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Vivarium

Volume 16

1978

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2013


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VIVARIUM

AN

INTERNATIONAL

JOURNAL

FOR

THE

PHILOS-

OPHY

AND

INTELLECTUAL

LIFE

OF

THE

MIDDLE

AGES

AND

RENAISSANCE

vivarium

s

devoted n

particular

o

the

profane

side

of

mediaeval

philosophy

nd

the

ntellectual

ife

f the

Middle

Ages

and

Renaissance.

editors

C.

J.

de

Vogel,

Utrecht)

L.

M. de

Rijk,

Leyden)

H.

A.

G.

Braakhuis,

Nijmegen)

F.

F.

Blok,

Amsterdam)

J.

IJsewijn,

Louvain).

Secretaryf heEditorial oard:Prof. . M. deRijk.

All

communications,

xcept

hose

f

business

ature,

hould

be

addressed

o

C.

H.

Kneepkens,

atholieke

Universiteit,

Erasmuslaan

0,

8.26,

Nijmegen,

he

Netherlands.

advisory

Marie-Therse

AJverny,

Paris-Poitiers)

Tullio

Gregory,

committee

(Rome)

Paul

Oskar

Kristeller,

New

York)

-

Jan

Pinborg,

(Copenhagen)

Albert

immermann,

Cologne).

publishers

E.

J.

Brill,

eiden,

The

Netherlands.

published

Twice

yearly,

ay

nd

November;

a

160

pages

yearly.

Contributions

ubmitted

o

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ong

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ontinuously

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hey

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ree

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5/165

CONTENTS

OF

VOLUME

XVI

(1978)

Calvin bower Boethius and Nicomachus An Essay

Chapei

Hill

,

Concerning

he

Sources

of

De institu-

N.C.,

U.S.A.

tione

musica

1

ELIZABETH arger

Consequences

t

nconsequences

e

a

sup-

Paris

position

vide

dans la

logique

d Ockham.

46

E. p.

bos

Mental

Verbs

n

Terminist

Logic

{John

Leiden

Buridan Albert

f

Saxony,

Marsilius

of

Inghen)

56

Olga WEI ers

Contribution

l histoire

des termes

na-

Voorburg

N.L.)

tura

naturans

et

natura naturata*

us-

qu Spinoza

70

l.

M. de

rij

On

Ancient

and

Mediaeval

Semantics

Leiden and

Metaphysics(2)

81

c. h.

KNEEPKENS

Master

Guido

and his

View

on Govern

Nijmegen

ment:

On

Twelfth

Century

Linguistic

Thought

108

m.

L.

fuehrer Wisdom

and

Eloquence

in Nicholas

of

Minneapolis

Cusa s Idiota de

sapientia

and

de

Minn.,

U.S.A.

mente

142

book reviews 156


6/165

Vivarium,

VI,

i

(1978)

Boethius and

Nicomachus:

An

Essay Concerning

the

Sources

of

De institutionemusica

CALVIN BOWER

Remarquonsue

cette

partie

e la

science

musicale,

ar

suite

d'un

vice

de

la nomenclature

recque,

ggrev

ncore

ar

les auteurs

u

moyen-ge,

t

perdue endant

es

sicles,

mle t

confonduevec celle

des

tons u

chelles etransposition.e chaos, j mpntrableBoce, uVie sicle

de notre

re,

'a

commenc

se dbrouiller

ue depuis

e milieu e

XVIIIe.1

These

go,

sentences

are

representative

of

Franois

of

most

Auguste

scholarship

Gevaert,

concerning

written a

Boethius'

century

ago,

are

representative

f

most

scholarship concerning

Boethius'

De

institutionemusica

during

the

last one

hundred

years.

While

the

prejudice

of recensior

rgo

deterior

as been forsaken n

most

areas

of classical

and medieval

studies,

it

seems

to

hang

on with

dogged

tenacity

in critical

literature

concerning

Boethius'

musical

treatise.

The onlymuscological studywhichhas expressly challengedGevaert

is

Henri

Potiron's

Boce,

Thoricien

e la

Musique grecque

Paris, i960).

But

Potiron's

study

s

principally

xpository

n

character

nd

does not

systematically

deal

with

the

question

of

Boethius' sources.

The

most

thorough study

concerning

sources

of

Boethius1

musical

treatise,

Ubaldo

Pizzani's

Studi sulle

fonti

del "De

Institutione

Musica

"

di

Boezio

2

adopts

Gevaers

attitude

to the

point

of even

citing

the

century-old

work

concerning

fundamental

theoretical

matters.

The

attitude

eads him

to a

somewhat

distorted

picture

of

the

relationship

betweenBoethius and his sources.3Hence the presentstudy.

I

begin

this

nquiry

with

two

underlying ssumptions.

First,

Boethius

1

Franois uguste

evaert,

istoire

t

horie

e

a

musique

e

V

ntiquit,

and

1875-81,

ol.

,

p.

128.

z

In: bacns

erudrn,

6

(1965),

-164.

3

I cannot

gree

with

.

J.

de

Vogel's

rief

ssessment

f

Potiron'snd

Pizzani's

(not

Pizzano,

orrected

n:

Vivarium,

0

1972), 7)

worksBoethiana

in: Viva-

rium,

(197 )52-53).

As

will

become

videntn

the

present

tudy,

izzanidoes

not

suggest

hat

Boethiusbased

his

workon

a Latin

source,

ut

rather n

variousGreek

works,

ne

of

which

was

translatedntoLatin.Thusboth

Pizzani

and

Potiron

gree

hat

Boethius

s not

a

homo

nius

ibri. otiron's

omments

onBoethius'ources ake heformfopinionsxpressed, hereas izzanibuilds

arguments

nd

theses.Pizzani's

arguments

bviously

vershadow otiron's

opinions.

I


7/165

was

principally

a translatorwhen

putting together

the De

institutione

musica. The

treatise seems to follow

the De

institutionerithmetica

n

the chronologyof Boethius' works,4 nd the arithmeticaltreatise is

recognized

to

be a translation of

Nicomachus

of Gerasa

)

piOpjTix).5

assodorus

referred o Boethius' works

on the various

artes

n the

following

erms:

Translationibus

nim tais

Pythagoras

musicus,

tolomaeus

stronomus

leguntur

tali,

Nicomachus

rithmeticus,

eometricus

uclides

udiuntur

Ausonii.

Boethius'

method

of

composing

n his

early

works

s

that

of

compiling

through

translation

with

some

commentary;

the arithmetical

reatise

and the ogicalworks7clearlydemonstrate hispoint.Thus in nquiring

into

Boethius'

sources

I

am

trying

o determine

which Greek

treatise

Boethius

was

translating

when he

compiled

his musical

treatise.

My

second

assumption

is

that Boethius

was

a conscientious

and

competent

translator.

Boethius characterized

his

approach

to trans-

lating

the

mathematical

works

as

adhering

o

the

strictest

aw

of

trans-

lation,

but

adding

for

the sake of

elucidation,

sometimes

condensing

when

his

source

became

too

diffuse,

nd

supplying

harts

and

diagrams

for the

sake

of

clarity:

.

. . artissima

emet

pse

ranslations

ege

onstringo,

ed

paululum

iberius

evagatus

lieno

tineri,

on

vestigiis,

nsisto.

am et

ea,

quae

de

numeris

Nicomacho

iffusius

isputata

unt,

moderata

revitate

ollegi

t

quae

4

See

S.

Brandt,

ntstehungszeit

nd

eitliche

olge

der Werke

on

Boethius,

n:

Philologus,

2

(1903),

152-154.

.

P.

McKinlay,

tylistic

ests

nd the

hronology

of

the

works

f

Boethius,

n:

Harvard

tudies

n

Classical

Philology,

8

(1907b

123-156,

hallenges

randt's iew

that the

arithmetical

nd

musical

reatises

belong

ogether

nd

are

Boethius'

irst orks.

ut

McKinlay's

tylistic

tatistics

can be

used

to

prove

Brandt's

iew

f imilar

ubject

matter

n the

wo reatises

is

compared,

s I

will

how

n a

forthcoming

tudy.

