
and
viola),
and
ops
9 and 10
occur
before
op
8
in
aggregate
3
(vln.
1, viola,
cello).
In its
customary
ompositional
role,
the
cyclic
source
provides
for
structural
unity
of
pitch
combinations
and
control
over
pitch-class
urnover.
The initial
presentation
of the PS
(mm. 2-6), immediately
following
the one-measure
ntroduction,
uses durations o
pro-
ject
the
"reciprocal"
PS
values,
beginning
with
11
and shrink-
ing
to
2
at the
midpoint.
These are measured n
eighth
notes
changing
to
sixteenths above the condensed
score
in
Example
15;
the final
duration
(m.
6)
is
nine rather than
eleven
six-
teenths.
The
pitch
structureof each chord
s based on the
repe-
titions indicated
below the score: chords
"a,
b, h,
i"
display
source,
so
that,
for
example,
op
7
occurs before
op
6 in
aggre-
gate
2
(cello
and
viola),
and
ops
9 and 10
occur
before
op
8
in
aggregate
3
(vln.
1, viola,
cello).
In its
customary
ompositional
role,
the
cyclic
source
provides
for
structural
unity
of
pitch
combinations
and
control
over
pitch-class
urnover.
The initial
presentation
of the PS
(mm. 2-6), immediately
following
the one-measure
ntroduction,
uses durations o
pro-
ject
the
"reciprocal"
PS
values,
beginning
with
11
and shrink-
ing
to
2
at the
midpoint.
These are measured n
eighth
notes
changing
to
sixteenths above the condensed
score
in
Example
15;
the final
duration
(m.
6)
is
nine rather than
eleven
six-
teenths.
The
pitch
structureof each chord
s based on the
repe-
titions indicated
below the score: chords
"a,
b, h,
i"
display
source,
so
that,
for
example,
op
7
occurs before
op
6 in
aggre-
gate
2
(cello
and
viola),
and
ops
9 and 10
occur
before
op
8
in
aggregate
3
(vln.
1, viola,
cello).
In its
customary
ompositional
role,
the
cyclic
source
provides
for
structural
unity
of
pitch
combinations
and
control
over
pitch-class
urnover.
The initial
presentation
of the PS
(mm. 2-6), immediately
following
the one-measure
ntroduction,
uses durations o
pro-
ject
the
"reciprocal"
PS
values,
beginning
with
11
and shrink-
ing
to
2
at the
midpoint.
These are measured n
eighth
notes
changing
to
sixteenths above the condensed
score
in
Example
15;
the final
duration
(m.
6)
is
nine rather than
eleven
six-
teenths.
The
pitch
structureof each chord
s based on the
repe-
titions indicated
below the score: chords
"a,
b, h,
i"
display
single-interval
tructures,
and the others
employ
intervallical-
ternations.
Toward the end of m.
2,
and
again
in m.
3,
two
notes of the chords are
altered,
temporarilydisrupting
he in-
tervallic scheme.
Though
the basic chord
structure
changes
with each new
duration,
he
interval
izes
do
not
project
a PS of
their
own,
as
they might
have done
by forming
chords of
elev-
enths, sevenths, fifths, thirds,
and
seconds,
for
example.
How-
ever,
the
registral
span
of the chords effects a
general, unsys-
tematic
contraction-expansion
hat
parallels
he
PS,
reaching
a
narrow
point
at the shortest duration:chord e
spans
two oc-
taves
plus
a
fifth,
in
contrast to chords a
and
b
(three
octaves
plus
a
tritone)
and chord
(four
octaves
plus
a
fifth).
The
size of
vertical
intervals also
reaches
a
maximumnear the
end,
with
single-interval
tructures,
and the others
employ
intervallical-
ternations.
Toward the end of m.
2,
and
again
in m.
3,
two
notes of the chords are
altered,
temporarilydisrupting
he in-
tervallic scheme.
Though
the basic chord
structure
changes
with each new
duration,
he
interval
izes
do
not
project
a PS of
their
own,
as
they might
have done
by forming
chords of
elev-
enths, sevenths, fifths, thirds,
and
seconds,
for
example.
How-
ever,
the
registral
span
of the chords effects a
general, unsys-
tematic
contraction-expansion
hat
parallels
he
PS,
reaching
a
narrow
point
at the shortest duration:chord e
spans
two oc-
taves
plus
a
fifth,
in
contrast to chords a
and
b
(three
octaves
plus
a
tritone)
and chord
(four
octaves
plus
a
fifth).
The
size of
vertical
intervals also
reaches
a
maximumnear the
end,
with
single-interval
tructures,
and the others
employ
intervallical-
ternations.
Toward the end of m.
2,
and
again
in m.
3,
two
notes of the chords are
altered,
temporarilydisrupting
he in-
tervallic scheme.
Though
the basic chord
structure
changes
with each new
duration,
he
interval
izes
do
not
project
a PS of
their
own,
as
they might
have done
by forming
chords of
elev-
enths, sevenths, fifths, thirds,
and
seconds,
for
example.
How-
ever,
the
registral
span
of the chords effects a
general, unsys-
tematic
contraction-expansion
hat
parallels
he
PS,
reaching
a
narrow
point
at the shortest duration:chord e
spans
two oc-
taves
plus
a
fifth,
in
contrast to chords a
and
b
(three
octaves
plus
a
tritone)
and chord
(four
octaves
plus
a
fifth).
The
size of
vertical
intervals also
reaches
a
maximumnear the
end,
with
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 31/41
72
Music
Theory
Spectrum
2
Music
Theory
Spectrum
2
Music
Theory
Spectrum
Example
15.
In
re con moto et
al,
mm.
2-6,
condensedscore.
xample
15.
In
re con moto et
al,
mm.
2-6,
condensedscore.
xample
15.
In
re con moto et
al,
mm.
2-6,
condensedscore.
J=
11
=
11
=
11
combination
cycle:
5
combination
cycle:
5
combination
cycle:
5
7
311i
=
2
3
5
311i
=
2
3
5
311i
=
2
3
5
a.
b.
c.
d.
e.
f.
g. h.
i.
5/5
7/7
7/6
5/6 4/5 4/9
5/8
10/10
11/1
a.
b.
c.
d.
e.
f.
g. h.
i.
5/5
7/7
7/6
5/6 4/5 4/9
5/8
10/10
11/1
a.
b.
c.
d.
e.
f.
g. h.
i.
5/5
7/7
7/6
5/6 4/5 4/9
5/8
10/10
11/1
interval
10 in chord h and 11
in
chord
representing
n
increase
over the verticalintervals used in
previous
chords
(intervals
4
through9).
Several
adjacent
sonorities n this initial
presentation
of the
PS
are connected
through
a
cyclic
derivational
process
resem-
bling
the
linkages
between
interval-5
and
-7
structures
n mm.
8-13 of
The Fourth
of July
(Ex.
12a).
First,
the interval-5
ycle
of
chord
a
(Ex. 15) "wraps
around,"
or
connects
at the
top,
to
its
inverse,
the interval-7
cycle
of
chord
b,
as between
mm.
8
and 10
or 12
and 13
of The Fourth
of July.
Then the 7/6
cycle
of
chordc
similarly
wraps
around
o the
5/6
cycle
in chord
d.
Lines
1 and 2 of
Figure
6 summarize hese
connections,
withbrackets
and chord
labels
indicatingpositions
of chords within
the
cy-
cles. Because
two
connected
sonorities exhibit converse
regis-
tral
distributions f
pitch-class
order,
the lower
notes in the mu-
sic are those of the outer
portions
of
the
cycle
as it
is
notated
in
the
figure,
and the
higher
notes
appeartogether
in the center.
In line
1,
chord a
encompasses
the first seven elements
of the
interval-5
cycle
and b
spans
the other five
plus
two
repetitions
interval
10 in chord h and 11
in
chord
representing
n
increase
over the verticalintervals used in
previous
chords
(intervals
4
through9).
Several
adjacent
sonorities n this initial
presentation
of the
PS
are connected
through
a
cyclic
derivational
process
resem-
bling
the
linkages
between
interval-5
and
-7
structures
n mm.
8-13 of
The Fourth
of July
(Ex.
12a).
First,
the interval-5
ycle
of
chord
a
(Ex. 15) "wraps
around,"
or
connects
at the
top,
to
its
inverse,
the interval-7
cycle
of
chord
b,
as between
mm.
8
and 10
or 12
and 13
of The Fourth
of July.
Then the 7/6
cycle
of
chordc
similarly
wraps
around
o the
5/6
cycle
in chord
d.
Lines
1 and 2 of
Figure
6 summarize hese
connections,
withbrackets
and chord
labels
indicatingpositions
of chords within
the
cy-
cles. Because
two
connected
sonorities exhibit converse
regis-
tral
distributions f
pitch-class
order,
the lower
notes in the mu-
sic are those of the outer
portions
of
the
cycle
as it
is
notated
in
the
figure,
and the
higher
notes
appeartogether
in the center.
In line
1,
chord a
encompasses
the first seven elements
of the
interval-5
cycle
and b
spans
the other five
plus
two
repetitions
interval
10 in chord h and 11
in
chord
representing
n
increase
over the verticalintervals used in
previous
chords
(intervals
4
through9).
Several
adjacent
sonorities n this initial
presentation
of the
PS
are connected
through
a
cyclic
derivational
process
resem-
bling
the
linkages
between
interval-5
and
-7
structures
n mm.
8-13 of
The Fourth
of July
(Ex.
12a).
First,
the interval-5
ycle
of
chord
a
(Ex. 15) "wraps
around,"
or
connects
at the
top,
to
its
inverse,
the interval-7
cycle
of
chord
b,
as between
mm.
8
and 10
or 12
and 13
of The Fourth
of July.
Then the 7/6
cycle
of
chordc
similarly
wraps
around
o the
5/6
cycle
in chord
d.
Lines
1 and 2 of
Figure
6 summarize hese
connections,
withbrackets
and chord
labels
indicatingpositions
of chords within
the
cy-
cles. Because
two
connected
sonorities exhibit converse
regis-
tral
distributions f
pitch-class
order,
the lower
notes in the mu-
sic are those of the outer
portions
of
the
cycle
as it
is
notated
in
the
figure,
and the
higher
notes
appeartogether
in the center.
In line
1,
chord a
encompasses
the first seven elements
of the
interval-5
cycle
and b
spans
the other five
plus
two
repetitions
(pc
1 and
6,
enclosed in
parentheses).
Line 2 illustrates
a 7/6
cycle concluding
with its
first
pc repetition(PCL
=
11)
at
pc
6,
which
s the bass
note
common to chords
b,
c,
and d. The
wrap-
around
between
chords
c
and d
includes
a
point
of intersection
at
pcs
8, 3,
and
9,
the
upper
three notes
of
both chords.
Pc
0,
which
would
complete
the
aggregate
on line
2 of
Figure
6,
appears
not in chord d but as the lowest note of the subse-
quent
sonority
and of
every
chordfor the remainder f the
pas-
sage.
This
implied
continuationof the
cycle connecting
hordsc
and d thus
extends a
wraparound
process
that includes
cyclic
linkages
between
a and b and
between c and
d
in the
upper
reg-
isters and common-tone
inkages
between b and c and between
c
and d in
the lower
registers.
With
the
arrivalof the
pc
0 bass
"anchor" n chord
e,
subsequent
chords do
not
wrap
around
but
continue the
low-register
common-tone
connections,
with
some association
between
adjacencies.
Chord
e,
for
example,
exhibits
the 4/5
cycle
notated on line
3 of
Figure
6,
which
con-
nects
in
the bass clef to four common
tones
of chord f.
Thus
f
begins identically
to e on line 3 of the
figure
with
pcs
(pc
1 and
6,
enclosed in
parentheses).
Line 2 illustrates
a 7/6
cycle concluding
with its
first
pc repetition(PCL
=
11)
at
pc
6,
which
s the bass
note
common to chords
b,
c,
and d. The
wrap-
around
between
chords
c
and d
includes
a
point
of intersection
at
pcs
8, 3,
and
9,
the
upper
three notes
of
both chords.
Pc
0,
which
would
complete
the
aggregate
on line
2 of
Figure
6,
appears
not in chord d but as the lowest note of the subse-
quent
sonority
and of
every
chordfor the remainder f the
pas-
sage.
This
implied
continuationof the
cycle connecting
hordsc
and d thus
extends a
wraparound
process
that includes
cyclic
linkages
between
a and b and
between c and
d
in the
upper
reg-
isters and common-tone
inkages
between b and c and between
c
and d in
the lower
registers.
With
the
arrivalof the
pc
0 bass
"anchor" n chord
e,
subsequent
chords do
not
wrap
around
but
continue the
low-register
common-tone
connections,
with
some association
between
adjacencies.
Chord
e,
for
example,
exhibits
the 4/5
cycle
notated on line
3 of
Figure
6,
which
con-
nects
in
the bass clef to four common
tones
of chord f.
Thus
f
begins identically
to e on line 3 of the
figure
with
pcs
(pc
1 and
6,
enclosed in
parentheses).
