THE CONTRIBUTIONS OF

THE CONTRIBUTIONS OF

JOSEPH SAUVEUR (1653-1716)

TO ACOUSrpICS

VOLTlME I

Presented by

Robert E Maxham

To fulfill tne disserta tion requirement for the degree of

Doctor of Phtlosophy

Department of Musicology and Music History

Thesis Director Erich P Schwandt

Eastman School of Music

of the

University of Rochester

March 1976

iA i 2

i

~ ) ~ )

vrrA

Robert 11nxham was born in Erie Pennsyl vnnla on

March 1 1947 and attended pnrochial schools there radushy

sting from Cathedral preparatory School with honors 1n

ItJGb liaving obtained his bachelors deproe with

distinction at the Eastman School of fIIusic in 1969 he

studied philosophy and theology at st Marks Seminary

in Erie bull Charles Horromeo Seminary in Philadelphia

where he won the Hanna Cusick Ryan Prize for General

Excellence in Bunrlamental Theology and St Mary 1 s

Seminary and University in Baltimore tie returned to the

stlJdy of musiC receiving the Master of Arts degree in

Musicology from the strnan School in 1975 ~he next

year was spent in preparation in absentia of the

present study

ii

PREFACE

Although Joseph Sauveur (1653-1716) has with some

justification been named as the founder of the modern

science of acoustics a science to which he contributed

not only clarificatory terminology ingenious scales and

systems of measurement brilliant insights and wellshy

reasoned principles but the very name itself his work

has been neglected in recent times The low estate to

which his fortune has fallen is grimly illustrated by the

fact that Groves Dictionary not containing an article

devoted exclusively to him includes an article on

acoustics which does not mention his name even in passing

That a more thorough investigation of Sauveurs

works may provide a basis for further exploration of the

performance practices of the period during which he lived

is suggested by Erich Schwandt 1 s study of the tempos of

dances of the French court as they are indicated by

-Michel LAffilard Schwandt contends that LAffilard

misapplied Sauveurs scale for the measl~ement of temporal

duration and thus speci fied tempos which are twice too

fast

Sauveurs division of the octave into 43 a~d

further into 301 logarithmic degrees is mentioned in the

various works on the theory and practice of temperament

iii

written since his time A more tho~ou~h inveot1vntinn of

Sauveurs works should make possible a more just assessment

of his position in the history of that sctence or art-shy

temperinp the 1ust scale--to which he is I1811a] 1y

acknowled~ed to have h~en an i111nortant contrihlltor

rhe relationship of Srntvenr to tho the()rl~~t T~nnshy

Philippe Rameau ~hould also he illuminate~ by a closer

scrutiny of the works of Sauvcllr

It shall he the program of this study to trace

ttroughout Sauveurs five oub1ished Mfmo5res the developshy

ment (providing demonstrations where they are lacking or

unclear) of four of his most influential ideas the

chronometer or scale upon which teMporal ~urntions cnu16

be measured within a third (or a sixtieth of a second) of

time the division of the octave into 43 and further

301 equal p(lrtfl and the vnr10u8 henefi ts wrich nC(~-rlH~ fr0~~

snch a division the establishment of a tone with a 1Ptrgtl_

mined number of vibrations peT second as a fixed Ditch to

which all others could be related and which cou]n thus

serve as a standard for comparing the VqriOl~S standaTds

of pitch in use throughout the world an~ the ~rmon1c

series recognized by Sauveur as arisin~ frnm the vib~ation

of a string in aliquot parts The vRrious c 1aims which

have been mane concerning Sauveurs theories themselves

and thei r influence on th e works of at hels shall tr en be

more closely examined in the l1ght of the p-receding

exposition The exposition and analysis shall he

1v

accompanied by c ete trans tions of Sauveu~ls five

71Aemoires treating of acoustics which will make his works

available for the fipst time in English

Thanks are due to Dr Erich Schwandt whose dedishy

cation to the work of clarifying desi~nRtions of tempo of

donees of the French court inspiled the p-resent study to

Dr Joel Pasternack of the Department of Mathematics of

the University of Roc ster who pointed the way to the

solution of the mathematical problems posed by Sauveurs

exposition and to the Cornell University Libraries who

promptly and graciously provided the scientific writings

upon which the study is partly based

v

ABSTHACT

Joseph Sauveur was born at La Flampche on March 24

1653 Displayin~ an early interest in mechanics he was

sent to the Tesuit Collere at La Pleche and lA-ter

abandoning hoth the relipious and the medical professions

he devoted himsel f to the stl1dy of Mathematics in Paris

He became a hi~hly admired geometer and was admitted to

the lcad~mie of Paris in 1696 after which he turned to

the science of sound which he hoped to establish on an

equal basis with Optics To that end he published four

trea tises in the ires de lAc~d~mie in 1701 1702

1707 and 1711 (a fifth completed in 1713 was published

posthu~ously in 1716) in the first of which he presented

a corrprehensive system of notation of intervaJs sounds

Lonporal duratIon and harrnonlcs to which he propo-1od

adrlltions and developments in his later papers

The chronometer a se e upon which teMporal

r1llr~ltj OYJS could he m~asnred to the neapest thlrd (a lixtinth

of a second) of time represented an advance in conception

he~Tond the popLllar se e of Etienne Loulie divided slmnly

into inches which are for the most part incomrrensurable

with seco~ds Sauveurs scale is graduated in accordance

wit~1 the lavl that the period of a pendulum is proportional

to the square root of the length and was taken over by

vi

Michel LAffilard in 1705 and Louis-Leon Pajot in 1732

neither of whom made chan~es in its mathematical

structu-re

Sauveurs system of 43 rreridians 301 heptamerldians

nno 3010 decllmcridians the equal logarithmic units into

which he divided the octave made possible not only as

close a specification of pitch as could be useful for

acoustical purposes but also provided a satisfactory

approximation to the just scale degrees as well as to

15-comma mean t one t Th e correspondt emperamen ence 0 f

3010 to the loparithm of 2 made possible the calculation

of the number units in an interval by use of logarithmic

tables but Sauveur provided an additional rrethod of

bimodular computation by means of which the use of tables

could be avoided

Sauveur nroposed as am eans of determining the

frequency of vib~ation of a pitch a method employing the

phenomena of beats if two pitches of which the freshy

quencies of vibration are known--2524--beat four times

in a second then the first must make 100 vibrations in

that period while the other makes 96 since a beat occurs

when their pulses coincide Sauveur first gave 100

vibrations in a second as the fixed pitch to which all

others of his system could be referred but later adopted

256 which being a power of 2 permits identification of an

octave by the exuonent of the power of 2 which gives the

flrst pi tch of that octave

vii

AI thouph Sauveur was not the first to ohsArvc tUl t

tones of the harmonic series a~e ei~tte(] when a strinr

vibrates in aliquot parts he did rive R tahJ0 AxrrAssin~

all the values of the harmonics within th~ compass of

five octaves and thus broupht order to earlinr Bcnttered

observations He also noted that a string may vibrate

in several modes at once and aoplied his system a1d his

observations to an explanation of the 1eaninr t0nes of

the morine-trumpet and the huntinv horn His vro~ks n]so

include a system of solmization ~nrl a treatm8nt of vihrntshy

ing strtnTs neither of which lecpived mnch attention

SaUVe1)r was not himself a music theorist a r c1

thus Jean-Philippe Remean CRnnot he snid to have fnlshy

fiJ led Sauveurs intention to found q scIence of fwrvony

Liml tinp Rcollstics to a science of sonnn Sanveur (~1 r

however in a sense father modern aCo11stics and provi r 2

a foundation for the theoretical speculations of otners

viii

bull bull bull

bull bull bull

CONTENTS

INTRODUCTION BIOGRAPHIC ~L SKETCH ND CO~middot8 PECrrUS 1

C-APTER I THE MEASTTRE 1 TENT OF TI~JE middot 25

CHAPTER II THE SYSTEM OF 43 A1fD TiiE l~F ASTTR1~MT~NT OF INTERVALS bull bull bull bull bull bull bull middot bull bull 42middot middot middot

CHAPTER III T1-iE OVERTONE SERIES 94 middot bull bull bull bull bullmiddot middot bull middot ChAPTER IV THE HEIRS OF SAUVEUR 111middot bull middot bull middot bull middot bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull middot WOKKS CITED bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull bull 154

ix

LIST OF ILLUSTKATIONS

1 Division of the Chronometer into thirds of time 37bull

2 Division of the Ch~onometer into thirds of time 38bull

3 Correspondence of the Monnchord and the Pendulum 74

4 CommuniGation of vihrations 98

5 Jodes of the fundamental and the first five harmonics 102

x

LIST OF TABLES

1 Len~ths of strings or of chron0meters (Mersenne) 31

2 Div~nton of the chronomptol 3nto twol ftl of R

n ltcond bull middot middot middot middot bull ~)4

3 iJivision of the chronometer into th 1 r(l s 01 t 1n~e 00

4 Division of the chronometer into thirds of ti ~le 3(middot 5 Just intonation bull bull

6 The system of 12 7 7he system of 31 middot 8 TIle system of 43 middot middot 9 The system of 55 middot middot c

10 Diffelences of the systems of 12 31 43 and 55 to just intonation bull bull bull bull bull bull bull bullbull GO

11 Corqparl son of interval size colcul tnd hy rltio and by bimodular computation bull bull bull bull bull bull 6R

12 Powers of 2 bull bull middot 79middot 13 rleciprocal intervals

Values from Table 13 in cents bull Sl

torAd notes for each final in 1 a 1) G 1~S

I) JlTrY)nics nne vibratIons p0r Stcopcl JOr

J 7 bull The cvc]e of 43 and ~-cornma te~Der~m~~t n-~12 The cycle of 43 and ~-comma te~perament bull LCv

b

19 Chromatic application of the cycle of 43 bull bull 127

xi

INTRODUCTION BIOGRAPHICAL SKETCH AND CONSPECTUS

Joseph Sauveur was born on March 24 1653 at La

F1~che about twenty-five miles southwest of Le Mans His

parents Louis Sauveur an attorney and Renee des Hayes

were according to his biographer Bernard Ie Bovier de

Fontenelle related to the best families of the district rrl

Joseph was through a defect of the organs of the voice 2

absolutely mute until he reached the age of seven and only

slowly after that acquired the use of speech in which he

never did become fluent That he was born deaf as well is

lBernard le Bovier de Fontenelle rrEloge de M Sauvellr Histoire de lAcademie des SCiences Annee 1716 (Ams terdam Chez Pierre lTorti er 1736) pp 97 -107 bull Fontenelle (1657-1757) was appointed permanent secretary to the Academie at Paris in 1697 a position which he occushypied until 1733 It was his duty to make annual reports on the publications of the academicians and to write their obitua-ry notices (Hermann Scherchen The Natu-re of Music trans William Mann (London Dennis Dobson Ltd 1950) D47) Each of Sauveurs acoustical treatises included in the rlemoires of the Paris Academy was introduced in the accompanying Histoire by Fontenelle who according to Scherchen throws new light on every discove-ryff and t s the rna terial wi th such insi ght tha t one cannot dis-Cuss Sauveurs work without alluding to FOYlteneJles reshyorts (Scherchen ibid) The accounts of Sauvenrs lite

L peG2in[ in Die Musikin Geschichte und GerenvJart (s v ltSauveuy Joseph by Fritz Winckel) and in Bio ranrile

i verselle des mu cien s et biblio ra hie el ral e dej

-----~-=-_by F J Fetis deuxi me edition Paris CLrure isation 1815) are drawn exclusively it seems

fron o n ten elle s rr El 0 g e bull If

2 bull par 1 e defau t des or gan e s d e la 11 0 ~ x Jtenelle ElogeU p 97

1

2

alleged by SCherchen3 although Fontenelle makes only

oblique refepences to Sauveurs inability to hear 4

3Scherchen Nature of Music p 15

4If Fontenelles Eloge is the unique source of biographical information about Sauveur upon which other later bio~raphers have drawn it may be possible to conshynLplIo ~(l)(lIohotln 1ltltlorLtnTl of lit onllJ-~nl1tIl1 (inflrnnlf1 lilt

a hypothesis put iorwurd to explain lmiddotont enelle D remurllt that Sauveur in 1696 had neither voice nor ear and tlhad been reduced to borrowing the voice or the ear of others for which he rendered in exchange demonstrations unknown to musicians (nIl navoit ni voix ni oreille bullbullbull II etoit reduit a~emprunter la voix ou Itoreille dautrui amp il rendoit en echange des gemonstrations incorynues aux Musiciens 1t Fontenelle flEloge p 105) Fetis also ventured the opinion that Sauveur was deaf (n bullbullbull Sauveur was deaf had a false voice and heard no music To verify his experiments he had to be assisted by trained musicians in the evaluation of intervals and consonances bull bull

rSauveur etait sourd avait la voix fausse et netendait ~

rien a la musique Pour verifier ses experiences il etait ~ ohlige de se faire aider par des musiciens exerces a lappreciation des intervalles et des accords) Winckel however in his article on Sauveur in Muslk in Geschichte und Gegenwart suggests parenthetically that this conclusion is at least doubtful although his doubts may arise from mere disbelief rather than from an examination of the documentary evidence and Maury iI] his history of the Academi~ (L-F Maury LAncienne Academie des Sciences Les Academies dAutrefois (Paris Libraire Academique Didier et Cie Libraires-Editeurs 1864) p 94) merely states that nSauveur who was mute until the age of seven never acquired either rightness of the hearing or the just use of the voice (II bullbullbull Suuveur qui fut muet jusqua lage de sept ans et qui nacquit jamais ni la rectitude de loreille ni la iustesse de la voix) It may be added that not only 90es (Inn lon011 n rofrn 1 n from llfd nf~ the word donf in tho El OF~O alld ron1u I fpoql1e nt 1 y to ~lau v our fl hul Ling Hpuoeh H H ~IOlnmiddotl (]

of difficulty to him (as when Prince Louis de Cond~ found it necessary to chastise several of his companions who had mistakenly inferred from Sauveurs difficulty of speech a similarly serious difficulty in understanding Ibid p 103) but he also notes tha t the son of Sauveurs second marriage was like his father mlte till the age of seven without making reference to deafness in either father or son (Ibid p 107) Perhaps if there was deafness at birth it was not total or like the speech impediment began to clear up even though only slowly and never fully after his seventh year so that it may still have been necessary for him to seek the aid of musicians in the fine discriminashytions which his hardness of hearing if not his deafness rendered it impossible for him to make

3

Having displayed an early interest in muchine) unci

physical laws as they are exemplified in siphons water

jets and other related phenomena he was sent to the Jesuit

College at La Fleche5 (which it will be remembered was

attended by both Descartes and Mersenne6 ) His efforts

there were impeded not only by the awkwardness of his voice

but even more by an inability to learn by heart as well

as by his first master who was indifferent to his talent 7

Uninterested in the orations of Cicero and the poetry of

Virgil he nonetheless was fascinated by the arithmetic of

Pelletier of Mans8 which he mastered like other mathematishy

cal works he was to encounter in his youth without a teacher

Aware of the deficiencies in the curriculum at La 1

tleche Sauveur obtained from his uncle canon and grand-

precentor of Tournus an allowance enabling him to pursue

the study of philosophy and theology at Paris During his

study of philosophy he learned in one month and without

master the first six books of Euclid 9 and preferring

mathematics to philosophy and later to t~eology he turned

hls a ttention to the profession of medici ne bull It was in the

course of his studies of anatomy and botany that he attended

5Fontenelle ffEloge p 98


7Fontenelle Elogett p 98 Although fI-~ ltii(~ ilucl better under a second who discerned what he worth I 12 fi t beaucoup mieux sons un second qui demela ce qJ f 11 valoit

9 Ib i d p 99

4

the lectures of RouhaultlO who Fontenelle notes at that

time helped to familiarize people a little with the true

philosophy 11 Houhault s writings in which the new

philosophical spirit c~itical of scholastic principles

is so evident and his rigid methods of research coupled

with his precision in confining himself to a few ill1portnnt

subjects12 made a deep impression on Sauveur in whose

own work so many of the same virtues are apparent

Persuaded by a sage and kindly ecclesiastic that

he should renounce the profession of medicine in Which the

physician uhas almost as often business with the imagination

of his pa tients as with their che ets 13 and the flnancial

support of his uncle having in any case been withdrawn

Sauveur Uturned entirely to the side of mathematics and reshy

solved to teach it14 With the help of several influential

friends he soon achieved a kind of celebrity and being

when he was still only twenty-three years old the geometer

in fashion he attracted Prince Eugene as a student IS

10tTacques Rouhaul t (1620-1675) physicist and the fi rst important advocate of the then new study of Natural Science tr was a uthor of a Trai te de Physique which appearing in 1671 hecame tithe standard textbook on Cartesian natural studies (Scherchen Nature of Music p 25)

11 Fontenelle EIage p 99


13Fontenelle IrEloge p 100 fl bullbullbull Un medecin a presque aussi sauvant affaire a limagination de ses malades qUa leur poitrine bullbullbull

14Ibid bull n il se taurna entierement du cote des mathematiques amp se rasolut a les enseigner

15F~tis Biographie universelle sv nSauveur

5

An anecdote about the description of Sauveur at

this time in his life related by Fontenelle are parti shy

cularly interesting as they shed indirect Ii Ppt on the

character of his writings

A stranger of the hi~hest rank wished to learn from him the geometry of Descartes but the master did not know it at all He asked for a week to make arrangeshyments looked for the book very quickly and applied himself to the study of it even more fot tho pleasure he took in it than because he had so little time to spate he passed the entire ni~hts with it sometimes letting his fire go out--for it was winter--and was by morning chilled with cold without nnticing it

He read little hecause he had hardly leisure for it but he meditated a great deal because he had the talent as well as the inclination He withdtew his attention from profitless conversations to nlace it better and put to profit even the time of going and coming in the streets He guessed when he had need of it what he would have found in the books and to spare himself the trouble of seeking them and studyshying them he made things up16

If the published papers display a single-mindedness)

a tight organization an absence of the speculative and the

superfluous as well as a paucity of references to other

writers either of antiquity or of the day these qualities

will not seem inconsonant with either the austere simplicity

16Fontenelle Elope1I p 101 Un etranper de la premiere quaIl te vOlJlut apprendre de lui la geometrie de Descartes mais Ie ma1tre ne la connoissoit point- encore II demands hui~ joyrs pour sarranger chercha bien vite Ie livre se mit a letudier amp plus encore par Ie plaisir quil y prenoit que parce quil ~avoit pas de temps a perdre il y passoit les nuits entieres laissoit quelquefois ~teindre son feu Car c~toit en hiver amp se trouvoit Ie matin transi de froid sans sen ~tre apper9u

II lis~it peu parce quil nlen avoit guere Ie loisir mais il meditoit beaucoup parce ~lil en avoit Ie talent amp Ie gout II retiroit son attention des conversashy

tions inutiles pour la placer mieux amp mettoit a profit jusquau temps daller amp de venir par les rues II devinoit quand il en avoit besoin ce quil eut trouve dans les livres amp pour separgner la peine de les chercher amp de les etudier il se lesmiddotfaisoit

6

of the Sauveur of this anecdote or the disinclination he

displays here to squander time either on trivial conversashy

tion or even on reading It was indeed his fondness for

pared reasoning and conciseness that had made him seem so

unsuitable a candidate for the profession of medicine--the

bishop ~~d judged

LIII L hh w)uLd hnvu Loo Hluel tlouble nucetHJdllll-~ In 1 L with an understanding which was great but which went too directly to the mark and did not make turns with tIght rat iocination s but dry and conci se wbere words were spared and where the few of them tha t remained by absolute necessity were devoid of grace l

But traits that might have handicapped a physician freed

the mathematician and geometer for a deeper exploration

of his chosen field

However pure was his interest in mathematics Sauveur

did not disdain to apply his profound intelligence to the

analysis of games of chance18 and expounding before the

king and queen his treatment of the game of basset he was

promptly commissioned to develop similar reductions of

17 Ibid pp 99-100 II jugea quil auroit trop de peine a y reussir avec un grand savoir mais qui alloit trop directement au but amp ne p1enoit point de t rll1S [tvee dla rIi1HHHlIHlIorHl jllItI lIlnl HHn - cOlln11 011 1 (11 pnlo1nl

etoient epargnees amp ou Ie peu qui en restoit par- un necessite absolue etoit denue de prace

lSNor for that matter did Blaise Pascal (1623shy1662) or Pierre Fermat (1601-1665) who only fifty years before had co-founded the mathematical theory of probashyhility The impetus for this creative activity was a problem set for Pascal by the gambler the ChevalIer de Ivere Game s of chance thence provided a fertile field for rna thematical Ena lysis W W Rouse Ball A Short Account of the Hitory of Mathematics (New York Dover Publications 1960) p 285

guinguenove hoca and lansguenet all of which he was

successful in converting to algebraic equations19

In 1680 he obtained the title of master of matheshy

matics of the pape boys of the Dauphin20 and in the next

year went to Chantilly to perform experiments on the waters21

It was durinp this same year that Sauveur was first mentioned ~

in the Histoire de lAcademie Royale des Sciences Mr

De La Hire gave the solution of some problems proposed by

Mr Sauveur22 Scherchen notes that this reference shows

him to he already a member of the study circle which had

turned its attention to acoustics although all other

mentions of Sauveur concern mechanical and mathematical

problems bullbullbull until 1700 when the contents listed include

acoustics for the first time as a separate science 1I 23

Fontenelle however ment ions only a consuming int erest

during this period in the theory of fortification which

led him in an attempt to unite theory and practice to

~o to Mons during the siege of that city in 1691 where

flhe took part in the most dangerous operations n24

19Fontenelle Elopetr p 102

20Fetis Biographie universelle sv Sauveur

2lFontenelle nElove p 102 bullbullbull pour faire des experiences sur les eaux

22Histoire de lAcademie Royale des SCiences 1681 eruoted in Scherchen Nature of MusiC p 26 Mr [SiC] De La Hire donna la solution de quelques nroblemes proposes par Mr [SiC] Sauveur