5 Foreditionee ntroductiomsrtthmehcaeibri I, recensuit. Hoche,JLeipzig

1866.

See

also

Nicomachus

f

Gerasa

Introduction

o

Arithmetic,

rans,

nto

English

y

Martin

uther

D'Ooge

with tudies

n

Greek

rithmetic

y

Frank

Egleston

Robbins

nd Louis

Charles

Karpinski,

nn Arbor

938,

p.

132-137.

Cassiodorus

ariae

,

45,

4 (ed.

Mommsen,

GH,

Auct.

nt.

12,

p.

40).

7

Concerning

he

ogical

works

ee

L. M. de

Rij

,

Un the

hronology

noetmus

works

n

ogic

n:

Vivarium,

(1964),

-49,

25-162.

ecent

tudies

f

he

ogical

works

ave

argued

hat

Boethius

may

have translated

ven

more

han

prin-

cipal

source,

or

n some

cases his

commentaries

re

translations

f

glosses

n

Greek

reatises;

ee

J.

Bidez,

ohce t

Porphyre,

n:

Revue

Belge

de

philologie

t

d'histoire,

(1923),

89-201;

.

Minio-Paluello,

Latin

ommentary

?

ranslated

byBoethius)nthe riorAnalytics

nd ts

Greek

ources,

n:

Journal

f

Hellenic

Studies, 7 (1957), 3-102;James hiel,BoethiuscommentariesnAristotle,n:

Mediaeval

nd

Renaissance

tudies,

(1958),

16-244

nd

L.

M. de

Rijk,

Logica

Modernorum

,

Assen

1962,

8-39.

2


8/165

transcursa

elocius

ngustiorem

ntellegentiaeraestabant

ditum

mediocri

adiectione esera

i,

ut

aliquando

d evidentiamerum ostristiam

ormulis

ac

descriptionibus

teremur.

De

inst.

rith.,

raefatio,

,

28-5,

4)

8

In

a

brief

study

of the

relationship

between Boethius'

De

institutione

arithmetica

nd

Nicomachus'

treatise on

arithmetic,

Frank

Egleston

Robbins comments as

follows

A

comparison

f

the

two

books will

convince

he

reader

hat

Boethius

follows

icomachus

rom

irsto

ast,

xpanding

ere

nd

condensing

here,

as he

ays

n his

preface

hathe

will

do,

butnever

dding

nything

ssential,

either

riginal

r

derived

rom

ther

ources,

hat

departs

rom is model.

Boethius

expressed

his

concern for careful

translation

again

in the

In

IsagogenPorphyriicommenta10 nd the logical worksfurther ttest to

Boethius'

skill

as a

translator.

Arthur

Patch

McKinlay

sees

the

in-

fluence

f

translating

rom

Greek

to be

the

essential

element n

forming

Boethius1

style,

and he

characterizes

Boethius'

style

of

translationas

"literal".11

ince

Boethius'

knowledge

nd

understanding

f Greek and

his

ability

as

a

translator are

demonstrated n works

for

which his

sources

are

extant,

I

hold

that

his abilities did not forsake

him

when

he

wrote

concerning

he

art of

music.

Thus

I

assume

that

Boethius in

compiling

the

De

institutione

musica

followed the

"path"

if

not the

"footprints"

ofhis Greek source.

Since this

essay

is

somewhat

expository

n

nature,

its

organization

must

largely

followthat

of

Boethius'

treatise.

Seven

principal

sections

will

be

designated

as follows

I.

Pattern of

citation

in

the mathematical

works

II.

Books

I

and

II

III.

Book

III

IV. Book IV

V.

Unity

of

Books

I-IV

8

Page

and ine

itations

ollowinguotes

rcitations

f

Boethius'

mathematical

works

efer o

the

edition

f G.

Friedlein,

nicii

Manlii

Torquati

oetii

de

institutione

rithmetica

ibri

duo,

de

institutione usica ibri

quinqu

accedit

geometriauaefertur

oetii,

eipzig1867.

9

D'Ooge, op.

cit.,

. 132.

10

n

Isagogen orphyrii

ommenta,

d. S.

Brandt,

eipzig

1906,

p.

135, 5-10:

Secundus ie

arreptae xpositionis

abor

nostrae

eriem ranslationis

xpediet,

in

qua quidem

uereor e

subierim idi

nterpretisulpam,

um uerbum erbo

expressm

omparatumque

eddiderim.

uius

ncepti

atio st

quod

nhis

criptisinquibus erumognitiouaeritur,on uculentaerationisepos, ed ncorrupta

ueritas

xprimenda

st.

11

McKinlay,

p.

cit.,

p.

124,

127.

3


9/165

VI. Ncomachus

and

Ptolemy

VII. Book

V

and the

original scope

of De

institutionemusica

In that

my

conclusions

concerning

he first hreebooks are similarto

those of

Pizzani

and other

writers,

hese sections

may

be brief

and

concise.

Since

my

treatmentof Book IV stands in

sharp

contrast to

previous

scholarship,

that

section

must

be the most detailed and

extended.

I. Pattern

of

citation

n

the

mathematicalworks

An

initial

step

in

determining

he

relationship

between

Boethius

and his sources is a descriptionof a certain pattern of citing other

authors

that

appears

in the De institutione

rithmetica nd

De institu-

tione

musica.

The De

institutione rithmetica s a translation of

the

arithmetical

treatise

of

Nicomachus,

yet

at no time does Boethius

acknowledge

that fact

apart

from the referenceto

expanding

and

condensing

Nicomachus

in his

prefatio.

Nicomachus

is not

even men-

tioned

during

the

course of Book

I,

and

he

is

mentioned

only

three

times

in Book

II. The firsttwo citations of

Nicomachus

are

merely

"ut ait

Nicomachus"

clauses,

referring

o

unusual

words or accounts

recorded in Nicomachus' treatise (80, 5 and 114, 17-18). The third

citation

of Nicomachus

accredits

him with

having

discovered

a

unique

characteristic

of

arithmetical

proportionality,

characteristic

Nico-

machus

himself

tates has

escaped

the notice of other

writers.12 lmost

all references

o

sources

other than

Nicomachus

are

taken over

from

Nicomachus

himself,13

nd

thus Boethius'

treatise

basically predicates

knowledge

of no other treatise

except

that of Nicomachus.

The

signifi-

cant

pattern

to note is

that

Boethius

only

cites Nicomachus

when

some

aspect

of

general

mathematical

theory

or

language

is

unique

to

Nicomachus. So long as the text is consistentwith the general arith-

metical

and

philosophical

position

of

Neo-Pythagoreanism

which domi-

nates

Nicomachus'

text,

Boethius

apparently

saw

no

necessity

o refer

to

Nicomachus.

This

pattern

can be

further

ubstantiated

using

Book

V of

De insti-

tutione

musica,

for

the last

book of the musical

treatise s

clearly

based

12

Eisagoge

rithmetica

i,

23;

see

D'Ooge,

op.

cit.,

.

269,

n.

3.

13

Boethius

dds

one

important loss

concerning

ategories

n

ue vnsntuttone

arithmeticai,42, ccordingowhich rchytashe ythagoreanirstistinguished

the en

praedicamenta

licet

ubusdam

it

mbiguum),

nd

hat

lato

ndAristotle

followed

is distinction

139,

9-21).

4


10/165

on

Book

I of

Ptolemy's

Apfxovix.

4

The

following

itations of authors

occur

n Boethius'

fifth

ook

Chapter

3.

Aristoxenus,

Pythagoreans,

Ptolemy

4.

Aristoxenus,

Pythagoreans,

Ptolemy

5.

Ptolemy

7.

Pythagoreans

8.

Ptolemy,

Pythagoreans

9. Ptolemy

10.

Ptolemy

11.

Ptolemy

13. Aristoxenus

14.