Line 2 illustrates
a 7/6
cycle concluding
with its
first
pc repetition(PCL
=
11)
at
pc
6,
which
s the bass
note
common to chords
b,
c,
and d. The
wrap-
around
between
chords
c
and d
includes
a
point
of intersection
at
pcs
8, 3,
and
9,
the
upper
three notes
of
both chords.
Pc
0,
which
would
complete
the
aggregate
on line
2 of
Figure
6,
appears
not in chord d but as the lowest note of the subse-
quent
sonority
and of
every
chordfor the remainder f the
pas-
sage.
This
implied
continuationof the
cycle connecting
hordsc
and d thus
extends a
wraparound
process
that includes
cyclic
linkages
between
a and b and
between c and
d
in the
upper
reg-
isters and common-tone
inkages
between b and c and between
c
and d in
the lower
registers.
With
the
arrivalof the
pc
0 bass
"anchor" n chord
e,
subsequent
chords do
not
wrap
around
but
continue the
low-register
common-tone
connections,
with
some association
between
adjacencies.
Chord
e,
for
example,
exhibits
the 4/5
cycle
notated on line
3 of
Figure
6,
which
con-
nects
in
the bass clef to four common
tones
of chord f.
Thus
f
begins identically
to e on line 3 of the
figure
with
pcs
1
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 32/41
Interval
ycles
as
Compositional
esources 73
nterval
ycles
as
Compositional
esources 73
nterval
ycles
as
Compositional
esources 73
Figure
6.
Cyclic relationships
of
simultaneities
in
Example
15.
igure
6.
Cyclic relationships
of
simultaneities
in
Example
15.
igure
6.
Cyclic relationships
of
simultaneities
in
Example
15.
bassassass a.
sopr.
.
sopr.
.
sopr.
1.
int.-5cycle:
pc
1
6
.
int.-5cycle:
pc
1
6
.
int.-5cycle:
pc
1
6
11
4 9 2 7
0 5
I
11
4 9 2 7
0 5
I
11
4 9 2 7
0 5
I
10 3 8
(1)
(6)
0 3 8
(1)
(6)
0 3 8
(1)
(6)
b.
bass
.
bass
.
bass
bass
2. 7/6 comb.
cycle: pc
6 1
bass
2. 7/6 comb.
cycle: pc
6 1
bass
2. 7/6 comb.
cycle: pc
6 1
c.
sopr.
7 2
8
3 91 4
I
c.
sopr.
7 2
8
3 91 4
I
c.
sopr.
7 2
8
3 91 4
I
sopr.opr.opr.
d...
10
5
11
6
[0]
bass
\
10
5
11
6
[0]
bass
\
10
5
11
6
[0]
bass
\
- - - - A'- - - A'- - - A'
bass
-
3. 4/5 comb.
cycle: pc
0' 4
9
1
If.
/
/
f.
(bass clef) /
k
v
bass
-
3. 4/5 comb.
cycle: pc
0' 4
9
1
If.
/
/
f.
(bass clef) /
k
v
bass
-
3. 4/5 comb.
cycle: pc
0' 4
9
1
If.
/
/
f.
(bass clef) /
k
v
e.
sopr.
6 10 3 7
e.
sopr.
6 10 3 7
e.
sopr.
6 10 3 7
/
/
/
/
f.
(top
6
notes)
4. 4/9 comb.
cycle:
pc
9 1
10 2 11
3
/
/
/
/
f.
(top
6
notes)
4. 4/9 comb.
cycle:
pc
9 1
10 2 11
3
/
/
/
/
f.
(top
6
notes)
4. 4/9 comb.
cycle:
pc
9 1
10 2 11
3
<0,4,9,1>,
but
then
shifts
to
line
4,
with
pcs
<9,1>
overlap-
ping
to
become
the
beginning
of a
4/9
alternation n the
upper
register.
Other
portions
of
In
re,
while
by
no means
uniformly
con-
ceived,
maintain ome
degree
of
cyclicunderpinning.
Common
to mostof the variationss the recurrence f a
sonority
Ivescalls
the
"Grit
Chord,"
which
s
frequently
used
to
articulate
he be-
ginning
of a
unit
of
the PS.58 n variation
1,
for
example,
meter
changes
project
the
PS,
so that the
beginning
of
each
measure
signals
a new PS unit.59
Measures
in
this variation hat
do not
58Ives,
Memos,
101.
59The
PS determines he
numberof
beats
per
measure n variation
1,
begin-
ning
with
6
meter
(2 beats), (3
beats),
15
(5 beats),
and
so forth.See
Rinehart,
Ives's
Compositional
Idioms,"
50-51.
<0,4,9,1>,
but
then
shifts
to
line
4,
with
pcs
<9,1>
overlap-
ping
to
become
the
beginning
of a
4/9
alternation n the
upper
register.
Other
portions
of
In
re,
while
by
no means
uniformly
con-
ceived,
maintain ome
degree
of
cyclicunderpinning.
Common
to mostof the variationss the recurrence f a
sonority
Ivescalls
the
"Grit
Chord,"
which
s
frequently
used
to
articulate
he be-
ginning
of a
unit
of
the PS.58 n variation
1,
for
example,
meter
changes
project
the
PS,
so that the
beginning
of
each
measure
signals
a new PS unit.59
Measures
in
this variation hat
do not
58Ives,
Memos,
101.
59The
PS determines he
numberof
beats
per
measure n variation
1,
begin-
ning
with
6
meter
(2 beats), (3
beats),
15
(5 beats),
and
so forth.See
Rinehart,
Ives's
Compositional
Idioms,"
50-51.
<0,4,9,1>,
but
then
shifts
to
line
4,
with
pcs
<9,1>
overlap-
ping
to
become
the
beginning
of a
4/9
alternation n the
upper
register.
Other
portions
of
In
re,
while
by
no means
uniformly
con-
ceived,
maintain ome
degree
of
cyclicunderpinning.
Common
to mostof the variationss the recurrence f a
sonority
Ivescalls
the
"Grit
Chord,"
which
s
frequently
used
to
articulate
he be-
ginning
of a
unit
of
the PS.58 n variation
1,
for
example,
meter
changes
project
the
PS,
so that the
beginning
of
each
measure
signals
a new PS unit.59
Measures
in
this variation hat
do not
58Ives,
Memos,
101.
59The
PS determines he
numberof
beats
per
measure n variation
1,
begin-
ning
with
6
meter
(2 beats), (3
beats),
15
(5 beats),
and
so forth.See
Rinehart,
Ives's
Compositional
Idioms,"
50-51.
begin
with the Grit Chord
(GC) begin
with
its
literal
comple-
ment
(GCC).
Example
16
gives
the
opening
of variation
1,
dis-
playing
GC on the first beats
of
mm.
7
and 9 and GCC
on the
firstbeat of m. 8 and on the second beat
c m.
9.
The two
sonorities
appear together
in
this
type
of
pairing
(without
change
in
pc content)
in most
subsequent
appearances,
the
only
possible
variable
being
the
registral
distribution
of the
pitch
classes
of
GCC.
In
keeping
with the
cyclic
nature
of
other structural
spects
of
the
work,
GC is formed from
eight
notes,
in
registral
order
from
low
to
high,
of the
7/6
combination
cycle beginning
on
pc
0.
Upward
stems
in
Figure
7
extract GC
from a
7/6
cycle
that
continues
to
the
point
of
aggregate
completion,
one
element
past
the
repetition
of
pc
0
(PCL
=
11).
GCC
(downward
tems
in
the
figure)
then
contains
the
remainingpitch
classesof
the
begin
with the Grit Chord
(GC) begin
with
its
literal
comple-
ment
(GCC).
Example
16
gives
the
opening
of variation
1,
dis-
playing
GC on the first beats
of
mm.
7
and 9 and GCC
on the
firstbeat of m. 8 and on the second beat
c m.
9.
The two
sonorities
appear together
in
this
type
of
pairing
(without
change
in
pc content)
in most
subsequent
appearances,
the
only
possible
variable
being
the
registral
distribution
of the
pitch
classes
of
GCC.
In
keeping
with the
cyclic
nature
of
other structural
spects
of
the
work,
GC is formed from
eight
notes,
in
registral
order
from
low
to
high,
of the
7/6
combination
cycle beginning
on
pc
0.
Upward
stems
in
Figure
7
extract GC
from a
7/6
cycle
that
continues
to
the
point
of
aggregate
completion,
one
element
past
the
repetition
of
pc
0
(PCL
=
11).
GCC
(downward
tems
in
the
figure)
then
contains
the
remainingpitch
classesof
the
begin
with the Grit Chord
(GC) begin
with
its
literal
comple-
ment
(GCC).
Example
16
gives
the
opening
of variation
1,
dis-
playing
GC on the first beats
of
mm.
7
and 9 and GCC
on the
firstbeat of m. 8 and on the second beat
c m.
9.
The two
sonorities
appear together
in
this
type
of
pairing
(without
change
in
pc content)
in most
subsequent
appearances,
the
only
possible
variable
being
the
registral
distribution
of the
pitch
classes
of
GCC.
In
keeping
with the
cyclic
nature
of
other structural
spects
of
the
work,
GC is formed from
eight
notes,
in
registral
order
from
low
to
high,
of the
7/6
combination
cycle beginning
on
pc
0.
Upward
stems
in
Figure
7
extract GC
from a
7/6
cycle
that
continues
to
the
point
of
aggregate
completion,
one
element
past
the
repetition
of
pc
0
(PCL
=
11).
GCC
(downward
tems
in
the
figure)
then
contains
the
remainingpitch
classesof
the
sopr.opr.opr.
m
(
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 33/41
74
Music
TheorySpectrum
4
Music
TheorySpectrum
4
Music
TheorySpectrum
Example
16. In
re
con moto et
al,
variation
1,
mm.
7-9.
xample
16. In
re
con moto et
al,
variation
1,
mm.
7-9.
xample
16. In
re
con moto et
al,
variation
1,
mm.
7-9.
inf
c' b; .
.
v
mf
mf
inf
c' b; .
.
v
mf
mf
inf
c' b; .
.
v
mf
mf
)
mf
[GC]
mf
[GC]
mf
[GC]
[GC,
GCC]
[GC, GCC]
[GC,
GCC]
[GC, GCC]
[GC,
GCC]
[GC, GCC]
GCC]GCC]GCC]
cycle,
including
a
pc
3
that
GC
omits from ts
otherwise
contigu-
ous
extraction,
and
excluding
the redundant
pc
0. The
cyclic
source does
not
determinethe
registralordering
of
GCC.
Portions of the
variationsthat
are
not restatements
of GC
and GCC
may
further
perpetuate
a
connection
with a
cyclic
source
through
consistentrestatementof the
primary
ntervals.
Measure
9,
for
example
(Ex.
16),
can
be subdivided
almost
ex-
clusively
into
tritones
(interval 6), starting
with
those that
are
inherent
in
GC
and
GCC,
and
continuing
through
the
struc-
tures
n the
remainder
of the
measure.Each circled
and abeled
dyad
in
Example
16
highlights
an occurrenceof a tritonebe-
tween
registrally
and/or
emporally
associated
pitches.
Follow-
ing
the
completion
of the
aggregate
rom
GC/GCC,
the
dyads
reiterate
pitch
classes from
the
first
part
of the
measure,
effect-
ing
a redistribution f those same intervals:
dyads
, h,
and
g
are
pitch-class
equivalent
o
a, b,
and
c,
respectively.
Figure
8
plots
each
dyad
on the
7/6
cycle,
including
he
three
reiterations,
he
cycle,
including
a
pc
3
that
GC
omits from ts
otherwise
contigu-
ous
extraction,
and
excluding
the redundant
pc
0. The
cyclic
source does
not
determinethe
registralordering
of
GCC.
Portions of the
variationsthat
are
not restatements
of GC
and GCC
may
further
perpetuate
a
connection
with a
cyclic
source
through
consistentrestatementof the
primary
ntervals.
Measure
9,
for
example
(Ex.
16),
can
be subdivided
almost
ex-
clusively
into
tritones
(interval 6), starting
with
those that
are
inherent
in
GC
and
GCC,
and
continuing
through
the
struc-
tures
n the
remainder
of the
measure.Each circled
and abeled
dyad
in
Example
16
highlights
an occurrenceof a tritonebe-
tween
registrally
and/or
emporally
associated
pitches.
Follow-
ing
the
completion
of the
aggregate
rom
GC/GCC,
the
dyads
reiterate
pitch
classes from
the
first
part
of the
measure,
effect-
ing
a redistribution f those same intervals:
dyads
, h,
and
g
are
pitch-class
equivalent
o
a, b,
and
c,
respectively.
Figure
8
plots
each
dyad
on the
7/6
cycle,
including
he
three
reiterations,
he
cycle,
including
a
pc
3
that
GC
omits from ts
otherwise
contigu-
ous
extraction,
and
excluding
the redundant
pc
0. The
cyclic
source does
not
determinethe
registralordering
of
GCC.
Portions of the
variationsthat
are
not restatements
of GC
and GCC
may
further
perpetuate
a
connection
with a
cyclic
source
through
consistentrestatementof the
primary
ntervals.
Measure
9,
for
example
(Ex.
