Symmetry as a Compositional Determinant 01

8/9/2019 Symmetry as a Compositional Determinant 01

1/43

Symmetry as a Compositional

Determinant

copyright Larry J. Solomon, 1973, revised 22

A description and analysis of the various types of symmetry that occur in music

!"stract

#re$ace

%. %ntrod&ction

%%. De$initions

%%%. 'e$lection%(. )ranslation and 'otation

(. !nalytical *ethods

(%. !nalysis o$ +nspecialied -ors

/ach0s %nvention o. 1

(%%. !nalysis o$ -ors ith %ntensive !pplications

/arto0sMusic for Strings, Percussion & Celesta, and -e"ern0s Variations for

Piano, 4p. 27

(%%%. 5&adrate )rans$ormations

%6. Some #sychological Considerations

6. /i"liography

!"stract

Symmetry is shown to be the major determining factor in composition. Some of the

compositional parameters that are demonstrated to be symmetry operations are: all

aspects of serialized composition, all contrapuntal operations (including imitation,canon, rounds, cancrizans, melodic inversion, invertible counterpoint, augmentation,

and diminution, and cantus firmus composition), all musical forms (including all

sectional, contrapuntal forms, and arch forms), isorhythm and isomelos, ostinati and

passacaglia, mirror chords, planing and fauxbourdon, vibrato, scale formation, invertible

counterpoint, meter and pulse, timbre, trills and other ornaments, lberti bass and other

accompaniment figurations, antiphony, the circle of fifths, and pitch itself.

!efinitions, descriptions, and mathematical formulations of the different types of

symmetry are provided, and each of the major types is explored with examples.

"arallels are shown in nature and other art forms. n analytical methodology is

developed, and specific wor#s are examined to demonstrate intensive applications.
http://solomonsmusic.net/diss.htm#Abstracthttp://solomonsmusic.net/diss.htm#Prefacehttp://solomonsmusic.net/diss1.htmhttp://solomonsmusic.net/diss2.htmhttp://solomonsmusic.net/diss3.htmhttp://solomonsmusic.net/diss4.htmhttp://solomonsmusic.net/diss5.htmhttp://solomonsmusic.net/bachin1.htmhttp://solomonsmusic.net/bachin1.htmhttp://solomonsmusic.net/diss7.htmhttp://solomonsmusic.net/diss8.htmhttp://solomonsmusic.net/diss9.htmhttp://solomonsmusic.net/diss10.htmhttp://solomonsmusic.net/dissbib.htmhttp://solomonsmusic.net/diss.htm#Abstracthttp://solomonsmusic.net/diss.htm#Abstracthttp://solomonsmusic.net/diss.htm#Prefacehttp://solomonsmusic.net/diss.htm#Prefacehttp://solomonsmusic.net/diss1.htmhttp://solomonsmusic.net/diss1.htmhttp://solomonsmusic.net/diss2.htmhttp://solomonsmusic.net/diss2.htmhttp://solomonsmusic.net/diss3.htmhttp://solomonsmusic.net/diss3.htmhttp://solomonsmusic.net/diss4.htmhttp://solomonsmusic.net/diss4.htmhttp://solomonsmusic.net/diss5.htmhttp://solomonsmusic.net/diss5.htmhttp://solomonsmusic.net/bachin1.htmhttp://solomonsmusic.net/bachin1.htmhttp://solomonsmusic.net/bachin1.htmhttp://solomonsmusic.net/diss7.htmhttp://solomonsmusic.net/diss7.htmhttp://solomonsmusic.net/diss8.htmhttp://solomonsmusic.net/diss8.htmhttp://solomonsmusic.net/diss9.htmhttp://solomonsmusic.net/diss9.htmhttp://solomonsmusic.net/diss10.htmhttp://solomonsmusic.net/dissbib.htm


2/43

$hese include %arto#&sMusic for Strings, Percussion, and Celestaand 'ebern&s

Variations for Piano, p. *.

Some new transformations are developed, called the +uadrate transformations, which

are - degree rotations of a basic set, exchanging the time and pitch dimensions. n

essay on these also appeared inPerspectives of New Music, */, under the title "NewSymmetric Transformations". 0hapter 11 (on !efinitions of musical symmetry) was also

published in theJournal of Transfigural Mathematics(%erlin) 2ol. /3/ (*45).

chapter is also devoted to the possible psychological effects of musical symmetry.

Preface

$he first edition was originally published as a dissertation for completion of a "h! in

music at 'est 2irginia 6niversity in */. $he original title wasSymmetry as aDeterminant of Musical Composition. $his new edition preserves the original ideas and

adds some new materials and discoveries. $heNew Transformations names have been

transposed so that the +uadrate "rime (+") is the form that reflects around an ascending

diagonal, thereby switching time and pitch axes. $his ma#es the +uadrates more

consistent with their own transformations and with the "rime form. 7athematical

descriptions of these forms have been added.