Ptolemy

16.

Aristoxenus

17.

Archytas

18.

Ptolemy, Archytas,

Aristoxenus

Ptolemy

is

clearly

cited

more

than

any

other

source,

yet

he

is never

identified s

the

author

upon

which the

text

is

based.

Boethius attrib-

utes

theories

to

specific

authors

only a)

where differences

f

opinion

between Pythagoreans and Ptolemy arise, b) where theories of the

Pythagoreans

and

Ptolemy

can

be used

to refute

heories of Aristox-

enus,

or

c)

where a

certain

theory

is of a

personal

nature

and not

necessarily

generally

ccepted.

So

long

as

theory

n

question

s consist-

ent with

the

basic

tenets of

Pythagorean-Ptolemaic

musical

thought,

Boethius

gives

no

citation

whatsoever.

Furthermore,

he mere

citation

of a

name does not

imply

that

Boethius used

that

author's work as a

direct

source;

his

source

for

theories of

Aristoxenus,

Archytas,

and

even the

Pythagoreans

n Book V

is

simply Ptolemy's

treatise.

If thesegeneralprinciplescan be established as governing he pat-

ternof citation n

the

arithmetical

reatise

and

last book of the

musical

treatise,

extsforwhich

the

Greek

source

is

still

extant,

the

same

prin-

ciples

should

be

equally

valid in

the

first

four

books for

which no

complete

source

is

extant.

II.

Books I and II

Cassiodorus'

reference

to the

source

of

the

musical

treatise

as

14For edition ee Ingemar ring, ie HarmonielehreesKlaudios tolemaios

Gtebprg

Gteborgs

gskolas

Arsskrift

XXVI)

1930.

shall

atinize his

treatise

s

Harmonica.

5


11/165

"Pythagoras

mus

us"

15

s

impossible

to

interpret

s

any

indicative

reference.

Although

the

references to

"Nicomachus

arithmeticus",

"geometricusEuclides", and "Ptolomaeus astronomus"may be taken

to refer

to

the sources of

the

treatises

on

these

respective

arts,

the

reference

o

Pythagoras

must

merely

be

a

rhetorical citation of the

attitude

characterized

by

the

musical treatise.

The

sole distraction

from orthodox

Pythagoreanism

n

the musical

work is the

theory

of

Ptolemy,

and

although Ptolemy

s critical of

Pythagoreans

concerning

several

matters,

he remains

faithful

o the

crucial

Pythagorean

doc-

trine

of

expressing

ntervals as

proportions

s

opposed

to

the Aristox-

enian

method

of

using

unrelated

quantities.

Moreover,

Ptolemy's

insistence on superparticularproportionsthroughouthis tetrachord

divisions

carries

one

Pythagorean

principle

further han traditional

Pythagoreanism.

Nicomachus of

Gerasa

has

long

been considered

the

source

of the

essentiallyPythagorean

theory

found

n

Books

I

and II.1

Citations

of authors

and

works

n Books

I

and

II are as follows:

Book

I.

Chapter

i.

Plato,

Cicero,

Statius

2.

Aristotle

3.

Ptolemy

4. Ptolemy,De institutione rithmetica

9.

Pythagoreans

12.

Albinus

20.

Nicomachus

24.

Albinus

27.

Cicero

30.

Plato

31.

Nicomachus

32.

Nicomachus

Book II.

Chapter

2.

Pythagoras

3.

Pythagoras

4.

De institutione

rithmetica

7.

De institutione

rithmetica

12. De institutione

rithmetica

14.

*

De institutione

rithmetica

15

See

above

n. 6.

16

See

e.g.,

W.

Miekley,

e

Boethnibri e

musica

fontibus,

ena

898;

M.

Cappuyns, obee,

n:

Dictionaire 'histoire

t de

gographiecclsiastiques,(Paris191 ), col.364; P. CoTircelle,ateLatinWritersndtheir reekources,

trans.H. E.

Wedeck,

ambridge,

assachusetts

969,

p.

278;

Pizzani,

p.

cit.,

pp.

10-66.

6


12/165

15-

De institutione

rithmetica

17.

De institutione

rithmetica

18. Nicomachus

19.

Eubulides,

Hippasus

20.

Nicomachus

27. Pythagoreans,

Nicomachus,

Ptolemy

31.

Aristoxenus

Nicomachus

is

cited more than

any

other author

in Books

I-IV,

and

he

appears

to

predominateparticularly

n the

first wo.

The most cited

source,

however,

s the De institutione

rithmetica,

work

which Boe-

thius translatedfromNicomachus; thedependenceofthe arithmetical

treatisethus further

oints

to Nicomachus as

the source

for t least

the

first

wo

books.

The

only

extant musical work

of Nicomachus

is the

'Apjxovixv

Yx^tpStov,17

brief

work which

can

be

considered

a minimal

intro-

duction

to

Pythagorean

musical

thought.

Definitions

by

Boethius

of

such

terms as

sound,

interval, consonance,

dissonance,

and

types

of

voice seem

to be taken

quite literally

from this brief

treatise

by

Nicomachus.18On

the other

hand,

theories attributed

to

Nicomachus

in Book I, 20, 31,and 32, as well as Book II, 20 and 27, cannot be found

in

any

extant work of Nicomachus.

Nicomachus'

Enchiridion

s a workaddressed to

a

noble

lady,

written

on her

request (JanS.

237, 15),

and in the

introductory

entences

Nico-

machus

acknowledges

that this

exposition

of music

is limited

(JanS.

238, 6-7).

Furthermore,

e

promises

more

complete

musical

treatise,

an

],

s

soon

as he has the time to

compose

it

(JanS.

238,

6ff.).

This

promise

is

repeated throughout

the work

in

conjunction

with

specific spects

of musical

theory:

n

chapter

3

Nicomachus

promises

1) more concerning he harmonyof the spheres (JanS. 242, uff.) ; at

the end of

chapter 9

he

promises

2)

more

concerning

the

addition

of

notes,

their

nventors,

nd the times

and circumstances

f

their nven-

17

Ed. Karl

von

Jan,

Musici

scriptoresraeci, eipzig

1895,

p.

235-265

ref-

erences o

this dition ill e

ndicated

ith

JanS."

nd

page

and ine

number).

I shall atinize

his

reatise

s

Enchiridion.or translationee

Flora R.

Levin,

Nicomachus

f

Gerasa,

Manual

of

Harmonics: ranslation

nd

Commentary,

h.

D.

diss.,

Columbia

niversity,

ew York

1967.

18

For

comparisons

fvarious

arallel

assages

etween

oethius nd

Enchiri-

dion eePizzani, p.cit., p.35-62.MyforthcomingranslationfDe institutione

musica

Yale University,

usic

heory

n

Translation

eries)

willdocument

ach

suchdefinitionaken rom

icomachus.

7


13/165

tions

(JanS.

260,

i2f.),

as

well

as

3)

a

division

of the

monochord

following

Pythagorean

principles

JanS.

260,

I2ff.)

chapter

12

prom-

ises 4) more concerning

musical

proportions JanS. 261, 18),

as

well

as

5)

further

discussion

of the

octave,

its

merit,

nd

that it consists

of

five

tones

and two semitones rather

than

six tones

(JanS.

264,

iff.).

Nicomachus'

Enchiridion

concludes

with

an

apology

for

the

brevity

of the

work,

and

again

promises

a

much

more

complete

work

on

music

(JanS.

265).

A

comparison

of these

promises

with the

chapters

of

Boethius'

treatise

which

cite Nicomachus

clearly

indicates

that Boethius

must

have

had

access

to

the more

extended work.

Book

I,

20

cites

Nico-

machus concerningthe additions of stringsto the lyre, giving their

inventors

and

the circumstancesof

their

nvention. This

discussion

s

not

found

n the

Enchiridion but

it

fulfills

romise

no.

2.

Book

1,

31

and

32

cite

Nicomachus

concerning heory

of consonance and

the merits

of

various

consonances,

especially

the

diapason.