16),
can
be subdivided
almost
ex-
clusively
into
tritones
(interval 6), starting
with
those that
are
inherent
in
GC
and
GCC,
and
continuing
through
the
struc-
tures
n the
remainder
of the
measure.Each circled
and abeled
dyad
in
Example
16
highlights
an occurrenceof a tritonebe-
tween
registrally
and/or
emporally
associated
pitches.
Follow-
ing
the
completion
of the
aggregate
rom
GC/GCC,
the
dyads
reiterate
pitch
classes from
the
first
part
of the
measure,
effect-
ing
a redistribution f those same intervals:
dyads
, h,
and
g
are
pitch-class
equivalent
o
a, b,
and
c,
respectively.
Figure
8
plots
each
dyad
on the
7/6
cycle,
including
he
three
reiterations,
he
tritone that occurs
within
GCC
(dyad d),
and
the tritone n
the
second violin
(dyade)
formedfrom
pc
6
of
GCC
and
pc
0 on the
next beat. This
segmentation
amasses five of the
six
available
tritones,
avoidingthroughout
he measure
the
pcs
3 and
9
that
are
inherently
absent
from
GC.
The most
striking
evidence of
interval
cycles
in
Ives's
music
comes in
the
form
of a
compositional
"model" hat seems
con-
tinuously
o have
occupied
his
interest.
His
earliest
nspirations
toward the
structure
of
the model
are
suggestedby
several
ac-
counts in Memos
of
experiments
he
conducted
together
with
his
father, including,
for
example,
recollections
of
chord
suc-
cessions
constructed
from
"3rds all and
over,
then 3rds
and
2nds,
then
3rds and
4ths,
then
3rds
and 4ths and
5ths,
etc."60
Such
patterns
of
gradual
change
in
intervalsize evolved
into
a
model of chord
succession,
the broad outlines of which main-
tain
some
degree
of
uniformity
n a
variety
of musicalcontexts.
Most
generally,
the
model
is
comprised
of
successiveverticali-
ties formed from
single-interval
r
combination
ycles,
with the
sizes
of
the
generating
ntervals
gradually ncreasing
r decreas-
ing
in
established
increments.
The
pattern typicallydisplays
a
symmetrical
tructure
by reversing
tself
following
he arrival t
a
high
or
low
point
of interval size. If the numberof voices in
the chords
remains
constant-including
octave
doublings
for
smaller
cardinalities-the
expansion-contraction rocess
may
be
displayed
as
a
registral
"wedge"
shape
outlined
by
the
verti-
cal
span
of
each
chord. If
the chord
voicings
are
flexible,
how-
ever,
the
process may
occur within sonorities
that exhibit no
significant hanges
nvertical
span, only
internal
changes
n in-
tervallicstructure.
The
prime-number
eries
in In re con moto
et al is a variant
of the
model,
with the
pattern
of
changedetermining
aspects
other than
intervallicstructure.
Of
course,
the first
setting
of
the PS
shown in
Example
15 does
exhibit
a
general
pattern
of
intervallic
change
in
support
of
the durational
pattern.
The
tritone that occurs
within
GCC
(dyad d),
and
the tritone n
the
second violin
(dyade)
formedfrom
pc
6
of
GCC
and
pc
0 on the
next beat. This
segmentation
amasses five of the
six
available
tritones,
avoidingthroughout
he measure
the
pcs
3 and
9
that
are
inherently
absent
from
GC.
The most
striking
evidence of
interval
cycles
in
Ives's
music
comes in
the
form
of a
compositional
"model" hat seems
con-
tinuously
o have
occupied
his
interest.
His
earliest
nspirations
toward the
structure
of
the model
are
suggestedby
several
ac-
counts in Memos
of
experiments
he
conducted
together
with
his
father, including,
for
example,
recollections
of
chord
suc-
cessions
constructed
from
"3rds all and
over,
then 3rds
and
2nds,
then
3rds and
4ths,
then
3rds
and 4ths and
5ths,
etc."60
Such
patterns
of
gradual
change
in
intervalsize evolved
into
a
model of chord
succession,
the broad outlines of which main-
tain
some
degree
of
uniformity
n a
variety
of musicalcontexts.
Most
generally,
the
model
is
comprised
of
successiveverticali-
ties formed from
single-interval
r
combination
ycles,
with the
sizes
of
the
generating
ntervals
gradually ncreasing
r decreas-
ing
in
established
increments.
The
pattern typicallydisplays
a
symmetrical
tructure
by reversing
tself
following
he arrival t
a
high
or
low
point
of interval size. If the numberof voices in
the chords
remains
constant-including
octave
doublings
for
smaller
cardinalities-the
expansion-contraction rocess
may
be
displayed
as
a
registral
"wedge"
shape
outlined
by
the
verti-
cal
span
of
each
chord. If
the chord
voicings
are
flexible,
how-
ever,
the
process may
occur within sonorities
that exhibit no
significant hanges
nvertical
span, only
internal
changes
n in-
tervallicstructure.
The
prime-number
eries
in In re con moto
et al is a variant
of the
model,
with the
pattern
of
changedetermining
aspects
other than
intervallicstructure.
Of
course,
the first
setting
of
the PS
shown in
Example
15 does
exhibit
a
general
pattern
of
intervallic
change
in
support
of
the durational
pattern.
The
tritone that occurs
within
GCC
(dyad d),
and
the tritone n
the
second violin
(dyade)
formedfrom
pc
6
of
GCC
and
pc
0 on the
next beat. This
segmentation
amasses five of the
six
available
tritones,
avoidingthroughout
he measure
the
pcs
3 and
9
that
are
inherently
absent
from
GC.
The most
striking
evidence of
interval
cycles
in
Ives's
music
comes in
the
form
of a
compositional
"model" hat seems
con-
tinuously
o have
occupied
his
interest.
His
earliest
nspirations
toward the
structure
of
the model
are
suggestedby
several
ac-
counts in Memos
of
experiments
he
conducted
together
with
his
father, including,
for
example,
recollections
of
chord
suc-
cessions
constructed
from
"3rds all and
over,
then 3rds
and
2nds,
then
3rds and
4ths,
then
3rds
and 4ths and
5ths,
etc."60
Such
patterns
of
gradual
change
in
intervalsize evolved
into
a
model of chord
succession,
the broad outlines of which main-
tain
some
degree
of
uniformity
n a
variety
of musicalcontexts.
Most
generally,
the
model
is
comprised
of
successiveverticali-
ties formed from
single-interval
r
combination
ycles,
with the
sizes
of
the
generating
ntervals
gradually ncreasing
r decreas-
ing
in
established
increments.
The
pattern typicallydisplays
a
symmetrical
tructure
by reversing
tself
following
he arrival t
a
high
or
low
point
of interval size. If the numberof voices in
the chords
remains
constant-including
octave
doublings
for
smaller
cardinalities-the
expansion-contraction rocess
may
be
displayed
as
a
registral
"wedge"
shape
outlined
by
the
verti-
cal
span
of
each
chord. If
the chord
voicings
are
flexible,
how-
ever,
the
process may
occur within sonorities
that exhibit no
significant hanges
nvertical
span, only
internal
changes
n in-
tervallicstructure.
The
prime-number
eries
in In re con moto
et al is a variant
of the
model,
with the
pattern
of
changedetermining
aspects
other than
intervallicstructure.
Of
course,
the first
setting
of
the PS
shown in
Example
15 does
exhibit
a
general
pattern
of
intervallic
change
in
support
of
the durational
pattern.
The
60Ives, Memos,
120.
0Ives, Memos,
120.
0Ives, Memos,
120.
V-_-
-I
I
-_-
-I
I
-_-
-I
I
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 34/41
Interval
ycles
as
Compositional
esources 75
nterval
ycles
as
Compositional
esources 75
nterval
ycles
as
Compositional
esources 75
Figure
7. 7/6 combination
cycle
(portion),
GC/GCC extraction.
igure
7. 7/6 combination
cycle
(portion),
GC/GCC extraction.
igure
7. 7/6 combination
cycle
(portion),
GC/GCC extraction.
I
I I
I
7 1
8 2 9
I
I I
I
7 1
8 2 9
I
I I
I
7 1
8 2 9
I
(SC 8-29)(SC 8-29)(SC 8-29)
3 10
10
10
10
10
10 4 11 5 011 5 011 5 0
GCC:CC:CC:
Figure
8.
Tritones
within
7/6 combination
cycle.
GC
I
I
I
I
I
Figure
8.
Tritones
within
7/6 combination
cycle.
GC
I
I
I
I
I
Figure
8.
Tritones
within
7/6 combination
cycle.
GC
I
I
I
I
I
a
rI
7/6: 0 7 1
f
a
rI
7/6: 0 7 1
f
a
rI
7/6: 0 7 1
f
b
I
I
8 2
h
b
I
I
8 2
h
b
I
I
8 2
h
9
more standard version mixes
cycles
of
single
intervals with
closelyassociated combinations o intensify he gradualnature
of
the
process.
This is
displayed
n
the
early
choralwork Proces-
sional
("Let
there
be
Light,"
1901)
and n a broader ense
in
the
still earlier
Psalm
24,
the
beginning
of which s shown n Exam-
ple
8.
The
growth process
in the
Psalm,
which s
displayed
n
a
linear,
not
vertical
orm,
is
observable n the nonmodular ums
of
the
x/y
values on which each
verse
is
based,
treating every
intervallic
repetition
as
a
combination
cycle.
The chromatic
scale of verse 1 is sum 2
(x/y
=
1/1;
1
+
1
=
2),
the whole tone
of verse2 is sum 4 (x/y = 2/2;2 + 2 = 4), and so on, compiling
the
following progression
of sums in
the
first
seven verses:
more standard version mixes
cycles
of
single
intervals with
closelyassociated combinations o intensify he gradualnature
of
the
process.
This is
displayed
n
the
early
choralwork Proces-
sional
("Let
there
be
Light,"
1901)
and n a broader ense
in
the
still earlier
Psalm
24,
the
beginning
of which s shown n Exam-
ple
8.
The
growth process
in the
Psalm,
which s
displayed
n
a
linear,
not
vertical
orm,
is
observable n the nonmodular ums
of
the
x/y
values on which each
verse
is
based,
treating every
intervallic
repetition
as
a
combination
cycle.
The chromatic
scale of verse 1 is sum 2
(x/y
=
1/1;
1
+
1
=
2),
the whole tone
of verse2 is sum 4 (x/y = 2/2;2 + 2 = 4), and so on, compiling
the
following progression
of sums in
the
first
seven verses:
more standard version mixes
cycles
of
single
intervals with
closelyassociated combinations o intensify he gradualnature
of
the
process.
This is
displayed
n
the
early
choralwork Proces-
sional
("Let
there
be
Light,"
1901)
and n a broader ense
in
the
still earlier
Psalm
24,
the
beginning
of which s shown n Exam-
ple
8.
The
growth process
in the
Psalm,
which s
displayed
n
a
linear,
not
vertical
orm,
is
observable n the nonmodular ums
of
the
x/y
values on which each
verse
is
based,
treating every
intervallic
repetition
as
a
combination
cycle.
The chromatic
scale of verse 1 is sum 2
(x/y
=
1/1;
1
+
1
=
2),
the whole tone
of verse2 is sum 4 (x/y = 2/2;2 + 2 = 4), and so on, compiling
the
following progression
of sums in
the
first
seven verses:
1
2
3
4
5 6
72
3
4
5 6
72
3
4
5 6
7
combination
cycle:
1/1 2/2 3/3 4/3
5/5 6/5
7/7
ombination
cycle:
1/1 2/2 3/3 4/3
5/5 6/5
7/7
ombination
cycle:
1/1 2/2 3/3 4/3
5/5 6/5
7/7
2 4
6
7
10 11 14
4
6
7
10 11 14
4
6
7
10 11 14
c
I
3 10
4
g
c
I
3 10
4
g
c
I
3 10
4
g
GCC
I
I
4 11 5
I
I
d
d
GCC
I
I
4 11 5
I
I
d
d
GCC
I
I
4 11 5
I
I
d
d
I
*
I
(SC
4-29)
*repetition
(PCL
=
11)
I
*
I
(SC
4-29)
*repetition
(PCL
=
11)
I
*
I
(SC
4-29)
*repetition
(PCL
=
11)
I
0 6
I
e
I
0 6
I
e
I
0 6
I
e
Versions
of
the
pattern
n other later works
might
add,
for ex-
ample, a 1/2cyclebetween 1/1 and 2/2to fill in the gapbetween
sums 2
and
4,
though,
of
course,
some
potential
fill-ins
for
in-
stance,
6/6)
will have
undesirably
ow
PCL values.
Included
among
ater
settings
of
the model are an
attempt
at
integrating
patterns
of
pitch
and
rhythm
n
Over
he Pavements
(1906-13)61
and
a
programmatic
association
of the model's
"wedge" shape
with a
specific
scenario in Tone Roads No.
3
(1915).62
Nowhere is the
model
more
pervasive,
however,
than
61See the
bassoon,
clarinet,
and
trumpet parts,
mm. 81-92.
(Rinehart,
"Ives's
Compositional
Idioms,"
44-46,
91-93.)
62Ives
escribesthe scenario n
Memos,
64;
the
model
is most noticeable n
mm. 24-26
but is
present
elsewhere in the
piece
in variant orms. Some other
occurrences
of
the model are:
CentralPark in the Dark
(1906),
mm.