Chapter I. Introduction

$o understand the very nature of creation one must ac#nowledge that there was no light before the 8ord said: 98et there be 8ight9. nd since

there was not yet light, the 8ord&s omniscience embraced a vision of it which only is omnipotence could call forth.... creator has a visionof something which has not existed before this vision. nd a creator has the power to bring this vision to life. . . .;rnold Schoenberg,90omposition with $welve $ones9 from Style and Idea or, if Eim li#es Eane, it does

not necessarily follow that Eane li#es Eim. Similarly, if Eim is a 0hristian, it does not

necessarily follow that a 0hristian is Eim.

mathematical operation or transformation that results in the same figure as the

original or its mirror image is called a symmetry operation. $hese operations includereflection, rotation, and translation, which will be defined. $he set of all operations that

leave a figure unchanged is called the symmetry groupfor that figure.

$he purpose here is to demonstrate that symmetry is a general determinant of musical

composition> a composition being an arrangement of parts to form a whole. 7usical

symmetry will be mathematically defined as a congruence that results from the

operations of reflection, rotation, or translation. $hese may be applied to any parameter

of a musical composition. Stated in another way, the purpose is to demonstrate how

postoperative congruences define the arrangement and combination of the parts of

musical wor#s.

Symmetry can be recognized in the human body, in snowfla#es, in beehives, and in

geometric figures. 1t is an object of scientific study in botany and crystallography. Such

study has led to a complex but important system of classifying and relating organisms

and structures. ow does a flower, a seashell, or a millipede relate to music, if at allA

ow diverse, related, and pervasive are manifestations of symmetry in existing musicA

0an they be defined and classifiedA 1s there a method or procedure for recognizing this

type of organizationA 'hat may be the psychological effects or reasons for employing

symmetry in a compositionA $he answers to these and other =uestions are posed herein.

$he word 9symmetry9 has a common root of origin with 9syndicate>4namely the Free#prefixsyn', meaning together, or fromsym#i#tein, to fall together. ne of the first

Dnglish uses of the word in print occurs in a boo# on architecture by Shute4in @B/:

90oncerning ye proportion and simetry to vse the accustomed terme of the arts of the

forenamed columbes. ;7urray, E., /BB


4/43

0urrently, the term 9symmetry9 is used in all of the sciences and arts. 1t has similar

meanings in geometry and biological science, namely the arrangement of pairs of parts

which when joined by a line can be bisected by a line or a

?igure . drawing by 8eonardo da 2inci showing symmetry in the human body

plane. $his type is called ilateral symmetryand is evidenced in the human form.

$he same type of symmetry is found in the morphology of many animals and plants,

both living and extinct.

?igure . %ilateral symmetry in an oa# lea# and an extinct trilobite

Hadial symmetry also occurs in nature as a form whose features are e=uidistant from a

point, and thereby can be rotated.

?igure /. Hadial symmetry in the pine cone and flower


5/43

nother type of symmetry occurs as reiterating parts as those in fern leaves and

beehives.

?igure G. $ranslational symmetry in leaf arrangement, the millipeded, and a beehive

$hese forms do not have axes but are translational along a line.

Symmetry occurs not only in the direct creations of nature, but in man&s own. 1t is

especially apparent in the visual arts, including architecture. ?rom theParthenon of

1ctinus to Haphael&s School of (thens to 8e 0orbusier&s)oncham# Church, symmetry is

a strong organizational feature and has been the object of considerable study in arthistory. ermann 'eyl&s boo# on Symmetrydescribes its various manifestations in art,

mathematics, and in nature. 'eyl&s boo# serves as a model for the extensions into music

that are the subject of this study.

1n music, the word 9symmetry9 is most often used in its application to phrases and the

dimensions of musical forms. lthough ugo 8eichtentritt does not define the term, he

implies that e=ual lengths are symmetrical ;8eichtentritt, @


6/43

I / I /. Simple triple is a symmetrical meter because it is not divisible. symmetrical

meters would include @35 (divided into / I , or I /), *35, etc.

"aul ?ontaine offers a very different use of the word: 9$he word symmetric is arbitrarily

used to describe a phrase e=ual in length to an even number of measures. $he reference

is to length only and not to melodic balances or imbalances within a phrase.9 ;?ontaine, * this points out their fundamental

relationship.

$here may be occurences of - degree rotations in music which result in an axis of four4

fold symmetry. 0onsider the following visual form:

?igure . visual form having - degree, four fold symmetry

1f we revolve this about the point -, the figure goes into itself four times, once every -degrees of rotation. $he transformations may be defined as follows for -, 5- and *-

degree positions respectively.

T 2/,y3 4 2y '/3

T2/,y3 4 2'/,'y3

T 2/,y3 4 2'y,/3

$he use of this operation upon the variables of pitch and time results in the mutual

exchange of their functions. $his creates new transformations which are reserved for

discussion in a later chapter.