Again

no

such

dis-

cussion

s

found

n

the Enchiridion

but

such

a

presentation

would

result

from

promise

no.

5.

The

theory

attributed

to Nicomachus

in Book

II,

18, 20,

and

27

is

likewise

missing

rom he

Enchiridion but

the

discussion

of consonances

found

n

these

chapters

continues

that of

promise

no.

5.

Promise no. 1 is fulfilledn Book I, 27, with no specificreference o

Nicomachus;

for

this

chapter

presents

a

more

accurate

picture

of

celestial

harmony

han

that

found

n the Enchiridion

19Books

I and

II

as

a whole

represent

he reference

o more

concerning

musical

proportions

of

promise

no.

4.

Only

the

promise

of a monochord

division

according

to

Pythagorean

principles promise

no.

3)

is

not

found

n Books

I and

II.

The

obvious source

for the

first

chapters

of Book

II is

Boethius'

De

institutione

rithmetica.

his work

s,

in

fact,

such

an

integral

part

of the De institutionemusica that the musical work

appears

to be a

direct

continuation

of the

arithmetical

treatise.

The arithmetical

treatise,

however,

s

nothing

more

than

a translation

of

Nicomachus'

treatise

on

the same

subject.

Just

as

Boethius'

musical

treatise

is

dependent

on

his

arithmetical

work,

o

Nicomachus'

Eic


14/165

Book

II is

such

a

logical

and

necessary

outgrowth

f

Book

I,

and

since

Nicomachus

through

both cited and

uncited

sources

as

well as

through

his

Eisagoge

arithmetica

is without doubt

the

prevailingsourcefor he

theorypresented

n Books I and

II,

it seems

inescapable

to

conclude

that the

more

extended musical work

promised

by

Nico-

machus served

as

the

principal

source fromwhich Boethius

translated

and

compiled

at

least the

first two books of De

institutione

musica.

The fact that

Nicomachus is cited

only

where

some

theorypeculiar

to

him

is

discussed

is

consistent with

Boethius'

general

use

of citations.

Boethius'

source for

theoriesof

Eubulides,

Hippasus,

and Aristoxenus

was

the work of

Nicomachus;

his citations of

Cicero,

Statius,

and

Albinuswereadditionsof a well read Roman. The act ofworking rom

one

source

is

betrayed

n

Boethius'

second

citation

of Albinus

Sed nobis n alieno

pere

non

erit nmorandum.

. 26

(219,

1-2)

The

meaning

s

obvious

Let

us

quit lingering

n

another

work and

get

back

to

our

central

source,

namely

Nicomachus.

III. Book

III

Citations

of

authors

and

works

n

Book III are as follows:

Chapter 1. Aristoxenus,De institutione rithmetica

3.

Aristoxenus

5.

Philolaus

8.

Philolaus

9.

Archy

as

Nicomachus

emerges

as the

source

for

the

first

wo

books

fromboth

citations and

uncited

passages;

but

Nicomachus is not

cited

again

throughout

De

institutione

musica and

no

single

extant source

has

been discovered forBooks III and IV. I propose to argue concerning

the

sources

of

these

two

books

based

on

the

following

hree

criteria:

1)

sources

that

can be

identified s

possible

sources for hort

passages

or

definitions;

)

cross

references

etween

Books

I

and

II

and Books

III

and

IV which

demonstrate

dependencies

between the

books;

and

3)

the

general

tone of

Books III

and

IV

in

relation to

other

ources

and

the

remainder f

the

treatise.

No extant

sources

can be

found from

which

passages

of Book

III

266-282)mayrepresentxtracts rom he ostEisagoge see Jan, p.225-232);

yet

many

uestions

oncerning

he

text,

ontents,

nd

manuscript

radition f

these

ragments

emain

o be

answered.

9


15/165

were

taken.

Moreover,

the

authors

cited in

the book

are

unlikely

sources

for the whole

of Book III. All

citations of Aristoxenus

are

refutations f his position that a semitone s half of a tone.

The

two

references

o

Philolaus,

though

not

refutations,

an be considered ittle

more

than

interesting

nsertions,21

or the

theory

of

these references

is m

no

way

central to the basic

contents

of Book

III.

Finally,

the

citation of

Archytas

efers

o a

basic

geometric

xiom

which s

necessary

to

prove

certain

argumentspresented

n the

treatise;

but

the axiom is

presented

here

as

being inadequate

for

proving

the

particular

point.22

If one

compares

these references o those other than to Nicomachus

in

the first

two books

and to those other

than to

Ptolemy

in

Book

V,

theyare seen to be presented n exactlythesame manner.Thus it must

be concluded

that

Boethius used

no

primary

ources for the

specific

citations

occuring

n Book III. Rather Book

III

was

based on some

Greek

work which also

made

reference

o these authors.

An

examination

of the

interrelationships

etween

Book

III and the

first

wo

books

is

the

key

to

determine

he source

of the third

book.

De institutione

rithmetica

s cited in the

very

first

hapter

of this book

(269, 9-10),

and a

dependence

on the mathematical

theory

f

the arith-

metical treatise

and Book

II remains evident

throughout:

the six

continuoustones of II, 31 are cited and reexamined n III, 3 (273, 22-

23)

;

the number

containing

the

comma

discussed

in

II,

31

is recalled

in

III,

4

(275,

13-14)

and

finally,

he axioms

concerningproportions

presented

n

II,

9

are cited

and

used

to

prove arguments

concerning

the size

of

the semitone n

III,

12

(288, 9

and

290,

7-9).

In

short,

the

contents

of Book

III

would

be

incomprehensible

were

it not

for the

mathematical foundation

f

Book

II.

Furthermore,

ook

III

completes

discussions

which were

promised

in earlier

passages:

I,

16

(202,

18)

promised

conclusive

arguments

that the tone

could

not be divided

into

half,

a

proof

that is found n III, 1-2; II,

29 (262,

13) anticipated

and

cited

the

proof

found n

III,

1 that

the semitone

ies between

18

17

and

17

16 and Aristoxenus'

concept

of six

tones

completing

he dia-

pason

found

n

III,

3

completes

the reference o this

doctrine

found n

II,

31

(267,

3-5).

Thus

just

as the first

wo books

are

necessary

for

an

intelligible

reading

of Book

III,

so Book

III

completes

theory

only

anticipated

in

Books

I and II.

21

Concerning

he

uthenticity

nd

significance

f hese

xtracts

rom

hilolaus

seeWalterBurkert,ore ndSciencen Ancientythagoreanism,rans. dwin

L.

Minar,

r.,

Cambridge,

assachusetts

972,

p.

394-400.

22

Cf.

Burkert,

p.

cit.,

p. 442-447.

10


16/165

The

general

tone of

Book

III

is

completely Pythagorean,

and

no

theories

presented

as valid in

this

book are

exceptional

to theories

of the first wo books. Since the positionof Book III iswholly onsist-

ent with the basic

tenets

of

Pythagorean

theory,

no citation of

its

primary

ource

was

necessary.

Given the

Pythagorean

tone

of

the book

and its

logical dependence

on the

previous

books and the arithmetical

treatise,

ne

must

conclude

that it

is

a

continuation

of the same

source

which

was

used

for

he first

wo

books,

Nicomachus'

Eisagoge

musica 23

IV.

Book

IV

No

student

of De

institutione

musica can

deny

that Book

IV

is

the

most difficult

art

of thetreatiseto read and to relate to other ources.

Pizzani

has called

this book a

''wild

forest",24

nd

he

sees

it

as the

only

place

in

the

entire

work where

Boethius

tries

to

use

disparate

sources

and,

in

the

end,

contaminates

the

basic

integrity

f

the work.25 aw-

rence

Gushee

is

equally

critical

of Book

IV,

arguing

that

in

this book

Boethius'

"dependence

on

sources

of

differing oints

of

view, content,

or

age

lead the

reader

(and

the

author?)

into

obscurity

and contra-

diction'

26

t

seems,

however,

hat with

patience,

a

glance

at

Boethius'

earlyworksas a group,and closerexamination,one can perceiveBook

IV

as

a

unified

whole and

read

its

theory

as

logically

related

to

the

remainderof

the

work.