1-10 and
subsequent
repetitions
of the ten-bar
string
pattern;
Soliloquy (1907),
mm.
6-
7;
Robert
Browning
Overture
1908-12), linearly
in the
upper
strings,
mm.
Versions
of
the
pattern
n other later works
might
add,
for ex-
ample, a 1/2cyclebetween 1/1 and 2/2to fill in the gapbetween
sums 2
and
4,
though,
of
course,
some
potential
fill-ins
for
in-
stance,
6/6)
will have
undesirably
ow
PCL values.
Included
among
ater
settings
of
the model are an
attempt
at
integrating
patterns
of
pitch
and
rhythm
n
Over
he Pavements
(1906-13)61
and
a
programmatic
association
of the model's
"wedge" shape
with a
specific
scenario in Tone Roads No.
3
(1915).62
Nowhere is the
model
more
pervasive,
however,
than
61See the
bassoon,
clarinet,
and
trumpet parts,
mm. 81-92.
(Rinehart,
"Ives's
Compositional
Idioms,"
44-46,
91-93.)
62Ives
escribesthe scenario n
Memos,
64;
the
model
is most noticeable n
mm. 24-26
but is
present
elsewhere in the
piece
in variant orms. Some other
occurrences
of
the model are:
CentralPark in the Dark
(1906),
mm.
1-10 and
subsequent
repetitions
of the ten-bar
string
pattern;
Soliloquy (1907),
mm.
6-
7;
Robert
Browning
Overture
1908-12), linearly
in the
upper
strings,
mm.
Versions
of
the
pattern
n other later works
might
add,
for ex-
ample, a 1/2cyclebetween 1/1 and 2/2to fill in the gapbetween
sums 2
and
4,
though,
of
course,
some
potential
fill-ins
for
in-
stance,
6/6)
will have
undesirably
ow
PCL values.
Included
among
ater
settings
of
the model are an
attempt
at
integrating
patterns
of
pitch
and
rhythm
n
Over
he Pavements
(1906-13)61
and
a
programmatic
association
of the model's
"wedge" shape
with a
specific
scenario in Tone Roads No.
3
(1915).62
Nowhere is the
model
more
pervasive,
however,
than
61See the
bassoon,
clarinet,
and
trumpet parts,
mm. 81-92.
(Rinehart,
"Ives's
Compositional
Idioms,"
44-46,
91-93.)
62Ives
escribesthe scenario n
Memos,
64;
the
model
is most noticeable n
mm. 24-26
but is
present
elsewhere in the
piece
in variant orms. Some other
occurrences
of
the model are:
CentralPark in the Dark
(1906),
mm.
1-10 and
subsequent
repetitions
of the ten-bar
string
pattern;
Soliloquy (1907),
mm.
6-
7;
Robert
Browning
Overture
1908-12), linearly
in the
upper
strings,
mm.
GC:
7/6:
pc
GC:
7/6:
pc
GC:
7/6:
pc
I
6
verse:erse:erse:
I
sums:ums:ums:
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 35/41
76 Music
Theory
Spectrum
6 Music
Theory
Spectrum
6 Music
Theory
Spectrum
in the
song
On the
Antipodes (1915-23),
which,
according
to
the
composer,
is based on a "chordal
cycle
for
a
symphony."63
The "chordal
cycle"
is an extensive intervallic contraction-
expansion pattern
that
occurs three times in the
song, appar-
ently realizing
ideas that were
initially
conceived for
the
incom-
plete
Universe
Symphony (1911-28).64
In both
the
song
and the
Symphony,
Ives
expounds upon
universal themes related to the
forces of nature and
processes
of
life,
and seems to
regard
the
intervallic model
as a
symbol
of some of these cosmic
powers.
The
subjects
of the
composer's
text
for the
song
are
the
"antip-
odal"
aspects
of
nature,
described as
paradoxical
extremes:
nature is both "relentless" and
"kind,"
nature is man's "en-
emy,"
but also his
"friend,"
and so forth. The text concludes
that nature
is
"nothing
but
atomic cosmic
cycles" revolving
be-
tween
the
many antipodes,
advancing
a
cyclic
view
of natural
evolution that is
appropriately
mirrored
by
the
many cyclic
pitch
constructions
in the musical
setting.65
Ives uses
the "chordal
cycle"
at the
beginning
and end
(the
"antipodes")
of the
song,
and in the center. The first and
sec-
ond
statements,
which
are identical
in
pitch-class
content,
pro-
ject
the structure outlined in
Figure
9,
progressing
from
perfect
fifths
to a semitonal cluster in the center of the
symmetrical pat-
tern and then
returning
to fifths.66The
upper-case
letter
labels,
119-136, 312-330,
384-390;
The Fourth
of July (1911-13),
m. 20. Some of
these are
described n Nachum
Schoffman,
"Serialism n the
Worksof Charles
Ives,"
Tempo
138
(September
1981),
21-32.
63Charles ves, NineteenSongs (Bryn Mawr, Pa.: Merion Music, 1935),
[52].
64See,
for
example, page
q3039
of the
Universe
Symphony
sketches
(Kirkpatrick,
Catalogue,
27).
65For urther
commentary
on the
text,
see
Hitchcock,
Ives:A
Survey
of
the
Music, 18-20,
and Nachum
Schoffman,
"The
Songs
of
Charles
Ives"
(Ph.D.
dissertation,
Hebrew
University
of
Jerusalem,
1977),
233.
66Also,
the final
statement
of
chord
A
is
preceded by
a chord of stacked
interval
11s;
this has
been omitted from
Figure
9
because
it is absent from the
in the
song
On the
Antipodes (1915-23),
which,
according
to
the
composer,
is based on a "chordal
cycle
for
a
symphony."63
The "chordal
cycle"
is an extensive intervallic contraction-
expansion pattern
that
occurs three times in the
song, appar-
ently realizing
ideas that were
initially
conceived for
the
incom-
plete
Universe
Symphony (1911-28).64
In both
the
song
and the
Symphony,
Ives
expounds upon
universal themes related to the
forces of nature and
processes
of
life,
and seems to
regard
the
intervallic model
as a
symbol
of some of these cosmic
powers.
The
subjects
of the
composer's
text
for the
song
are
the
"antip-
odal"
aspects
of
nature,
described as
paradoxical
extremes:
nature is both "relentless" and
"kind,"
nature is man's "en-
emy,"
but also his
"friend,"
and so forth. The text concludes
that nature
is
"nothing
but
atomic cosmic
cycles" revolving
be-
tween
the
many antipodes,
advancing
a
cyclic
view
of natural
evolution that is
appropriately
mirrored
by
the
many cyclic
pitch
constructions
in the musical
setting.65
Ives uses
the "chordal
cycle"
at the
beginning
and end
(the
"antipodes")
of the
song,
and in the center. The first and
sec-
ond
statements,
which
are identical
in
pitch-class
content,
pro-
ject
the structure outlined in
Figure
9,
progressing
from
perfect
fifths
to a semitonal cluster in the center of the
symmetrical pat-
tern and then
returning
to fifths.66The
upper-case
letter
labels,
119-136, 312-330,
384-390;
The Fourth
of July (1911-13),
m. 20. Some of
these are
described n Nachum
Schoffman,
"Serialism n the
Worksof Charles
Ives,"
Tempo
138
(September
1981),
21-32.
63Charles ves, NineteenSongs (Bryn Mawr, Pa.: Merion Music, 1935),
[52].
64See,
for
example, page
q3039
of the
Universe
Symphony
sketches
(Kirkpatrick,
Catalogue,
27).
65For urther
commentary
on the
text,
see
Hitchcock,
Ives:A
Survey
of
the
Music, 18-20,
and Nachum
Schoffman,
"The
Songs
of
Charles
Ives"
(Ph.D.
dissertation,
Hebrew
University
of
Jerusalem,
1977),
233.
66Also,
the final
statement
of
chord
A
is
preceded by
a chord of stacked
interval
11s;
this has
been omitted from
Figure
9
because
it is absent from the
in the
song
On the
Antipodes (1915-23),
which,
according
to
the
composer,
is based on a "chordal
cycle
for
a
symphony."63
The "chordal
cycle"
is an extensive intervallic contraction-
expansion pattern
that
occurs three times in the
song, appar-
ently realizing
ideas that were
initially
conceived for
the
incom-
plete
Universe
Symphony (1911-28).64
In both
the
song
and the
Symphony,
Ives
expounds upon
universal themes related to the
forces of nature and
processes
of
life,
and seems to
regard
the
intervallic model
as a
symbol
of some of these cosmic
powers.
The
subjects
of the
composer's
text
for the
song
are
the
"antip-
odal"
aspects
of
nature,
described as
paradoxical
extremes:
nature is both "relentless" and
"kind,"
nature is man's "en-
emy,"
but also his
"friend,"
and so forth. The text concludes
that nature
is
"nothing
but
atomic cosmic
cycles" revolving
be-
tween
the
many antipodes,
advancing
a
cyclic
view
of natural
evolution that is
appropriately
mirrored
by
the
many cyclic
pitch
constructions
in the musical
setting.65
Ives uses
the "chordal
cycle"
at the
beginning
and end
(the
"antipodes")
of the
song,
and in the center. The first and
sec-
ond
statements,
which
are identical
in
pitch-class
content,
pro-
ject
the structure outlined in
Figure
9,
progressing
from
perfect
fifths
to a semitonal cluster in the center of the
symmetrical pat-
tern and then
returning
to fifths.66The
upper-case
letter
labels,
119-136, 312-330,
384-390;
The Fourth
of July (1911-13),
m. 20. Some of
these are
described n Nachum
Schoffman,
"Serialism n the
Worksof Charles
Ives,"
Tempo
138
(September
1981),
21-32.
63Charles ves, NineteenSongs (Bryn Mawr, Pa.: Merion Music, 1935),
[52].
64See,
for
example, page
q3039
of the
Universe
Symphony
sketches
(Kirkpatrick,
Catalogue,
27).
65For urther
commentary
on the
text,
see
Hitchcock,
Ives:A
Survey
of
the
Music, 18-20,
and Nachum
Schoffman,
"The
Songs
of
Charles
Ives"
(Ph.D.
dissertation,
Hebrew
University
of
Jerusalem,
1977),
233.
66Also,
the final
statement
of
chord
A
is
preceded by
a chord of stacked
interval
11s;
this has
been omitted from
Figure
9
because
it is absent from the
which
generally adopt
the
composer's
markings
in
the
sketches,67
how that a
complete symmetrical
arrangement
s
disruptedwhen chordI (3/1)isreplacedbyL (2/2)in thesecond
half.
This
does
not, however,
disrupt
he
regularprogression
f
sums
from 14 to
2
and
back,
which
skipsonly
8
and 12.68 n-
deed,
the
variety
of
sums
portrays
he
diversity
of
cycles
used;
these
range through
most
of
the
possible
n
values
(omitting
only
0
and
8,
inferable
rom
the
absence of
sums 12 and
8),
and
including
the wide
range
of PCL values
listed on the
bottom
line
of
the
figure.
The intervals hemselvesare
mostly
he famil-
iar structures
ound in other realizations
of the
model,
includ-
ingthe circleof fifths(chordsA andD), cyclesof 7/6 and 6/5(B
and
C),
and
the
half-step
clusterat
the
midpoint
K).
Chords
E
and J are octatonic
collections,
and L is whole-tone.
To main-
tain
the
regularity
of
changes
in
sums,
the succession
even
in-
cludes a less
familiar tructure
with a PCL
of
only
4 at chord
G,
a whole-tone
subset
(4-25
[0,2,6,8]).
The
realizations
of this model at the
beginning
and
center of
the
song
exhibit substantial
variations in the
ranges
of the
chords
and
do
not
consistently
project
the
complete
cycles
of
final
"crystallized"
version of the
pattern
(mm.
28-34)
and is not
part
of the
symmetrically
elated series of sums.
Figure
9 reflects
observations common
to
several
analyses
of this work:
Rinehart,
"Ives's
Compositional
dioms," 71-86;
Domenick
Argento,
"A Di-
gest Analysis
of
Ives's
'On
the
Antipodes,'
"
Student
Musicologists
t Minne-
sota
6
(1975-76),
192-200; Hitchcock,
Ives: A
Survey
of
the
Music,
18-20;
Schoffman, "The Songs of Ives," 209-234; and Schoffman, "Serialism n
Ives,"
28-29.
By
contrast,
the
emphasis
here is
placed
on the
cyclic
natureof
each
chord
construction,
not
merely
on
its intervallic ontent.
67Kirkpatrick, atalogue,
210:
q2908, q3048.
Ives
makes a distinction
with
superscripts)
between
separate
uses
of
the same intervalsand
does not use the
letter L.
68The
uccession is
provided
with
even finer
gradations
n
the version
in-
tended for the
Universe
Symphony
by
theinclusionof chordswith
quarter-tone
intervals.
(Kirkpatrick,Catalogue,
27:
q3039.)
which
generally adopt
the
composer's
markings
in
the
sketches,67
how that a
complete symmetrical
arrangement
s
disruptedwhen chordI (3/1)isreplacedbyL (2/2)in thesecond
half.