23Scherchen Nature of Music p 26 The dates of other references to Sauveur in the Ilistoire de lAcademie Hoyale des Sciences given by Scherchen are 1685 and 1696

24Fetis Biographie universelle s v Sauveur1f

8

In 1686 he had obtained a professorship of matheshy

matics at the Royal College where he is reported to have

taught his students with great enthusiasm on several occashy

25 olons and in 1696 he becnme a memhAI of tho c~~lnln-~

of Paris 1hat his attention had by now been turned to

acoustical problems is certain for he remarks in the introshy

ductory paragraphs of his first M~moire (1701) in the

hadT~emoires de l Academie Royale des Sciences that he

attempted to write a Treatise of Speculative Music26

which he presented to the Royal College in 1697 He attribshy

uted his failure to publish this work to the interest of

musicians in only the customary and the immediately useful

to the necessity of establishing a fixed sound a convenient

method for doing vmich he had not yet discovered and to

the new investigations into which he had pursued soveral

phenomena observable in the vibration of strings 27

In 1703 or shortly thereafter Sauveur was appointed

examiner of engineers28 but the papers he published were

devoted with but one exception to acoustical problems

25 Pontenelle Eloge lip 105

26The division of the study of music into Specu1ative ~hJsi c and Practical Music is one that dates to the Greeks and appears in the titles of treatises at least as early as that of Walter Odington and Prosdocimus de Beldemandis iarvard Dictionary of Music 2nd ed sv Musical Theory and If Greece

27 J Systeme General des ~oseph Sauveur Interval~cs shydes Sons et son application a taus les Systemes ct a 7()~lS ]cs Instruments de NIusi que Histoire de l Acnrl6rnin no 1 ( rl u ~l SCiences 1701 (Amsterdam Chez Pierre Mortl(r--li~)6) pp 403-498 see Vol II pp 1-97 below

28Fontenel1e iloge p 106

9

It has been noted that Sauveur was mentioned in

1681 1685 and 1696 in the Histoire de lAcademie 29 In

1700 the year in which Acoustics was first accorded separate

status a full report was given by Fontene1le on the method

SU1JveUr hnd devised fol the determination of 8 fl -xo(l p tch

a method wtl1ch he had sought since the abortive aLtempt at

a treatise in 1696 Sauveurs discovery was descrihed by

Scherchen as the first of its kind and for long it was

recognized as the surest method of assessing vibratory

frequenci es 30

In the very next year appeared the first of Sauveurs

published Memoires which purported to be a general system

of intervals and its application to all the systems and

instruments of music31 and in which according to Scherchen

several treatises had to be combined 32 After an introducshy

tion of several paragraphs in which he informs his readers

of the attempts he had previously made in explaining acousshy

tical phenomena and in which he sets forth his belief in

LtlU pOBulblJlt- or a science of sound whl~h he dubbol

29Each of these references corresponds roughly to an important promotion which may have brought his name before the public in 1680 he had become the master of mathematics of the page-boys of the Dauphin in 1685 professo~ at the Royal College and in 1696 a member of the Academie


31 Systeme General des Interva1les des Sons et son application ~ tous les Systemes et a tous les I1struments de Musique

32Scherchen Nature of MusiC p 31

10

Acoustics 33 established as firmly and capable of the same

perfection as that of Optics which had recently received

8110h wide recoenition34 he proceeds in the first sectIon

to an examination of the main topic of his paper--the

ratios of sounds (Intervals)

In the course of this examination he makes liboral

use of neologism cOining words where he feels as in 0

virgin forest signposts are necessary Some of these

like the term acoustics itself have been accepted into

regular usage

The fi rRt V[emoire consists of compressed exposi tory

material from which most of the demonstrations belonging

as he notes more properly to a complete treatise of

acoustics have been omitted The result is a paper which

might have been read with equal interest by practical

musicians and theorists the latter supplying by their own

ingenuity those proofs and explanations which the former

would have judged superfluous

33Although Sauveur may have been the first to apply the term acou~~ticsft to the specific investigations which he undertook he was not of course the first to study those problems nor does it seem that his use of the term ff AC 011 i t i c s n in 1701 wa s th e fi r s t - - the Ox f OT d D 1 c t i ()n1 ry l(porL~ occurroncos of its use H8 oarly us l(jU~ (l[- James A H Murray ed New En lish Dictionar on 1Iistomiddotrl shycal Principles 10 vols Oxford Clarendon Press 1888-1933) reprint ed 1933

34Since Newton had turned his attention to that scie)ce in 1669 Newtons Optics was published in 1704 Ball History of Mathemati cs pp 324-326

11

In the first section35 the fundamental terminology

of the science of musical intervals 1s defined wIth great

rigor and thoroughness Much of this terminology correshy

nponds with that then current althol1ph in hln nltnrnpt to

provide his fledgling discipline with an absolutely precise

and logically consistent vocabulary Sauveur introduced a

great number of additional terms which would perhaps have

proved merely an encumbrance in practical use

The second section36 contains an explication of the

37first part of the first table of the general system of

intervals which is included as an appendix to and really

constitutes an epitome of the Memoire Here the reader

is presented with a method for determining the ratio of

an interval and its name according to the system attributed

by Sauveur to Guido dArezzo

The third section38 comprises an intromlction to

the system of 43 meridians and 301 heptameridians into

which the octave is subdivided throughout this Memoire and

its successors a practical procedure by which the number

of heptameridians of an interval may be determined ~rom its

ratio and an introduction to Sauveurs own proposed

35sauveur nSyst~me Genera111 pp 407-415 see below vol II pp 4-12

36Sauveur Systeme General tf pp 415-418 see vol II pp 12-15 below

37Sauveur Systeme Genltsectral p 498 see vobull II p ~15 below

38 Sallveur Syst-eme General pp 418-428 see

vol II pp 15-25 below

12

syllables of solmization comprehensive of the most minute

subdivisions of the octave of which his system is capable

In the fourth section39 are propounded the division

and use of the Echometer a rule consisting of several

dl vldod 1 ines which serve as seal es for measuJing the durashy

tion of nOlln(lS and for finding their lntervnls nnd

ratios 40 Included in this Echometer4l are the Chronome lot f

of Loulie divided into 36 equal parts a Chronometer dividBd

into twelfth parts and further into sixtieth parts (thirds)

of a second (of ti me) a monochord on vmich all of the subshy

divisions of the octave possible within the system devised

by Sauveur in the preceding section may be realized a

pendulum which serves to locate the fixed soundn42 and

scales commensurate with the monochord and pendulum and

divided into intervals and ratios as well as a demonstrashy

t10n of the division of Sauveurs chronometer (the only

actual demonstration included in the paper) and directions

for making use of the Echometer

The fifth section43 constitutes a continuation of

the directions for applying Sauveurs General System by

vol 39Sauveur Systeme General pp

II pp 26-33 below 428-436 see

40Sauveur Systeme General II p 428 see vol II p 26 below

41Sauveur IISysteme General p 498 see vol II p 96 beow for an illustration

4 2 (Jouveur C tvys erne G 1enera p 4 31 soo vol Il p 28 below

vol 43Sauveur Syst~me General pp


13

means of the Echometer in the study of any of the various

established systems of music As an illustration of the

method of application the General System is applied to

the regular diatonic system44 to the system of meun semlshy

tones to the system in which the octave is divided into

55 parta45 and to the systems of the Greeks46 and

ori ontal s 1

In the sixth section48 are explained the applicashy

tions of the General System and Echometer to the keyboards

of both organ and harpsichord and to the chromatic system

of musicians after which are introduced and correlated

with these the new notes and names proposed by Sauveur

49An accompanying chart on which both the familiar and

the new systems are correlated indicates the compasses of

the various voices and instruments

In section seven50 the General System is applied

to Plainchant which is understood by Sauveur to consist

44Sauveur nSysteme General pp 438-43G see vol II pp 34-37 below

45Sauveur Systeme General pp 441-442 see vol II pp 39-40 below

I46Sauveur nSysteme General pp 442-444 see vol II pp 40-42 below


I

48Sauveur Systeme General n pp 447-456 see vol II pp 45-53 below

49 Sauveur Systeme General p 498 see

vol II p 97 below

50 I ISauveur Systeme General n pp 456-463 see


14

of that sort of vo cal music which make s us e only of the

sounds of the diatonic system without modifications in the

notes whether they be longs or breves5l Here the old

names being rejected a case is made for the adoption of

th e new ones which Sauveur argues rna rk in a rondily

cOHlprohonulhle mannor all the properties of the tUlIlpolod

diatonic system n52

53The General System is then in section elght

applied to music which as opposed to plainchant is

defined as the sort of melody that employs the sounds of

the diatonic system with all the possible modifications-shy

with their sharps flats different bars values durations

rests and graces 54 Here again the new system of notes

is favored over the old and in the second division of the

section 55 a new method of representing the values of notes

and rests suitable for use in conjunction with the new notes

and nruooa 1s put forward Similarly the third (U visionbtl

contains a proposed method for signifying the octaves to

5lSauveur Systeme General p 456 see vol II p 53 below

52Sauveur Systeme General p 458 see vol II

p 55 below 53Sauveur Systeme General If pp 463-474 see


54Sauveur Systeme Gen~ral p 463 see vol II p 60 below

55Sauveur Systeme I I seeGeneral pp 468-469 vol II p 65 below

I I 56Sauveur IfSysteme General pp 469-470 see vol II p 66 below

15

which the notes of a composition belong while the fourth57

sets out a musical example illustrating three alternative

methot1s of notating a melody inoluding directions for the

precise specifioation of its meter and tempo58 59Of Harmonics the ninth section presents a

summary of Sauveurs discoveries about and obsepvations

concerning harmonies accompanied by a table60 in which the

pitches of the first thirty-two are given in heptameridians

in intervals to the fundamental both reduced to the compass

of one octave and unreduced and in the names of both the

new system and the old Experiments are suggested whereby

the reader can verify the presence of these harmonics in vishy

brating strings and explanations are offered for the obshy

served results of the experiments described Several deducshy

tions are then rrade concerning the positions of nodes and

loops which further oxplain tho obsorvod phonom(nn 11nd

in section ten6l the principles distilled in the previous

section are applied in a very brief treatment of the sounds

produced on the marine trumpet for which Sauvellr insists

no adequate account could hitherto have been given

57Sauveur Systeme General rr pp 470-474 see vol II pp 67-70 below

58Sauveur Systeme Gen~raln p 498 see vol II p 96 below

59S8uveur Itsysteme General pp 474-483 see vol II pp 70-80 below

60Sauveur Systeme General p 475 see vol II p 72 below

6lSauveur Systeme Gen~ral II pp 483-484 Bee vo bull II pp 80-81 below

16

In the eleventh section62 is presented a means of

detormining whether the sounds of a system relate to any

one of their number taken as fundamental as consonances

or dissonances 63The twelfth section contains two methods of obshy

tain1ng exactly a fixed sound the first one proposed by

Mersenne and merely passed on to the reader by Sauveur

and the second proposed bySauveur as an alternative

method capable of achieving results of greater exactness

In an addition to Section VI appended to tho

M~moire64 Sauveur attempts to bring order into the classishy

fication of vocal compasses and proposes a system of names

by which both the oompass and the oenter of a voice would

be made plain

Sauveurs second Memoire65 was published in the

next year and consists after introductory passages on

lithe construction of the organ the various pipe-materials

the differences of sound due to diameter density of matershy

iul shapo of the pipe and wind-pressure the chElructor1ntlcB


63Sauveur Systeme General If pp 488-493 see vol II pp 84-89 below


65 Joseph Sauveur Application des sons hnrmoniques a 1a composition des Jeux dOrgues Memoires de lAcademie Hoale des Sciences Annee 1702 (Amsterdam Chez Pierre ~ortier 1736 pp 424-451 see vol II pp 98-127 below

17

of various stops a rrl dimensions of the longest and shortest

organ pipes66 in an application of both the General System

put forward in the previous Memoire and the theory of harshy

monics also expounded there to the composition of organ

stops The manner of marking a tuning rule (diapason) in Iaccordance with the general monochord of the first Momoiro

and of tuning the entire organ with the rule thus obtained

is given in the course of the description of the varlous

types of stops As corroboration of his observations

Sauveur subjoins descriptions of stops composed by Mersenne

and Nivers67 and concludes his paper with an estima te of

the absolute range of sounds 68

69The third Memoire which appeared in 1707 presents

a general method for forming the tempered systems of music

and lays down rules for making a choice among them It

contains four divisions The first of these70 sets out the

familiar disadvantages of the just diatonic system which

result from the differences in size between the various inshy

tervuls due to the divislon of the ditone into two unequal

66scherchen Nature of Music p 39

67 Sauveur II Application p 450 see vol II pp 123-124 below

68Sauveu r IIApp1ication II p 451 see vol II pp 124-125 below

69 IJoseph Sauveur Methode generale pour former des

systemes temperas de Musique et de celui quon rloit snlvoH ~enoi res de l Academie Ho ale des Sciences Annee 1707

lmsterdam Chez Pierre Mortier 1736 PP 259282~ see vol II pp 128-153 below

70Sauveur Methode Generale pp 259-2()5 see vol II pp 128-153 below

18

rltones and a musical example is nrovided in which if tho

ratios of the just diatonic system are fnithfu]1y nrniorvcd

the final ut will be hipher than the first by two commAS

rrho case for the inaneqllBcy of the s tric t dIn ton 10 ~ys tom

havinr been stat ad Sauveur rrooeeds in the second secshy

tl()n 72 to an oxposl tlon or one manner in wh teh ternpernd

sys terns are formed (Phe til ird scctinn73 examines by means

of a table74 constructed for the rnrrnose the systems which

had emerged from the precedin~ analysis as most plausible

those of 31 parts 43 meriltiians and 55 commas as well as

two--the just system and thnt of twelve equal semitones-shy

which are included in the first instance as a basis for

comparison and in the second because of the popula-rity

of equal temperament due accordi ng to Sauve) r to its

simp1ici ty In the fa lJrth section75 several arpurlents are

adriuced for the selection of the system of L1~) merIdians

as ttmiddote mos t perfect and the only one that ShOl11d be reshy

tained to nrofi t from all the advan tages wrdch can be

71Sauveur U thode Generale p 264 see vol II p 13gt b(~J ow

72Sauveur Ii0thode Gene-rale II pp 265-268 see vol II pp 135-138 below

7 ) r bull t ] I J [1 11 V to 1 rr IVI n t1 00 e G0nera1e pp 26f3-2B sC(~

vol II nne 138-J47 bnlow

4 1f1gtth -l)n llvenr lle Ot Ie Generale pp 270-21 sen

vol II p 15~ below

75Sauvel1r Me-thode Generale If np 278-2S2 see va II np 147-150 below

19

drawn from the tempored systems in music and even in the

whole of acoustics76

The fourth MemOire published in 1711 is an

answer to a publication by Haefling [siC] a musicologist

from Anspach bull bull bull who proposed a new temperament of 50

8degrees Sauveurs brief treatment consists in a conshy

cise restatement of the method by which Henfling achieved

his 50-fold division his objections to that method and 79

finally a table in which a great many possible systems

are compared and from which as might be expected the

system of 43 meridians is selected--and this time not on~y

for the superiority of the rna thematics which produced it

but also on account of its alleged conformity to the practice

of makers of keyboard instruments

rphe fifth and last Memoire80 on acoustics was pubshy

lished in 171381 without tne benefit of final corrections

76 IISauveur Methode Generale p 281 see vol II

p 150 below

77 tToseph Sauveur Table geneTale des Systemes tem-Ell

per~s de Musique Memoires de lAcademie Ro ale des Sciences Anne6 1711 (Amsterdam Chez Pierre Mortier 1736 pp 406shy417 see vol II pp 154-167 below

78scherchen Nature of Music pp 43-44

79sauveur Table gen~rale p 416 see vol II p 167 below

130Jose)h Sauveur Rapport des Sons ciet t) irmiddot r1es Q I Ins trument s de Musique aux Flampches des Cordc~ nouv( ~ shyC1~J8rmtnatl()n des Sons fixes fI Memoires de 1 t Acd~1i e f() deS Sciences Ann~e 1713 (Amsterdam Chez Pie~f --r~e-~~- 1736) pp 433-469 see vol II pp 168-203 belllJ

81According to Scherchen it was cOlrL-l~-tgt -1 1shy

c- t bull bull bull did not appear until after his death in lf W0n it formed part of Fontenelles obituary n1t_middot~~a I (S~

20

It is subdivided into seven sections the first82 of which

sets out several observations on resonant strings--the material

diameter and weight are conside-red in their re1atlonship to

the pitch The second section83 consists of an attempt

to prove that the sounds of the strings of instruments are

1t84in reciprocal proportion to their sags If the preceding

papers--especially the first but the others as well--appeal

simply to the readers general understanning this section

and the one which fol1ows85 demonstrating that simple

pendulums isochronous with the vibrati~ns ~f a resonant

string are of the sag of that stringu86 require a familshy

iarity with mathematical procedures and principles of physics

Natu re of 1~u[ic p 45) But Vinckel (Tgt1 qusik in Geschichte und Gegenllart sv Sauveur cToseph) and Petis (Biopraphie universelle sv SauvGur It ) give the date as 1713 which is also the date borne by t~e cony eXR~ined in the course of this study What is puzzling however is the statement at the end of the r~erno ire th it vIas printed since the death of IIr Sauveur~ who (~ied in 1716 A posshysible explanation is that the Memoire completed in 1713 was transferred to that volume in reprints of the publi shycations of the Academie

82sauveur Rapport pp 4~)Z)-1~~B see vol II pp 168-173 below

83Sauveur Rapporttr pp 438-443 see vol II Pp 173-178 below

04 n3auvGur Rapport p 43B sec vol II p 17~)

how

85Sauveur Ranport tr pp 444-448 see vol II pp 178-181 below

86Sauveur ftRanport I p 444 see vol II p 178 below

21

while the fourth87 a method for finding the number of

vibrations of a resonant string in a secondn88 might again

be followed by the lay reader The fifth section89 encomshy

passes a number of topics--the determination of fixed sounds

a table of fixed sounds and the construction of an echometer

Sauveur here returns to several of the problems to which he

addressed himself in the M~mo~eof 1701 After proposing

the establishment of 256 vibrations per second as the fixed

pitch instead of 100 which he admits was taken only proshy90visionally he elaborates a table of the rates of vibration

of each pitch in each octave when the fixed sound is taken at

256 vibrations per second The sixth section9l offers

several methods of finding the fixed sounds several more

difficult to construct mechanically than to utilize matheshy

matically and several of which the opposite is true The 92last section presents a supplement to the twelfth section

of the Memoire of 1701 in which several uses were mentioned

for the fixed sound The additional uses consist generally

87Sauveur Rapport pp 448-453 see vol II pp 181-185 below

88Sauveur Rapport p 448 see vol II p 181 below

89sauveur Rapport pp 453-458 see vol II pp 185-190 below




22

in finding the number of vibrations of various vibrating

bodies includ ing bells horns strings and even the

epiglottis

One further paper--devoted to the solution of a

geometrical problem--was published by the Academie but

as it does not directly bear upon acoustical problems it

93hus not boen included here

It can easily be discerned in the course of

t~is brief survey of Sauveurs acoustical papers that

they must have been generally comprehensible to the lay-shy94non-scientific as well as non-musical--reader and

that they deal only with those aspects of music which are

most general--notational systems systems of intervals

methods for measuring both time and frequencies of vi shy

bration and tne harmonic series--exactly in fact

tla science superior to music u95 (and that not in value

but in logical order) which has as its object sound

in general whereas music has as its object sound

in so fa r as it is agreeable to the hearing u96 There

93 II shyJoseph Sauveur Solution dun Prob]ome propose par M de Lagny Me-motres de lAcademie Royale des Sciencefi Annee 1716 (Amsterdam Chez Pierre Mort ier 1736) pp 33-39