But

I

must

first xamine

Pizzani's thesis con-

cerning

he

structure

f

Book IV.

Pizzani

describes the

fourth

book in

the

following

erms:

chapters

1-2 are

a

"faithful

translation" of

Sectio

canonist

chapters

3-4

are

taken from a

Latin

source,

namely

Mutianus' translation

of

Gauden-

tius;28

hapters

5-12

are

derived

from

ome

unknown

source;29

hapter

23Pizzani, p.cit., p. 83-87, grees hatNicomachus as the ource orBook

III.

24

Pizzani,

p.

cit.,

.

9:

"Senza

dire

poi

che,

e si

esclude

uella

selva

selvaggia

che il

quarto

ibro,

a

novit

ispetto

l

trattato

ritmeticoi

reduce,

n

sostanza,

all'assunzione i

due n

luogo

di

una

sola

fonte

rincipale

.."

25

Pizzani,

p.

cit.,

. 87:

".

. .

non

i

periti

i

contaminareel

olo

quarto

ibro

un

materiale

ratto

nequivocabilmente

a

fonti

iverse".

2e

Lawrence

A.

Gushee,

uestions

f

Genre

n

Medieval

Treatises

n

Music,

n:

Gattungen

erMusik n

Einzeldarstellungen,

edenkschrift

eo

Schrade,

rste

Folge,

Bern

nd

Munich

973,

p. 365-433, . 380.

27

Pizzani,

p.

cit.,

.

88: "I

due

primi

apitoli

on

sono

che una

fedele

radu-

zionedei

primi

tto

paragrafi

i

un'opera

he non

appartiene Nicomaco,

a

Sectio anonis.."

28

Pizzani,

p.

cit.,

p. 89-105.

2e

Pizzani,

p.

cit.,

.

122.

II


17/165

13

is

based

on Nicomachus' lost

musical

treatise;30

nd

chapters

14-18

are

taken

from

Ptolemy's

Harmonica

31

Before examiningeach of these claims I would examine the broad

implications

of the

argument

hat

Book IV

is taken from

wide

variety

of

sources.

Even Pizzani

recognizes

this set of

eighteen

hapters

as

the

only

place

in

the

musical treatise

where

Boethius

uses

more

than one

principal

source.

In

fact,

this

would be the

sole

example

in

all the

early

works

of

Boethius,

the

mathematical

works and the

logical

works,

n

which

the

author

set aside his

basic

technique

of

ystematically

working

through

one

particular

source at a

time

and

embarked

upon

a

course

of

coordinating

nd

reconciling

at

least five

different

ources.

Such

a

process s simplynotBoethius' moduscomponendi. oethiusdoes some-

times

bring

n

references o

works other than the one

with which

he is

working,

but such

references re

consistently assages

which reinforce

or

complement

he

principal

ource,

and

the variant

source

s

generally

cited.32

No

extended

passage

in Boethius'

early

corpus

attempts

to

coordinate

such

a

variety

of

sources

as

Pizzani

and Gushee

would

have

us

believe

concerning

Book IV

of

the

musical

treatise.

Thus from

he

perspective

of

Boethius'

general technique

of

compiling

works

through-

out

his

early

career,

the

thesis

that Book IV

is

a

pasticcio

of variant

sources seemsvery unlikely.

The

weekness

of

this

position

s

further

emonstrated

upon

examin-

ation

of

its

particulars.

will

thus

examine each

part

of

the

"disparate

sources"

hypothesis

in

relation

to the

six sections

of

Book

IV:

A.

introduction,

.

notation,

C. monochord

division,

D.

fixed nd

movable

notes,

F.

modal

theory,

G.

intervallic

tests.

A. Introduction

Chapters

1-2 of Book

IV are

certainly

drawn

from

Sectio

canonist

yet

I would hesitate to describe

them as

a

"faithful

translation".

In the

first

place,

the end of

each

axiom,

presentedonly

geometrically

n Sectiocanonis is

expanded

to include an arithmetical

statement

of

the

argument.

But

much

more

significant

han

these

arithmetical

additions

are

two

rather

substantial

differences

etween

the

text

as

found

n Boethius

and Sectio

canonis itself.

The first

on-

30

Pizzani,

p.

cit.,

p.

121

i

22.

31

Pizzani,

p.

cit.,

p. 124-139.

32

See

e.g.

references

o

Cicero

n

De institutions

usica

,

i

(185,

"1?)

nd

27

(219, 19-25).

33

For edition

ee

JanS.

1

13-166.

or

corrective oteson

Jan's

edition

nd

a

cleartranslationfthework, ee ThomasJ.Matheisen, n Annotatedrans

lation

of

Euclid*

Division

of

a

Monochordin:

Journal

f

Music

Theory,

9

(I975)

236-258.

12


18/165

cerns

the

origins

of

high

and low

sounds,

and

the

passages

are as

follows

Tcvk tvifjaecovt xv uuxvTepacjiv,

8k

paiTspai,

al

l

[xvruxvTEpat,

vr-

pou

rotoai

o

pdyyoD,

l

8k

paiTe-

pat

PapUTpou,

vayxatov

lv

ur-

pou

vai,

7ret7rep

x

cuxvoTpov

al

7uXetvcv

uyxsivrat

tvrjaecov,

o

8

ixpuTepovq

'

patOTpwv

al

Xaaavcv

yxetvTai

tvrjaecov.

are

o

v

uTpou

Sovro,

to

k

PapuTpouTctTetvofzvourpoa-

aet

xiVTjaeco

uyxveiv

Seovto.

(JanS.148, -149, )

Sedomnismotus abetnse tumvelo-

citatela umetiam

arditatem.

i

igi-

tur it ardus

npellendo

otus

ravior

reddituronus.Nam

ut tar itas

prxi-

ma stationi

st,

ta

gravitas ontigua

taciturnitati.

elox

vero motus

cu-

tam

voculam

praestat.

Praeterea

uae

gravis

st intentione

crescit d

medium,

uae

vero

acuta,

remissione

escrescit

d

medium.

(301,17-23)

-

Two

aspects

ofhis

'

'translation"

require

comment

1)

the

Latin

version,

with

the

exception

of one

short

sentence,

s

considerably

condensed;

the

entire

cvayxaiov

lause

is omitted n the

Latin and

the

final entence

(cgts

praeterea)

is

greatly

abbreviated.

Secondly,

the Greek terms

7tuxvoxvTepo

nd

patTspo

o some form

of

tcc/

nd

paSu^.

Yet a

more

significant

'infidelity'

in the

transmission

f the Sectio

canonis text is

found in

the definitionof

consonant

and

dissonant

sounds:

xal

to

xv

ufxcpcovou

xtav paiv tjv

e

jxotv

CoiouvTa,

o

k

iavou


19/165

which

mingle,

while dissonant

sounds

axe

those which do not. Boethius'

text

expands

this definition

by describing

onsonant

sounds

as both

"pleasant

and

intermingled",

nd adds the

phrase "when struck

at

the same

time".

Again

one

might

rgue

that the new words

were added

by

Boethius,

but such

an

argument

would

give

Boethius

considerable

theoretical

powers.

The

wording

of this definition

s

very

close to

that

found

n

Book

I,

27,

where the

terms

"permixtum

.

. et

suavem" and

"simul

pulsi"

likewise

appear:

Quotens

nimduo nervi no

graviore

ntenduntur

imulque ulsi

reddunt

permixtum

uodammodo

t suavem

onum,

uaeque

voces

n unum

uasi

coniunctae

oalescunt;

uncfit

a,

quae

dicitur

onsonantia.

220,

2-7)

The effect fthe additional words in the Latin text clearlymakes the

definition

of consonance

presented

in Book IV consistent

with that

found

n Book

I.