14/43

7any other types of rotational symmetry occur in nature but seem to have little

application in music. Qotable among these are the hexagonal six fold symmetry of

snowfla#es, flowers, and the cells of beehives. 1n three dimensions this principle is

stri#ingly represented in the science of crystallography where complex systems of

nomenclature and classification have been developed for various levels and

manifestations of symmetry in natural crystals.

nother symmetry operation is exemplified by what some artists call 9infinite rapport9

and what will here be called translation. $he following ornamental figure will serve as

an example:

?igure -. $ranslational symmetry in an ornament

RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR

RRRRRRRRRRR

$here is a pattern in this which is repeated at a regular spatial rhythm. Similar

manifestations which are common in nature are called metamerisms by zoologists. $he

legs of a centipede or the leaflets of a fern are examples. f course the pattern does not

continue infinitely in its physical manifestation in any of these examples, but one may

call them potentially infinite. n exact image may be moved linearly upon the original

and result in congruences at regular intervals. $his operation may be precisely defined.

T2/,y3 4 2/7na,y) for horizontal translation

T2/,y34 2/,y7na3 for vertical translation

Since the operations are iterative, i.e., an operation performed upon itself yields a

similar expression, they may be expressed:

T 2/, y3 4 2/7a, y3

T 2/, y 3 4 2/, y 7 a3

$ranslational symmetry is often combined with reflective symmetry in music.

Dxample 5. ?igures with translational and reflective symmetry

a. b.


15/43

%oth examples 5a and 5b exhibit translational symmetry at an interval of four eighths,

i.e., T2/,y34 2/78,y3. 'hileexample 5b has reflective symmetry as shown by the axes,

example 5a does not.

f course, it is possible to employ the translation operation in the tonal dimension as

well as the temporal dimension. Such translations are normally combined, i.e.,bothtonal and temporal translation simultaneously, as in the following.

Dxample . $ranslational symmetry in 0hopin&stude, p. -, Qo.

$he combined iterative transformation may be expressed:

T2/,y34 2/7a,y7!3

Spatial dilation or contraction is a special type of symmetry transformation which will

here be called an automor#hism. $wo photographs of the same image in different sized

prints have a point for point correspondence. 1n geometry they would be called similar

figures. musical idea maintains a similarity if it undergoes a temporal or tonal

automorphism. ugmentation and diminution are traditional examples of this operation.

Helativity of size is a well #nown phenomenon in the art of perspective and is a

phenomenon of our perception. ;Fregory, G*4B/


16/43

$his figure shows the vertical4horizontal orientation of growth in maple leaves. Qotice

the difference in the size of the leaves a and d, but they are similar in shape. $his

illustrates automorphism across a horizontal axis. 1f leaf a is made larger by the

proper amount, or leaf d smaller, their symmetry can be shown by reflection. 8eaves

b and c grow larger on the bottom than on the top half, and their symmetry can be

shown after a similar automorphism across a horizontal axis. 1n fact, symmetry can be

demonstrated for the entire figure if an automorphism is performed on one side of the

horizontal axis. 1f any point above the axis,/0,y0, is expanded by the proper multiplier,

/0,y1will result. $he transformation may be generalized to:

T2/,y342/,c0y3where y N -, (/,c1y) where y OM -> c0,c19:

1n our example, c14 y0;y1. $he transformation can also apply to x, a horizontal dilation.

T 2/, y3 4 2c0/,y3 where /


17/43

Symmetry is then demonstrated by reflection about the axis. 1f yMpitch, the following is

also illustrative:

Dxample . Heflective symmetry after a vertical dilation.

Dxpanding the intervals after the axis by a factor of will yield symmetry by reflection

again.

utomorphisms are not enough by themselves to confirm symmetry. $hey must be

combined with some previously described operation, such as reflection or translation.

utomorphisms are, therefore, called auxiliary transformations. $he operation is

normally carried out before an accompanying primary transformation, i.e., those

previously described.

stri#ing non4linear type of automorphism occurs in the form of shells of the

chambered nautilus and other animals and plants which can leave a record of their

growth patterns.

?igure . !iagram of the structure of the nautilus. shell.

$he symmetry here is rotational combined with an automorphism. Dach chamber is an

exact copy of another except for size. Similarly, in music, such dilations exist in the

e=ual tempered scale since the semitone becomes increasingly expanded with respect to

fre=uency difference the higher we go in pitch.

?igure /. utomorphic rotation in the e=ual tempered scale.


18/43

ll of these symmetry operations, then, can be combined to form our general definition

of symmetry. $he reflection and 5- degree rotation are both of the form:

T 2/, y 3 4 27'/, 7'y 3

1n order to include automorphisms on either side of the vertical axis we need:

2c0/, c1y3 where /9:, y9:

T2/,y34 c0, c1, c>, c89:

2c>/, c8y3 where /9:, y b,b,b/,bGO-

T 2/, y 3 4 (b/x, c/y) where x, y NM - > c,c,c/,cGO-

(bGx, cGy) where y NM - N x

$he primary transformations may be generalized to include the - degree rotations.

S02/, y3 4 27'/7al, 7'y7a13


19/43

S12/, y 3 4 27'y7a0, 7'/7a13

Symmetry e/ists, therefore, if a congruence results after at least one of the #rimary

o#erations or after a com!ination of au/iliary and #rimary o#erations.

%y this definition, the following figure can be shown to possess svmmetry.

?igure G. figure having some symmetry.

?irst, the automorphism transformation is applied to the four =uadrants independently to

give:

?igure @. n automorphism of figure G.