94Fontenelle relates however that Sauveur had little interest in the new geometries of infinity which were becoming popular very rapidly but that he could employ them where necessary--as he did in two sections of the fifth MeMoire (Fontenelle ffEloge p 104)



23

is no attempt anywhere in the corpus to ground a science

of harmony or to provide a basis upon which the merits

of one style or composition might be judged against those

of another style or composition

The close reasoning and tight organization of the

papers become the object of wonderment when it is discovered

that Sauveur did not write out the memoirs he presented to

th(J Irnrlomle they being So well arranged in hill hond Lhlt

Ile had only to let them come out ngrl

Whether or not he was deaf or even hard of hearing

he did rely upon the judgment of a great number of musicians

and makers of musical instruments whose names are scattered

throughout the pages of the texts He also seems to have

enjoyed the friendship of a great many influential men and

women of his time in spite of a rather severe outlook which

manifests itself in two anecdotes related by Fontenelle

Sauveur was so deeply opposed to the frivolous that he reshy

98pented time he had spent constructing magic squares and

so wary of his emotions that he insisted on closjn~ the

mi-tr-riLtge contr-act through a lawyer lest he be carrIed by

his passions into an agreement which might later prove

ur 3Lli table 99

97Fontenelle Eloge p 105 If bullbullbull 81 blen arr-nngees dans sa tete qu il n avoit qu a les 10 [sirsnrttir n

98 Ibid p 104 Mapic squares areiumbr- --qni 3

_L vrhicn Lne sums of rows columns and d iaponal ~l _ YB

equal Ball History of Mathematics p 118

99Fontenelle Eloge p 104

24

This rather formidable individual nevertheless

fathered two sons by his first wife and a son (who like

his father was mute until the age of seven) and a daughter

by a second lOO

Fontenelle states that although Ur Sauveur had

always enjoyed good health and appeared to be of a robust

Lompor-arncn t ho wai currlod away in two days by u COI1post lon

1I101of the chost he died on July 9 1716 in his 64middotth year

100Ib1d p 107

101Ibid Quolque M Sauveur eut toujours Joui 1 dune bonne sante amp parut etre dun temperament robuste

11 fut emporte en dellx iours par nne fIllxfon de poitr1ne 11 morut Ie 9 Jul11ct 1716 en sa 64me ann6e

CHAPTER I

THE MEASUREMENT OF TI~I~E

It was necessary in the process of establ j~Jhlng

acoustics as a true science superior to musicu for Sauveur

to devise a system of Bcales to which the multifarious pheshy

nomena which constituted the proper object of his study

might be referred The aggregation of all the instruments

constructed for this purpose was the Echometer which Sauveur

described in the fourth section of the Memoire of 1701 as

U a rule consisting of several divided lines which serve as

scales for measuring the duration of sounds and for finding

their intervals and ratios I The rule is reproduced at

t-e top of the second pInte subioin~d to that Mcmn i re2

and consists of six scales of ~nich the first two--the

Chronometer of Loulie (by universal inches) and the Chronshy

ometer of Sauveur (by twelfth parts of a second and thirds V l

)-shy

are designed for use in the direct measurement of time The

tnird the General Monochord 1s a scale on ihich is

represented length of string which will vibrate at a given

1 l~Sauveur Systeme general II p 428 see vol l

p 26 below

2 ~ ~ Sauveur nSysteme general p 498 see vol I ~

p 96 below for an illustration

3 A third is the sixtieth part of a secon0 as tld

second is the sixtieth part of a minute

25

26

interval from a fundamental divided into 43 meridians

and 301 heptameridians4 corresponding to the same divisions

and subdivisions of the octave lhe fourth is a Pendulum

for the fixed sound and its construction is based upon

tho t of the general Monochord above it The fi ftl scal e

is a ru1e upon which the name of a diatonic interval may

be read from the number of meridians and heptameridians

it contains or the number of meridians and heptflmerldlans

contained can be read from the name of the interval The

sixth scale is divided in such a way that the ratios of

sounds--expressed in intervals or in nurnhers of meridians

or heptameridians from the preceding scale--can be found

Since the third fourth and fifth scales are constructed

primarily for use in the measurement tif intervals they

may be considered more conveniently under that head while

the first and second suitable for such measurements of

time as are usually made in the course of a study of the

durat10ns of individual sounds or of the intervals between

beats in a musical comnosltion are perhaps best

separated from the others for special treatment

The Chronometer of Etienne Loulie was proposed by that

writer in a special section of a general treatise of music

as an instrument by means of which composers of music will be able henceforth to mark the true movement of their composition and their airs marked in relation to this instrument will be able to be performed in

4Meridians and heptameridians are the small euqal intervals which result from Sauveurs logarithmic division of the octave into 43 and 301 parts

27

their absenQe as if they beat the measure of them themselves )

It is described as composed of two parts--a pendulum of

adjustable length and a rule in reference to which the

length of the pendulum can be set

The rule was

bull bull bull of wood AA six feet or 72 inches high about ten inches wide and almost an inch thick on a flat side of the rule is dravm a stroke or line BC from bottom to top which di vides the width into two equal parts on this stroke divisions are marked with preshycision from inch to inch and at each point of division there is a hole about two lines in diaMeter and eight lines deep and these holes are 1)Tovlded wi th numbers and begin with the lowest from one to seventy-two

I have made use of the univertal foot because it is known in all sorts of countries

The universal foot contains twelve inches three 6 lines less a sixth of a line of the foot of the King

5 Et ienne Lou1 iEf El ~m en t S ou nrinc -t Dt- fJ de mn s i que I

ml s d nns un nouvel ordre t ro= -c ~I ~1i r bull bull bull Avee ] e oS ta~n e 1a dC8cription et llu8ar~e (111 chronom~tto Oil l Inst YlllHlcnt de nouvelle invention par 1 e moyen dl1qllc] In-- COtrnosi teurs de musique pourront d6sormais maTq1J2r In verit3hle mouverlent rie leurs compositions f et lctlrs olJvYao~es naouez par ranport h cst instrument se p()urrort cx6cutcr en leur absence co~~e slils en hattaient ~lx-m6mes 1q mPsu~e (Paris C Ballard 1696) pp 71-92 Le Chrono~et~e est un Instrument par le moyen duque1 les Compositeurs de Musique pourront desormais marquer Je veriLable movvement de leur Composition amp leur Airs marquez Dar rapport a cet Instrument se pourront executer en leur ab3ence comme sils en battoient eux-m-emes 1a Iftesure The qlJ ota tion appears on page 83

6Ibid bull La premier est un ~e e de bois AA haute de 5ix pieurods ou 72 pouces Ihrge environ de deux Douces amp opaisse a-peu-pres d un pouee sur un cote uOt de la RegIe ~st tire un trai t ou ligna BC de bas (~n oFl()t qui partage egalement 1amp largeur en deux Sur ce trait sont IT~rquez avec exactitude des Divisions de pouce en pouce amp ~ chaque point de section 11 y a un trou de deux lignes de diamette environ l de huit li~nes de rrrOfnnr1fllr amp ces trons sont cottez paf chiffres amp comrncncent par 1e plus bas depuls un jusqufa soixante amp douze

Je me suis servi du Pied universel pnrce quill est connu cans toutes sortes de pays

Le Pied universel contient douze p0uces trois 1ignes moins un six1eme de ligna de pied de Roy

28

It is this scale divided into universal inches

without its pendulum which Sauveur reproduces as the

Chronometer of Loulia he instructs his reader to mark off

AC of 3 feet 8~ lines7 of Paris which will give the length

of a simple pendulum set for seoonds

It will be noted first that the foot of Paris

referred to by Sauveur is identical to the foot of the King

rt) trro(l t 0 by Lou 11 () for th e unj vo-rHll foo t 18 e qua t od hy

5Loulie to 12 inches 26 lines which gi ves three universal

feet of 36 inches 8~ lines preoisely the number of inches

and lines of the foot of Paris equated by Sauveur to the

36 inches of the universal foot into which he directs that

the Chronometer of Loulie in his own Echometer be divided

In addition the astronomical inches referred to by Sauveur

in the Memoire of 1713 must be identical to the universal

inches in the Memoire of 1701 for the 36 astronomical inches

are equated to 36 inches 8~ lines of the foot of Paris 8

As the foot of the King measures 325 mm9 the universal

foot re1orred to must equal 3313 mm which is substantially

larger than the 3048 mm foot of the system currently in

use Second the simple pendulum of which Sauveur speaks

is one which executes since the mass of the oscillating

body is small and compact harmonic motion defined by

7A line is the twelfth part of an inch

8Sauveur Rapport n p 434 see vol II p 169 below

9Rosamond Harding The Ori ~ins of Musical Time and Expression (London Oxford University Press 1938 p 8

29

Huygens equation T bull 2~~ --as opposed to a compounn pe~shydulum in which the mass is distrihuted lO Third th e period

of the simple pendulum described by Sauveur will be two

seconds since the period of a pendulum is the time required 11

for a complete cycle and the complete cycle of Sauveurs

pendulum requires two seconds

Sauveur supplies the lack of a pendulum in his

version of Loulies Chronometer with a set of instructions

on tho correct use of the scale he directs tho ronclol to

lengthen or shorten a simple pendulum until each vibration

is isochronous with or equal to the movement of the hand

then to measure the length of this pendulum from the point

of suspension to the center of the ball u12 Referring this

leneth to the first scale of the Echometer--the Chronometer

of Loulie--he will obtain the length of this pendultLn in Iuniversal inches13 This Chronometer of Loulie was the

most celebrated attempt to make a machine for counting

musical ti me before that of Malzel and was Ufrequently

referred to in musical books of the eighte3nth centuryu14

Sir John Hawkins and Alexander Malcolm nbo~h thought it

10Dictionary of PhYSiCS H J Gray ed (New York John Wil ey amp Sons Inc 1958) s v HPendulum

llJohn Backus The Acollstlcal FoundatIons of Muic (New York W W Norton amp Company Inc 1969) p 25

12Sauveur trSyst~me General p 432 see vol ~ p 30 below

13Ibid bull

14Hardlng 0 r i g1nsmiddot p 9 bull

30

~ 5 sufficiently interesting to give a careful description Ill

Nonetheless Sauveur dissatisfied with it because the

durations of notes were not marked in any known relation

to the duration of a second the periods of vibration of

its pendulum being flro r the most part incommensurable with

a secondu16 proceeded to construct his own chronometer on

the basis of a law stated by Galileo Galilei in the

Dialogo sopra i due Massimi Slstemi del rTondo of 1632

As to the times of vibration of oodies suspended by threads of different lengths they bear to each other the same proportion as the square roots of the lengths of the thread or one might say the lengths are to each other as the squares of the times so that if one wishes to make the vibration-time of one pendulum twice that of another he must make its suspension four times as long In like manner if one pendulum has a suspension nine times as long as another this second pendulum will execute three vibrations durinp each one of the rst from which it follows thll t the lengths of the 8uspendinr~ cords bear to each other tho (invcrne) ratio [to] the squares of the number of vishybrations performed in the same time 17

Mersenne bad on the basis of th is law construc ted

a table which correlated the lengths of a gtendulum and half

its period (Table 1) so that in the fi rst olumn are found

the times of the half-periods in seconds~n the second

tt~e square of the corresponding number fron the first

column to whic h the lengths are by Galileo t slaw

151bid bull

16 I ISauveur Systeme General pp 435-436 seD vol

r J J 33 bel OVI bull

17Gal ileo Galilei nDialogo sopra i due -a 8tr~i stCflli del 11onoo n 1632 reproduced and transl at-uri in

fl f1Yeampsv-ry of Vor1d SCience ed Dagobert Runes (London Peter Owen 1962) pp 349-350

31

TABLE 1

TABLE OF LENGTHS OF STRINGS OR OFl eHRONOlYTE TERS

[FROM MERSENNE HARMONIE UNIVEHSELLE]

I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016

f)1B71middot25 625 tJ ~ shy ~~

26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865

proportional and in the third the lengths of a pendulum

with the half-periods indicated in the first column

For example if you want to make a chronomoter which ma rks the fourth of a minute (of an hOllr) with each of its excursions the 15th number of ttlC first column yenmich signifies 15 seconds will show in th~ third [column] that the string ought to be Lfe~t lonr to make its exc1rsions each in a quartfr emiddotf ~ minute or if you wish to take a sma1ler 8X-1iiijlIC

because it would be difficult to attach a stY niT at such a height if you wi sh the excursion to last

32

2 seconds the 2nd number of the first column shows the second number of the 3rd column namely 14 so that a string 14 feet long attached to a nail will make each of its excursions 2 seconds 18

But Sauveur required an exnmplo smallor still for

the Chronometer he envisioned was to be capable of measurshy

ing durations smaller than one second and of measuring

more closely than to the nearest second

It is thus that the chronometer nroposed by Sauveur

was divided proportionally so that it could be read in

twelfths of a second and even thirds 19 The numbers of

the points of division at which it was necessary for

Sauveur to arrive in the chronometer ruled in twelfth parts

of a second and thirds may be determined by calculation

of an extension of the table of Mersenne with appropriate

adjustments

If the formula T bull 2~ is applied to the determinashy

tion of these point s of di vision the constan ts 2 1 and r-

G may be represented by K giving T bull K~L But since the

18Marin Mersenne Harmonie unive-sel1e contenant la theorie et 1s nratiaue de la mnSi(l1Je ( Paris 1636) reshyprint ed Edi tions du Centre ~~a tional de In Recherche Paris 1963)111 p 136 bullbullbull pflr exe~ple si lon veut faire vn Horologe qui marque Ie qllar t G r vne minute ci neura pur chacun de 8es tours Ie IS nOiehre 0e la promiere colomne qui signifie ISH monstrera dins 1[ 3 cue c-Jorde doit estre longue de 787-1 p01H ire ses tours t c1tiCUn d 1 vn qua Tmiddott de minute au si on VE1tt prendre vn molndrc exenple Darce quil scroit dtff~clle duttachcr vne corde a vne te11e hautelJr s1 lon VC11t quo Ie tour d1Jre 2 secondes 1 e 2 nombre de la premiere columne monstre Ie second nambre do la 3 colomne ~ scavoir 14 de Borte quune corde de 14 pieds de long attachee a vn clou fera chacun de ses tours en 2

19As a second is the sixtieth part of a minute so the third is the sixtieth part of a second

33

length of the pendulum set for seconds is given as 36

inches20 then 1 = 6K or K = ~ With the formula thus

obtained--T = ~ or 6T =L or L = 36T2_-it is possible

to determine the length of the pendulum in inches for

each of the twelve twelfths of a second (T) demanded by

the construction (Table 2)

All of the lengths of column L are squares In

the fourth column L2 the improper fractions have been reshy

duced to integers where it was possible to do so The

values of L2 for T of 2 4 6 8 10 and 12 twelfths of

a second are the squares 1 4 9 16 25 and 36 while

the values of L2 for T of 1 3 5 7 9 and 11 twelfths

of a second are 1 4 9 16 25 and 36 with the increments

respectively

Sauveurs procedure is thus clear He directs that

the reader to take Hon the first scale AB 1 4 9 16

25 36 49 64 and so forth inches and carry these

intervals from the end of the rule D to E and rrmark

on these divisions the even numbers 0 2 4 6 8 10

12 14 16 and so forth n2l These values correspond

to the even numbered twelfths of a second in Table 2

He further directs that the first inch (any univeYsal

inch would do) of AB be divided into quarters and

that the reader carry the intervals - It 2~ 3~ 4i 5-4-

20Sauveur specifies 36 universal inches in his directions for the construction of Loulies chronometer Sauveur Systeme General p 420 see vol II p 26 below

21 Ibid bull

34

TABLE 2

T L L2

(in integers + inc rome nt3 )

12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4

6t 7t and so forth over after the divisions of the

even numbers beginning at the end D and that he mark

on these new divisions the odd numbers 1 3 5 7 9 11 13

15 and so forthrr22 which values correspond to those

22Sauveur rtSysteme General p 420 see vol II pp 26-27 below

35

of Table 2 for the odd-numbered twelfths of u second

Thus is obtained Sauveurs fi rst CIlronome ter div ided into

twelfth parts of a second (of time) n23

The demonstration of the manner of dividing the

chronometer24 is the only proof given in the M~moire of 1701

Sauveur first recapitulates the conditions which he stated

in his description of the division itself DF of 3 feet 8

lines (of Paris) is to be taken and this represents the

length of a pendulum set for seconds After stating the law

by which the period and length of a pendulum are related he

observes that since a pendulum set for 1 6

second must thus be

13b of AC (or DF)--an inch--then 0 1 4 9 and so forth

inches will gi ve the lengths of 0 1 2 3 and so forth

sixths of a second or 0 2 4 6 and so forth twelfths

Adding to these numbers i 1-14 2t 3i and- so forth the

sums will be squares (as can be seen in Table 2) of

which the square root will give the number of sixths in

(or half the number of twelfths) of a second 25 All this

is clear also from Table 2

The numbers of the point s of eli vis ion at which it

WIlS necessary for Sauveur to arrive in his dlvis10n of the

chronometer into thirds may be determined in a way analogotls

to the way in which the numbe])s of the pOints of division

of the chronometer into twe1fths of a second were determined

23Sauveur Systeme General p 420 see vol II pp 26-27 below

24Sauveur Systeme General II pp 432-433 see vol II pp 30-31 below

25Ibid bull

36

Since the construction is described 1n ~eneral ternls but

11111strnted between the numbers 14 and 15 the tahle

below will determine the numbers for the points of

division only between 14 and 15 (Table 3)

The formula L = 36T2 is still applicable The

values sought are those for the sixtieths of a second between

the 14th and 15th twelfths of a second or the 70th 7lst

72nd 73rd 74th and 75th sixtieths of a second

TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100

These values of L1 as may be seen from their

equivalents in Column L are squares

Sauveur directs the reader to take at the rot ght

of one division by twelfths Ey of i of an inch and

divide the remainder JE into 5 equal parts u26

( ~ig1Jr e 1)

26 Sauveur Systeme General p 420 see vol II p 27 below

37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

In the figure P and PI represent two consecutive points

of di vision for twelfths of a second PI coincides wi th r and P with ~ The points 1 2 3 and 4 represent the

points of di vision of crE into 5 equal parts One-fourth

inch having been divided into 25 small equal parts

Sauveur instructs the reader to

take one small pa -rt and carry it after 1 and mark K take 4 small parts and carry them after 2 and mark A take 9 small parts and carry them after 3 and mark f and finally take 16 small parts and carry them after 4 and mark y 27

This procedure has been approximated in Fig 1 The four

points K A fA and y will according to SauvenT divide

[y into 5 parts from which we will obtain the divisions

of our chronometer in thirds28

Taking P of 14 (or ~g of a second) PI will equal

15 (or ~~) rhe length of the pendulum to P is 4igg 5625and to PI is -roo Fig 2 expanded shows the relative

positions of the diVisions between 14 and 15

The quarter inch at the right having been subshy

700tracted the remainder 100 is divided into five equal

parts of i6g each To these five parts are added the small

- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39

parts obtained by dividing a quarter inch into 25 equal

parts in the quantities 149 and 16 respectively This

addition gives results enumerated in Table 4

TABLE 4

IN11EHVAL INrrERVAL ADDEND NEW NAME PENDULln~ NAItiE LENGTH L~NGTH

tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100

The four lengths thus constructed correspond preshy

cisely to the four found previously by us e of the formula

and set out in Table 3

It will be noted that both P and p in r1ig 2 are squares and that if P is set equal to A2 then since the

difference between the square numbers representing the

lengths is consistently i (a~ can be seen clearly in

rabl e 2) then PI will equal (A+~)2 or (A2+ A+t)

represerting the quarter inch taken at the right in

Ftp 2 A was then di vided into f 1 ve parts each of

which equa Is g To n of these 4 parts were added in

40

2 nturn 100 small parts so that the trinomial expressing 22 An n

the length of the pendulum ruled in thirds is A 5 100

The demonstration of the construction to which

Sauveur refers the reader29 differs from this one in that

Sauveur states that the difference 6[ is 2A + 1 which would

be true only if the difference between themiddot successive

numbers squared in L of Table 2 were 1 instead of~ But

Sauveurs expression A2+ 2~n t- ~~ is equivalent to the

one given above (A2+ AS +l~~) if as he states tho 1 of

(2A 1) is taken to be inch and with this stipulation

his somewhat roundabout proof becomes wholly intelligible

The chronometer thus correctly divided into twelfth

parts of a second and thirds is not subject to the criticism

which Sauveur levelled against the chronometer of Loulie-shy

that it did not umark the duration of notes in any known

relation to the duration of a second because the periods

of vibration of its pendulum are for the most part incomshy

mensurable with a second30 FonteneJles report on

Sauveurs work of 1701 in the Histoire de lAcademie31

comprehends only the system of 43 meridians and 301

heptamerldians and the theory of harmonics making no

29Sauveur Systeme General pp432-433 see vol II pp 39-31 below

30 Sauveur uSysteme General pp 435-436 see vol II p 33 below

31Bernard Le Bovier de Fontenelle uSur un Nouveau Systeme de Musique U Histoire de l 1 Academie Royale des SCiences Annee 1701 (Amsterdam Chez Pierre Mortier 1736) pp 159-180