Nicomachus

was

the

source

of

that

book,

and Nico-

machus'

extant definition

f

consonance

is

strikingly

imilar to that

found

both

in Book

I and here in Book

IV

:

a(x


20/165

other

than

a

diagram

of

the

notation. The

presence

of

Latin

names

for

the notes

in

chapter

3

leads

Pizzani

to

argue

that there

must have

been a Latin sourcefor V, 3-4. The facts that 1) the Latin names used

by

Boethius

are the

same

as those

used

by

Martianus

Capella,

and

2)

Boethius

did

know

(and

had

cited)

theory

n

Latin

through

he

works

of

Albinus,

are

"eloquent

coincidences"

leading

Pizzani to state that

a Latin

source

for

these

chapters

is

an

"incontrovertible

given".36

Pizzani sees

parallels

between a

passage

in Gaudentius

and

chapter

3

37

thus

he

argues

that

the Latin

translation

of

Gaudentius

by

Mutianus

was the

source.38

There are

indeed

parallels

albeit loose-

between

the two

passages

cited

by

Pizzani,

but

to describe

them as

"perfect

textual correspondence"39s a misleading exaggeration.The Gauden-

tius

text refers o

signs

for

only

18

sounds,

while

Boethius'

text

pre-

sents 28

signs.

Boethius' text

refers o

writing

he

signs

over

a

metered

verse,

a

reference

wholly

absent

from

Gaudentius.

Moreover,

Boethius

presents

igns

forthe

Lydian

mode,

while

Gaudentius

begins

with

the

Hypolydian.

It

is difficult o maintain

that Gaudentius

is

in

any way

a

direct source

for

the

chapters

n

question.

Rather

than

leap

to

speculative

conclusions

concerning

the

third

and

fourth

chapters

of

Book

IV

I

would

answer

the

following

four

questions: 1) Why reintroduce he names ofthe notes in both Greek

and Latin in

this

place

in the treatise

2)

Why

introduceGreek nota-

36

Pizzani,

op.

cit.,

p.

94:

"Di

fronte coincidenze

anto

eloquenti

on

pi

legittimo

arlare

di

ipotesi:

'influsso

u tutto

l

brano

di una fonte

atina si

impone

omme ato di

fatto

ncontrovertibile".

37

Pizzani,

op.

cit.,

p.

98;

for

edition f

Gaudentius ee

JanS.

317-356.

The

parallel

assages

ited

by

Pizzani

re

as follows

'ExpvjaavTo

ol

7raXaii

vfiaai]

p

TT)

o7)jjL

xa

Yp^fxaai,oxocXoujxvotnrjxeoi[zouaixo,uepl vvvjTov. tv

(xouatxcv

7){xetcv

xOeat

rfovz

izl

aet

&v


21/165

tion?

3)

What are

possible

sources of the notation?

4)

What is the

relationship

of these

chapters

to the whole?

1)

The answerto the first uestion is given in the first entenceof

chapter

3

Restt

nunc

quoniam

umus

nervum ecundum

raedictas

onsonantias

per

regulam

divisuri,

uoniamque

necessrios

onos tribus

generibus

cantilenae

xhibebit

sta

partitio,

musicas nterim

otas

pponere,

t,

cum

divisam

inem

sdem notulis

igna

erimus;

uod

unicuique

nomen

it,

facillime

ossit

agnosci.

308,

18-24)

The

treatise

has

arrived at the

point

of division

of the monochord.

Such

a

division

requires

that the notes be named in

all three

genera.

But no mentionofthe names of any noteshas occurredsince Book I,

26,

for

no

notes

are named

in either Books

II

or

III It indeed seems

necessary

at

this

point

to review the names

of the

notes,

so

that

they

may

be

fresh

n the mind

of the reader as he studies

the division

of the

monochord.

Moreover,

t

would

seem

necessary

for the Latin

reader

to

have

some

translation

of the

Greek

names

in order to

understand

the

logic

of

the

system

as

a

whole.

Thus

the mere

presence

of

Latin

names

per

se

does not

argue

for a Latin source

of the

passage.

2) The introduction f Greek notation into this treatise is perhaps

the most

perplexing

uestion

with

regard

to the work.

Extant

theoreti-

cal

works

treating

notation

are all in

the Aristoxenian

tradition,

a

tradition

wholly

foreign

o Boethius' treatise.

Treatises

of a

specula-

tive

character

such

as

the Sectio canonis

and

Ptolemy's

Harmonica

seem

to

avoid

the

use of

notation;

and even

Aristoxenus

himself riti-

cizes

those

theorists

who would base

their theoretical

considerations

on

notation.40

et

in the first

enturies

A.D.

the use of

notation

seems

to

have

been

in the

air;

with the

exceptions

of

Ptolemy,

Clenides,

and Nicomachus' briefEnchiridion,everytreatise uses notation, es-

pecially

in

describing

he

transpositions

of

the modes.

As

it

happens,

notation

does

become

a

crucial

factor n

explicating

the modes

later

in

Book

IV,

and

this seems

to be

the most

acceptable

justification

for

the use

of

notation

in the work.

Another

reason

for

using

notation

is

the

very

convenience

of

the

system

in the

elementary

monochord

division

which

follows

these

chapters.

Thus

the

principles

of

notation

are

fittingly

ntroduced

here at

the

beginning

of

Book

IV. Yet

there

are

indications

n these

chapters

on

notation

which reveal

a tradition

10

The

Harmonics

f

Aristoxenus,

d. with

rans,

y

Henry

.

Macran,

Oxford

1902:

Book

I,

40.

16


22/165

other

than

that

found n

the

extant treatises

using

notation,

but

these

indications

must

be discussed

under

the

question

of

possible

sources.

3) Althoughno singlework can be cited as the obvious source of

Boethius'

notational

discussion,

at

least three

treatises

may

be

cited

as

possible

texts

fromwhich

the

notational

theory

was drawn.

Gauden-

tius'

Harmonica

ntroductio,

ellermann's

Anonymus

II,

and

Alypius'

Isagoge

all

present

notational

theory

in the

form

of

sentences de-

scribing

altered Greek letters

representing

various notes of

a

specific

ancient

mode;

moreover,

Alypius presents

the

Lydian

mode

first,41

and

Anonymus

II

presents

a

discussion

of

only

the

Lydian

mode,

very

much

like

Boethius.42The

primacy

of the

Lydian

mode in

these

Aristoxenian reatises s further estified o by Bachius, who,without

any

specific

eference o the

Lydian,

uses

that

mode for

ll

his illustra-

tions

of

musiceli ntervals.43 he

terminology

orthe lichanoi

and

para-

netai in

chapter

3

further

inks the

descriptions

of the notes

to the

tradition

of these

treatises,

for,

imilar

to the notational

treatises,

he

names

lichanoi and

paranetai

are not

used at

all,

and

the

notes

are

merely

cited

as

hypaton

enharmonios,

hypaton

chromatice,

or

meson

chromatice,

meson

diatonos,

or

hyperboleon

nharmonios,

yperboleon

chromatice.

Finally,

the

use

of the

term

hyperdorian

s

equivalent

to

mixolydianin the notational charts at the end of Book IV further

reflects

he

terminology

f the

notational

treatises

343,

addenda).

Yet minor

differencesn both the

descriptions

of the

symbols

and

the

symbols

themselves raise

serious

doubt as

to

whether

ny

of

the

sources mentioned re

the

actual source of

Boethius'

third

nd fourth

chapters.44

erhaps

an even

more

serious

question

is raised

by

the fact

that the three

genera

are

integrated

into one

list

in

Boethius'

text,

while

the

notational

treatises

explicate

each

genus

as

a

separate

entity.

Therefore

t

seems

unlikely

that

any

of the

treatises cited

served as a

source for the

passage

in

question.

They

all

represent

a tradition of

musical

theory

ssentially

opposed

to

that

found

n the De

institutione

41

For edition

ee

JanS.357-406.