This
does
not, however,
disrupt
he
regularprogression
f
sums
from 14 to
2
and
back,
which
skipsonly
8
and 12.68 n-
deed,
the
variety
of
sums
portrays
he
diversity
of
cycles
used;
these
range through
most
of
the
possible
n
values
(omitting
only
0
and
8,
inferable
rom
the
absence of
sums 12 and
8),
and
including
the wide
range
of PCL values
listed on the
bottom
line
of
the
figure.
The intervals hemselvesare
mostly
he famil-
iar structures
ound in other realizations
of the
model,
includ-
ingthe circleof fifths(chordsA andD), cyclesof 7/6 and 6/5(B
and
C),
and
the
half-step
clusterat
the
midpoint
K).
Chords
E
and J are octatonic
collections,
and L is whole-tone.
To main-
tain
the
regularity
of
changes
in
sums,
the succession
even
in-
cludes a less
familiar tructure
with a PCL
of
only
4 at chord
G,
a whole-tone
subset
(4-25
[0,2,6,8]).
The
realizations
of this model at the
beginning
and
center of
the
song
exhibit substantial
variations in the
ranges
of the
chords
and
do
not
consistently
project
the
complete
cycles
of
final
"crystallized"
version of the
pattern
(mm.
28-34)
and is not
part
of the
symmetrically
elated series of sums.
Figure
9 reflects
observations common
to
several
analyses
of this work:
Rinehart,
"Ives's
Compositional
dioms," 71-86;
Domenick
Argento,
"A Di-
gest Analysis
of
Ives's
'On
the
Antipodes,'
"
Student
Musicologists
t Minne-
sota
6
(1975-76),
192-200; Hitchcock,
Ives: A
Survey
of
the
Music,
18-20;
Schoffman, "The Songs of Ives," 209-234; and Schoffman, "Serialism n
Ives,"
28-29.
By
contrast,
the
emphasis
here is
placed
on the
cyclic
natureof
each
chord
construction,
not
merely
on
its intervallic ontent.
67Kirkpatrick, atalogue,
210:
q2908, q3048.
Ives
makes a distinction
with
superscripts)
between
separate
uses
of
the same intervalsand
does not use the
letter L.
68The
uccession is
provided
with
even finer
gradations
n
the version
in-
tended for the
Universe
Symphony
by
theinclusionof chordswith
quarter-tone
intervals.
(Kirkpatrick,Catalogue,
27:
q3039.)
which
generally adopt
the
composer's
markings
in
the
sketches,67
how that a
complete symmetrical
arrangement
s
disruptedwhen chordI (3/1)isreplacedbyL (2/2)in thesecond
half.
This
does
not, however,
disrupt
he
regularprogression
f
sums
from 14 to
2
and
back,
which
skipsonly
8
and 12.68 n-
deed,
the
variety
of
sums
portrays
he
diversity
of
cycles
used;
these
range through
most
of
the
possible
n
values
(omitting
only
0
and
8,
inferable
rom
the
absence of
sums 12 and
8),
and
including
the wide
range
of PCL values
listed on the
bottom
line
of
the
figure.
The intervals hemselvesare
mostly
he famil-
iar structures
ound in other realizations
of the
model,
includ-
ingthe circleof fifths(chordsA andD), cyclesof 7/6 and 6/5(B
and
C),
and
the
half-step
clusterat
the
midpoint
K).
Chords
E
and J are octatonic
collections,
and L is whole-tone.
To main-
tain
the
regularity
of
changes
in
sums,
the succession
even
in-
cludes a less
familiar tructure
with a PCL
of
only
4 at chord
G,
a whole-tone
subset
(4-25
[0,2,6,8]).
The
realizations
of this model at the
beginning
and
center of
the
song
exhibit substantial
variations in the
ranges
of the
chords
and
do
not
consistently
project
the
complete
cycles
of
final
"crystallized"
version of the
pattern
(mm.
28-34)
and is not
part
of the
symmetrically
elated series of sums.
Figure
9 reflects
observations common
to
several
analyses
of this work:
Rinehart,
"Ives's
Compositional
dioms," 71-86;
Domenick
Argento,
"A Di-
gest Analysis
of
Ives's
'On
the
Antipodes,'
"
Student
Musicologists
t Minne-
sota
6
(1975-76),
192-200; Hitchcock,
Ives: A
Survey
of
the
Music,
18-20;
Schoffman, "The Songs of Ives," 209-234; and Schoffman, "Serialism n
Ives,"
28-29.
By
contrast,
the
emphasis
here is
placed
on the
cyclic
natureof
each
chord
construction,
not
merely
on
its intervallic ontent.
67Kirkpatrick, atalogue,
210:
q2908, q3048.
Ives
makes a distinction
with
superscripts)
between
separate
uses
of
the same intervalsand
does not use the
letter L.
68The
uccession is
provided
with
even finer
gradations
n
the version
in-
tended for the
Universe
Symphony
by
theinclusionof chordswith
quarter-tone
intervals.
(Kirkpatrick,Catalogue,
27:
q3039.)
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 36/41
Interval
ycles
as
Compositional
esources
77nterval
ycles
as
Compositional
esources
77nterval
ycles
as
Compositional
esources
77
Figure
9. On
the
Antipodes,
chordal
cycle.
igure
9. On
the
Antipodes,
chordal
cycle.
igure
9. On
the
Antipodes,
chordal
cycle.
label: A B C D E FG H I
7 6 5 5 4 3 4 2 1
7 7
6
5 5 4 2 3 3
sums: 14 13 11
10
9
7
6
5 4
PCL:
12 11 12 12
8 8
4 6
6
label: A B C D E FG H I
7 6 5 5 4 3 4 2 1
7 7
6
5 5 4 2 3 3
sums: 14 13 11
10
9
7
6
5 4
PCL:
12 11 12 12
8 8
4 6
6
label: A B C D E FG H I
7 6 5 5 4 3 4 2 1
7 7
6
5 5 4 2 3 3
sums: 14 13 11
10
9
7
6
5 4
PCL:
12 11 12 12
8 8
4 6
6
J
2
1
3
8
J
2
1
3
8
J
2
1
3
8
K JL
1
2
2
1 1 2
K JL
1
2
2
1 1 2
K JL
1
2
2
1 1 2
H
2
3
H
2
3
H
2
3
G F E
4 3 4
2
4
5
G F E
4 3 4
2
4
5
G F E
4 3 4
2
4
5
2
34
5
6
7
9
12 8 6 6 4
8 8
2
34
5
6
7
9
12 8 6 6 4
8 8
2
34
5
6
7
9
12 8 6 6 4
8 8
D C B
5 5 6
5 6 7
D C B
5 5 6
5 6 7
D C B
5 5 6
5 6 7
A
7
7
A
7
7
A
7
7
10 11 13 14
12 12 11
12
10 11 13 14
12 12 11
12
10 11 13 14
12 12 11
12
eachintervallic tructure.69ome of the
changes
are motivated
by
registral
concerns: the realizations
generally
outline
contracting-expanding edge shapes,
with the
exception
of the
centralcluster. Also
influencing
nconsistencies
are the choices
of
pitch
classes
in the
upper
voice,
which unfold
the
aggregate
with one
repetition.70
The final realization of the
model,
reproduced
as
Example
17 with chord
labels
added,
maximizes the
registral span
of
each
chord with
pitch-classrepetitions
hat
may
extend
beyond
the PCLs.
Accompanying
a vocal line that asks the climactic
textual
question,71
he
four-hand
piano part presents
chords
structured
according
to the
outline
of
Figure
9,
except
that I
and L
exchange positions.Unique
to this realization s the
pc
0,
which,
as the lowest
voice of
every
chord
(doubled
by
an
op-
tional
organ pedal),
serves as the
point
of
departure
or each
cycle
in
the
pattern.
In
addition,
many
of
the sonorities exhibit
or
imply
an
upward-directed yclic
return o
the
pc-0
point
of
origin,
similar
to the
"wraparounds"
n TheFourth
of July (Ex. 12)
and In re
69The core
of
these
versions
s
reproduced
n
Argento,
"A
Digest Analy-
sis,"
201-203 and
Schoffman,
"Serialism n
Ives,"
28.
70This
bservation s
made
in
Schoffman,
"The
Songs
of
Ives,"
216-217.
71Thevocal line is
divided into two
parts,
both of which
produce aggre-
gates.
See
Hitchcock,
Ives: A
Survey
of
the
Music,
19-20. The
aggregate
n
the
upper
ine is formed
by adjacent
[0,1,4]
trichords.
eachintervallic tructure.69ome of the
changes
are motivated
by
registral
concerns: the realizations
generally
outline
contracting-expanding edge shapes,
with the
exception
of the
centralcluster. Also
influencing
nconsistencies
are the choices
of
pitch
classes
in the
upper
voice,
which unfold
the
aggregate
with one
repetition.70
The final realization of the
model,
reproduced
as
Example
17 with chord
labels
added,
maximizes the
registral span
of
each
chord with
pitch-classrepetitions
hat
may
extend
beyond
the PCLs.
Accompanying
a vocal line that asks the climactic
textual
question,71
he
four-hand
piano part presents
chords
structured
according
to the
outline
of
Figure
9,
except
that I
and L
exchange positions.Unique
to this realization s the
pc
0,
which,
as the lowest
voice of
every
chord
(doubled
by
an
op-
tional
organ pedal),
serves as the
point
of
departure
or each
cycle
in
the
pattern.
In
addition,
many
of
the sonorities exhibit
or
imply
an
upward-directed yclic
return o
the
pc-0
point
of
origin,
similar
to the
"wraparounds"
n TheFourth
of July (Ex. 12)
and In re
69The core
of
these
versions
s
reproduced
n
Argento,
"A
Digest Analy-
sis,"
201-203 and
Schoffman,
"Serialism n
Ives,"
28.
70This
bservation s
made
in
Schoffman,
"The
Songs
of
Ives,"
216-217.
71Thevocal line is
divided into two
parts,
both of which
produce aggre-
gates.
See
Hitchcock,
Ives: A
Survey
of
the
Music,
19-20. The
aggregate
n
the
upper
ine is formed
by adjacent
[0,1,4]
trichords.
eachintervallic tructure.69ome of the
changes
are motivated
by
registral
concerns: the realizations
generally
outline
contracting-expanding edge shapes,
with the
exception
of the
centralcluster. Also
influencing
nconsistencies
are the choices
of
pitch
classes
in the
upper
voice,
which unfold
the
aggregate
with one
repetition.70
The final realization of the
model,
reproduced
as
Example
17 with chord
labels
added,
maximizes the
registral span
of
each
chord with
pitch-classrepetitions
hat
may
extend
beyond
the PCLs.
Accompanying
a vocal line that asks the climactic
textual
question,71
he
four-hand
piano part presents
chords
structured
according
to the
outline
of
Figure
9,
except
that I
and L
exchange positions.Unique
to this realization s the
pc
0,
which,
as the lowest
voice of
every
chord
(doubled
by
an
op-
tional
organ pedal),
serves as the
point
of
departure
or each
cycle
in
the
pattern.
In
addition,
many
of
the sonorities exhibit
or
imply
an
upward-directed yclic
return o
the
pc-0
point
of
origin,
similar
to the
"wraparounds"
n TheFourth
of July (Ex. 12)
and In re
69The core
of
these
versions
s
reproduced
n
Argento,
"A
Digest Analy-
sis,"
201-203 and
Schoffman,
"Serialism n
Ives,"
28.
70This
bservation s
made
in
Schoffman,
"The
Songs
of
Ives,"
216-217.
71Thevocal line is
divided into two
parts,
both of which
produce aggre-
gates.
See
Hitchcock,
Ives: A
Survey
of
the
Music,
19-20. The
aggregate
n
the
upper
ine is formed
by adjacent
[0,1,4]
trichords.
con moto etal (Ex. 15). ChordsA andD, forexample,are com-
plete
interval-7 and interval-5
cycles
that returnto
pc
0 when
continued one
step
further,
mplying
a
linkage
to
the bass note
of the
ensuing verticality.
The same is true of the 6/5
cycle
of
chord
C:
the final
upper
interval 5
(C#6-F#6,
m.
29 beat
1)
would be followed
by
interval
6
to
return
to
pc
0.
In
other
sonorities the return to
pc
0
occurs within the
chord,
either
as
the
upper
voice
or
as
a
result
of
extensive
repetition
of a small
PCL. Chord B
(7/6)
contains
pc
0 in both
top
and bottom
voices, its
only
pc duplication.
The
arpeggiated
tatementof H
(PCL
=
6)
in m. 30
continues far
beyond
its first
pc repetition
(E
3), cyclingthrough
several
pc
duplications
before
returning
to
pc
0
as the
highest
note. Chords hat
repeat
within
an
octave
and that
may
therefore contain several nstances
of
pc
0 are G
(4/2,
PCL
=
4),
I
(3/1,
PCL
=
6),
J
(1/2,
PCL
=
8),
and L
(2/2,
PCL
=
6);
of
these,
only
G contains
pc
0 in the
top
voice,
al-
though
each includes
pc
0
at last twice in inner
voices.