41

mention of the Echometer or any of its scales nevertheless

it was the first practical instrument--the string lengths

required by Mersennes calculations made the use of

pendulums adiusted to them awkward--which took account of

the proportional laws of length and time Within the next

few decades a number of theorists based thei r wri tings

on (~l1ronornotry llPon tho worl of Bauvour noLnbly Mtchol

LAffilard and Louis-Leon Pajot Cheva1ier32 but they

will perhaps best be considered in connection with

others who coming after Sauveur drew upon his acoustical

discoveries in the course of elaborating theories of

music both practical and speculative

32Harding Origins pp 11-12

CHAPTER II

THE SYSTEM OF 43 AND THE MEASUREMENT OF INTERVALS

Sauveurs Memoire of 17011 is concerned as its

title implies principally with the elaboration of a system

of measurement classification nomenclature and notation

of intervals and sounds and with examples of the supershy

imposition of this system on existing systems as well as

its application to all the instruments of music This

program is carried over into the subsequent papers which

are devoted in large part to expansion and clarification

of the first

The consideration of intervals begins with the most

fundamental observation about sonorous bodies that if

two of these

make an equal number of vibrations in the sa~e time they are in unison and that if one makes more of them than the other in the same time the one that makes fewer of them emits a grave sound while the one which makes more of them emits an acute sound and that thus the relation of acute and grave sounds consists in the ratio of the numbers of vibrations that both make in the same time 2

This prinCiple discovered only about seventy years

lSauveur Systeme General

2Sauveur Syst~me Gen~ral p 407 see vol II p 4-5 below

42

43

earlier by both Mersenne and Galileo3 is one of the

foundation stones upon which Sauveurs system is built

The intervals are there assigned names according to the

relative numbers of vibrations of the sounds of which they

are composed and these names partly conform to usage and

partly do not the intervals which fall within the compass

of one octave are called by their usual names but the

vnTlous octavos aro ~ivon thnlr own nom()nCl11tllro--thono

more than an oc tave above a fundamental are designs ted as

belonging to the acute octaves and those falling below are

said to belong to the grave octaves 4 The intervals

reaching into these acute and grave octaves are called

replicas triplicas and so forth or sub-replicas

sub-triplicas and so forth

This system however does not completely satisfy

Sauveur the interval names are ambiguous (there are for

example many sizes of thirds) the intervals are not

dOllhled when their names are dOllbled--n slxth for oxnmplo

is not two thirds multiplying an element does not yield

an acceptable interval and the comma 1s not an aliquot

part of any interval Sauveur illustrates the third of

these difficulties by pointing out the unacceptability of

intervals constituted by multiplication of the major tone

3Rober t Bruce Lindsay Historical Intrcxiuction to The flheory of Sound by John William Strutt Baron Hay] eiGh 1

1877 (reprint ed New York Dover Publications 1945)

4Sauveur Systeme General It p 409 see vol IIJ p 6 below

44

But the Pythagorean third is such an interval composed

of two major tones and so it is clear here as elsewhere

too t the eli atonic system to which Sauveur refers is that

of jus t intona tion

rrhe Just intervuls 1n fact are omployod by

Sauveur as a standard in comparing the various temperaments

he considers throughout his work and in the Memoire of

1707 he defines the di atonic system as the one which we

follow in Europe and which we consider most natural bullbullbull

which divides the octave by the major semi tone and by the

major and minor tone s 5 so that it is clear that the

diatonic system and the just diatonic system to which

Sauveur frequently refers are one and the same

Nevertheless the system of just intonation like

that of the traditional names of the intervals was found

inadequate by Sauveur for reasons which he enumerated in

the Memo ire of 1707 His first table of tha t paper

reproduced below sets out the names of the sounds of two

adjacent octaves with numbers ratios of which represhy

sent the intervals between the various pairs o~ sounds

24 27 30 32 36 40 45 48 54 60 64 72 80 90 98

UT RE MI FA SOL LA 8I ut re mi fa sol la s1 ut

T t S T t T S T t S T t T S

lie supposes th1s table to represent the just diatonic

system in which he notes several serious defects

I 5sauveur UMethode Generale p 259 see vol II p 128 below

7

45

The minor thirds TS are exact between MISOL LAut SIrej but too small by a comma between REFA being tS6

The minor fourth called simply fourth TtS is exact between UTFA RESOL MILA SOLut Slmi but is too large by a comma between LAre heing TTS

A melody composed in this system could not he aTpoundTues be

performed on an organ or harpsichord and devices the sounns

of which depend solely on the keys of a keyboa~d without

the players being able to correct them8 for if after

a sound you are to make an interval which is altered by

a commu--for example if after LA you aroto rise by a

fourth to re you cannot do so for the fourth LAre is

too large by a comma 9 rrhe same difficulties would beset

performers on trumpets flut es oboes bass viols theorbos

and gui tars the sound of which 1s ruled by projections

holes or keys 1110 or singers and Violinists who could

6For T is greater than t by a comma which he designates by c so that T is equal to tc Sauveur 1I~~lethode Generale p 261 see vol II p 130 below

7 Ibid bull

n I ~~uuveur IIMethodo Gonopule p 262 1~(J vol II p bull 1)2 below Sauveur t s remark obvlously refer-s to the trHditIonal keyboard with e]even keys to the octnvo and vvlthout as he notes mechanical or other means for making corrections Many attempts have been rrade to cOfJstruct a convenient keyboard accomrnodatinr lust iYltO1ation Examples are given in the appendix to Helmshyholtzs Sensations of Tone pp 466-483 Hermann L F rielmhol tz On the Sen sations of Tone as a Physiological Bnsis for the Theory of Music trans Alexander J Ellis 6th ed (New York Peter Smith 1948) pp 466-483

I9 I Sauveur flIethode Generaletr p 263 see vol II p 132 below

I IlOSauveur Methode Generale p 262 see vol II p 132 below

46

not for lack perhaps of a fine ear make the necessary

corrections But even the most skilled amont the pershy

formers on wind and stringed instruments and the best

11 nvor~~ he lnsists could not follow tho exnc t d J 11 ton c

system because of the discrepancies in interval s1za and

he subjoins an example of plainchant in which if the

intervals are sung just the last ut will be higher than

the first by 2 commasll so that if the litany is sung

55 times the final ut of the 55th repetition will be

higher than the fi rst ut by 110 commas or by two octaves 12

To preserve the identity of the final throughout

the composition Sauveur argues the intervals must be

changed imperceptibly and it is this necessity which leads

13to the introduc tion of t he various tempered ays ternf

After introducing to the reader the tables of the

general system in the first Memoire of 1701 Sauveur proshy

ceeds in the third section14 to set out ~is division of

the octave into 43 equal intervals which he calls

llSauveur trlllethode Generale p ~64 se e vol II p 133 below The opposite intervals Lre 2 4 4 2 rising and 3333 falling Phe elements of the rising liitervals are gi ven by Sauveur as 2T 2t 4S and thos e of the falling intervals as 4T 23 Subtractinp 2T remalns to the ri si nr in tervals which is equal to 2t 2c a1 d 2t to the falling intervals so that the melody ascends more than it falls by 20

12Ibid bull

I Il3S auveur rAethode General e p 265 SE e vol bull II p 134 below

14 ISauveur Systeme General pp 418-428 see vol II pp 15-26 below

47

meridians and the division of each meridian into seven

equal intervals which he calls Ifheptameridians

The number of meridians in each just interval appears

in the center column of Sauveurs first table15 and the

number of heptameridians which in some instances approaches

more nearly the ratio of the just interval is indicated

in parentheses on th e corresponding line of Sauveur t s

second table

Even the use of heptameridians however is not

sufficient to indicate the intervals exactly and although

Sauveur is of the opinion that the discrepancies are too

small to be perceptible in practice16 he suggests a

further subdivision--of the heptameridian into 10 equal

decameridians The octave then consists of 43

meridians or 301 heptameridja ns or 3010 decal11eridians

rihis number of small parts is ospecially well

chosen if for no more than purely mathematical reasons

Since the ratio of vibrations of the octave is 2 to 1 in

order to divide the octave into 43 equal p~rts it is

necessary to find 42 mean proportionals between 1 and 217


16The discrepancy cannot be tar than a halfshyLcptarreridian which according to Sauveur is only l vib~ation out of 870 or one line on an instrument string 7-i et long rr Sauveur Systeme General If p 422 see vol II p 19 below The ratio of the interval H7]~-irD ha~J for tho mantissa of its logatithm npproxlrnuto] y

G004 which is the equivalent of 4 or less than ~ of (Jheptaneridian

17Sauveur nM~thode Gen~ralen p 265 seo vol II p 135 below

48

The task of finding a large number of mean proportionals

lIunknown to the majority of those who are fond of music

am uvery laborious to others u18 was greatly facilitated

by the invention of logarithms--which having been developed

at the end of the sixteenth century by John Napier (1550shy

1617)19 made possible the construction of a grent number

01 tOtrll)o ramon ts which would hi therto ha va prOBon Lod ~ ront

practical difficulties In the problem of constructing

43 proportionals however the values are patticularly

easy to determine because as 43 is a prime factor of 301

and as the first seven digits of the common logarithm of

2 are 3010300 by diminishing the mantissa of the logarithm

by 300 3010000 remains which is divisible by 43 Each

of the 43 steps of Sauveur may thus be subdivided into 7-shy

which small parts he called heptameridians--and further

Sllbdlvlded into 10 after which the number of decnmoridlans

or heptameridians of an interval the ratio of which

reduced to the compass of an octave 1s known can convenshy

iently be found in a table of mantissas while the number

of meridians will be obtained by dividing vhe appropriate

mantissa by seven

l8~3nUVelJr Methode Generale II p 265 see vol II p 135 below

19Ball History of Mathematics p 195 lltholl~h II t~G e flrnt pllbJ ic announcement of the dl scovery wnn mado 1 tI ill1 ]~lr~~t~ci LO[lrlthmorum Ganon1 s DeBcriptio pubshy1 j =~hed in 1614 bullbullbull he had pri vutely communicated a summary of his results to Tycho Brahe as early as 1594 The logarithms employed by Sauveur however are those to a decimal base first calculated by Henry Briggs (156l-l63l) in 1617

49

The cycle of 301 takes its place in a series of

cycles which are sometime s extremely useful fo r the purshy

20poses of calculation lt the cycle of 30103 jots attribshy

uted to de Morgan the cycle of 3010 degrees--which Is

in fact that of Sauveurs decameridians--and Sauveurs

cycl0 01 001 heptamerldians all based on the mllnLlsln of

the logarithm of 2 21 The system of decameridlans is of

course a more accurate one for the measurement of musical

intervals than cents if not so convenient as cents in

certain other ways

The simplici ty of the system of 301 heptameridians

1s purchased of course at the cost of accuracy and

Sauveur was aware that the logarithms he used were not

absolutely exact ubecause they are almost all incommensurshy

ablo but tho grnntor the nurnbor of flputon tho

smaller the error which does not amount to half of the

unity of the last figure because if the figures stricken

off are smaller than half of this unity you di sregard

them and if they are greater you increase the last

fif~ure by 1 1122 The error in employing seven figures of

1lO8ari thms would amount to less than 20000 of our 1 Jheptamcridians than a comma than of an507900 of 6020600

octave or finally than one vibration out of 86n5800

~OHelmhol tz) Sensatlons of Tone p 457

21 Ibid bull

22Sauveur Methode Generale p 275 see vol II p 143 below

50

n23which is of absolutely no consequence The error in

striking off 3 fir-ures as was done in forming decameridians

rtis at the greatest only 2t of a heptameridian 1~3 of a 1comma 002l of the octave and one vibration out of

868524 and the error in striking off the last four

figures as was done in forming the heptameridians will

be at the greatest only ~ heptamerldian or Ii of a

1 25 eomma or 602 of an octave or lout of 870 vlbration

rhls last error--l out of 870 vibrations--Sauveur had

found tolerable in his M~moire of 1701 26

Despite the alluring ease with which the values

of the points of division may be calculated Sauveur 1nshy

sists that he had a different process in mind in making

it Observing that there are 3T2t and 2s27 in the

octave of the diatonic system he finds that in order to

temper the system a mean tone must be found five of which

with two semitones will equal the octave The ratio of

trIO tones semltones and octaves will be found by dlvldlnp

the octave into equal parts the tones containing a cershy

tain number of them and the semi tones ano ther n28

23Sauveur Methode Gen6rale II p ~75 see vol II pp 143-144 below

24Sauveur Methode GenEsectrale p 275 see vol II p 144 below

25 Ibid bull


2Where T represents the maior tone t tho minor tone S the major semitone ani s the minor semitone

28Sauveur MEthode Generale p 265 see vol II p 135 below

51

If T - S is s (the minor semitone) and S - s is taken as

the comma c then T is equal to 28 t 0 and the octave

of 5T (here mean tones) and 2S will be expressed by

128t 7c and the formula is thus derived by which he conshy

structs the temperaments presented here and in the Memoire

of 1711

Sau veul proceeds by determining the ratios of c

to s by obtaining two values each (in heptameridians) for

s and c the tone 28 + 0 has two values 511525 and

457575 and thus when the major semitone s + 0--280287-shy

is subtracted from it s the remainder will assume two

values 231238 and 177288 Subtracting each value of

s from s + 0 0 will also assume two values 102999 and

49049 To obtain the limits of the ratio of s to c the

largest s is divided by the smallest 0 and the smallest s

by the largest c yielding two limiting ratlos 29

31 5middot 51 to l4~ or 17 and 1 to 47 and for a c of 1 s will range

between l~ and 4~ and the octave 12s+70 will 11e30 between

2774 and 6374 bull For simplicity he settles on the approximate

2 2limits of 1 to between 13 and 43 for c and s so that if

o 1s set equal to 1 s will range between 2 and 4 and the

29Sauveurs ratios are approximations The first Is more nearly 1 to 17212594 (31 equals 7209302 and 5 437 equals 7142857) the second 1 to 47144284

30Computing these ratios by the octave--divlding 3010300 by both values of c ard setting c equal to 1 he obtains s of 16 and 41 wlth an octave ranging between 29-2 an d 61sect 7 2

35 35

52

octave will be 31 43 and 55 With a c of 2 s will fall

between 4 and 9 and the octave will be 62748698110

31 or 122 and so forth

From among these possible systems Sauveur selects

three for serious consideration

lhe tempored systems amount to thon e which supposo c =1 s bull 234 and the octave divided into 3143 and 55 parts because numbers which denote the parts of the octave would become too great if you suppose c bull 2345 and so forth Which is contrary to the simplicity Which is necessary in a system32

Barbour has written of Sauveur and his method that

to him

the diatonic semi tone was the larger the problem of temperament was to decide upon a definite ratio beshytween the diatonic and chromatic semitones and that would automat1cally gi ve a particular di vision of the octave If for example the ratio is 32 there are 5 x 7 + 2 x 4 bull 43 parts if 54 there are 5 x 9 + 2 x 5 - 55 parts bullbullbullbull Let us sen what the limlt of the val ue of the fifth would be if the (n + l)n series were extended indefinitely The fifth is (7n + 4)(12n + 7) octave and its limit as n~ct-) is 712 octave that is the fifth of equal temperament The third similarly approaches 13 octave Therefore the farther the series goes the better become its fifths the Doorer its thirds This would seem then to be inferior theory33

31 f I ISauveur I Methode Generale ff p 267 see vol II p 137 below He adds that when s is a mu1tiple of c the system falls again into one of the first which supposes c equal to 1 and that when c is set equal to 0 and s equal to 1 the octave will equal 12 n --which is of course the system of equal temperament

2laquo I I Idlluveu r J Methode Generale rr p 2f)H Jl vo1 I p 1~1 below

33James Murray Barbour Tuning and rTenrre~cr1Cnt (Eu[t Lac lng Michigan state College Press 196T)----P lG~T lhe fLr~lt example is obviously an error for if LLU ratIo o S to s is 3 to 2 then 0 is equal to 1 and the octave (123 - 70) has 31 parts For 43 parts the ratio of S to s required is 4 to 3

53

The formula implied in Barbours calculations is

5 (S +s) +28 which is equlvalent to Sauveur t s formula

12s +- 70 since 5 (3 + s) + 2S equals 7S + 55 and since

73 = 7s+7c then 7S+58 equals (7s+7c)+5s or 128+70

The superparticular ratios 32 43 54 and so forth

represont ratios of S to s when c is equal to 1 and so

n +1the sugrested - series is an instance of the more genshyn

eral serie s s + c when C is equal to one As n increases s

the fraction 7n+4 representative of the fifthl2n+7

approaches 127 as its limit or the fifth of equal temperashy11 ~S4

mont from below when n =1 the fraction equals 19

which corresponds to 69473 or 695 cents while the 11mitshy

7lng value 12 corresponds to 700 cents Similarly

4s + 2c 1 35the third l2s + 70 approaches 3 of an octave As this

study has shown however Sauveur had no intention of

allowing n to increase beyond 4 although the reason he

gave in restricting its range was not that the thirds

would otherwise become intolerably sharp but rather that

the system would become unwieldy with the progressive

mUltiplication of its parts Neverthelesf with the

34Sauveur doe s not cons ider in the TJemoi -re of 1707 this system in which S =2 and c = s bull 1 although he mentions it in the Memoire of 1711 as having a particularshyly pure minor third an observation which can be borne out by calculating the value of the third system of 19 in cents--it is found to have 31578 or 316 cents which is close to the 31564 or 316 cents of the minor third of just intonation with a ratio of 6

5

35 Not l~ of an octave as Barbour erroneously states Barbour Tuning and Temperament p 128

54

limitation Sauveur set on the range of s his system seems

immune to the criticism levelled at it by Barbour

It is perhaps appropriate to note here that for

any values of sand c in which s is greater than c the

7s + 4cfrac tion representing the fifth l2s + 7c will be smaller

than l~ Thus a1l of Suuveurs systems will be nngative-shy

the fifths of all will be flatter than the just flfth 36

Of the three systems which Sauveur singled out for

special consideration in the Memoire of 1707 the cycles

of 31 43 and 55 parts (he also discusses the cycle of

12 parts because being very simple it has had its

partisans u37 )--he attributed the first to both Mersenne

and Salinas and fi nally to Huygens who found tile

intervals of the system exactly38 the second to his own

invention and the third to the use of ordinary musicians 39

A choice among them Sauveur observed should be made

36Ib i d p xi

37 I nSauveur I J1ethode Generale p 268 see vol II p 138 below

38Chrlstlan Huygen s Histoi-re des lt)uvrap()s des J~env~ 1691 Huy~ens showed that the 3-dl vLsion does

not differ perceptibly from the -comma temperamenttI and stated that the fifth of oUP di Vision is no more than _1_ comma higher than the tempered fifths those of 1110 4 comma temperament which difference is entIrely irrpershyceptible but which would render that consonance so much the more perfect tf Christian Euygens Novus Cyc1us harnonicus II Opera varia (Leyden 1924) pp 747 -754 cited in Barbour Tuning and Temperament p 118

39That the system of 55 was in use among musicians is probable in light of the fact that this system corshyresponds closely to the comma variety of meantone

6

55

partly on the basis of the relative correspondence of each

to the diatonic system and for this purpose he appended

to the Memoire of 1707 a rable for comparing the tempered

systems with the just diatonic system40 in Which the

differences of the logarithms of the various degrees of

the systems of 12 31 43 and 55 to those of the same

degrees in just intonation are set out

Since cents are in common use the tables below

contain the same differences expressed in that measure

Table 5 is that of just intonation and contains in its

first column the interval name assigned to it by Sauveur41

in the second the ratio in the third the logarithm of

the ratio given by Sauveur42 in the fourth the number

of cents computed from the logarithm by application of

the formula Cents = 3986 log I where I represents the

ratio of the interval in question43 and in the fifth

the cents rounded to the nearest unit (Table 5)

temperament favored by Silberman Barbour Tuning and Temperament p 126

40 u ~ I Sauveur Methode Genera1e pp 270-211 see vol II p 153 below

41 dauveur denotes majnr and perfect intervals by Roman numerals and minor or diminished intervals by Arabic numerals Subscripts have been added in this study to di stinguish between the major (112) and minor (Ill) tones and the minor 72 and minimum (71) sevenths

42Wi th the decimal place of each of Sauveur s logarithms moved three places to the left They were perhaps moved by Sauveur to the right to show more clearly the relationship of the logarithms to the heptameridians in his seventh column