S0 is then applied to reflect this about an axis, yielding a congruence. lthough figure lG

is symmetric, its order of symmetry is low.

The order of symmetry is #ro#ortional to the num!er of #rimary transformations that

can result in a congruence and is less than this amount !y an increment for each #artial

transformation necessary for congruence.

$he automorphism performed above is an example of such a partial transformation. 1n a

musical %, the order of events in the da ca#o section is normally not the reverse of

the first section. 1n this respect it is similar to the symmetry of the word !! as distinct

from that of 77. Qotice that 77 possesses point for point symmetry after the

operation of reflection about an axis of vertical bisection, but !! does not. lthough

!! is less symmetric than 77, it is more symmetric than 9fly,9 because a partial

transformation can be performed on !! before it may be reflected, namely a 5-degree rotation of one of the !&s.


20/43

$he word 77 has a symmetry of order one, whereas !! is less than one. $he word

9fly9 has a symmetry order of zero. 1f no partial transformation is performed on !!,

one out of three of its letters will be congruent upon reflection. 1ts degree of symmetry

is said to be // per cent, or just //. The degree of symmetry is determined !y the

#ro#ortion of congruent #oints or #arts after an o#eration. $herefore, after both the

partial and reflective transformations of !! previously mentioned, the degree ofsymmetry is --. $he tabulation of !!&s symmetry may be represented:

"DH$1QS H!DH !DFHDD

Heflection //

"artial rotation and reflection N --

S!, however, has only a symmetry order , degree //.

1t should be noted that !! is here examined for point for point symmetry, but if !!

is regarded as a simple three element figure in which the letters are construed as units

($he shapes of the letters become inconse=uential) its symmetry may be generalized as

identical to that of 77, i.e., order , degree --. "oint for point analysis is generally

more thorough, however, and is preferred when possible. 0onsider the following:

Dxample . Symmetry types in music

a. # < f 9 # b. ## < ff 9 #

Dxamplea and d are symmetric by order , degree --. Dxample b is symmetric

after a partial temporal automorphism across theff axis> the order is N, degree --.

Dxample c must have two automorphisms, one of pitch and the other of time, to yielda lesser order of symmetry than b, designated NN, --. $hese may be tabulated as

follows:

"8 may be reflected about either of the letters or to yield symmetry degree @.

Qote that the degree here is not @-, because separate operations must be carried out for

each letter. 1 may be reflected about a vertical bisector to yield order , degree @-,

or it may&be reflected about a horizontal bisector to yield order , degree --. $hen two

or more order designations are #ossi!le, the one with the highest degree is assigned.

$o clarify the preceding exposition the reader may wish to determine the order and

degree of symmetry in the following, point for point:


21/43

. $H"

. SS

/.

G. "H

@.

B. % 0

*. # < ff 9 ##

5.

nswers:

H!DH !DFHDD "DH$1QS


22/43

. @ 2ertical reflection about $ or

. -- Hotation, 5- degrees

/. / -- 2ertical and horizontal reflection, 5- degree rotation.

G. - -

@. *@ $ranslation

B. @- 2ertical reflection*. N -- "artial dynamic automorphism and temporal reflection.

5. N -- "artial automorphism across second bar and temporal

reflection.

. -- $onal and temporal reflection

-. -- $ranslation and $emporal reflection.

Some distinction exists between real and tonal congruences, referring to real and tonal

answers in fugue, se=uences, and generally in diatonic compensations. congruence is

to be considered real here if the intervallic correspondences are identical and tonal if

changes are made to conform to a superceding diatonic or similar framewor#. 1n either

case, the congruences are strict if there is no departure outside of tonal variation. 1f, forexample, a canonic voice is in tonal imitation rather than real imitation, it is still

considered in strict congruence. 1f, however, the canonic voice even briefly brea#s the

pattern of imitation, the congruence is no longer strict. 1n this sense, 9strictness9

corresponds to symmetric degree.

Chapter %%%. 'e$lection

$his chapter is devoted to demonstrating how reflective symmetry is variously

manifested in musical composition and how these manifestations fall into similar and

dissimilar classes. 'e have seen that the reflection operation involves the turning of a

figure about a linear axis which will result in a congruence. 7athematically, the

operation has been defined as:

T2/,y34 2'/,y3or

T2/,y34 2/,'y3

Tonal Reflection in the Elements of Music

tonal axis in music, or more specifically an axis of pitch class, may or may not have

temporal dimensions. 1t is, therefore, appropriate to classify these accordingly. Qon4

temporal tonal symmetry is especially applicable to unordered pitch class sets, e.g.,

chords and scales and other elements of music. $hese are notably independent.

97irror chords9 occur in both modern and traditional styles. $he ordinary diminished

triad can be reflected about its central tone to show its symmetry, as in example /a.

Dxample /. Some common chords showing reflective and nonreflective properties


23/43


24/43

$he operation of involution is our reflection operation. 1t is being used by anson to

create new sonorities for composition which are related to the original sonority in a

logical way. 1t should be noted, however, that a major chord is not 9symmetric to a

minor chord9> i.e., a major chord is simply not symmetric. $he major and minor chords

are the only asymmetric triads of the standard four. $he other two, the diminished and

augmented are built upon single intervals, and are, therefore, tonally symmetric in eithersimple or compound forms. $he major and minor chords, however, contain mixed

intervals and are not symmetric in themselves, but they are symmetric as a combination.