The re-
maining
chords
are
E
(5/4,
PCL
=
8),
which
cycles pastpc
0,
F
(4/3, PCL =
8),
which
stops
short of a returnto the
starting
point,
and
K,
the
four-octave semitonal cluster that
includes
several nstances of
pc
0,
but with
pc
1
in the
upper
voice.
The
series
of
cycles
based
on
successively
smaller ntervals
that return
to
the same
pitch-class
anchor
projects
the
idea ex-
pressed
in
the text
of
"atomic,
cosmic
cycles"
as
a
"spiral"pro-
gressing
inward from
cycles
of
larger
intervals
to those with
con moto etal (Ex. 15). ChordsA andD, forexample,are com-
plete
interval-7 and interval-5
cycles
that returnto
pc
0 when
continued one
step
further,
mplying
a
linkage
to
the bass note
of the
ensuing verticality.
The same is true of the 6/5
cycle
of
chord
C:
the final
upper
interval 5
(C#6-F#6,
m.
29 beat
1)
would be followed
by
interval
6
to
return
to
pc
0.
In
other
sonorities the return to
pc
0
occurs within the
chord,
either
as
the
upper
voice
or
as
a
result
of
extensive
repetition
of a small
PCL. Chord B
(7/6)
contains
pc
0 in both
top
and bottom
voices, its
only
pc duplication.
The
arpeggiated
tatementof H
(PCL
=
6)
in m. 30
continues far
beyond
its first
pc repetition
(E
3), cyclingthrough
several
pc
duplications
before
returning
to
pc
0
as the
highest
note. Chords hat
repeat
within
an
octave
and that
may
therefore contain several nstances
of
pc
0 are G
(4/2,
PCL
=
4),
I
(3/1,
PCL
=
6),
J
(1/2,
PCL
=
8),
and L
(2/2,
PCL
=
6);
of
these,
only
G contains
pc
0 in the
top
voice,
al-
though
each includes
pc
0
at last twice in inner
voices.
The re-
maining
chords
are
E
(5/4,
PCL
=
8),
which
cycles pastpc
0,
F
(4/3, PCL =
8),
which
stops
short of a returnto the
starting
point,
and
K,
the
four-octave semitonal cluster that
includes
several nstances of
pc
0,
but with
pc
1
in the
upper
voice.
The
series
of
cycles
based
on
successively
smaller ntervals
that return
to
the same
pitch-class
anchor
projects
the
idea ex-
pressed
in
the text
of
"atomic,
cosmic
cycles"
as
a
"spiral"pro-
gressing
inward from
cycles
of
larger
intervals
to those with
con moto etal (Ex. 15). ChordsA andD, forexample,are com-
plete
interval-7 and interval-5
cycles
that returnto
pc
0 when
continued one
step
further,
mplying
a
linkage
to
the bass note
of the
ensuing verticality.
The same is true of the 6/5
cycle
of
chord
C:
the final
upper
interval 5
(C#6-F#6,
m.
29 beat
1)
would be followed
by
interval
6
to
return
to
pc
0.
In
other
sonorities the return to
pc
0
occurs within the
chord,
either
as
the
upper
voice
or
as
a
result
of
extensive
repetition
of a small
PCL. Chord B
(7/6)
contains
pc
0 in both
top
and bottom
voices, its
only
pc duplication.
The
arpeggiated
tatementof H
(PCL
=
6)
in m. 30
continues far
beyond
its first
pc repetition
(E
3), cyclingthrough
several
pc
duplications
before
returning
to
pc
0
as the
highest
note. Chords hat
repeat
within
an
octave
and that
may
therefore contain several nstances
of
pc
0 are G
(4/2,
PCL
=
4),
I
(3/1,
PCL
=
6),
J
(1/2,
PCL
=
8),
and L
(2/2,
PCL
=
6);
of
these,
only
G contains
pc
0 in the
top
voice,
al-
though
each includes
pc
0
at last twice in inner
voices.
The re-
maining
chords
are
E
(5/4,
PCL
=
8),
which
cycles pastpc
0,
F
(4/3, PCL =
8),
which
stops
short of a returnto the
starting
point,
and
K,
the
four-octave semitonal cluster that
includes
several nstances of
pc
0,
but with
pc
1
in the
upper
voice.
The
series
of
cycles
based
on
successively
smaller ntervals
that return
to
the same
pitch-class
anchor
projects
the
idea ex-
pressed
in
the text
of
"atomic,
cosmic
cycles"
as
a
"spiral"pro-
gressing
inward from
cycles
of
larger
intervals
to those with
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 37/41
78
Music
Theory
Spectrum
Example
17.
On the
Antipodes,
mm. 28-34.
78
Music
Theory
Spectrum
Example
17.
On the
Antipodes,
mm. 28-34.
78
Music
Theory
Spectrum
Example
17.
On the
Antipodes,
mm. 28-34.
Largo maestoso
28 $h
Largo maestoso
28 $h
Largo maestoso
28 $h
r-3 -
L_-
-3 -
L_-
-3 -
L_-
Man
we ask
you
Is Na
- ture
noth-ing
but a
-
tom
-
ic cos
-
mic
cy
- cles _____
loco"
*
f
I
$n/
E--Ip
i= =
fftt8
I
cresc.
it
-i
'
"
i
4
t"bi
Largo
maestoso
cresc.
(Org.
ad
lib.)
(16'
and
32' only)
Man
we ask
you
Is Na
- ture
noth-ing
but a
-
tom
-
ic cos
-
mic
cy
- cles _____
loco"
*
f
I
$n/
E--Ip
i= =
fftt8
I
cresc.
it
-i
'
"
i
4
t"bi
Largo
maestoso
cresc.
(Org.
ad
lib.)
(16'
and
32' only)
Man
we ask
you
Is Na
- ture
noth-ing
but a
-
tom
-
ic cos
-
mic
cy
- cles _____
loco"
*
f
I
$n/
E--Ip
i= =
fftt8
I
cresc.
it
-i
'
"
i
4
t"bi
Largo
maestoso
cresc.
(Org.
ad
lib.)
(16'
and
32' only)
(Org.Ped.)
_
-
-
[A
B
C D E F
G
(Org.Ped.)
_
-
-
[A
B
C D E F
G
(Org.Ped.)
_
-
-
[A
B
C D E F
G
H
smaller units
of
repetition.
Figure
10 illustrates
this
process,
which
primarily
reflects the
changes
in
interval
sizes-as
reflected
by
the sums-not the sizes
of the PCLs
or the
cardi-
nalities of
complete
combinations. Chord
A,
cycle
7/7,
forms
the
outer
layer
of the
spiral,
and its return
o
pc
0 after a com-
plete
revolution coincides with the
beginning
of the
next mem-
ber
of
the chord
succession,
the
7/6
cycle
of chord
B,
on the
next
inner
layer.
Each
successive
layer corresponds
to each
subsequent
chord,
following
he
process
of
gradual
eduction
n
sums,
so that each member of the chordal
cycle
is
represented
by
one revolutionin the
spiral.
The chords
spiral
"inward"
n
the first
half
of
the
pattern,
and then reverse
direction
o return
to the
outer
layer
at
the
conclusionof
the chord
sequence.
smaller units
of
repetition.
Figure
10 illustrates
this
process,
which
primarily
reflects the
changes
in
interval
sizes-as
reflected
by
the sums-not the sizes
of the PCLs
or the
cardi-
nalities of
complete
combinations. Chord
A,
cycle
7/7,
forms
the
outer
layer
of the
spiral,
and its return
o
pc
0 after a com-
plete
revolution coincides with the
beginning
of the
next mem-
ber
of
the chord
succession,
the
7/6
cycle
of chord
B,
on the
next
inner
layer.
Each
successive
layer corresponds
to each
subsequent
chord,
following
he
process
of
gradual
eduction
n
sums,
so that each member of the chordal
cycle
is
represented
by
one revolutionin the
spiral.
The chords
spiral
"inward"
n
the first
half
of
the
pattern,
and then reverse
direction
o return
to the
outer
layer
at
the
conclusionof
the chord
sequence.
smaller units
of
repetition.
Figure
10 illustrates
this
process,
which
primarily
reflects the
changes
in
interval
sizes-as
reflected
by
the sums-not the sizes
of the PCLs
or the
cardi-
nalities of
complete
combinations. Chord
A,
cycle
7/7,
forms
the
outer
layer
of the
spiral,
and its return
o
pc
0 after a com-
plete
revolution coincides with the
beginning
of the
next mem-
ber
of
the chord
succession,
the
7/6
cycle
of chord
B,
on the
next
inner
layer.
Each
successive
layer corresponds
to each
subsequent
chord,
following
he
process
of
gradual
eduction
n
sums,
so that each member of the chordal
cycle
is
represented
by
one revolutionin the
spiral.
The chords
spiral
"inward"
n
the first
half
of
the
pattern,
and then reverse
direction
o return
to the
outer
layer
at
the
conclusionof
the chord
sequence.
The
overall formof On the
Antipodes
s further
ied
to the
chordal
cycle
through
a
"composing
out"
process
that
distrib-
utes
the sonorities
nto the texture
muchas the
interval-5
ycle
structures
he introduction
of
In
re con moto
et
al
(Ex.
14).
As
explainedby Argento,
the
passages
between
statements
of the
chord
are
entirely
based
on the structures
of the chords
them-
selves,
so that almost
every aspect
of the
song
is connectedto
the
cyclicpattern.72
n the
two measures
mmediately
ollowing
the
initial
presentation
of the
chords,
for
example,
the
A,
B,
and C sonorities
account
for most
of
the
pitch
materials
of the
72"A
Digest
Analysis,"
198-200.
Only
mm. 14-17 are not derived
from the
chordal
cycle.
The
overall formof On the
Antipodes
s further
ied
to the
chordal
cycle
through
a
"composing
out"
process
that
distrib-
utes
the sonorities
nto the texture
muchas the
interval-5
ycle
structures
he introduction
of
In
re con moto
et
al
(Ex.
14).
As
explainedby Argento,
the
passages
between
statements
of the
chord
are
entirely
based
on the structures
of the chords
them-
selves,
so that almost
every aspect
of the
song
is connectedto
the
cyclicpattern.72
n the
two measures
mmediately
ollowing
the
initial
presentation
of the
chords,
for
example,
the
A,
B,
and C sonorities
account
for most
of
the
pitch
materials
of the
72"A
Digest
Analysis,"
198-200.
Only
mm. 14-17 are not derived
from the
chordal
cycle.
The
overall formof On the
Antipodes
s further
ied
to the
chordal
cycle
through
a
"composing
out"
process
that
distrib-
utes
the sonorities
nto the texture
muchas the
interval-5
ycle
structures
he introduction
of
In
re con moto
et
al
(Ex.
14).
As
explainedby Argento,
the
passages
between
statements
of the
chord
are
entirely
based
on the structures
of the chords
them-
selves,
so that almost
every aspect
of the
song
is connectedto
the
cyclicpattern.72
n the
two measures
mmediately
ollowing
the
initial
presentation
of the
chords,
for
example,
the
A,
B,
and C sonorities
account
for most
of
the
pitch
materials
of the
72"A
Digest
Analysis,"
198-200.
Only
mm. 14-17 are not derived
from the
chordal
cycle.
, <29 _f-^ 3----, <29 _f-^ 3----, <29 _f-^ 3----
r--
30
cresc.
I 1 30 I
r--
30
cresc.
I 1 30 I
r--
30
cresc.
I 1 30 I
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 38/41
Interval
ycles
as
Compositional
esources
79nterval
ycles
as
Compositional
esources
79nterval
ycles
as
Compositional
esources
79
Example
17.
(cont'd.)
xample
17.
(cont'd.)
xample
17.
(cont'd.)
L
J
K
J
K
J
K
accompaniment.
Example
18 locates
A andB in
m.
5
and
C
in
m. 6
with
indications
of
order
positions
within the
three
cycles
notated
beneath
the score.
Because
the
registral
placement
of
each
pitch
class
generallycorresponds
to
that
of the
original
chord
sequence,
these
occurrences
project
a kind
of
varied
ar-
peggiation
of
the source
sonorities.
Often,
this
produces
a tri-
chordal
distribution
f the
chords,
highlighting
single
trichord
type;
any
three
adjacentpitch
classes
in the B
and
C chords
of
Example
18,
for
example,
form
a
3-5
[0,1,6]
trichord,
and
these
are
prominently
displayed
n mm.
5 and
6,
observable
as
adjacent ops.
The
remaining
uncircled)
notes
then
fill out
the
accompaniment.
Example
18 locates
A andB in
m.
5
and
C
in
m. 6
with
indications
of
order
positions
within the
three
cycles
notated
beneath
the score.
Because
the
registral
placement
of
each
pitch
class
generallycorresponds
to
that
of the
original
chord
sequence,
these
occurrences
project
a kind
of
varied
ar-
peggiation
of
the source
sonorities.
Often,
this
produces
a tri-
chordal
distribution
f the
chords,
highlighting
single
trichord
type;
any
three
adjacentpitch
classes
in the B
and
C chords
of
Example
18,
for
example,
form
a
3-5
[0,1,6]
trichord,
and
these
are
prominently
displayed
n mm.
5 and
6,
observable
as
adjacent ops.