43John Backus Acoustical Foundations p 292

56

TABLE 5

JUST INTONATION

INTERVAL HATIO LOGARITHM CENTS CENTS (ROl1NDED)

VII 158 2730013 lonn1B lOHH q D ) bull ~~)j ~~1 ~) 101 gt2 10JB

1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112

The first column of Table 6 gives the name of the

interval the second the number of parts of the system

of 12 which are given by Sauveur44 as constituting the

corresponding interval in the third the size of the

number of parts given in the second column in cents in

trIo fourth column tbo difference between the size of the

just interval in cents (taken from Table 5)45 and the

size of the parts given in the third column and in the

fifth Sauveurs difference calculated in cents by

44Sauveur Methode Gen~ra1e If pp 270-271 see vol II p 153 below

45As in Sauveurs table the positive differences represent the amount ~hich must be added to the 5ust deshyrrees to obtain the tempered ones w11i1e the ne[~nt tva dtfferences represent the amounts which must he subshytracted from the just degrees to obtain the tempered one s

57

application of the formula cents = 3986 log I but

rounded to the nearest cent

rABLE 6

SAUVE1TRS INTRVAL PAHllS CENTS DIFFERENCE DIFFEltENCE

VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12

It will be noted that tithe interval and it s comshy

plement have the same difference except that in one it

is positlve and in the other it is negative tl46 The sum

of differences of the tempered second to the two of just

intonation is as would be expected a comma (about

22 cents)

The same type of table may be constructed for the

systems of 3143 and 55

For the system of 31 the values are given in

Table 7

46Sauveur Methode Gen~rale tr p 276 see vol II p 153 below

58

TABLE 7

THE SYSTEM OF 31

SAUVEURS INrrEHVAL PARTS CENTS DIFFERENCE DI FliE rn~NGE

VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4

The small discrepancies of one cent between

Sauveurs calculation and those in the fourth column result

from the rounding to cents in the calculations performed

in the computation of the values of the third and fourth

columns

For the system of 43 the value s are given in

Table 8 (Table 8)

lhe several discrepancies appearlnt~ in thln tnblu

are explained by the fact that in the tables for the

systems of 12 31 43 and 55 the logarithms representing

the parts were used by Sauveur in calculating his differshy

encss while in his table for the system of 43 he employed

heptameridians instead which are rounded logarithms rEha

values of 6 V and IV are obviously incorrectly given by

59

Sauveur as can be noted in his table 47 The corrections

are noted in brackets

TABLE 8

THE SYSTEM OF 43

SIUVETJHS INlIHVAL PAHTS CJt~NTS DIFFERgNCE DIFFil~~KNCE

VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)

The values of the various differences are

collected in Table 10 of which the first column contains

the name of the interval the second third fourth and

fifth the differences from the fourth columns of

(ables 6 7 8 and 9 respectively The differences of

~)auveur where they vary from those of the third columns

are given in brackets In the column for the system of

43 the corrected values of Sauveur are given where they

[~re appropriate in brackets

47 IISauveur Methode Generale p 276 see vol I~ p 145 below

60

TABLE 9

ThE SYSTEM OF 55

SAUVKITHS rNlll~HVAL PAHT CEWllS DIPIIJ~I~NGE Dl 111 11I l IlNUE

VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS

INTERVAL 12 31 43 55

VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61

Sauveur notes that the differences for each intershy

val are largest in the extreme systems of the three 31

43 55 and that the smallest differences occur in the

fourths and fifths in the system of 55 J at the thirds

and sixths in the system of 31 and at the minor second

and major seventh in the system of 4348

After layin~ out these differences he f1nally

proceeds to the selection of a system The principles

have in part been stated previously those systems are

rejected in which the ratio of c to s falls outside the

limits of 1 to l and 4~ Thus the system of 12 in which

c = s falls the more so as the differences of the

thirds and sixths are about ~ of a comma 1t49

This last observation will perhaps seem arbitrary

Binee the very system he rejects is often used fiS a

standard by which others are judged inferior But Sauveur

was endeavoring to achieve a tempered system which would

preserve within the conditions he set down the pure

diatonic system of just intonation

The second requirement--that the system be simple-shy

had led him previously to limit his attention to systems

in which c = 1

His third principle

that the tempered or equally altered consonances do not offend the ear so much as consonances more altered

48Sauveur nriethode Gen~ral e ff pp 277-278 se e vol II p 146 below

49Sauveur Methode Generale n p 278 see vol II p 147 below

62

mingled wi th others more 1ust and the just system becomes intolerable with consonances altered by a comma mingled with others which are exact50

is one of the very few arbitrary aesthetic judgments which

Sauveur allows to influence his decisions The prinCiple

of course favors the adoption of the system of 43 which

it will be remembered had generally smaller differences

to LllO 1u~t nyntom tllun thonn of tho nyntcma of ~l or )-shy

the differences of the columns for the systems of 31 43

and 55 in Table 10 add respectively to 94 80 and 90

A second perhaps somewhat arbitrary aesthetic

judgment that he aJlows to influence his reasoning is that

a great difference is more tolerable in the consonances with ratios expressed by large numbers as in the minor third whlch is 5 to 6 than in the intervals with ratios expressed by small numhers as in the fifth which is 2 to 3 01

The popularity of the mean-tone temperaments however

with their attempt to achieve p1re thirds at the expense

of the fifths WJuld seem to belie this pronouncement 52

The choice of the system of 43 having been made

as Sauveur insists on the basis of the preceding princishy

pIes J it is confirmed by the facility gained by the corshy

~c~)()nd(nce of 301 heptar1oridians to the 3010300 whIch 1s

the ~antissa of the logarithm of 2 and even more from

the fa ct t1at

)oSal1veur M~thode Generale p 278 see vol II p 148 below

51Sauvenr UMethocle Generale n p 279 see vol II p 148 below

52Barbour Tuning and Temperament p 11 and passim

63

the logari thm for the maj or tone diminished by ~d7 reduces to 280000 and that 3010000 and 280000 being divisible by 7 subtracting two major semitones 560000 from the octa~e 2450000 will remain for the value of 5 tone s each of wh ic h W(1) ld conseQuently be 490000 which is still rllvJslhle hy 7 03

In 1711 Sauveur p11blished a Memolre)4 in rep] y

to Konrad Benfling Nho in 1708 constructed a system of

50 equal parts a description of which Was pubJisheci in

17J 055 ane] wl-dc) may accordinr to Bnrbol1T l)n thcl1r~ht

of as an octave comnosed of ditonic commas since

122 ~ 24 = 5056 That system was constructed according

to Sauveur by reciprocal additions and subtractions of

the octave fifth and major third and 18 bused upon

the principle that a legitimate system of music ought to

have its intervals tempered between the just interval and

n57that which he has found different by a comma

Sauveur objects that a system would be very imperfect if

one of its te~pered intervals deviated from the ~ust ones

53Sauveur Methode Gene~ale p 273 see vol II p 141 below

54SnuvelJr Tahle Gen~rn1e II

55onrad Henfling Specimen de novo S110 systernate musieo ImiddotITi seellanea Berolinensia 1710 sec XXVIII

56Barbour Tuninv and Tempera~ent p 122 The system of 50 is in fact according to Thorvald Kornerup a cyclical system very close in its interval1ic content to Zarlino s ~-corruna meantone temperament froIT whieh its greatest deviation is just over three cents and 0uite a bit less for most notes (Ibid p 123)

57Sauveur Table Gen6rale1I p 407 see vol II p 155 below

64

even by a half-comma 58 and further that although

Ilenflinr wnnts the tempered one [interval] to ho betwoen

the just an d exceeding one s 1 t could just as reasonabJ y

be below 59

In support of claims and to save himself the trolJhle

of respondi ng in detail to all those who might wi sh to proshy

pose new systems Sauveur prepared a table which includes

nIl the tempered systems of music60 a claim which seems

a bit exaggerated 1n view of the fact that

all of ttl e lY~ltcmn plven in ~J[lllVOllrt n tnhJ 0 nx(~opt

l~~ 1 and JS are what Bosanquet calls nogaLivo tha t 1s thos e that form their major thirds by four upward fifths Sauveur has no conception of II posishytive system one in which the third is formed by e1 ght dOvmwa rd fi fths Hence he has not inclllded for example the 29 and 41 the positive countershyparts of the 31 and 43 He has also failed to inshyclude the set of systems of which the Hindoo 22 is the type--22 34 56 etc 61

The positive systems forming their thirds by 8 fifths r

dowl for their fifths being larger than E T LEqual

TemperamentJ fifths depress the pitch bel~w E T when

tuned downwardsrt so that the third of A should he nb

58Ibid bull It appears rrom Table 8 however th~t ~)auveur I s own II and 7 deviate from the just II~ [nd 72

L J

rep ecti vely by at least 1 cent more than a half-com~a (Ii c en t s )

59~au veur Table Generale II p 407 see middotvmiddotfl GP Ibb-156 below

60sauveur Table Generale p 409~ seE V()-~ II p -157 below The table appe ar s on p 416 see voi 11

67 below

61 James Murray Barbour Equal Temperament Its history fr-on- Ramis (1482) to Rameau (1737) PhD dJ snt rtation Cornell TJnivolslty I 19gt2 p 246

65

which is inconsistent wi~h musical usage require a

62 separate notation Sauveur was according to Barbour

uflahlc to npprecinto the splondid vn]uo of tho third)

of the latter [the system of 53J since accordinp to his

theory its thirds would have to be as large as Pythagorean

thi rds 63 arei a glance at the table provided wi th

f)all VCllr t s Memoire of 171164 indi cates tha t indeed SauveuT

considered the third of the system of 53 to be thnt of 18

steps or 408 cents which is precisely the size of the

Pythagorean third or in Sauveurs table 55 decameridians

(about 21 cents) sharp rather than the nearly perfect

third of 17 steps or 385 cents formed by 8 descending fifths

The rest of the 25 systems included by Sauveur in

his table are rejected by him either because they consist

of too many parts or because the differences of their

intervals to those of just intonation are too Rro~t bull

flhemiddot reasoning which was adumbrat ed in the flemoire

of 1701 and presented more fully in those of 1707 and

1711 led Sauveur to adopt the system of 43 meridians

301 heptameridians and 3010 decameridians

This system of 43 is put forward confident1y by

Sauveur as a counterpart of the 360 degrees into which the

circle ls djvlded and the 10000000 parts into which the

62RHlIT Bosanquet Temperament or the di vision

of the Octave Musical Association Proceedings 1874shy75 p 13

63Barbour Tuning and Temperament p 125

64Sauveur Table Gen6rale p 416 see vol II p 167 below

66

whole sine is divided--as that is a uniform language

which is absolutely necessary for the advancement of that

science bull 65

A feature of the system which Sauveur describes

but does not explain is the ease with which the rntios of

intervals may be converted to it The process is describod

661n tilO Memolre of 1701 in the course of a sories of

directions perhaps directed to practical musicians rathor

than to mathematicians in order to find the number of

heptameridians of an interval the ratio of which is known

it is necessary only to add the numbers of the ratio

(a T b for example of the ratio ~ which here shall

represent an improper fraction) subtract them (a - b)

multiply their difference by 875 divide the product

875(a of- b) by the sum and 875(a - b) having thus been(a + b)

obtained is the number of heptameridians sought 67

Since the number of heptamerldians is sin1ply the

first three places of the logarithm of the ratio Sauveurs

II

65Sauveur Table Generale n p 406 see vol II p 154 below

66~3auveur

I Systeme Generale pp 421-422 see vol pp 18-20 below

67 Ihici bull If the sum is moTO than f1X ~ i rIHl I~rfltlr jtlln frlf dtfferonce or if (A + B)6(A - B) Wltlet will OCCllr in the case of intervals of which the rll~io 12 rreater than ~ (for if (A + B) = 6~ - B) then since

v A 7 (A + B) = 6A - 6B 5A =7B or B = 5) or the ratios of intervals greater than about 583 cents--nesrly tb= size of t ne tri tone--Sau veur recommends tha t the in terval be reduced by multipLication of the lower term by 248 16 and so forth to its complement or duplicate in a lower octave

67

process amounts to nothing less than a means of finding

the logarithm of the ratio of a musical interval In

fact Alexander Ellis who later developed the bimodular

calculation of logarithms notes in the supplementary

material appended to his translation of Helmholtzs

Sensations of Tone that Sauveur was the first to his

knowledge to employ the bimodular method of finding

68logarithms The success of the process depends upon

the fact that the bimodulus which is a constant

Uexactly double of the modulus of any system of logashy

rithms is so rela ted to the antilogari thms of the

system that when the difference of two numbers is small

the difference of their logarithms 1s nearly equal to the

bimodulus multiplied by the difference and di vided by the

sum of the numbers themselves69 The bimodulus chosen

by Sauveur--875--has been augmented by 6 (from 869) since

with the use of the bimodulus 869 without its increment

constant additive corrections would have been necessary70

The heptameridians converted to c)nt s obtained

by use of Sau veur I s method are shown in Tub1e 11

68111i8 Appendix XX to Helmhol tz 3ensatli ons of Ton0 p 447

69 Alexander J Ellis On an Improved Bi-norlu1ar ~ethod of computing Natural Bnd Tabular Logarithms and Anti-Logari thIns to Twel ve or Sixteen Places with very brief IabIes Proceedings of the Royal Society of London XXXI Febrmiddotuary 1881 pp 381-2 The modulus of two systems of logarithms is the member by which the logashyrithms of one must be multiplied to obtain those of the other

70Ellis Appendix XX to Helmholtz Sensattons of Tone p 447

68

TABLE 11

INT~RVAL RATIO SIZE (BYBIMODULAR

JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1

In this table the size of the interval calculated by

means of the bimodu1ar method recommended by Sauveur is

seen to be very close to that found by other means and

the computation produces a result which is exact for the 71perfect fourth which has a ratio of 43 as Ellis1s

method devised later was correct for the Major Third

The system of 43 meridians wi th it s variolls

processes--the further di vision into 301 heptame ridlans

and 3010 decameridians as well as the bimodular method of

comput ing the number of heptameridians di rt9ctly from the

ratio of the proposed interva1--had as a nncessary adshy

iunct in the wri tings of Sauveur the estSblishment of

a fixed pitch by the employment of which together with

71The striking difference noticeable between the tritone calculated by the bimodu1ar method and that obshytaIned from seven-place logarlthms occurs when the second me trion of reducing the size of a ra tio 1s employed (tho

I~ )rutlo of the tritone is given by Sauveur as 32) The

tritone can also be obtained by addition o~ the elements of 2Tt each of which can be found by bimodu1ar computashytion rEhe resul t of this process 1s a tritone of 594 cents only 4 cents sharp

69

the system of 43 the name of any pitch could be determined

to within the range of a half-decameridian or about 02

of a cent 72 It had been partly for Jack of such n flxod

tone the t Sauveur 11a d cons Idered hi s Treat i se of SpeculA t t ve

Munic of 1697 so deficient that he could not in conscience

publish it73 Having addressed that problem he came forth

in 1700 with a means of finding the fixed sound a

description of which is given in the Histoire de lAcademie

of the year 1700 Together with the system of decameridshy

ians the fixed sound placed at Sauveurs disposal a menns

for moasuring pitch with scientific accuracy complementary I

to the system he put forward for the meaSurement of time

in his Chronometer

Fontenelles report of Sauveurs method of detershy

mining the fixed sound begins with the assertion that

vibrations of two equal strings must a1ways move toshygether--begin end and begin again--at the same instant but that those of two unequal strings must be now separated now reunited and sepa~ated longer as the numbers which express the inequality of the strings are greater 74

72A decameridian equals about 039 cents and half a decameridi an about 019 cents

73Sauveur trSyst~me Generale p 405 see vol II p 3 below

74Bernard Ie Bovier de 1ontene1le Sur la LntGrminatlon d un Son Iilixe II Histoire de l Academie Roynle (H~3 Sciences Annee 1700 (Amsterdam Pierre Mortier 1l~56) p 185 n les vibrations de deux cordes egales

lvent aller toujour~ ensemble commencer finir reCOITLmenCer dans Ie meme instant mai s que celles de deux

~ 1 d i t I csraes lnega es o_ven etre tantotseparees tantot reunies amp dautant plus longtems separees que les

I nombres qui expriment 11inegal1te des cordes sont plus grands II

70

For example if the lengths are 2 and I the shorter string

makes 2 vibrations while the longer makes 1 If the lengths

are 25 and 24 the longer will make 24 vibrations while

the shorte~ makes 25

Sauveur had noticed that when you hear Organs tuned

am when two pipes which are nearly in unison are plnyan

to[~cthor tnere are certain instants when the common sOllnd

thoy rendor is stronrer and these instances scem to locUr

75at equal intervals and gave as an explanation of this

phenomenon the theory that the sound of the two pipes

together must have greater force when their vibrations

after having been separated for some time come to reunite

and harmonize in striking the ear at the same moment 76

As the pipes come closer to unison the numberS expressin~

their ratio become larger and the beats which are rarer

are more easily distinguished by the ear

In the next paragraph Fontenelle sets out the deshy

duction made by Sauveur from these observations which

made possible the establishment of the fixed sound

If you took two pipes such that the intervals of their beats should be great enough to be measshyured by the vibrations of a pendulum you would loarn oxnctly frorn tht) longth of tIll ponltullHTl the duration of each of the vibrations which it

75 Ibid p 187 JlQuand on entend accorder des Or[ues amp que deux tuyaux qui approchent de 1 uni nson jouent ensemble 11 y a certains instans o~ le son corrrnun qutils rendent est plus fort amp ces instans semblent revenir dans des intervalles egaux

76Ibid bull n le son des deux tuyaux e1semble devoita~oir plus de forge 9uand leurs vi~ratio~s apr~s avoir ete quelque terns separees venoient a se r~unir amp saccordient a frapper loreille dun meme coup

71

made and consequently that of the interval of two beats of the pipes you would learn moreover from the nature of the harmony of the pipes how many vibrations one made while the other made a certain given number and as these two numbers are included in the interval of two beats of which the duration is known you would learn precisely how many vibrations each pipe made in a certain time But there is certainly a unique number of vibrations made in a ~iven time which make a certain tone and as the ratios of vtbrations of all the tones are known you would learn how many vibrations each tone maknfJ In

r7 middotthl fl gl ven t 1me bull

Having found the means of establishing the number

of vibrations of a sound Sauveur settled upon 100 as the

number of vibrations which the fixed sound to which all

others could be referred in comparison makes In one

second

Sauveur also estimated the number of beats perceivshy

able in a second about six in a second can be distinguished

01[11] y onollph 78 A grenter numbor would not bo dlnshy

tinguishable in one second but smaller numbers of beats

77Ibid pp 187-188 Si 1 on prenoit deux tuyaux tel s que les intervalles de leurs battemens fussent assez grands pour ~tre mesures par les vibrations dun Pendule on sauroit exactemen t par la longeur de ce Pendule quelle 8eroit la duree de chacune des vibrations quil feroit - pnr conseqnent celIe de l intcrvalle de deux hn ttemens cics tuyaux on sauroit dai11eurs par la nfltDre de laccord des tuyaux combien lun feroit de vibrations pcncant que l autre en ferol t un certain nomhre dcterrnine amp comme ces deux nombres seroient compri s dans l intervalle de deux battemens dont on connoitroit la dlJree on sauroit precisement combien chaque tuyau feroit de vibrations pendant un certain terns Or crest uniquement un certain nombre de vibrations faites dans un tems ci(ternine qui fai t un certain ton amp comme les rapports des vibrations de taus les tons sont connus on sauroit combien chaque ton fait de vibrations dans ce meme terns deterrnine u

78Ibid p 193 nQuand bullbullbull leurs vi bratlons ne se rencontr01ent que 6 fois en une Seconde on distinguoit ces battemens avec assez de facilite

72

in a second Vlould be distinguished with greater and rreater

ease This finding makes it necessary to lower by octaves

the pipes employed in finding the number of vibrations in

a second of a given pitch in reference to the fixed tone

in order to reduce the number of beats in a second to a

countable number

In the Memoire of 1701 Sauvellr returned to the

problem of establishing the fixed sound and gave a very

careful ctescription of the method by which it could be

obtained 79 He first paid tribute to Mersenne who in

Harmonie universelle had attempted to demonstrate that

a string seventeen feet long and held by a weight eight

pounds would make 8 vibrations in a second80--from which

could be deduced the length of string necessary to make

100 vibrations per second But the method which Sauveur

took as trle truer and more reliable was a refinement of

the one that he had presented through Fontenelle in 1700

Three organ pipes must be tuned to PA and pa (UT

and ut) and BOr or BOra (SOL)81 Then the major thlrd PA

GAna (UTMI) the minor third PA go e (UTMlb) and

fin2l1y the minor senitone go~ GAna (MlbMI) which

79Sauveur Systeme General II pp 48f3-493 see vol II pp 84-89 below

80IJIersenne Harmonie univergtsel1e 11117 pp 140-146

81Sauveur adds to the syllables PA itA GA SO BO LO and DO lower case consonants ~lich distinguish meridians and vowels which d stingui sh decameridians Sauveur Systeme General p 498 see vol II p 95 below