Dxample B. $onal symmetry 9 resulting from combinations of major and minor chords.

1t can be shown that anson&s isometric and enharmonic involutions can result only

from operations upon tonally symmetric structures, the enharmonic involutions being

simply special cases of isometric forms. $he result after the operation will also

necessarily be symmetric. $he involution operation can be considered a reversal of

interval direction. $herefore, if the interval content of the original set is symmetric

about a tonal axis, a reversal of direction (high4low) will result in a congruence having

the 9same #ind of sound9 or even the same tones. pitch class set which is asymmetric,

however, will not result in a congruence after the same operation since reversal will

change the distribution of intervals.

n automorphism may result from an expansion of the intervals in a set on either side

of the tonal axis .

Dxample *. utomorphisms in tonally reflective structures.

1n these examples, the intervals on one side of the axis tone may be expanded or

contracted by an amount that will ma#e them e=uivalent to those on the opposite side.

'ith the reflective operation, then, a congruence results. Such a set is used in %arto#&s

ourth String *uartet, both in an ordered and unordered way. $his set is properly

divided into two parts labeled x and y.

Dxample 5. Symmetric pitch class sets from %arto#&sourth String *uartet(after "erle and others:Feorge "erle, 9Symmetric ?ormations in the String +uartets of %ela %arto#,9Music )e6iew, L21 (@@).


25/43

Score: 0opyright / by %oosey and aw#es, 1nc.

$he y set is an automorphism of x, and both are symmetric in themselves or in

combination, i.e., the total set, . $hese sets are transposable.

Scales are also unordered pitch class sets which may have symmetric tonal content.

Dssential to a scale is the order of tones and the scale&s intervallic content. Scale does

not necessarily carry the implication of tonality either, e. g., the chromatic scale and the

whole tone scale, although some do> e. g., the 0 major scale. tonal center may serve as

an axis of reflective symmetry, an in the !orian mode, which has an axis of the pitch

class !.

Dxample . $onal reflection in the !orian mode.


26/43

$he 9alternating9 scale, a synthetic scale, has similar properties, but it has no axis as

part of the scale.

Dxample -. $onal reflection in the 9alternating9 octatonic scale, with an axis outside of the scale.

$he axis shown above is only one of the many possible in this scale> that is, an axis may

be placed between any two notes of the scale, and it will retain symmetry. $he choice of

a tonal center here is not a re=uisite. scale such as this also possesses a high degree of

translational symmetry. $he intervallic order repeats after every two tones. Dvery scale

which repeats its intervallic order will have some degree of translational symmetry.

"olymodality can lead to interesting symmetric forms. lthough 0 major is not

reflective in itself, if it is combined with "hrygian on 0, the result is symmetric. 8ydian

may be combined with 8ocrian and eolian with 7ixolydian to obtain similar results. 1t

is inconse=uential that the finals be held in common but are shown as follows for

simplicity.

Dxample . 0ombinations of modes resulting in symmetric pitch class sets.


27/43

Tonal Reflection in Composition

0anons in contrary motion exhibit tonal reflection in the highest degree. Dxamples are

numerous, but to name a few, they occur in %ach&sMusical ?ffering,the @old!erg

Variationsand the(rt of ugue. Some of the greatest tour de forcesutilizing symmetric

operations are found in the wor#s of E.S. %ach.

%ach was fond of generating entire wor#s through the use of these operations. $he

following is a spectacular example.

Dxample . 9$rias armonica9 in E.S. %ach&s handwriting, an eight part puzzle canon in contrary motion.

$his eight part canon in contrary motion is notated in its most concise form in %ach&s

script, shown in the above example, and it contains all the necessary information, with

its description, for its realization.

Dxample /. realization of 9$rias armonica9 by symmetric operations.


28/43

$he axis of tonal reflection here is D3Db, outside of the scale, dividing the tonal motion

between the two choirs. $ranslation is also performed at a time interval of two =uarters

within each choir and at an interval of one =uarter at the fifth between choirs. nother

example is a canon from The Musical ?ffering, notated and realized as follows:

Dxample G. "uzzle canon as notated in E.S. %ach&s handwriting, from The Musical ?ffering,and itsrealization (in part). Qote the upside down treble clef.


29/43

Qormally, this type of canon is written out fully, but the above examples illustrate that

the full notation is not essential. 7ost of these shorthand notations directing theperformer to carry out symmetric operations have become obscure, but a few remain

with us. repeat sign, for instance, eliminates the need for notating large sections of

music repeatedly, and it directs the performer to carry out a large scale temporal

translation.

!allapicolla&s *uaderno di (nnali!eracontains a fully notated canon in contrary

motion in the twelve tone style. ne of the most straightforward uses of contrary motion

as tonal reflection is the concluding statement of the first movement of %arto#&s Music

for Strings, Percussion and Celesta.