The
remaining
uncircled)
notes
then
fill out
the
accompaniment.
Example
18 locates
A andB in
m.
5
and
C
in
m. 6
with
indications
of
order
positions
within the
three
cycles
notated
beneath
the score.
Because
the
registral
placement
of
each
pitch
class
generallycorresponds
to
that
of the
original
chord
sequence,
these
occurrences
project
a kind
of
varied
ar-
peggiation
of
the source
sonorities.
Often,
this
produces
a tri-
chordal
distribution
f the
chords,
highlighting
single
trichord
type;
any
three
adjacentpitch
classes
in the B
and
C chords
of
Example
18,
for
example,
form
a
3-5
[0,1,6]
trichord,
and
these
are
prominently
displayed
n mm.
5 and
6,
observable
as
adjacent ops.
The
remaining
uncircled)
notes
then
fill out
the
measure
with
pitch-class
reiterations,
often
recalling
the
pri-
mary
ntervals
of
the source
sonority.
Derivations
n the
remainderof
the
song
follow
similar
pro-
cedures,
either
continuing
he
varied
arpeggiations
r
redistrib-
uting
the sourcemore
extensively,
while
continuing
o
uphold
distinctive
features
of
each
structure.
These
sources
appear
n
their
original
order
(Fig.
9),
so that
the
entire
song
projects
an
expanded,
elaborated
macrocosm
of
the
chordal
cycle.73
The
series
of derivations
elaborates
roughly
the first
half
of
the se-
measure
with
pitch-class
reiterations,
often
recalling
the
pri-
mary
ntervals
of
the source
sonority.
Derivations
n the
remainderof
the
song
follow
similar
pro-
cedures,
either
continuing
he
varied
arpeggiations
r
redistrib-
uting
the sourcemore
extensively,
while
continuing
o
uphold
distinctive
features
of
each
structure.
These
sources
appear
n
their
original
order
(Fig.
9),
so that
the
entire
song
projects
an
expanded,
elaborated
macrocosm
of
the
chordal
cycle.73
The
series
of derivations
elaborates
roughly
the first
half
of
the se-
measure
with
pitch-class
reiterations,
often
recalling
the
pri-
mary
ntervals
of
the source
sonority.
Derivations
n the
remainderof
the
song
follow
similar
pro-
cedures,
either
continuing
he
varied
arpeggiations
r
redistrib-
uting
the sourcemore
extensively,
while
continuing
o
uphold
distinctive
features
of
each
structure.
These
sources
appear
n
their
original
order
(Fig.
9),
so that
the
entire
song
projects
an
expanded,
elaborated
macrocosm
of
the
chordal
cycle.73
The
series
of derivations
elaborates
roughly
the first
half
of
the se-
73See
Argento,
"A
Digest Analysis."
3See
Argento,
"A
Digest Analysis."
3See
Argento,
"A
Digest Analysis."
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 39/41
80
Music
TheorySpectrum
0
Music
TheorySpectrum
0
Music
TheorySpectrum
Figure
10.
"Spiral"
f
combination
ycles.
igure
10.
"Spiral"
f
combination
ycles.
igure
10.
"Spiral"
f
combination
ycles.
Example
18.
On
the
Antipodes,
mm.
5-6.
Allegro
r--
3
3
Andante
Example
18.
On
the
Antipodes,
mm.
5-6.
Allegro
r--
3
3
Andante
Example
18.
On
the
Antipodes,
mm.
5-6.
Allegro
r--
3
3
Andante
quence,
chords
A
through
K
plus
a return
of
J,
between
the
statements
of the chordal
cycle
at
the
beginning
and
midpoint,
and then elaborates
the
remainder
of
the
sequence,
chords L
back
to
A,
preceding
the final statement
of
the
chordal
cycle.
(Thereturn o A initiates he finalstatementof thechords.) Cy-
clic
principles
hus
permeate
musical
relationships
n the verti-
cal,
or chordal
dimension,
and in the horizontal
dimension
as
the
"cyclic"palindromic
structureof the chord
pattern,
all
of
which extends
from small to
large
structural
evels.
By
impart-
ing
these
qualities
to
the
song,
Ives
likewise characterizes
na-
ture
as ordered and
logical despitecomplexities
andcontradic-
tions that
seemingly defy explanation.
Below
the
complicated
quence,
chords
A
through
K
plus
a return
of
J,
between
the
statements
of the chordal
cycle
at
the
beginning
and
midpoint,
and then elaborates
the
remainder
of
the
sequence,
chords L
back
to
A,
preceding
the final statement
of
the
chordal
cycle.
(Thereturn o A initiates he finalstatementof thechords.) Cy-
clic
principles
hus
permeate
musical
relationships
n the verti-
cal,
or chordal
dimension,
and in the horizontal
dimension
as
the
"cyclic"palindromic
structureof the chord
pattern,
all
of
which extends
from small to
large
structural
evels.
By
impart-
ing
these
qualities
to
the
song,
Ives
likewise characterizes
na-
ture
as ordered and
logical despitecomplexities
andcontradic-
tions that
seemingly defy explanation.
Below
the
complicated
quence,
chords
A
through
K
plus
a return
of
J,
between
the
statements
of the chordal
cycle
at
the
beginning
and
midpoint,
and then elaborates
the
remainder
of
the
sequence,
chords L
back
to
A,
preceding
the final statement
of
the
chordal
cycle.
(Thereturn o A initiates he finalstatementof thechords.) Cy-
clic
principles
hus
permeate
musical
relationships
n the verti-
cal,
or chordal
dimension,
and in the horizontal
dimension
as
the
"cyclic"palindromic
structureof the chord
pattern,
all
of
which extends
from small to
large
structural
evels.
By
impart-
ing
these
qualities
to
the
song,
Ives
likewise characterizes
na-
ture
as ordered and
logical despitecomplexities
andcontradic-
tions that
seemingly defy explanation.
Below
the
complicated
7/6)
p
0
1
2
3 4 5 6 7 8 9
10
11
/)
pc
7
2
8 3
9 4
10 5
11 6 0 7
/6)
p
0
1
2
3 4 5 6 7 8 9
10
11
/)
pc
7
2
8 3
9 4
10 5
11 6 0 7
/6)
p
0
1
2
3 4 5 6 7 8 9
10
11
/)
pc
7
2
8 3
9 4
10 5
11 6 0 7
op
0 1 2
3
4
5 6
7
8 9
(6/5) C:
pc
1 7 0
6 11 5 10
4
9 3
op
0 1 2
3
4
5 6
7
8 9
(6/5) C:
pc
1 7 0
6 11 5 10
4
9 3
op
0 1 2
3
4
5 6
7
8 9
(6/5) C:
pc
1 7 0
6 11 5 10
4
9 3
10 11
8 2
10 11
8 2
10 11
8 2
surface
filled with
antipodal
contrasts
s
a coherent
design
that
can be
understood as
"nothing
but.
. .
cycles."74
The
association,
in
On
the
Antipodes,
of
cyclicpitch
deriva-
74Quoted
rom
the
last
line of
the text
(see
Ex.
17).
This
interpretation
s
adapted
from
Hitchcock
(Ives:
A
Survey
of
the
Music,
17-18)
and Schoffman
("The
Songs
of
Ives,"
233).
surface
filled with
antipodal
contrasts
s
a coherent
design
that
can be
understood as
"nothing
but.
. .
cycles."74
The
association,
in
On
the
Antipodes,
of
cyclicpitch
deriva-
74Quoted
rom
the
last
line of
the text
(see
Ex.
17).
This
interpretation
s
adapted
from
Hitchcock
(Ives:
A
Survey
of
the
Music,
17-18)
and Schoffman
("The
Songs
of
Ives,"
233).
surface
filled with
antipodal
contrasts
s
a coherent
design
that
can be
understood as
"nothing
but.
. .
cycles."74
The
association,
in
On
the
Antipodes,
of
cyclicpitch
deriva-
74Quoted
rom
the
last
line of
the text
(see
Ex.
17).
This
interpretation
s
adapted
from
Hitchcock
(Ives:
A
Survey
of
the
Music,
17-18)
and Schoffman
("The
Songs
of
Ives,"
233).
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 40/41
Interval
ycles
as
Compositional
esources
81
nterval
ycles
as
Compositional
esources
81
nterval
ycles
as
Compositional
esources
81
tions
with natureand
natural
processes
realizes
a
philosophical
stance
common
to much
of Ives'smusic
and
thought.
Themes
relatedto nature,includingspecificpictorial magesandrefer-
ences
to
abstractnatural
orces,
appear
with
familiar
egularity
in the
texts
and
concepts
of his
compositions
and in the
philo-
sophical
positions
of his
writings,
ultimately
connecting
to
a
line
of Transcendentalist
hinking
n which nature
reflectsa
di-
vine
presence.75
n the same
specific
way
that a work
such as
On
the
Antipodes
displays
an
underlying
unity
of
cyclic
pitch
con-
struction,
his view
recognizes
a
cyclic
unity
n nature
embodied
in
planetary
orbits
andthe
resulting
cyclic passage
of time
over
the courseof a dayor year, aswell as in the "lifecycles"of liv-
ing
organisms.
Ives meant
to
make his most
extensive
state-
ment
on such
natural
laws
in the Universe
Symphony,
which
would
"tracewith tonal
imprints
he ... evolution
of all
life,
in
nature
of
humanity
rom the
great
roots
of life to the
spiritual
eternities."76
The
cyclic
structures
notated
on the
sketch
page
of the
Symphony
shown
in
Example
11 above
wereintended
to
represent
the
"body
of the
earth,"
and
subsequent
musical
re-
alizations
of these
structures
would
then
depict
the
formation
of the "rocks,trees andmountains"out of the initialrawmate-
rials.77
As the
Symphony
evolves,
the seminal
function
of the
cyclicpitch
structures
parallels
a
cyclic
underpinning
or natural
laws.
Implicit
n this
perspective,
and
an
important
aspect
of Ives's
cosmic
musical
metaphor,
is
a
juxtaposition
of the
concepts
of
evolution
and
revolution,
for
the
concept
of a
cyclic
revolution
does not
necessarily imply
a
complete
and
literal
return
to
a
point
of
origin,
exclusive
of
any
evolution
or
growth.
Rather,
a
75Ives's
adherence
to
"Transcendentalist"
eliefs,
as
expressed
in his Es-
says
before
A
Sonata,
recognizes
the
importance
of
nature
along
with other
central
hemes,
though
these do not
necessarily
orm a
coherent
"philosophy."
See
Burkholder,
Charles
ves: TheIdeasBehind the
Music,
20-32.
76Kirkpatrick,
atalogue,
27.
7Memos,
107.
tions
with natureand
natural
processes
realizes
a
philosophical
stance
common
to much
of Ives'smusic
and
thought.
Themes
relatedto nature,includingspecificpictorial magesandrefer-
ences
to
abstractnatural
orces,
appear
with
familiar
egularity
in the
texts
and
concepts
of his
compositions
and in the
philo-
sophical
positions
of his
writings,
ultimately
connecting
to
a
line
of Transcendentalist
hinking
n which nature
reflectsa
di-
vine
presence.75
n the same
specific
way
that a work
such as
On
the
Antipodes
displays
an
underlying
unity
of
cyclic
pitch
con-
struction,
his view
recognizes
a
cyclic
unity
n nature
embodied
in
planetary
orbits
andthe
resulting
cyclic passage
of time
over
the courseof a dayor year, aswell as in the "lifecycles"of liv-
ing
organisms.
Ives meant
to
make his most
extensive
state-
ment
on such
natural
laws
in the Universe
Symphony,
which
would
"tracewith tonal
imprints
he ... evolution
of all
life,
in
nature
of
humanity
rom the
great
roots
of life to the
spiritual
eternities."76
The
cyclic
structures
notated
on the
sketch
page
of the
Symphony
shown
in
Example
11 above
wereintended
to
represent
the
"body
of the
earth,"
and
subsequent
musical
re-
alizations
of these
structures
would
then
depict
the
formation
of the "rocks,trees andmountains"out of the initialrawmate-
rials.77
As the
Symphony
evolves,
the seminal
function
of the
cyclicpitch
structures
parallels
a
cyclic
underpinning
or natural
laws.
Implicit
n this
perspective,
and
an
important
aspect
of Ives's
cosmic
musical
metaphor,
is
a
juxtaposition
of the
concepts
of
evolution
and
revolution,
for
the
concept
of a
cyclic
revolution
does not
necessarily imply
a
complete
and
literal
return
to
a
point
of
origin,
exclusive
of
any
evolution
or
growth.
Rather,
a
75Ives's
adherence
to
"Transcendentalist"
eliefs,
as
expressed
in his Es-
says
before
A
Sonata,
recognizes
the
importance
of
nature
along
with other
central
hemes,
though
these do not
necessarily
orm a
coherent
"philosophy."
See
Burkholder,
Charles
ves: TheIdeasBehind the
Music,
20-32.
76Kirkpatrick,
atalogue,
27.
7Memos,
107.
tions
with natureand
natural
processes
realizes
a
philosophical
stance
common
to much
of Ives'smusic
and
thought.