73

has a ratio of 24 to 25 A beating will occur at each

25th vibra tion of the sha rper one GAna (MI) 82

To obtain beats at each 50th vibration of the highshy

est Uemploy a mean g~ca between these two pipes po~ and

GAna tt so that ngo~ gSJpound and GAna bullbullbull will make in

the same time 48 59 and 50 vibrationSj83 and to obtain

beats at each lOath vibration of the highest the mean ga~

should be placed between the pipes g~ca and GAna and the v

mean gu between go~ and g~ca These five pipes gose

v Jgu g~~ ga~ and GA~ will make their beats at 96 97

middot 98 99 and 100 vibrations84 The duration of the beats

is me asured by use of a pendulum and a scale especially

rra rke d in me ridia ns and heptameridians so tha t from it can

be determined the distance from GAna to the fixed sound

in those units

The construction of this scale is considered along

with the construction of the third fourth fifth and

~l1xth 3cnlnn of tho Chronomotor (rhe first two it wI]l

bo remembered were devised for the measurement of temporal

du rations to the nearest third The third scale is the

General Monochord It is divided into meridians and heptashy

meridians by carrying the decimal ratios of the intervals

in meridians to an octave (divided into 1000 pa~ts) of the

monochord The process is repeated with all distances

82Sauveur Usysteme Gen~Yal p 490 see vol II p 86 helow

83Ibid bull The mean required is the geometric mean

84Ibid bull

v

74

halved for the higher octaves and doubled for the lower

85octaves The third scale or the pendulum for the fixed

sound employed above to determine the distance of GAna

from the fixed sound was constructed by bringing down

from the Monochord every other merldian and numbering

to both the left and right from a point 0 at R which marks

off 36 unlvornul inches from P

rphe reason for thi s division into unit s one of

which is equal to two on the Monochord may easily be inshy

ferred from Fig 3 below

D B

(86) (43) (0 )

Al Dl C11---------middot-- 1middotmiddot---------middotmiddotmiddotmiddotmiddotmiddot--middot I _______ H ____~

(43) (215)

Fig 3

C bisects AB an d 01 besects AIBI likewi se D hi sects AC

und Dl bisects AlGI- If AB is a monochord there will

be one octave or 43 meridians between B and C one octave

85The dtagram of the scal es on Plat e II (8auveur Systfrrne ((~nera1 II p 498 see vol II p 96 helow) doe~ not -~101 KL KH ML or NM divided into 43 pnrtB_ Snl1velHs directions for dividing the meridians into heptamAridians by taking equal parts on the scale can be verifed by refshyerence to the ratlos in Plate I (Sauveur Systerr8 General p 498 see vol II p 95 below) where the ratios between the heptameridians are expressed by numhers with nearly the same differences between heptamerldians within the limits of one meridian

75

or 43 more between C and D and so forth toward A If

AB and AIBI are 36 universal inches each then the period

of vibration of AIBl as a pendulum will be 2 seconds

and the half period with which Sauveur measured~ will

be 1 second Sauveur wishes his reader to use this

pendulum to measure the time in which 100 vibrations are

mn(Je (or 2 vibrations of pipes in tho ratlo 5049 or 4

vibratlons of pipes in the ratio 2524) If the pendulum

is AIBI in length there will be 100 vihrations in 1

second If the pendulu111 is AlGI in length or tAIBI

1 f2 d ithere will be 100 vibrations in rd or 2 secons s nee

the period of a pendulum is proportional to the square root

of its length There will then be 100-12 vibrations in one 100

second (since 2 =~ where x represents the number of

2

vibrations in one second) or 14142135 vibrations in one

second The ratio of e vibrations will then be 14142135

to 100 or 14142135 to 1 which is the ratio of the tritone

or ahout 21i meridians Dl is found by the same process to

mark 43 meridians and from this it can be seen that the

numhers on scale AIBI will be half of those on AB which

is the proportion specified by Sauveur

rrne fifth scale indicates the intervals in meridshy

lans and heptameridJans as well as in intervals of the

diatonic system 1I86 It is divided independently of the

f ~3t fonr and consists of equal divisionsJ each

86Sauveur s erne G 1 U p 1131 see 1 II ys t enera Ll vo p 29 below

76

representing a meridian and each further divisible into

7 heptameridians or 70 decameridians On these divisions

are marked on one side of the scale the numbers of

meridians and on the other the diatonic intervals the

numbers of meridians and heptameridians of which can be I I

found in Sauveurs Table I of the Systeme General

rrhe sixth scale is a sCale of ra tios of sounds

nncl is to be divided for use with the fifth scale First

100 meridians are carried down from the fifth scale then

these pl rts having been subdivided into 10 and finally

100 each the logarithms between 100 and 500 are marked

off consecutively on the scale and the small resulting

parts are numbered from 1 to 5000

These last two scales may be used Uto compare the

ra tios of sounds wi th their 1nt ervals 87 Sauveur directs

the reader to take the distance representinp the ratIo

from tbe sixth scale with compasses and to transfer it to

the fifth scale Ratios will thus be converted to meridians

and heptameridia ns Sauveur adds tha t if the numberS markshy

ing the ratios of these sounds falling between 50 and 100

are not in the sixth scale take half of them or double

themn88 after which it will be possible to find them on

the scale

Ihe process by which the ratio can be determined

from the number of meridians or heptameridians or from

87Sauveur USysteme General fI p 434 see vol II p 32 below

I I88Sauveur nSyst~me General p 435 seo vol II p 02 below

77

an interval of the diatonic system is the reverse of the

process for determining the number of meridians from the

ratio The interval is taken with compasses on the fifth

scale and the length is transferred to the sixth scale

where placing one point on any number you please the

other will give the second number of the ratio The

process Can be modified so that the ratio will be obtainoo

in tho smallest whole numbers

bullbullbull put first the point on 10 and see whether the other point falls on a 10 If it doe s not put the point successively on 20 30 and so forth lJntil the second point falls on a 10 If it does not try one by one 11 12 13 14 and so forth until the second point falls on a 10 or at a point half the way beshytween The first two numbers on which the points of the compasses fall exactly will mark the ratio of the two sounds in the smallest numbers But if it should fallon a half thi rd or quarter you would ha ve to double triple or quadruple the two numbers 89

Suuveur reports at the end of the fourth section shy

of the Memoire of 1701 tha t Chapotot one of the most

skilled engineers of mathematical instruments in Paris

has constructed Echometers and that he has made one of

them from copper for His Royal Highness th3 Duke of

Orleans 90 Since the fifth and sixth scale s could be

used as slide rules as well as with compas5es as the

scale of the sixth line is logarithmic and as Sauveurs

above romarl indicates that he hud had Echometer rulos

prepared from copper it is possible that the slide rule

89Sauveur Systeme General I p 435 see vol II

p 33 below

90 ISauveur Systeme General pp 435-436 see vol II p 33 below

78

which Cajori in his Historz of the Logarithmic Slide Rule91

reports Sauveur to have commissioned from the artisans Gevin

am Le Bas having slides like thos e of Seth Partridge u92

may have been musical slide rules or scales of the Echo-

meter This conclusion seems particularly apt since Sauveur

hnd tornod his attontion to Acoustlcnl problems ovnn boforo

hIs admission to the Acad~mie93 and perhaps helps to

oxplnin why 8avere1n is quoted by Ca10ri as wrl tlnf in

his Dictionnaire universel de mathematigue at de physique

that before 1753 R P Pezenas was the only author to

discuss these kinds of scales [slide rules] 94 thus overshy

looking Sauveur as well as several others but Sauveurs

rule may have been a musical one divided although

logarithmically into intervals and ratios rather than

into antilogaritr~s

In the Memoire of 171395 Sauveur returned to the

subject of the fixed pitch noting at the very outset of

his remarks on the subject that in 1701 being occupied

wi th his general system of intervals he tcok the number

91Florian Cajori flHistory of the L)gari thmic Slide Rule and Allied Instruments n String Figure3 and other Mono~raphs ed IN W Rouse Ball 3rd ed (Chelsea Publishing Company New York 1969)

92Ib1 d p 43 bull


94Saverien Dictionnaire universel de mathematigue et de physi~ue Tome I 1753 art tlechelle anglolse cited in Cajori History in Ball String Figures p 45 n bull bull seul Auteur Franyois qui a1 t parltS de ces sortes dEchel1es

95Sauveur J Rapport It

79

100 vibrations in a seoond only provisionally and having

determined independently that the C-SOL-UT in practice

makes about 243~ vibrations per second and constructing

Table 12 below he chose 256 as the fundamental or

fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216

With this fixed sound the octaves can be convenshy

iently numbered by taking the power of 2 which represents

the number of vibrations of the fundamental of each octave

as the nmnber of that octave

The intervals of the fundamentals of the octaves

can be found by multiplying 3010300 by the exponents of

the double progression or by the number of the octave

which will be equal to the exponent of the expression reshy

presenting the number of vibrations of the various func1ashy

mentals By striking off the 3 or 4 last figures of this

intervalfI the reader will obtain the meridians for the 96interval to which 1 2 3 4 and so forth meridians

can be added to obtain all the meridians and intervals

of each octave

96 Ibid p 454 see vol II p 186 below

80

To render all of this more comprehensible Sauveur

offers a General table of fixed sounds97 which gives

in 13 columns the numbers of vibrations per second from

8 to 65536 or from the third octave to the sixteenth

meridian by meridian 98

Sauveur discovered in the course of his experiments

with vibra ting strings that the same sound males twice

as many vibrations with strings as with pipes and con-

eluded that

in strings you mllDt take a pllssage and a return for a vibration of sound because it is only the passage which makes an impression against the organ of the ear and the return makes none and in organ pipes the undulations of the air make an impression only in their passages and none in their return 99

It will be remembered that even in the discllssion of

pendullLTlS in the Memoi re of 1701 Sauveur had by implicashy

tion taken as a vibration half of a period lOO

rlho th cory of fixed tone thon and thB te-rrnlnolopy

of vibrations were elaborated and refined respectively

in the M~moire of 1713

97 Sauveur Rapport lip 468 see vol II p 203 below

98The numbers of vibrations however do not actually correspond to the meridians but to the d8cashyneridians in the third column at the left which indicate the intervals in the first column at the left more exactly

99sauveur uRapport pp 450-451 see vol II p 183 below

lOOFor he described the vibration of a pendulum 3 feet 8i lines lon~ as making one vibration in a second SaUVe1Jr rt Systeme General tI pp 428-429 see vol II p 26 below

81

The applications which Sauveur made of his system

of measurement comprising the echometer and the cycle

of 43 meridians and its subdivisions were illustrated ~

first in the fifth and sixth sections of the Memoire of

1701

In the fifth section Sauveur shows how all of the

varIous systems of music whether their sounas aro oxprossoc1

by lithe ratios of their vibrations or by the different

lengths of the strings of a monochord which renders the

proposed system--or finally by the ratios of the intervals

01 one sound to the others 101 can be converted to corshy

responding systems in meridians or their subdivisions

expressed in the special syllables of solmization for the

general system

The first example he gives is that of the regular

diatonic system or the system of just intonation of which

the ratios are known

24 27 30 32 36 40 ) 484

I II III IV v VI VII VIII

He directs that four zeros be added to each of these

numhors and that they all be divided by tho ~Jmulle3t

240000 The quotient can be found as ratios in the tables

he provides and the corresponding number of meridians

a~d heptameridians will be found in the corresponding

lOlSauveur Systeme General p 436 see vol II pp 33-34 below

82

locations of the tables of names meridians and heptashy

meridians

The Echometer can also be applied to the diatonic

system The reader is instructed to take

the interval from 24 to each of the other numhors of column A [n column of the numhers 24 i~ middot~)O and so forth on Boule VI wi th compasses--for ina l~unce from 24 to 32 or from 2400 to 3200 Carry the interval to Scale VI02

If one point is placed on 0 the other will give the

intervals in meridians and heptameridians bull bull bull as well

as the interval bullbullbull of the diatonic system 103

He next considers a system in which lengths of a

monochord are given rather than ratios Again rntios

are found by division of all the string lengths by the

shortest but since string length is inversely proportional

to the number of vibrations a string makes in a second

and hence to the pitch of the string the numbers of

heptameridians obtained from the ratios of the lengths

of the monochord must all be subtracted from 301 to obtain

tne inverses OT octave complements which Iru1y represent

trIO intervals in meridians and heptamerldlnns which corshy

respond to the given lengths of the strings

A third example is the system of 55 commas Sauveur

directs the reader to find the number of elements which

each interval comprises and to divide 301 into 55 equal


l03Sauveur Systeme General p 439 see vol II p 37 below

83

26parts The quotient will give 555 as the value of one

of these parts 104 which value multiplied by the numher

of parts of each interval previously determined yields

the number of meridians or heptameridians of each interval

Demonstrating the universality of application of

hll Bystom Sauveur ploceoda to exnmlne with ltn n1ct

two systems foreign to the usage of his time one ancient

and one orlental The ancient system if that of the

Greeks reported by Mersenne in which of three genres

the diatonic divides its fourths into a semitone and two tones the chromatic into two semitones and a trisemitone or minor third and the Enharmonic into two dieses and a ditone or Major third 105

Sauveurs reconstruction of Mersennes Greek system gives

tl1C diatonic system with steps at 0 28 78 and 125 heptashy

meridians the chromatic system with steps at 0 28 46

and 125 heptameridians and the enharmonic system with

steps at 0 14 28 and 125 heptameridians In the

chromatic system the two semi tones 0-28 and 28-46 differ

widely in size the first being about 112 cents and the

other only about 72 cents although perhaps not much can

be made of this difference since Sauveur warns thnt

104Sauveur Systeme Gen6ralII p 4middot11 see vol II p 39 below

105Nlersenne Harmonie uni vers ell e III p 172 tiLe diatonique divise ses Quartes en vn derniton amp en deux tons Ie Ghromatique en deux demitons amp dans vn Trisnemiton ou Tierce mineure amp lEnharmonic en deux dieses amp en vn diton ou Tierce maieure

84

each of these [the genres] has been d1 vided differently

by different authors nlD6

The system of the orientalsl07 appears under

scrutiny to have been composed of two elements--the

baqya of abou t 23 heptamerldl ans or about 92 cen ts and

lOSthe comma of about 5 heptamerldlans or 20 cents

SnUV0Ul adds that

having given an account of any system in the meridians and heptameridians of our general system it is easy to divide a monochord subsequently according to 109 that system making use of our general echometer

In the sixth section applications are made of the

system and the Echometer to the voice and the instruments

of music With C-SOL-UT as the fundamental sound Sauveur

presents in the third plate appended to tpe Memoire a

diagram on which are represented the keys of a keyboard

of organ or harpsichord the clef and traditional names

of the notes played on them as well as the syllables of

solmization when C is UT and when C is SOL After preshy

senting his own system of solmization and notes he preshy

sents a tab~e of ranges of the various voices in general

and of some of the well-known singers of his day followed

106Sauveur II Systeme General p 444 see vol II p 42 below

107The system of the orientals given by Sauveur is one followed according to an Arabian work Edouar translated by Petis de la Croix by the Turks and Persians

lOSSauveur Systeme General p 445 see vol II p 43 below

I IlO9Sauveur Systeme General p 447 see vol II p 45 below

85

by similar tables for both wind and stringed instruments

including the guitar of 10 frets

In an addition to the sixth section appended to

110the Memoire Sauveur sets forth his own system of

classification of the ranges of voices The compass of

a voice being defined as the series of sounds of the

diatonic system which it can traverse in sinping II

marked by the diatonic intervals III he proposes that the

compass be designated by two times the half of this

interval112 which can be found by adding 1 and dividing

by 2 and prefixing half to the number obtained The

first procedure is illustrated by V which is 5 ~ 1 or

two thirds the second by VI which is half 6 2 or a

half-fourth or a fourth above and third below

To this numerical designation are added syllables

of solmization which indicate the center of the range

of the voice

Sauveur deduces from this that there can be ttas

many parts among the voices as notes of the diatonic system

which can be the middles of all possible volces113

If the range of voices be assumed to rise to bis-PA (UT)

which 1s c and to descend to subbis-PA which is C-shy

110Sauveur nSyst~me General pp 493-498 see vol II pp 89-94 below

lllSauveur Systeme General p 493 see vol II p 89 below

l12Ibid bull

II p

113Sauveur

90 below

ISysteme General p 494 see vol

86

four octaves in all--PA or a SOL UT or a will be the

middle of all possible voices and Sauveur contends that

as the compass of the voice nis supposed in the staves

of plainchant to be of a IXth or of two Vths and in the

staves of music to be an Xlth or two Vlthsnl14 and as

the ordinary compass of a voice 1s an Xlth or two Vlths

then by subtracting a sixth from bis-PA and adrllnp a

sixth to subbis-PA the range of the centers and hence

their number will be found to be subbis-LO(A) to Sem-GA

(e) a compass ofaXIXth or two Xths or finally

19 notes tll15 These 19 notes are the centers of the 19

possible voices which constitute Sauveurs systeml16 of

classification

1 sem-GA( MI)

2 bull sem-RA(RE) very high treble

3 sem-PA(octave of C SOL UT) high treble or first treble

4 DO( S1)

5 LO(LA) low treble or second treble

6 BO(G RE SOL)

7 SO(octave of F FA TIT)

8 G(MI) very high counter-tenor

9 RA(RE) counter-tenor

10 PA(C SOL UT) very high tenor

114Ibid 115Sauveur Systeme General p 495 see vol

II pp 90-91 below l16Sauveur Systeme General tr p 4ltJ6 see vol

II pp 91-92 below

87

11 sub-DO(SI) high tenor

12 sub-LO(LA) tenor

13 sub-BO(sub octave of G RE SOL harmonious Vllth or VIIlth

14 sub-SOC F JA UT) low tenor

15 sub-FA( NIl)

16 sub-HAC HE) lower tenor

17 sub-PA(sub-octave of C SOL TIT)

18 subbis-DO(SI) bass

19 subbis-LO(LA)

The M~moire of 1713 contains several suggestions

which supplement the tables of the ranges of voices and

instruments and the system of classification which appear

in the fifth and sixth chapters of the M6moire of 1701

By use of the fixed tone of which the number of vlbrashy

tions in a second is known the reader can determine

from the table of fixed sounds the number of vibrations

of a resonant body so that it will be possible to discover

how many vibrations the lowest tone of a bass voice and

the hif~hest tone of a treble voice make s 117 as well as

the number of vibrations or tremors or the lip when you whistle or else when you play on the horn or the trumpet and finally the number of vibrations of the tones of all kinds of musical instruments the compass of which we gave in the third plate of the M6moires of 1701 and of 1702118

Sauveur gives in the notes of his system the tones of

various church bells which he had drawn from a Ivl0rno 1 re

u117Sauveur Rapnort p 464 see vol III

p 196 below

l18Sauveur Rapport1f p 464 see vol II pp 196-197 below

88

on the tones of bells given him by an Honorary Canon of

Paris Chastelain and he appends a system for determinshy

ing from the tones of the bells their weights 119

Sauveur had enumerated the possibility of notating

pitches exactly and learning the precise number of vibrashy

tions of a resonant body in his Memoire of 1701 in which

he gave as uses for the fixed sound the ascertainment of

the name and number of vibrations 1n a second of the sounds

of resonant bodies the determination from changes in

the sound of such a body of the changes which could have

taken place in its substance and the discovery of the

limits of hearing--the highest and the lowest sounds

which may yet be perceived by the ear 120

In the eleventh section of the Memoire of 1701

Sauveur suggested a procedure by which taking a particshy

ular sound of a system or instrument as fundamental the

consonance or dissonance of the other intervals to that

fundamental could be easily discerned by which the sound

offering the greatest number of consonances when selected

as fundamental could be determined and by which the

sounds which by adjustment could be rendered just might

be identified 121 This procedure requires the use of reshy

ciprocal (or mutual) intervals which Sauveur defines as

119Sauveur Rapport rr p 466 see vol II p 199 below



89

the interval of each sound of a system or instrument to

each of those which follow it with the compass of an

octave 122

Sauveur directs the ~eader to obtain the reciproshy

cal intervals by first marking one af~er another the

numbers of meridians and heptameridians of a system in

two octaves and the numbers of those of an instrument

throughout its whole compass rr123 These numbers marked

the reciprocal intervals are the remainders when the numshy

ber of meridians and heptameridians of each sound is subshy

tracted from that of every other sound

As an example Sauveur obtains the reciprocal

intervals of the sounds of the diatonic system of just

intonation imagining them to represent sounds available

on the keyboard of an ordinary harpsiohord

From the intervals of the sounds of the keyboard

expressed in meridians

I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39

VIII 9 IX 10 X 11 XI XII 13 XIII 14 XIV 43 46 50 54 57 61 64 68 (l) 75 79 82

he constructs a table124 (Table 13) in which when the

l22Sauveur Systeme Gen6ral II pp t184-485 see vol II p 81 below

123Sauveur Systeme GeniJral p 485 see vol II p 81 below

I I 124Sauveur Systeme GenertllefI p 487 see vol II p 83 below

90

sounds in the left-hand column are taken as fundamental

the sounds which bear to it the relationship marked by the

intervals I 2 II 3 and so forth may be read in the

line extending to the right of the name

TABLE 13

RECIPHOCAL INT~RVALS

Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI

Old names UT d RE b MI FA d SOL d U b 51 VT

New names PA pi RA go GA SO sa BO ba LO de DO FA

UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43

It will be seen that the original octave presented

b ~ bis that of C C D E F F G G A B B and C

since 3 meridians represent the chromatic semitone and 4

91

the diatonic one whichas Barbour notes was considered

by Sauveur to be the larger of the two 125 Table 14 gives

the values in cents of both the just intervals from

Sauveurs table (Table 13) and the altered intervals which

are included there between brackets as well as wherever

possible the names of the notes in the diatonic system

TABLE 14

VALUES FROM TABLE 13 IN CENTS

INTERVAL MERIDIANS CENTS NAME

(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )

The names were assigned in Table 14 on the assumpshy

tion that 3 meridians represent the chromatic semitone


92

and 4 the diatonic semi tone and with the rreatest simshy

plicity possible--8 meridians was thus taken as 3 meridians

or a chromatic semitone--lower than 11 meridians or Eb

With Table 14 Sauveurs remarks on the selection may be

scrutinized

If RA or LO is taken for the final--D or A--all

the tempered diatonic intervals are exact tr 126_-and will

be D Eb E F F G G A Bb B e ell and D for the

~ al D d- A Bb B e ell D DI Lil G GYf 1 ~ on an ~ r c

and A for the final on A Nhen another tone is taken as

the final however there are fewer exact diatonic notes

Bbbwith Ab for example the notes of the scale are Ab

cb Gbbebb Dbb Db Ebb Fbb Fb Bb Abb Ab with

values of 0 112 223 304 419 502 614 725 809 921

1004 1116 and 1200 in cents The fifth of 725 cents and

the major third of 419 howl like wolves

The number of altered notes for each final are given

in Table 15

TABLE 15

ALT~dED NOYES FOt 4GH FIi~AL IN fLA)LE 13

C v rtil D Eb E F Fil G Gtt A Bb B

2 5 0 5 2 3 4 1 6 1 4 3

An arrangement can be made to show the pattern of

finals which offer relatively pure series

126SauveurI Systeme General II p 488 see vol

II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6

The number of altered notes is thus seen to increase as

the finals ascend by fifths and having reached a

maximum of six begins to decrease after G as the flats

which are substituted for sharps decrease in number the

finals meanwhile continuing their ascent by fifths

The method of reciplocal intervals would enable

a performer to select the most serviceable keys on an inshy

strument or in a system of tuning or temperament to alter

those notes of an instrument to make variolJs keys playable

and to make the necessary adjustments when two instruments

of different tunings are to be played simultaneously

The system of 43 the echometer the fixed sound

and the method of reciprocal intervals together with the

system of classification of vocal parts constitute a

comprehensive system for the measurement of musical tones

and their intervals

CHAPTER III

THE OVERTONE SERIES

In tho ninth section of the M6moire of 17011

Sauveur published discoveries he had made concerning

and terminology he had developed for use in discussing

what is now known as the overtone series and in the

tenth section of the same Mernoire2 he made an application

of the discoveries set forth in the preceding chapter

while in 1702 he published his second Memoire3 which was

devoted almost wholly to the application of the discovershy

ies of the previous year to the construction of organ

stops

The ninth section of the first M~moire entitled

The Harmonics begins with a definition of the term-shy

Ira hatmonic of the fundamental [is that which makes sevshy

eral vibrations while the fundamental makes only one rr4 -shy

which thus has the same extension as the ~erm overtone

strictly defined but unlike the term harmonic as it

lsauveur Systeme General 2 Ibid pp 483-484 see vol II pp 80-81 below

3 Sauveur Application II

4Sauveur Systeme General9 p 474 see vol II p 70 below

94

95

is used today does not include the fundamental itself5

nor does the definition of the term provide for the disshy

tinction which is drawn today between harmonics and parshy

tials of which the second term has Ifin scientific studies

a wider significance since it also includes nonharmonic

overtones like those that occur in bells and in the comshy

plex sounds called noises6 In this latter distinction

the term harmonic is employed in the strict mathematical

sense in which it is also used to denote a progression in

which the denominators are in arithmetical progression

as f ~ ~ ~ and so forth

Having given a definition of the term Ifharmonic n

Sauveur provides a table in which are given all of the

harmonics included within five octaves of a fundamental

8UT or C and these are given in ratios to the vibrations

of the fundamental in intervals of octaves meridians

and heptameridians in di~tonic intervals from the first

sound of each octave in diatonic intervals to the fundashy

mental sOlJno in the new names of his proposed system of

solmization as well as in the old Guidonian names

5Harvard Dictionary 2nd ed sv Acoustics u If the terms [harmonic overtones] are properly used the first overtone is the second harmonie the second overtone is the third harmonic and so on

6Ibid bull

7Mathematics Dictionary ed Glenn James and Robert C James (New York D Van Nostrand Company Inc 1949)s bull v bull uHarmonic If

8sauveur nSyst~me Gen~ral II p 475 see vol II p 72 below

96

The harmonics as they appear from the defn--~ tior

and in the table are no more than proportions ~n~ it is

Juuveurs program in the remainder of the ninth sect ton

to make them sensible to the hearing and even to the

slvht and to indicate their properties 9 Por tlLl El purshy

pose Sauveur directs the reader to divide the string of

(l lillHloctlord into equal pnrts into b for intlLnnco find

pllleklrtl~ tho open ntrlnp ho will find thflt 11 wllJ [under

a sound that I call the fundamental of that strinplO

flhen a thin obstacle is placed on one of the points of

division of the string into equal parts the disturbshy

ance bull bull bull of the string is communicated to both sides of

the obstaclell and the string will render the 5th harshy

monic or if the fundamental is C E Sauveur explains

tnis effect as a result of the communication of the v1brashy

tions of the part which is of the length of the string

to the neighboring parts into which the remainder of the

ntring will (11 vi de i taelf each of which is elt11101 to tllO

r~rst he concludes from this that the string vibrating

in 5 parts produces the 5th ha~nonic and he calls

these partial and separate vibrations undulations tneir

immObile points Nodes and the midpoints of each vibrashy

tion where consequently the motion is greatest the

9 bull ISauveur Systeme General p 476 see vol II

p 73 below

I IlOSauveur Systeme General If pp 476-477 S6B

vol II p 73 below

11Sauveur nSysteme General n p 477 see vol p 73 below

97

bulges12 terms which Fontenelle suggests were drawn

from Astronomy and principally from the movement of the

moon 1113

Sauveur proceeds to show that if the thin obstacle

is placed at the second instead of the first rlivlsion

hy fifths the string will produce the fifth harmonic

for tho string will be divided into two unequal pn rts

AG Hnd CH (Ilig 4) and AC being the shortor wll COntshy

municate its vibrations to CG leaving GB which vibrashy

ting twice as fast as either AC or CG will communicate

its vibrations from FG to FE through DA (Fig 4)

The undulations are audible and visible as well

Sauveur suggests that small black and white paper riders

be attached to the nodes and bulges respectively in orcler

tnat the movements of the various parts of the string mirht

be observed by the eye This experiment as Sauveur notes

nad been performed as early as 1673 by John iJallls who

later published the results in the first paper on muslshy

cal acoustics to appear in the transactions of the society

( t1 e Royal Soci cty of London) und er the ti t 1 e If On the SIremshy

bJing of Consonant Strings a New Musical Discovery 14

- J-middotJ[i1 ontenelle Nouveau Systeme p 172 (~~ ~llVel)r

-lla) ces vibrations partia1es amp separees Ord1ulations eIlS points immobiles Noeuds amp Ie point du milieu de

c-iaque vibration oD parcon5equent 1e mouvement est gt1U8 grarr1 Ventre de londulation

-L3 I id tirees de l Astronomie amp liei ement (iu mouvement de la Lune II

Ii Groves Dictionary of Music and Mus c1 rtn3

ej s v S)und by LI S Lloyd

98

B

n

E

A c B

lig 4 Communication of vibrations

Wallis httd tuned two strings an octave apart and bowing

ttJe hipher found that the same note was sounderl hy the

oLhor strinr which was found to be vihratyening in two

Lalves for a paper rider at its mid-point was motionless16

lie then tuned the higher string to the twefth of the lower

and lIagain found the other one sounding thjs hi~her note

but now vibrating in thirds of its whole lemiddot1gth wi th Cwo

places at which a paper rider was motionless l6 Accordng

to iontenelle Sauveur made a report to t

the existence of harmonics produced in a string vibrating

in small parts and

15Ibid bull

16Ibid

99

someone in the company remembe~ed that this had appeared already in a work by Mr ~allis where in truth it had not been noted as it meri ted and at once without fretting vr Sauvellr renounced all the honor of it with regard to the public l

Sauveur drew from his experiments a series of conshy

clusions a summary of which constitutes the second half

of the ninth section of his first M6mnire He proposed

first that a harmonic formed by the placement of a thin

obstacle on a potential nodal point will continue to

sound when the thin obstacle is re-r1oved Second he noted

that if a string is already vibratin~ in five parts and

a thin obstacle on the bulge of an undulation dividing

it for instance into 3 it will itself form a 3rd harshy

monic of the first harmonic --the 15th harmon5_c of the

fundamental nIB This conclusion seems natnral in view

of the discovery of the communication of vibrations from

one small aliquot part of the string to others His

third observation--that a hlrmonic can he indllced in a

string either by setting another string nearby at the

unison of one of its harmonics19 or he conjectured by

setting the nearby string for such a sound that they can

lPontenelle N~uveau Syste11r- p~ J72-173 lLorsole 1-f1 Sauveur apports aI Acadcl-e cette eXnerience de de~x tons ~gaux su~les deux parties i~6~ales dune code eJlc y fut regue avec tout Je pla-isir Clue font les premieres decouvert~s Mals quelquun de la Compagnie se souvint que celIe-Ie etoi t deja dans un Ouvl--arre de rf WalliS o~ a la verit~ elle navoit ~as ~t~ remarqu~e con~~e elle meritoit amp auss1-t~t NT Sauv8nr en abandonna sans peine tout lhonneur ~ lega-rd du Public

p

18 Sauveur 77 below

ItS ysteme G Ifeneral p 480 see vol II

19Ibid bull

100

divide by their undulations into harmonics Wilich will be

the greatest common measure of the fundamentals of the

two strings 20__was in part anticipated by tTohn Vallis

Wallis describing several experiments in which harmonics

were oxcttod to sympathetIc vibration noted that ~tt hnd

lon~ since been observed that if a Viol strin~ or Lute strinp be touched wIth the Bowar Bann another string on the same or another inst~Jment not far from it (if in unison to it or an octave or the likei will at the same time tremble of its own accord 2

Sauveur assumed fourth that the harmonics of a

string three feet long could be heard only to the fifth

octave (which was also the limit of the harmonics he preshy

sented in the table of harmonics) a1 though it seems that

he made this assumption only to make cleare~ his ensuing

discussion of the positions of the nodal points along the

string since he suggests tha t harmonic s beyond ti1e 128th

are audible

rrhe presence of harmonics up to the ~S2nd or the

fIfth octavo having been assumed Sauveur proceeds to

his fifth conclusion which like the sixth and seventh

is the result of geometrical analysis rather than of

observation that

every di Vision of node that forms a harr1onic vhen a thin obstacle is placed thereupon is removed from

90 f-J Ibid As when one is at the fourth of the other

and trie smaller will form the 3rd harmonic and the greater the 4th--which are in union

2lJohn Wallis Letter to the Publisher concerning a new NIusical Discovery Phllosophical Transactions of the Royal Society of London vol XII (1677) p 839

101

the nearest node of other ha2~onics by at least a 32nd part of its undulation

This is easiJy understood since the successive

thirty-seconds of the string as well as the successive

thirds of the string may be expressed as fractions with

96 as the denominator Sauveur concludes from thIs that

the lower numbered harmonics will have considerah1e lenrth

11 rUU nd LilU lr nod lt1 S 23 whll e the hn rmon 1e 8 vV 1 t l it Ivll or

memhe~s will have little--a conclusion which seems reasonshy

able in view of the fourth deduction that the node of a

harmonic is removed from the nearest nodes of other harshy24monics by at least a 32nd part of its undulationn so

t- 1 x -1 1 I 1 _ 1 1 1 - 1t-hu t the 1 rae lons 1 ~l2middot ~l2 2 x l2 - 64 77 x --- - (J v df t)

and so forth give the minimum lengths by which a neighborshy

ing node must be removed from the nodes of the fundamental

and consecutive harmonics The conclusion that the nodes

of harmonics bearing higher numbers are packed more

tightly may be illustrated by the division of the string

1 n 1 0 1 2 ) 4 5 and 6 par t s bull I n III i g 5 th e n 11mbe r s

lying helow the points of division represent sixtieths of

the length of the string and the numbers below them their

differences (in sixtieths) while the fractions lying

above the line represent the lengths of string to those

( f22Sauveur Systeme Generalu p 481 see vol II pp 77-78 below


T24Sauveur Systeme General p 481 see vol LJ

pp 77-78 below

102

points of division It will be seen that the greatest

differences appear adjacent to fractions expressing

divisions of the diagrammatic string into the greatest

number of parts

3o

3110 l~ IS 30 10

10

Fig 5 Nodes of the fundamental and the first five harmonics

11rom this ~eometrical analysis Sauvcllr con JeeturO1

that if the node of a small harmonic is a neighbor of two

nodes of greater sounds the smaller one wi]l be effaced

25by them by which he perhaps hoped to explain weakness

of the hipher harmonics in comparison with lower ones

The conclusions however which were to be of

inunediate practical application were those which concerned

the existence and nature of the harmonics ~roduced by

musical instruments Sauveur observes tha if you slip

the thin bar all along [a plucked] string you will hear

a chirping of harmonics of which the order will appear

confused but can nevertheless be determined by the princishy

ples we have established26 and makes application of

25 IISauveur Systeme General p 482 see vol II p 79 below

26Ibid bull

10

103

the established principles illustrated to the explanation

of the tones of the marine trurnpet and of instruments

the sounds of which las for example the hunting horn

and the large wind instruments] go by leaps n27 His obshy

servation that earlier explanations of the leaping tones

of these instruments had been very imperfect because the

principle of harmonics had been previously unknown appears

to 1)6 somewhat m1sleading in the light of the discoverlos

published by Francis Roberts in 1692 28

Roberts had found the first sixteen notes of the

trumpet to be C c g c e g bb (over which he

d ilmarked an f to show that it needed sharpening c e

f (over which he marked I to show that the corresponding

b l note needed flattening) gtl a (with an f) b (with an

f) and c H and from a subse()uent examination of the notes

of the marine trumpet he found that the lengths necessary

to produce the notes of the trumpet--even the 7th 11th

III13th and 14th which were out of tune were 2 3 4 and

so forth of the entire string He continued explaining

the 1 eaps

it is reasonable to imagine that the strongest blast raises the sound by breaking the air vi thin the tube into the shortest vibrations but that no musical sound will arise unless tbey are suited to some ali shyquot part and so by reduplication exactly measure out the whole length of the instrument bullbullbullbull As a

27 n ~Sauveur Systeme General pp 483-484 see vol II p 80 below

28Francis Roberts A Discourse concerning the Musical Notes of the Trumpet and Trumpet Marine and of the Defects of the same Philosophical Transactions of the Royal Society XVII (1692) pp 559-563~

104

corollary to this discourse we may observe that the distances of the trumpet notes ascending conshytinually decreased in the proportion of 11 12 13 14 15 in infinitum 29

In this explanation he seems to have anticipated

hlUVOll r wno wrot e thu t

the lon~ wind instruments also divide their lengths into a kind of equal undulatlons If then an unshydulation of air bullbullbull 1s forced to go more quickly it divides into two equal undulations then into 3 4 and so forth according to the length of the instrument 3D

In 1702 Sauveur turned his attention to the apshy

plication of harmonics to the constMlction of organ stops

as the result of a conversatlon with Deslandes which made

him notice that harmonics serve as the basis for the comshy

position of organ stops and for the mixtures that organshy

ists make with these stops which will be explained in a I

few words u3l Of the Memoire of 1702 in which these

findings are reported the first part is devoted to a

description of the organ--its keyboards pipes mechanisms

and the characteristics of its various stops To this

is appended a table of organ stops32 in which are

arrayed the octaves thirds and fifths of each of five

octaves together with the harmoniC which the first pipe

of the stop renders and the last as well as the names

29 Ibid bull


31 Sauveur uApplicationn p 425 see vol II p 98 below

32Sauveur Application p 450 see vol II p 126 below

105

of the various stops A second table33 includes the

harmonics of all the keys of the organ for all the simple

and compound stops1I34

rrhe first four columns of this second table five

the diatonic intervals of each stop to the fundamental

or the sound of the pipe of 32 feet the same intervaJs

by octaves the corresponding lengths of open pipes and

the number of the harmonic uroduced In the remnincier

of the table the lines represent the sounds of the keys

of the stop Sauveur asks the reader to note that

the pipes which are on the same key render the harmonics in cOYlparison wi th the sound of the first pipe of that key which takes the place of the fundamental for the sounds of the pipes of each key have the same relation among themselves 8S the 35 sounds of the first key subbis-PA which are harmonic

Sauveur notes as well til at the sounds of all the

octaves in the lines are harmonic--or in double proportion

rrhe first observation can ea 1y he verified by

selecting a column and dividing the lar~er numbers by

the smallest The results for the column of sub-RE or

d are given in Table 16 (Table 16)

For a column like that of PI(C) in whiCh such

division produces fractions the first note must be conshy

sidered as itself a harmonic and the fundamental found

the series will appear to be harmonic 36


34Sauveur Anplication If p 434 see vol II p 107 below

35Sauveur IIApplication p 436 see vol II p 109 below

36The method by which the fundamental is found in

106

TABLE 16

SOUNDS OR HARMONICSsom~DS 9

9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96

Principally from these observotions he d~aws the

conclusion that the compo tion of organ stops is harronic

tha t the mixture of organ stops shollld be harmonic and

tflat if deviations are made flit is a spec1es of ctlssonance

this table san be illustrated for the instance of the column PI( Cir ) Since 24 is the next numher after the first 19 15 24 should be divided by 19 15 The q1lotient 125 is eqlJal to 54 and indi ca tes that 24 in t (le fifth harmonic ond 19 15 the fourth Phe divLsion 245 yields 4 45 which is the fundnr1fntal oj the column and to which all the notes of the collHnn will be harmonic the numbers of the column being all inte~ral multiples of 4 45 If howevpr a number of 1ines occur between the sounds of a column so that the ratio is not in its simnlest form as for example 86 instead of 43 then as intervals which are too larf~e will appear between the numbers of the haTrnonics of the column when the divlsion is atterrlpted it wIll be c 1 nn r tria t the n1Jmber 0 f the fv ndamental t s fa 1 sen no rrn)t be multiplied by 2 to place it in the correct octave

107

in the harmonics which has some relation with the disshy

sonances employed in music u37

Sauveur noted that the organ in its mixture of

stops only imitated

the harmony which nature observes in resonant bodies bullbullbull for we distinguish there the harmonics 1 2 3 4 5 6 as in bell~ and at night the long strings of the harpsichord~38

At the end of the Memoire of 1702 Sauveur attempted

to establish the limits of all sounds as well as of those

which are clearly perceptible observing that the compass

of the notes available on the organ from that of a pipe

of 32 feet to that of a nipe of 4t lines is 10 octaves

estimated that to that compass about two more octaves could

be added increasing the absolute range of sounds to

twelve octaves Of these he remarks that organ builders

distinguish most easily those from the 8th harmonic to the

l28th Sauveurs Table of Fixed Sounds subioined to his

M~moire of 171339 made it clear that the twelve octaves

to which he had referred eleven years earlier wore those

from 8 vibrations in a second to 32768 vibrations in a

second

Whether or not Sauveur discovered independently

all of the various phenomena which his theory comprehends


38sauveur Application pp 450-451 see vol II p 124 below

39Sauveur Rapnort p 468 see vol II p 203 below

108

he seems to have made an important contribution to the

development of the theory of overtones of which he is

usually named as the originator 40

Descartes notes in the Comeendiurn Musicae that we

never hear a sound without hearing also its octave4l and

Sauveur made a similar observation at the beginning of

his M~moire of 1701

While I was meditating on the phenomena of sounds it came to my attention that especially at night you can hear in long strings in addition to the principal sound other little sounds at the twelfth and the seventeenth of that sound 42