Dxample @. $onal reflection in the concluding statement of the first movement of %arto#&sMusic for

Strings, Percussion and Celesta. 0opyright / by %oosey and aw#es.


30/43

$he following are from %arto#&sifth String *uartet.

Dxample B. Dxcerpts from %arto#&sifth String *uartetshowing their axes of tonal reflection. 0opyright/ by %oosey and aw#es, 1nc.

a. $hird movement, measure G. b. ?irst movement,

measure *.

$hese examples are limited in that the inversions are stated simultaneously. 7elodicinversions of any transposition and temporal placement will contain an axis of tonal

symmetry.

Dxample *. $emporally displaced inversion, showing tonal reflection, from %arto#&sourth String*uartet. 0opyright / by %oosey and aw#es, 1nc.

ll of these examples are restricted in being of fixed registration. 1t is entirely possible

to mirror register changes, too. $herefore,it is necessary to ma#e some distinction

between the mirroring of pitches and that of pitch classes.


31/43


32/43

Hudolph Heti has abstracted large scale #ey changes and relates them to motivic cells.

is analysis of the #ey structure in the movements of %eethoven&s 9"atheti=ue9 sonata is

apropos..

Dxample /-. $onal reflection in the #ey structure of 9parallel9 movements of %eethoven&s 9"atheti=ue9sonata, after Hudolph Heti, Thematic Patterns in Sonatas of Aeetho6en(Q..: 7ac7illan, B*),B.

Spatial Reflection

$he various forms of symmetry in music may be classified according to their type of

axis and operation. xes ordinarily occur in time, space, or pitch> they may occur in

other dimensions, but the three most common classes of reflective symmetry, defined by

their axes, are temporal, tonal (pitch), and spatial. $he first two of these have been most

important compositionally.


33/43

Spatial reflection was exploited early in the antiphonal tradition of Syriac 0hant. $his

too# the form of groups of singers in opposite locations. $he reflection was temporal as

they responded antiphonally to one another. mbrose apparently developed a special

method of composition for this spatial music. $he clear division in space is often held as

essential to antiphonal chant.

$he 2enetian polychoral school later continued this tradition. 'illaert and the Fabrielis

developed compositional procedures utilizing cori s#eBBati.$he principles of imitative

counterpoint were abstracted to spatial imitation employing the translation operation.

$his was continued into the %aro=ue with the development of the concerto, but the

spatial dimension then became less significant.

7any contemporary composers have become interested in spatial composition,

foreshadowed by such wor#s as 2arese&s&esertsand 1ves&s The nanswered *uestion.

enry %rant and Carlheinz Stoc#hausen have tried to formulate compositional methods

for the use of space. 7any of these employ symmetric divisions.

Dxample /. Spatial plans by contemporary composers having reflective symmetry> a. placement ofperformers in relationship to an audience by enry %rant,9Space as an Dssential spect of 7usical0omposition,9 in Contem#orary Com#osers on Contem#orary Music (Qew or#: olt, Hinehart, and'inston,B*),> b. placement of sound sources to correlate with binaural perception by CarlheinzStoc#hausen, 97usic in Space,9 in&ie )eihe, 2 (@), **.

Space may also be used to divide music which is polytonal into distinct tonal choirs.

Such a division is used in 1ves&s 9"utnam&s 0amp9 from Three #laces in New ngland,

as well as in The nanswered *uestionand other of his orchestral wor#s. section in

9"utnam&s 0amp9 alludes to 1ves&s childhood recollection of the marching bands of localrival football teams, each playing their own march and trying to outplay the other as

they converged on the village green. $he two bands were playing in different #eys,

speeds and rhythms. Space is used in 9"utnam&s 0amp9 to separate the simultaneous

tonalities of reflected timbres.

$he recent interest in the compositional resources of timbre have led to its alliance with

spatial divisions and symmetry. Simple examples of this are found early in the history of

music, e.g., in responsorial chant. 1n the nineteenth century %erlioz employed spatial

divisions of timbre in his He=uiem and other wor#s, and more recent examples are

found in 2aughan 'illiams&santasia on a Theme !y Thomas Tallisand %arto#&sMusic

for Strings, Percussion and Celesta, the last of which is discussed in detail in a laterchapter.


34/43

Temporal Reflection

$emporal organization is critical for music, and the principles of temporal organization

have been highly developed. Heflective relationships are among the most common.0onsider the movement of tones in time.

Dxample /. Heflective patterns in time: a. from Stravins#y&sPetroushDa, second tableau> b. from0hopin&stude, p. -, Qo. .

$he second halves of /a and /b are retrogrades of their first halves. Qote, however,

that non4pitch symmetry is preserved after the operation of reflection of /a only with

respect to the order of notes, rather than in real time.

$he order of the tones is the principle aspect of symmetry here. 7any similar examples

may be found throughout the literature. ne of the most dramatic occurs at the climax

of the third movement of %arto#&sMusic for Strings, Percussion and Celestaacross bar

G5.

Dxample //. $emporal reflection at the climax of %arto#&sMusic for Strings, Percussion and Celesta,third movement. 0opyright / by %oosey and aw#es.