Themes
relatedto nature,includingspecificpictorial magesandrefer-
ences
to
abstractnatural
orces,
appear
with
familiar
egularity
in the
texts
and
concepts
of his
compositions
and in the
philo-
sophical
positions
of his
writings,
ultimately
connecting
to
a
line
of Transcendentalist
hinking
n which nature
reflectsa
di-
vine
presence.75
n the same
specific
way
that a work
such as
On
the
Antipodes
displays
an
underlying
unity
of
cyclic
pitch
con-
struction,
his view
recognizes
a
cyclic
unity
n nature
embodied
in
planetary
orbits
andthe
resulting
cyclic passage
of time
over
the courseof a dayor year, aswell as in the "lifecycles"of liv-
ing
organisms.
Ives meant
to
make his most
extensive
state-
ment
on such
natural
laws
in the Universe
Symphony,
which
would
"tracewith tonal
imprints
he ... evolution
of all
life,
in
nature
of
humanity
rom the
great
roots
of life to the
spiritual
eternities."76
The
cyclic
structures
notated
on the
sketch
page
of the
Symphony
shown
in
Example
11 above
wereintended
to
represent
the
"body
of the
earth,"
and
subsequent
musical
re-
alizations
of these
structures
would
then
depict
the
formation
of the "rocks,trees andmountains"out of the initialrawmate-
rials.77
As the
Symphony
evolves,
the seminal
function
of the
cyclicpitch
structures
parallels
a
cyclic
underpinning
or natural
laws.
Implicit
n this
perspective,
and
an
important
aspect
of Ives's
cosmic
musical
metaphor,
is
a
juxtaposition
of the
concepts
of
evolution
and
revolution,
for
the
concept
of a
cyclic
revolution
does not
necessarily imply
a
complete
and
literal
return
to
a
point
of
origin,
exclusive
of
any
evolution
or
growth.
Rather,
a
75Ives's
adherence
to
"Transcendentalist"
eliefs,
as
expressed
in his Es-
says
before
A
Sonata,
recognizes
the
importance
of
nature
along
with other
central
hemes,
though
these do not
necessarily
orm a
coherent
"philosophy."
See
Burkholder,
Charles
ves: TheIdeasBehind the
Music,
20-32.
76Kirkpatrick,
atalogue,
27.
7Memos,
107.
cyclically
conceived
structure
provides
an
underlying
cohesive
framework
withinwhich
nonrepetitive
elements
may grow
and
evolve, just as a time periodsuch as a daycanencompassvast
changes
within
its
cyclic
boundaries.78
These
principles
are
clearly displayed
n In re con
moto
et
al,
where
virtually
every
measure
participates
in a
pervasive
constructional
scheme
basedon the
Prime
Series
and articulated
with the Grit
Chord,
yet
the methods
of
projecting
these
unifying
hreads
are com-
plex
and diverse.
Nature,
according
o
Ives,
despite
its funda-
mental
cyclic
character,
"loves
analogy
and
hates
repetition,"
and
any
musical
reflection
of natural
processes
wouldtherefore
reconcilethe necessityof growthwith the universality f cyclic
return 79
Ultimately,
Ives's
cyclic pitch
derivations
reflect
principles
of
pitch
structure
that
are
richly
attractive
to
a
composer
searching
or nontonal methods
of
organization.
Extramusical
considerations,
such as
points
of action
provided
by
a
program
or
scenario,
might suggest
a structural
ramework
n the
ab-
sence
of
tonality,
but would not
provide
the
inherent
and
natu-
ral
logic
of the tonal
system.
With
cyclic
intervallic
repetitions
the techniques of pitch organization, including methods of
pitch-class
exhaustion,
are
given
a
logical
and natural
mpetus
from the
varioussubdivisionsof the octave
into
equal parts
and
the extension
of these
principles
to
produce
cyclic
combina-
78Audrey
Davidson makes
this
point
in
"Transcendental
Unity
in the
Worksof Charles
Ives,"
American
Quarterly
2
(Spring
1970),
35-44.
79CharlesE. Ives,
Essays before
A Sonata, TheMajority,and OtherWrit-
ings,
ed.
Howard
Boatwright
New
York:
Norton,
1970),
22.
"Repetition"
s
a
frequent
target
of criticism
n Ives's
writings,
especially
when
referring
o the
conventions
of
common-practice
onality.
In
his
essay
on
quarter-tone
har-
mony,
for
example,
he
criticizes
"the
drag
of
repetition
n
manyphases
of
art,"
finding
an
absence of an essential
"organic
low"
(Essays, 115).
The
repetitive
nature
of
many
of his
compositional
experiments,
however,
makes
t clearthat
he does
not
reject
all forms
of
repetition;
rather,
he
objects
to
easy
reliance
on
traditional
musical
materials-repetition
without
nspiration.
cyclically
conceived
structure
provides
an
underlying
cohesive
framework
withinwhich
nonrepetitive
elements
may grow
and
evolve, just as a time periodsuch as a daycanencompassvast
changes
within
its
cyclic
boundaries.78
These
principles
are
clearly displayed
n In re con
moto
et
al,
where
virtually
every
measure
participates
in a
pervasive
constructional
scheme
basedon the
Prime
Series
and articulated
with the Grit
Chord,
yet
the methods
of
projecting
these
unifying
hreads
are com-
plex
and diverse.
Nature,
according
o
Ives,
despite
its funda-
mental
cyclic
character,
"loves
analogy
and
hates
repetition,"
and
any
musical
reflection
of natural
processes
wouldtherefore
reconcilethe necessityof growthwith the universality f cyclic
return 79
Ultimately,
Ives's
cyclic pitch
derivations
reflect
principles
of
pitch
structure
that
are
richly
attractive
to
a
composer
searching
or nontonal methods
of
organization.
Extramusical
considerations,
such as
points
of action
provided
by
a
program
or
scenario,
might suggest
a structural
ramework
n the
ab-
sence
of
tonality,
but would not
provide
the
inherent
and
natu-
ral
logic
of the tonal
system.
With
cyclic
intervallic
repetitions
the techniques of pitch organization, including methods of
pitch-class
exhaustion,
are
given
a
logical
and natural
mpetus
from the
varioussubdivisionsof the octave
into
equal parts
and
the extension
of these
principles
to
produce
cyclic
combina-
78Audrey
Davidson makes
this
point
in
"Transcendental
Unity
in the
Worksof Charles
Ives,"
American
Quarterly
2
(Spring
1970),
35-44.
79CharlesE. Ives,
Essays before
A Sonata, TheMajority,and OtherWrit-
ings,
ed.
Howard
Boatwright
New
York:
Norton,
1970),
22.
"Repetition"
s
a
frequent
target
of criticism
n Ives's
writings,
especially
when
referring
o the
conventions
of
common-practice
onality.
In
his
essay
on
quarter-tone
har-
mony,
for
example,
he
criticizes
"the
drag
of
repetition
n
manyphases
of
art,"
finding
an
absence of an essential
"organic
low"
(Essays, 115).
The
repetitive
nature
of
many
of his
compositional
experiments,
however,
makes
t clearthat
he does
not
reject
all forms
of
repetition;
rather,
he
objects
to
easy
reliance
on
traditional
musical
materials-repetition
without
nspiration.
cyclically
conceived
structure
provides
an
underlying
cohesive
framework
withinwhich
nonrepetitive
elements
may grow
and
evolve, just as a time periodsuch as a daycanencompassvast
changes
within
its
cyclic
boundaries.78
These
principles
are
clearly displayed
n In re con
moto
et
al,
where
virtually
every
measure
participates
in a
pervasive
constructional
scheme
basedon the
Prime
Series
and articulated
with the Grit
Chord,
yet
the methods
of
projecting
these
unifying
hreads
are com-
plex
and diverse.
Nature,
according
o
Ives,
despite
its funda-
mental
cyclic
character,
"loves
analogy
and
hates
repetition,"
and
any
musical
reflection
of natural
processes
wouldtherefore
reconcilethe necessityof growthwith the universality f cyclic
return 79
Ultimately,
Ives's
cyclic pitch
derivations
reflect
principles
of
pitch
structure
that
are
richly
attractive
to
a
composer
searching
or nontonal methods
of
organization.
Extramusical
considerations,
such as
points
of action
provided
by
a
program
or
scenario,
might suggest
a structural
ramework
n the
ab-
sence
of
tonality,
but would not
provide
the
inherent
and
natu-
ral
logic
of the tonal
system.
With
cyclic
intervallic
repetitions
the techniques of pitch organization, including methods of
pitch-class
exhaustion,
are
given
a
logical
and natural
mpetus
from the
varioussubdivisionsof the octave
into
equal parts
and
the extension
of these
principles
to
produce
cyclic
combina-
78Audrey
Davidson makes
this
point
in
"Transcendental
Unity
in the
Worksof Charles
Ives,"
American
Quarterly
2
(Spring
1970),
35-44.
79CharlesE. Ives,
Essays before
A Sonata, TheMajority,and OtherWrit-
ings,
ed.
Howard
Boatwright
New
York:
Norton,
1970),
22.
"Repetition"
s
a
frequent
target
of criticism
n Ives's
writings,
especially
when
referring
o the
conventions
of
common-practice
onality.
In
his
essay
on
quarter-tone
har-
mony,
for
example,
he
criticizes
"the
drag
of
repetition
n
manyphases
of
art,"
finding
an
absence of an essential
"organic
low"
(Essays, 115).
The
repetitive
nature
of
many
of his
compositional
experiments,
however,
makes
t clearthat
he does
not
reject
all forms
of
repetition;
rather,
he
objects
to
easy
reliance
on
traditional
musical
materials-repetition
without
nspiration.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions
7/24/2019 Lambert, Interval Cycles, Spectrum
http://slidepdf.com/reader/full/lambert-interval-cycles-spectrum 41/41
82 Music
Theory Spectrum
2 Music
Theory Spectrum
2 Music
Theory Spectrum
tions.
Coupled
with
their
analogies
to forces of
nature,
the
cy-
clic
procedures provide
a
fertile area for
experimentation,
and
they
constitute a central
organizing principle
for some of Ives's
most
profound
musical
expressions.
ABSTRACT
In the
"experimental"
music of
Charles
Ives,
interval
patterns
and
particular
ntervallic
combinations often
serve as
primary
structural
forces.
Particularly revalent
are
cyclic
ntervallic
repetitions,
reflect-
tions.
Coupled
with
their
analogies
to forces of
nature,
the
cy-
clic
procedures provide
a
fertile area for
experimentation,
and
they
constitute a central
organizing principle
for some of Ives's
most
profound
musical
expressions.
ABSTRACT
In the
"experimental"
music of
Charles
Ives,
interval
patterns
and
particular
ntervallic
combinations often
serve as
primary
structural
forces.
Particularly revalent
are
cyclic
ntervallic
repetitions,
reflect-
tions.
Coupled
with
their
analogies
to forces of
nature,
the
cy-
clic
procedures provide
a
fertile area for
experimentation,
and
they
constitute a central
organizing principle
for some of Ives's
most
profound
musical
expressions.
ABSTRACT
In the
"experimental"
music of
Charles
Ives,
interval
patterns
and
particular
ntervallic
combinations often
serve as
primary
structural
forces.
Particularly revalent
are
cyclic
ntervallic
repetitions,
reflect-
ing
compositional
concerns for coherence
through repetition
and
pitch-class variety.
Ives
experiments
both with the familiar
nterval
cycles and with cycles of two alternating ntervals,or "combination
cycles."
Musical
usages
include
straightforward
yclic presentations
as
well
as
developments
of
cyclically generated
structural frame-
works.
In his
most
sophisticatedcyclicexperiments,including
In re
con moto et
al,
Onthe
Antipodes,
and the Universe
Symphony,cycles
and
cyclic
principles
are
central to
the
musicaland
metaphorical
ub-
stance,
mirroring
pervasive
elements
of
Ives's attitudes
toward art
and nature.
ing
compositional
concerns for coherence
through repetition
and
pitch-class variety.
Ives
experiments
both with the familiar
nterval
cycles and with cycles of two alternating ntervals,or "combination
cycles."
Musical
usages
include
straightforward
yclic presentations
as
well
as
developments
of
cyclically generated
structural frame-
works.
In his
most
sophisticatedcyclicexperiments,including
In re
con moto et
al,
Onthe
Antipodes,
and the Universe
Symphony,cycles
and
cyclic
principles
are
central to
the
musicaland
metaphorical
ub-
stance,
mirroring
pervasive
elements
of
Ives's attitudes
toward art
and nature.
ing
compositional
concerns for coherence
through repetition
and
pitch-class variety.
Ives
experiments
both with the familiar
nterval
cycles and with cycles of two alternating ntervals,or "combination
cycles."
Musical
usages
include
straightforward
yclic presentations
as
well
as
developments
of
cyclically generated
structural frame-
works.
In his
most
sophisticatedcyclicexperiments,including
In re
con moto et
al,
Onthe
Antipodes,
and the Universe
Symphony,cycles
and
cyclic
principles
are
central to
the
musicaland
metaphorical
ub-
stance,
mirroring
pervasive
elements
of
Ives's attitudes
toward art
and nature.
This content downloaded from 128.186.53.76 on Wed, 23 Sep 2015 16:49:40 UTCAll use subject to JSTOR Terms and Conditions