"

The

original

dition f

the

Greek

nonymi,

.

Bellermann,

nonymi

criptio

de

musica

Berlin 841),

has

been

uperseded

y

Dietmar

Najock,

Drei

anonyme

griechische

raktate ber

die

Musik,

Eine

kommentierte

euausgabe

es Beller-

mannschen

nonymus

Gttinger

usikwissenschaftliche

rbeiten,

and

2),

Gttingen972;

for se of

Lydianmode,

ee

pp.

116-110.

43

For edition ee

JanS.283-316;

or se of

notation

ee,

.g.pp. 294-295.

44

For

xample, lypius

escribes he

symbol

as

9)|eXir)Tixva0eiXxuo(ivov(JanS. 369,25), whereasBoethiusdescribes t as "ny inversum eductum"

(311,

14).

The

nstrumental

rite

hyperboleon

ccording

o

Alypius

s formed-

(JanS.

369,

16),

whereas

ccording

o

Boethius he

symbol

s written

(314).

17


23/165

musica,

and,

given

Boethius' manner of

citing

other

writers,

Boethius

would

have cited

any

theoristwho was not consistentwith the

Pytha-

gorean-Ptolemaicposition

of

the treatise.Thus Boethius' presentation

of notation

seems to reflect a

slightly

different radition than

that

found

n

Aristoxenian

works,

and is thus rooted in a text which is

no

longer

extant.

4)

The

question

of

relationship

of Book

IV,

3-4

to

the

treatise as a

whole has

been

partially

answered n

explainingwhy

the names of the

notes

and the notational

system

were

introduced

at this

particular

moment

in the

treatise.

These

chapters

are

obviously

linked with

both

the

division of the

monochord and the discussion

of the

modes

throughtheir review of the names of the notes and theirexposition

of

notation.

Yet these

chapters

seem to be related

equally

to Book I.

Although

chapter

3

presents system

of

naming

notes which resembles

the

notational

treatises,

it

presents

the

notes of the three

genera

integrated

into one list.

Moreover,

chapter

4

returns to the termi-

nology

of the notes which was

found

n

Book

I, 22,

and the

notes

as

presented

here add

up

to the

twenty-eight

which are

specifically

cited

in

I,

22 :

In

quibus

t similitudinemominum

t

differentiam

ernotabis

ut si nervi

similes n omnibus umeis, qui suntdissimiles,olligantur,iant imul

omnes

cto

et

viginti.215,16-19)

The

adherence

to

twenty-eight

otes in Book

IV,

carried

over

from

Book

I,

is

a

distinctlyPythagorean imprint

on

the

text;

for

twenty-

eight

s

a

perfect

number

n

Pythagorean

arithmetic.45

Therefore,

lthough

a

certain

disparate

element

is introduced

nto

De

institutione

musica in the

discussion

of Greek

notation,

that dis-

parate

element

s

presented

n a

way

consistent

with the treatise as

a

whole rather than with sources of notational theory. The question

remains

whether

the

hand

of Boethius

changed

this material

to make

it consistent

with

the remainderof

his

work,

or

whether

the

material

46

Concerningerfect

umbers ee

Nicomachus

isagoge

rithmetica

,

16,

and

Boethius

e

institutione

rithmetica

,

20

(41-42).

'Ooge,

op.

cit.,

.

209,

rans-

lates

Nicomachus

s follows: Now when

number,

omparing

ith tself

he

sum

and

combination

f

all the factors

whose

presence

t

will

admit,

neither

exceeds

hem

n multitude

or s exceeded

y

them,

hen

uch

a

number

s

properly

aid

to be

perfect,

s onewhich

s

equal

to ts

own

parts.

uch

numbers

axe 6

and

28.

.

.

Twenty-eight

as thefactors

alf,

ourth,

eventh,

ourteenth,

and

twenty-eighth,

hich re

14, 7, 4,2,

and

1

these

dded

together

ake

28,

andso neitherrethepartsgreaterhan hewholenor hewhole reaterhan

the

parts,

uttheir

omparison

s n

equality,

hich

s the

peculiar

uality

f

he

perfect

umber".

18


24/165

was

already

changed

in Ms Greek

source.

The immediate

response

to

this

question

is

that

Boethius

does not

play

such

a

strong

hand in his

earlyworks.

C.

Monochord

division: The

division

of the

monochord

formsthe

heart of

Book

IV,

and

indeed

if the

opening

sentences

of Book

IV are

to

be

taken

literally,

the

monochord

division forms

something

of

a

climax to the

treatise as

a

whole;

it is "the division of

the

rule

toward

which

our

whole

effort s directed"

Etsi

omnia,

uae

demonstranda

rant,

uperioris

ibri ractatione

igessi-

mus,

non

paenitet

amen

rursus adem brevitermemoriae

ecolligenda

praestare umquademdiversitateractatus, t his rursus d memoriamredeuntibusd

regulae

ivisionem

uo

totatendit

ntentio,

eniamus.

301,

7-12)

Moreover,

the

reader

must

assume

that

this

rule

is that

same

rule

introduced

n Book

I, 11,

the

rule discovered

by Pythagoras,

"con-

cerning

which

we will

speak

later,

by

which

we

measure

the

sizes and

sound

of

notes"

Itaque

invenit

egulam,

e

qua

posteriusoquemur

.

.

,

perquam magni-

tudines

ordarum

onumque

metimur.

198,

23-26)

In the Enchiridion

JanS

260,

I2ff.)

Nicomachus

had likewise

promised

a

division of

the

monochord to be

included

in

his

Eisagoge

and this

division s

likewise

associated with

the name of

Pythagoras.

Therefore,

if

the

principles

determining

he

division of Book IV

can

demonstrably

be

shown to

be

those

of

Nicomachus,

then

we

may

conclude

that

Nicomachus'

Eisagoge

is the

source for the heart of Book

IV.

In

his

principal

passage

concerning

the

monochord,

Nicomachus

promises

a

division

of the

canon

"rigorously

fashioned

n accordance

with the design of this master Pythagoras, not as Eratosthenes or

Thrasyllus

misinterpreted

t,

but

as

the

Locrian Timaeus understood

it,

whom Plato

followed,

up

to the

twenty-seventhmultiple"

46

xal

7Tpoaex7)a{xc0a

v

ou

eyopivou

avvo

aTocrofrjvxp^co

xal xart

ouX7)fxa

$

ou

iSaaxXou

uvrTeXeajziv7)v,%

'EpaTOcGvrj

7cap^xouaev

J

pauXXo,

XX*

>

Aoxp ^aio,

xal IIXTcov

rapY]xoXo'j-

07jaev,

c

ou

717.

JanS.

260,

12-17)

Yet

another

passage

from the

Enchiridion

mentions

the

monochord

division

to

come,

and

one

principleofPythagoras' approach is stated

46

Translation

y

Flora

Rose

Levin,

Nicomachus

f

Ger

sa Manual f

Harmonics

Translation

nd

Commentary

diss.

Columbia

niversity,

967), .

47.

*9


25/165

clearly:

Ptyhagoras

first

etermined he

division of

the diatonic

genus,

and

from this division

he

determinedthat of

the

chromatic

and

en-

harmonicgenera:

Ty)v

k

Tup0aotvcvdyxfl

ivI

cpuaixfl

*7u

apuTOCTOu

m

toctov

aT

touto

T

StaTOVtxv

vo

;

piaxe.

t

yp

xpv

TpTo

)XX*f*)00

P^

t

Skxtovov,

8k

8e

epo

[xv

tccTovtxc

aT

jxetvev,(jLOTOvet

k

vapjxovou(.

v

evapfxovc*)

l So

iia

1

r)XXY7)aavrp

StTOVOv

bar' vTtxetaOat

vap-

[xvLov

SiocTvp,xcov

'

aTcov

7rpxetv

xpwM-aTtxvH*pv


26/165

the

extent

that

they

all

add

up

to

equal

two

tones

and

a

semitone:

el xai

[ir]

X uo

vcov

al

yjjxitovou

vuxp

axiv,

XX*v acc

xov pav7]TaL

T8iaarj[xaTauaitvolalrj(jLLxovq>.JanS.262,18-21)

Before

comparing

hese

principles

f

division

fromNicomachus

with

the division

found

in

Boethius,

the

criticisms

of

Gushee

and

Pizzani

should

be examined. Gushee

suggests

that the

omission

of

the letter

"g"

in

the

monochord

lphabet

of

V,

11

mightpoint

to a Latin

source

for the

division.49Gushee

further

mplies

that Boethius

derived his

discussion

of the

monochord

from

tolemy,

but

that he did not

"clearly

or

unequivocally

adhere" to

the

sophisticated

instruments

presented

in Ptolemy.50 failto see howthepresenceor absence of the letter"g"

can

imply

a Latin

source

unless

one can find a

source

which

similarly

omits

"g".