$his bar also acts as an axis of symmetry on a large scale of the four movement

structure. 0ompared to a simple %, the order of symmetry in the above retrograde is

higher, namely order . $he degree of symmetry, however, is the same in both, --.


35/43

istorically, the use of cancrizans, or retrograde, is usually connected with medieval3

early renaissance (Gth4@th centuries) styles or with modern music, and almost

exclusively so. owever, this is somewhat of a misconception since this operation was

used throughout the history of music. $o illustrate, let us begin with 7achaut&sMa fin

est mon commencement et mon commencement ma fin.$he upper two voices are in

retrograded invertible counterpoint with an axis of reflection at bar , and thecountertenor is reversed from its midpoint at the same bar> these reversals are meant to

enhance the meaning of the text. $he degree of symmetry for the entire piece is the

highest possible, --, but the order is less than one due to the necessary partial

reflection of the upper two voices after the complete reflection about bar .

Dxample /G. $emporal reflection in 7achaut&sMa fin est mon commencement


36/43


37/43

$he precompositional cancrizans is probably most responsible for the severe restrictions

on harmonic and rhythmic motion. $he lines do not change register after the axis, and,

therefore, preserve the position of the bass line. %oth this example and the 7achaut

exhibit harmonic retrograde, i.e., the cancrizans is effected in the harmonic progression

as well as in the individual lines.

Dxample /@. harmonic reduction of %yrd&s&iliges &ominum, central portion, showing temporalreflection across the central axis.

retrograde canon occurs in E.S. %ach&s The Musical ?ffering, again using a condensed

form of notation which re=uires one of the performers to flip the page and read

bac#wards, a reflection operation. Qote the bac#ward facing symbols at the end.

Dxample /B. $he original notation of %ach&s retrograde canon from The Musical ?ffering,re=uiring aperformer to carry out a time reflection.

?rom the classical era, in aydn&s Sonata No. 8for 2iolin and "iano, theMenuetto at

ro6escio, the entire movement can be played forward or bac#ward with e=ual results.

$his is due to the cancrizans& structure, and it is particularly interesting because such

devices are not normally associated with the style.

Dxample /*. $emporal reflection in aydn&sMenuetto al ro6escio, from the Sonata No. 8 for Violin andPiano: piano part only, showing two parts which are mirrored. xis is measure .


38/43


39/43

Hetrograde motion is common in twelve tone music, both in terms of row variation and

in compositional construction. owever, even before the development of the twelve

tone method, Schoenberg used symmetrical constructs compositionally. ne of the most

stri#ing examples is the four part double canon fromPierre +unaire, Qo. 5. $he canon

is reversed, a cancrizans, from the tenth measure. $he complexity of this piece is

compounded by the piano part (not a part of the canon).

$he classic examples of symmetry in the serial style are found in the late wor#s of

'ebern.

Dxample /. Heflective orderings of pitch class sets in two of 'ebern&s rows as analyzed by "ierre%oulez,AouleB on Music Today(0ambridge, 7ass: arvard,*),*.

Heflective tonal landmar#s are found in most wor#s to some degree. ny wor# which

starts and ends in the same #ey has some symmetry, but not all do so. ?ranz 8iszt&s

ungarian )ha#sodies are asymmetric in this respect. 1t is remar#able, however, that so

many wor#s in the literature have this type of symmetry in common. Some will even

retrace the intermediate steps of modulation, for example:


40/43

1n this case, the #eys are arranged in retrograde.

musical % is symmetrical in the same way as a cancrizans. $he former, however, is

a more general construct. Simple ternary forms are common in music. nly a few types

need be mentioned here: the da ca#osong form, minuet and trio, scherzo, and certain

sonata4allegros. 0hopin&sNocturnesare fre=uently in simple ternary. ther forms mayshow several levels of reciprocity:

a. $he simple rondo: % %

Dxample: %eethoven&s)ondos

b. $he compound rondo: % %

aba aba aba

Dxample: 0lementi, piano Sonata No. E, last movement.

c. 0ompound ternary: %

aba cdc aba

or c

Dxample: %rahms, Sym#hony No. >, third movement.

d. Sonata rondo: % 0 %

#eys: 1 2 1 H 1 1 1

Dxample: 7ozart&s Trio in A!, C. @-, last mvmt.

ll of the above forms are examples of symmetrical forms, synonymous with sectional

retrogrades. nother of this type is % 0 % , not uncommon, which may be found in

%rahm&s Hhapsody in Db, p. , Qo. G and 0hopin&s 7azur#a, p. @B, Qo. . Slightly

more complex is the 9arch9 found in 0hopin&s 'altz, p. /G, Qo. : % 0 ! 0 % .

Hobert Schumann analyzed %erlioz&santastic Sym#hony, first movement, as an arch,

and Ddward $. 0one has later expanded upon this.

Dxample G-. Ddward $. 0one&s expanded analysis of Hobert Schumann&s on %erlioz&santasticSym#hony,first movement, from: Ddward $. 0one, 9Schumann mplified,9 in %erlioz:antastic

Sym#hony(Qew or#: Qorton, *), @.