Furthermore,

can

find

no

trace of

Ptolemy

in Boethius'

division of

the

monochord;

such

a

division of

the

entire

system

s

a

concern

wholly

foreign

o

Ptolemy.

One must not

let the

similarity

between the last

chapter

of

Book IV

and

a

passage

from

Ptolemy

ead

one

to assume

that the

heart

of Book

IV

is

influenced

by,

much

less

borrowed

from,

Ptolemy.51

Pizzani

considered the

possibility

that

Boethius'

division

might

be

based onthatwhichNicomachuspromisedforhisEisagoge But Pizzani

sees

three

difficulties ith

this

thesis

1.

The

mathematical

inconsistency

of

Boethius'

chromatic

genus;

2. The absence of

any

trace of the

divisions of

Thrasyllus

or

Era-

tosthenes;

3.

The

neglect,

n

Boethius'

division,

of the

number

27

mentioned

at Enchiridion

JanS.

261,

17.

Such problems ead Pizzani to conclude that even a hypotheticalderi-

added

together,hey

make

up

a

semitone.

semitone

ccording

o the

Pytha-

goreans

s

256:243,

proportion

hichwill

not

dmit

geometric

ivision

sing

wholenumbers.

49

Gushee,

p.

cit.

n.

26),

p.

380,

n.

47.

There s a certain

ymmetry

n

the

omitted

g's

,

for

hey

both

occur

ver

econd-to-highest

otes

n

enharmonic

tetrachords,

he firstn

the

hypaton

etrachordnd

the second

n

the

hyper-

boleon etrachord

334

addenda).Every

other

omparable

ote

n

enharmonic

tetrachords

as a

letter

ssigned

hereto,

hereas

hese wo

notes,

ne

"g"

and

the

other

gg"

are

missing.

uchan

inconsistency

ould

most

robably

rise

n

conjunction

ith

extual

ransmission.

50Gushee, p.cit., . 379.

51

Concerning

he last

chapter

f

Book

IV,

see

sectionF. Intervallic

ests,

PP-37-38.

21


27/165

vation

of

Boethius'

division

from

Nicomachus

is

impossible.52

izzani

is

perceptive

in

centering

on

these

three

questions, yet

he

does not

carry

them far

enough;

for

ultimately

these

traits

may

be used

as

evidence

to

argue

for

Nicomachus

as source

for

the division.

Nicomachus

had

set

forth

ne

overriding ythagorean principle

for

the

division

of

the

canon

in

the

Enchiridion the

primacy

of the

diatonic

genus.

One

does

not

have

to

read

very

far

nto theoretical iterature

o

sense

the

polemic

tone

in

Nicomachus' statementof

Pythagorean

prin-

ciples.

A monochord

division

begun

from the

Pythagorean

diatonic

genus

is

truly exceptional

in

ancient

musical

theory.

As

early

as

Aristoxenus,

the

Harmonists

Eratocles

in

particular

are

criticized

forbeing obsessed with the smallest of intervalsand the enharmonic

genus

to the exclusion

of the

chromatic nd diatonic

genera.53

lthough

Aristoxenus

nd

his followers o

present

wo

shades

of a diatonic

genus,

their

mphasis

remainson variations of the

enharmonic

nd

chromatic,

and their

mathematical

principles

are

unacceptable

to

any Pytha-

gorean.54

Yet

the

obsession with the

pyknon

the

lowest

segment

of

the enharmonic

and chromatic

tetrachords,

was

not

the franchise

of

Aristoxenianmusical

thought

for heorists

using Pythagorean

math-

ematical

principles

i.e.

proportions- equally

emphasized

the

enhar-

monicand chromaticgenerato theexclusionof the diatonicgenusorto

the

compromise

of

its

Pythagorean integrity.

Archytas

and

Didymus

had derived

divisions

of

the three

genera

which,

although expressed

n

proportions,

wholly

forsookthe

Pythagorean

tetrachord

onsisting

of

9:8/9:8/2

56

243.55

Ptolemy

does

include

the

Pythagorean

scale

as

one of

his

five shades

of

diatonic

genus,

but

the "diatonic

diatonic"

shade is

included

almost ike an

afterthought

nd

plays

no central role

in

his

derivation

of

genera

and shades. Thus

we come

to consider

the

52

Pizzani, p cit., p. 115-121p. 121 "Tutopertantooncorrefarci scludereanche na

potetica

erivazioneella ectio oeziana aNicomaco".

53

Aristoxenus,

lementa

,

2-3,

6;

ii,

35-36.

54

For Aristoxenus'

hades

f

he

genera,

ee

e.g.

Elementa

i,

48-52.

or

Pytha-

gorean

riticism,

ee

De inst mus.

ii,

3

(273-274).

55

The

division

f

Archytas

nd

Didymus

re

known

hrough

tolemy

armo-

nica

Archytas

Ptolemy

,

13)

Diatonic

28

27

8:7

9:8

Chromatic:

28:27 243:224

32:27

Enharmonic:

28:27

36:35 5:4

Didymus Ptolemyi,14)Diatonic: 16:15 10:9 9:8

Chromatic:

16:15

25:24

6:5

Enharmonic:

32:31 31:30 5:4

22


28/165

two

divisions mentioned

by

Nicomachus

in the

Enchiridion,

those of

Eratosthenes

and

Thrasyllus.

The

diatonic

genus

of

Eratosthenes did

consist

of

the

Pythagoreandiatonic,

but

his chromatic nd enharmonic

were

by

no means

related to

or

derived from he

diatonic,

but rather

were

related to each other

and the enharmonicwas

derived from he

chromatic.5

ur

knowledge

of

Thrasyllus'

division,

known

only

through

Theon of

Smyrna,57

s

extremely

ketchy.

Nevertheless we

can

affirm hat

Thrasyllus

based his division on the diatonic

genus,

and that the chromatic nd enharmonic

genera

were derived from he

diatonic.

Thrasyllus'

division,

n

fact,

resembles

hat found n

Boethius

more

than

any

other extant

division,

nd

Thrasyllus'

derivation

of the

chromatic ichanosmighteven seem more consistentthan that found

in Boethius.68

One

can

only

speculate concerning

Nicomachus' reasons

for

rejectingThrasyllus'

division;

based on the version

given by

Theon,

Thrasyllus'

mathematicswould

present

difficulties

o one

working

ut

details,

and

Thrasyllus'

divisionof the chromaticwas

probably

unsatis-

factory

o Nicomachus.59

Now

the broad

context of

Nicomachus'

citations of

Eratosthenes

and

Thrasyllus

can be

brought

nto focus: These

two

theorists

re

at

least

Pythagoreans, hey

followed he master

ouXrj^a

oSs

SiSatrxXou)t least to the extent ofusing the Pythagorean diatonic

genus,

but

they

somehow

misunderstood

(7tap)xou


29/165

semitone.

The

division of the Timaeus

was followed

by

Plato

up

to

the

twenty-seventh

multiple (eoa

7rocxaieixo0i7tX


30/165

putes

the

highest

nterval

of the chromatic

genus by

taking

the

arith-

metic

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