41/43

)heme !

pisode 8 8 8 8 8 8 8 8 8 8 8 8 )ransition

)heme / :Development o$ )heme !;/< 8 8 8 8 8 8 8 8 Dev. and 'ecap o$ )heme /

)ransition ith Cadential #hrase 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 'ecap ; =Dev o$ )h. !>

o$ Cad. #hr.

)heme ! 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 88 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 'ecap o$ )h. !

%ntrod&ction 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Coda

mea: 72 111 150 168 200

234 280 313 331 360 412 477

E.S. %ach used large scale reflective symmetry to organize many of his multi4movement

wor#s.

Dxample G. $he order of pieces in E.S. %ach&s The Musical ?ffering, showing temporal reflection.

ricercar ? canons trio sonata ? canons ricercar

@@@@@@@@@@@@@@@@@@@@@@@

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

$he movements of %ach&s cantatas are fre=uently arranged in a similar order.

Dxample G. Heflective arrangement of the movements of %ach&s Christ lag in Todes!anden, adaptedfrom Ferhard ertz, ea., %ach, Cantata No. 8(Qew or#: Qorton, B*),5@.

Sin$onia % %% %%% %( ( (% (%%I versus chorus duet solo =uartet solo duet chorus

c.f.: S,,$,% S, $ S,,$,% % S,$ S,,$,%

orch.: full cor.trb cont,2l.4 cont. 2l,2las, cont. cont. full

KJJJJJJJJJJJJJJJJJJK

KJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJK

KJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJK

ther cantatas of similar structure are Qos. @B, *5, -B, and G-, and in other of %ach&s

wor#s: the canonic variations on Von immel hochfor organ and the motetJesu, Meine

reude.

1n indemith&s one act operain und FurucD($here and %ac#) the action proceeds to

the midpoint, where the jealous husband shoots his wife, and then all goes in reverse,

not in perfect cancrizans, but, more or less section by section. $he music, too, follows

this plan. $heuga tertiain ? of indemith&s+udus Tonalisis a cancrizans with an axis

in measure /. $hePraeludiumandPostludiumof this wor# are retrograde inversions

of one another, and, on a large scale, the following plan is revealed.

Dxample G/. diagram of the order of the parts of indemith&s 8udus $onalis showing the overallreflective pattern> from: 1an Cemp,indemith (8ondon: xford, *-),G.

?ugue B (arch form) ?ugue * (arch form)?ugue @ (a a b form) ?ugue 5 (a a b form)

?ugue G (double fugue) ?ugue (1IH)?ugue / (repeat by retrograde) ?ugue - (repeat 1)

?ugue (Stretto fugue) ?ugue ll(canon)


42/43

?ugue 1 ($riple fugue) ?ugue (binary)"raeludium "ostludium

$hat these operations are still being used by contemporary composers is evidenced in

the wor#s of such widely different composers as Eohn 0age, 7ilton %abbitt, 8uigi

!allapiccola and others.

Dxample GG. utline of the Heflective plan of Eohn 0age&s Sonatas and 1nterPudes, from: !avid . "orter,9Heflective Symmetry in 7usic and 8iterature,9Pers#ecti6es of New Music,21113 (*-),G5.

Sonatas 1412

1nterlude 1

Sonatas 242111

1nterlude 11

1nterlude 111

Sonatas 1L4L11

1nterlude 12

Sonatas L1114L21

$he recurrence of a& section normally carries the connotation of related lengths.

owever, such a recurrence is not necessarily defined in terms of length. t times this is

stretched considerably to include relationships between sections of li#e content but of

very different lengths. separate category is needed to accomodate temporal length for

both sections of related and unrelated content.

$he sections of the following example are indicated to be unrelated in content, but the

length of the sections are related reflectively in time about the axis section, 0. $his

relationship may be appropriately called durational reciprocity.

Dxample G@. $he reflection of durations.

% 0 ! D

Qo.of measures: @ 5 - 5 @

t first glance, the next example may not seem to possess any durational reciprocity.

Dxample GB. reflective form with sectional durations specified

. ! / C C / !unit length of time: @ @ - K G B

owever, close inspection reveals an automorphism in the proportional lengths of the

sections across the axis.

Dxample G*. "roportional lengths of the form in example GB.

! / C C / !/ : : K : : /


43/43

$he third movement of %arto#&sourth String *uartetis an illustration of such an

automorphism. $he divisions between sections are clear and may be outlined in

numbers of measures, as follows:

Dxample G5. diagram of the durations of sections in measures, of the third movement of %arto#&s

ourth String *uartet.

! / !0 /0

/ K 5 K / K 5 K @ K 5 K 5 K 5 K

Sections and % are related by durational reciprocity after an automorphism:

/ : 5 : / U 5 : @ : 5

& and %& are obviously related in duration without the need for automorphism.

7usic may also have reflective temporal relationships in tempos, such as fast4slow4fastwhich commonly exist in the movements of a sonata, concerto, etc. $exture is

fre=uently governed reflectively in time when approaching and receding from climaxes,

in the same way as dynamics.

Symmetry as a Compositional Determinant 01

Documents

Transcript of Symmetry as a Compositional Determinant 01