It is true as well that Wallis and Roberts had antici shy

pated the discovery of Sauveur that strings will vibrate

in aliquot parts as has been seen But Sauveur brought

all these scattered observations together in a coherent

theory in which it was proposed that the harmonlc s are

sounded by strings the numbers of vibrations of which

in a given time are integral multiples of the numbers of

vibrations of the fundamental in that same time Sauveur

having devised a means of determining absolutely rather

40For example Philip Gossett in the introduction to Qis annotated translation of Rameaus Traite of 1722 cites Sauvellr as the first to present experimental evishydence for the exi stence of the over-tone serie s II ~TeanshyPhilippe Rameau Treatise on narrnony trans Philip Go sse t t (Ne w Yo r k Dov e r 1ub 1 i cat ion s Inc 1971) p xii

4lRene Descartes Comeendium Musicae (Rhenum 1650) nDe Octava pp 14-20


109

than relati vely the number of vibra tions eXfcuted by a

string in a second this definition of harmonics with

reference to numbers of vibrations could be applied

directly to the explanation of the phenomena ohserved in

the vibration of strings His table of harmonics in

which he set Ollt all the harmonics within the ranpe of

fivo octavos wlth their gtltches plvon in heptnmeYlrltnnB

brought system to the diversity of phenomena previolls1y

recognized and his work unlike that of Wallis and

Roberts in which it was merely observed that a string

the vibrations of which were divided into equal parts proshy

ducod the same sounds as shorter strIngs vlbrutlnr~ us

wholes suggested that a string was capable not only of

produc ing the harmonics of a fundamental indi vidlJally but

that it could produce these vibrations simultaneously as

well Sauveur thus claims the distinction of having

noted the important fact that a vibrating string could

produce the sounds corresponding to several of its harshy

monics at the same time43

Besides the discoveries observations and the

order which he brought to them Sauveur also made appli shy

ca tions of his theories in the explanation of the lnrmonic

structure of the notes rendered by the marine trumpet

various wind instruments and the organ--explanations

which were the richer for the improvements Sauveur made

through the formulation of his theory with reference to

43Lindsay Introduction to Rayleigh rpheory of Sound p xv

110

numbers of vibrations rather than to lengths of strings

and proportions

Sauveur aJso contributed a number of terms to the

s c 1enc e of ha rmon i c s the wo rd nha rmon i c s itself i s

one which was first used by Sauveur to describe phenomena

observable in the vibration of resonant bodIes while he

was also responsible for the use of the term fundamental ll

fOT thu tOllO SOllnoed by the vlbrution talcn 119 one 1n COttl shy

parisons as well as for the term Itnodes for those

pOints at which no motion occurred--terms which like

the concepts they represent are still in use in the

discussion of the phenomena of sound

CHAPTER IV

THE HEIRS OF SAUVEUR

In his report on Sauveurs method of determining

a fixed pitch Fontene11e speculated that the number of

beats present in an interval might be directly related

to its degree of consonance or dissonance and expected

that were this hypothesis to prove true it would tr1ay

bare the true source of the Rules of Composition unknown

until the present to Philosophy which relies almost enshy

tirely on the judgment of the earn1 In the years that

followed Sauveur made discoveries concerning the vibrashy

tion of strings and the overtone series--the expression

for example of the ratios of sounds as integral multip1es-shy

which Fontenelle estimated made the representation of

musical intervals

not only more nntl1T(ll tn thnt it 13 only tho orrinr of nllmbt~rs t hems elves which are all mul t 1 pI (JD of uni ty hllt also in that it expresses ann represents all of music and the only music that Nature gives by itself without the assistance of art 2

lJ1ontenelle Determination rr pp 194-195 Si cette hypothese est vraye e11e d~couvrira la v~ritah]e source d~s Regles de la Composition inconnue iusaua present a la Philosophie qui sen remettoit presque entierement au jugement de lOreille

2Bernard Ie Bovier de Pontenelle Sur 1 Application des Sons Harmoniques aux Jeux dOrgues hlstoire de lAcademie Royale des SCiences Anntsecte 1702 (Amsterdam Chez Pierre rtortier 1736) p 120 Cette

III

112

Sauveur had been the geometer in fashion when he was not

yet twenty-three years old and had numbered among his

accomplis~~ents tables for the flow of jets of water the

maps of the shores of France and treatises on the relationshy

ships of the weights of ~nrious c0untries3 besides his

development of the sCience of acoustics a discipline

which he has been credited with both naming and founding

It might have surprised Fontenelle had he been ahle to

foresee that several centuries later none of SallVeUT S

works wrnlld he available in translation to students of the

science of sound and that his name would be so unfamiliar

to those students that not only does Groves Dictionary

of Muslc and Musicians include no article devoted exclusshy

ively to his achievements but also that the same encyshy

clopedia offers an article on sound4 in which a brief

history of the science of acoustics is presented without

even a mention of the name of one of its most influential

founders

rrhe later heirs of Sauvenr then in large part

enjoy the bequest without acknowledging or perhaps even

nouvelle cOnsideration des rapnorts des Sons nest pas seulement plus naturelle en ce quelle nest que la suite meme des Nombres aui tous sont multiples de 1unite mais encore en ce quel1e exprime amp represente toute In Musique amp la seule Musi(1ue que la Nature nous donne par elle-merne sans Ie secours de lartu (Ibid p 120)

3bontenelle Eloge II p 104

4Grove t s Dictionary 5th ed sv Sound by Ll S Lloyd

113

recognizing the benefactor In the eighteenth century

however there were both acousticians and musical theorshy

ists who consciously made use of his methods in developing

the theories of both the science of sound in general and

music in particular

Sauveurs Chronometer divided into twelfth and

further into sixtieth parts of a second was a refinement

of the Chronometer of Louli~ divided more simply into

universal inches The refinements of Sauveur weTe incorshy

porated into the Pendulum of Michel LAffilard who folshy

lowed him closely in this matter in his book Principes

tr~s-faciles pour bien apprendre la musique

A pendulum is a ball of lead or of other metal attached to a thread You suspend this thread from a fixed point--for example from a nail--and you i~part to this ball the excursions and returns that are called Vibrations Each Vibration great or small lasts a certain time always equal but lengthening this thread the vibrations last longer and shortenin~ it they last for a shorter time

The duration of these vibrations is measured in Thirds of Time which are the sixtieth parts of a second just as the second is the sixtieth part of a minute and the ~inute is the sixtieth part of an hour Rut to pegulate the d11ration of these vibrashytions you must obtain a rule divided by thirds of time and determine the length of the pennuJum on this rule For example the vibrations of the Pendulum will be of 30 thirds of time when its length is equal to that of the rule from its top to its division 30 This is what Mr Sauveur exshyplains at greater length in his new principles for learning to sing from the ordinary gotes by means of the names of his General System

5lVlichel L t Affilard Principes tr~s-faciles Dour bien apprendre la musique 6th ed (Paris Christophe Ballard 1705) p 55

Un Pendule est une Balle de Plomb ou dautre metail attachee a un fil On suspend ce fil a un point fixe Par exemple ~ un cloud amp lon fait faire A cette Balle des allees amp des venues quon appelle Vibrations Chaque

114

LAffilards description or Sauveur1s first

Memoire of 1701 as new principles for leDrning to sing

from the ordinary notes hy means of his General Systemu6

suggests that LAffilard did not t1o-rollphly understand one

of the authors upon whose works he hasAd his P-rincinlea shy

rrhe Metrometer proposed by Loui 3-Leon Pai ot

Chevalier comte DOns-en-Bray7 intended by its inventor

improvement f li 8as an ~ on the Chronomet eT 0 I101L e emp1 oyed

the 01 vislon into t--tirds constructed hy ([luvenr

Everyone knows that an hour is divided into 60 minutAs 1 minute into 60 soconrls anrl 1 second into 60 thirds or 120 half-thirds t1is will r-i ve us a 01 vis ion suffici entl y small for VIhn t )e propose

You know that the length of a pEldulum so that each vibration should be of one second or 60 thirds must be of 3 feet 8i lines

In order to 1 earn the varion s lenr~ths for w-lich you should set the pendulum so t1jt tile durntions

~ibration grand ou petite dure un certnin terns toCjours egal mais en allonpeant ce fil CGS Vih-rntions durent plus long-terns amp en Ie racourcissnnt eJ10s dure~t moins

La duree de ces Vib-rattons se ieS1)pe P~ir tierces de rpeYls qui sont la soixantiEn-ne partie d une Seconde de ntffie que Ia Seconde est 1a soixanti~me naptie dune I1inute amp 1a Minute Ia so1xantirne pn rtie d nne Ieure ~Iais ponr repler Ia nuree oe ces Vlh-rltlons i1 f81Jt avoir une regIe divisee par tierces de rj18I1S ontcrmirAr la 1 0 n r e 11r d 1J eJ r3 u 1 e sur ] d i t erAr J e Pcd x e i () 1 e 1 e s Vihra t ion 3 du Pendule seYont de 30 t ierc e~~ e rems 1~ orOI~-e cor li l

r ~)VCrgtr (f e a) 8 bull U -J n +- rt(-Le1 OOU8J1 e~n] rgt O lre r_-~l uc a J

0 bull t d tmiddot t _0 30oepuls son extreml e en nau 1usqu a sa rn VlSlon bull Cfest ce oue TJonsieur Sauveur exnlirnJe nllJS au lon~ dans se jrinclres nouveaux pour appreJdre a c ter sur les Xotes orciinaires par les noms de son Systele fLener-al

7Louis-Leon Paiot Chevalipr cornto dOns-en-Rray Descrlptlon et u~)ae dun Metromntre ou ~aciline pour battre les lcsures et lea Temps de toutes sortes dAirs U

M~moires de 1 I Acad6mie Royale des Scie~ces Ann~e 1732 (Paris 1735) pp 182-195

8 Hardin~ Ori~ins p 12

115

of its vlbratlons should be riven by a lllilllhcr of thl-rds you must remember a principle accepted hy all mathematicians which is that two lendulums being of different lengths the number of vlbrations of these two Pendulums in a given time is in recipshyrical relation to the square roots of tfleir length or that the durations of the vibrations of these two Pendulums are between them as the squar(~ roots of the lenpths of the Pendulums

llrom this principle it is easy to cldllce the bull bull bull rule for calcula ting the length 0 f a Pendulum in order that the fulration of each vibrntion should be glven by a number of thirds 9

Pajot then specifies a rule by the use of which

the lengths of a pendulum can be calculated for a given

number of thirds and subJoins a table lO in which the

lengths of a pendulum are given for vibrations of durations

of 1 to 180 half-thirds as well as a table of durations

of the measures of various compositions by I~lly Colasse

Campra des Touches and NIato

9pajot Description pp 186-187 Tout Ie mond s9ait quune heure se divise en 60 minutes 1 minute en 60 secondes et 1 seconde en 60 tierces ou l~O demi-tierces cela nous donnera une division suffisamment petite pour ce que nous proposons

On syait aussi que 1a longueur que doit avoir un Pendule pour que chaque vibration soit dune seconde ou de 60 tierces doit etre de 3 pieds 8 li~neR et demi

POlrr ~

connoi tre

les dlfferentes 1onrrllcnlr~ flu on dot t don n () r it I on d11] e Ii 0 u r qu 0 1 a (ltn ( 0 d n rt I ~~ V j h rl t Ion 3

Dolt d un nOf1bre de tierces donne il fant ~)c D(Juvcnlr dun principe reltu de toutes Jas Mathematiciens qui est que deux Penc11Jles etant de dlfferente lon~neur Ie nombra des vibrations de ces deux Pendules dans lin temps donne est en raison reciproque des racines quarr~es de leur lonfTu8ur Oll que les duress des vibrations de CGS deux-Penoules Rant entre elles comme les Tacines quarrees des lon~~eurs des Pendules

De ce principe il est ais~ de deduire la regIe s~ivante psur calculer la longueur dun Pendule afin que 1a ~ur0e de chaque vibration soit dun nombre de tierces (-2~onna

lOIbid pp 193-195

116

Erich Schwandt who has discussed the Chronometer

of Sauveur and the Pendulum of LAffilard in a monograph

on the tempos of various French court dances has argued

that while LAffilard employs for the measurement of his

pendulum the scale devised by Sauveur he nonetheless

mistakenly applied the periods of his pendulum to a rule

divided for half periods ll According to Schwandt then

the vibration of a pendulum is considered by LAffilard

to comprise a period--both excursion and return Pajot

however obviously did not consider the vibration to be

equal to the period for in his description of the

M~trom~tr~ cited above he specified that one vibration

of a pendulum 3 feet 8t lines long lasts one second and

it can easily he determined that I second gives the half-

period of a pendulum of this length It is difficult to

ascertain whether Sauveur meant by a vibration a period

or a half-period In his Memoire of 1713 Sauveur disshy

cussing vibrating strings admitted that discoveries he

had made compelled him to talee ua passage and a return for

a vibration of sound and if this implies that he had

previously taken both excursions and returns as vibrashy

tions it can be conjectured further that he considered

the vibration of a pendulum to consist analogously of

only an excursion or a return So while the evidence

does seem to suggest that Sauveur understood a ~ibration

to be a half-period and while experiment does show that

llErich Schwandt LAffilard on the Prench Court Dances The Musical Quarterly IX3 (July 1974) 389-400

117

Pajot understood a vibration to be a half-period it may

still be true as Schwannt su~pests--it is beyond the purshy

view of this study to enter into an examination of his

argument--that LIAffilnrd construed the term vibration

as referring to a period and misapplied the perions of

his pendulum to the half-periods of Sauveurs Chronometer

thus giving rise to mlsunderstandinr-s as a consequence of

which all modern translations of LAffilards tempo

indications are exactly twice too fast12

In the procession of devices of musical chronometry

Sauveurs Chronometer apnears behind that of Loulie over

which it represents a great imnrovement in accuracy rhe

more sophisticated instrument of Paiot added little In

the way of mathematical refinement and its superiority

lay simply in its greater mechanical complexity and thus

while Paiots improvement represented an advance in execushy

tion Sauve11r s improvement represented an ac1vance in conshy

cept The contribution of LAffilard if he is to he

considered as having made one lies chiefly in the ~rAnter

flexibility which his system of parentheses lent to the

indication of tempo by means of numbers

Sauveurs contribution to the preci se measurement

of musical time was thus significant and if the inst~lment

he proposed is no lon~er in use it nonetheless won the

12Ibid p 395

118

respect of those who coming later incorporateci itA

scale into their own devic e s bull

Despite Sauveurs attempts to estabJish the AystArT

of 43 m~ridians there is no record of its ~eneral nCConshy

tance even for a short time among musicians As an

nttompt to approxmn te in n cyc11 cal tompernmnnt Ltln [lyshy

stern of Just Intonation it was perhans mo-re sucCO~t1fl]l

than wore the systems of 55 31 19 or 12--tho altnrnntlvo8

proposed by Sauveur before the selection of the system of

43 was rnade--but the suggestion is nowhere made the t those

systems were put forward with the intention of dupl1catinp

that of just intonation The cycle of 31 as has been

noted was observed by Huygens who calculated the system

logarithmically to differ only imperceptibly from that

J 13of 4-comma temperament and thus would have been superior

to the system of 43 meridians had the i-comma temperament

been selected as a standard Sauveur proposed the system

of 43 meridians with the intention that it should be useful

in showing clearly the number of small parts--heptamprldians

13Barbour Tuning and Temperament p 118 The

vnlu8s of the inte~~aJs of Arons ~-comma m~~nto~e t~mnershyLfWnt Hr(~ (~lv(n hy Bnrhour in cenl~8 as 00 ell 7fl D 19~middot Bb ~)1 0 E 3~36 F 503middot 11 579middot G 6 0 (4 17 l b J J

A 890 B 1007 B 1083 and C 1200 Those for- tIle cyshycle of 31 are 0 77 194 310 387 503 581 697 774 8vO 1006 1084 and 1200 The greatest deviation--for tle tri Lone--is 2 cents and the cycle of 31 can thus be seen to appIoximcl te the -t-cornna temperament more closely tlan Sauveur s system of 43 meridians approxirna tes the system of just intonation

119

or decameridians--in the elements as well as the larrer

units of all conceivable systems of intonation and devoted

the fifth section of his M~moire of 1701 to the illustration

of its udaptnbil ity for this purpose [he nystom willeh

approximated mOst closely the just system--the one which

[rave the intervals in their simplest form--thus seemed

more appropriate to Sauveur as an instrument of comparison

which was to be useful in scientific investigations as well

as in purely practical employments and the system which

meeting Sauveurs other requirements--that the comma for

example should bear to the semitone a relationship the

li~its of which we~e rigidly fixed--did in fact

approximate the just system most closely was recommended

as well by the relationship borne by the number of its

parts (43 or 301 or 3010) to the logarithm of 2 which

simplified its application in the scientific measurement

of intervals It will be remembered that the cycle of 301

as well as that of 3010 were included by Ellis amonp the

paper cycles14 _-presumnbly those which not well suited

to tuning were nevertheless usefUl in measurement and

calculation Sauveur was the first to snppest the llse of

small logarithmic parts of any size for these tasks and

was t~le father of the paper cycles based on 3010) or the

15logaritmn of 2 in particular although the divisIon of

14 lis Appendix XX to Helmholtz Sensations of Tone p 43

l5ElLis ascribes the cycle of 30103 iots to de Morgan and notes that Mr John Curwen used this in

120

the octave into 301 (or for simplicity 300) logarithmic

units was later reintroduced by Felix Sava~t as a system

of intervallic measurement 16 The unmodified lo~a~lthmic

systems have been in large part superseded by the syntem

of 1200 cents proposed and developed by Alexande~ EllisI7

which has the advantage of making clear at a glance the

relationship of the number of units of an interval to the

number of semi tones of equal temperament it contains--as

for example 1125 cents corresponds to lIt equal semi-

tones and this advantage is decisive since the system

of equal temperament is in common use

From observations found throughout his published

~ I bulllemOlres it may easily be inferred that Sauveur did not

put forth his system of 43 meridians solely as a scale of

musical measurement In the Ivrt3moi 1e of 1711 for exampl e

he noted that

setting the string of the monochord in unison with a C SOL UT of a harpsichord tuned very accnrately I then placed the bridge under the string putting it in unison ~~th each string of the harpsichord I found that the bridge was always found to be on the divisions of the other systems when they were different from those of the system of 43 18

It seem Clear then that Sauveur believed that his system

his IVLusical Statics to avoid logarithms the cycle of 3010 degrees is of course tnat of Sauveur1s decameridshyiuns I whiJ e thnt of 301 was also flrst sUrr~o~jted hy Sauv(]ur

16 d Dmiddot t i fiharvar 1C onary 0 Mus c 2nd ed sv Intershyvals and Savart II

l7El l iS Appendix XX to Helmholtz Sensatlons of Tone pn 446-451

18Sauveur uTable GeneraletI p 416 see vol II p 165 below

121

so accurately reflected contemporary modes of tuning tLat

it could be substituted for them and that such substitushy

tion would confer great advantages

It may be noted in the cou~se of evalllatlnp this

cJ nlm thnt Sauvours syntem 1 ike the cyc] 0 of r] cnlcll shy

luted by llily~ens is intimately re1ate~ to a meantone

temperament 19 Table 17 gives in its first column the

names of the intervals of Sauveurs system the vn] nos of shy

these intervals ate given in cents in the second column

the third column contains the differences between the

systems of Sauveur and the ~-comma temperament obtained

by subtracting the fourth column from the second the

fourth column gives the values in cents of the intervals

of the ~-comma meantone temperament as they are given)

by Barbour20 and the fifth column contains the names of

1the intervals of the 5-comma meantone temperament the exshy

ponents denoting the fractions of a comma by which the

given intervals deviate from Pythagorean tuning

19Barbour notes that the correspondences between multiple divisions and temperaments by fractional parts of the syntonic comma can be worked out by continued fractions gives the formula for calculating the nnmber part s of the oc tave as Sd = 7 + 12 [114d - 150 5 J VIrhere

12 (21 5 - 2d rr ~ d is the denominator of the fraction expressing ~he ~iven part of the comma by which the fifth is to be tempered and Wrlere 2ltdltll and presents a selection of correspondences thus obtained fit-comma 31 parts g-comma 43 parts

t-comrriU parts ~-comma 91 parts ~-comma 13d ports

L-comrr~a 247 parts r8--comma 499 parts n Barbour

Tuni n9 and remnerament p 126

20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA

NAMES CENTS DIFFERENCE CENTS NAMES

1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83

It will be noticed that the differences between

the system of Sauveur and the ~-comma meantone temperament

amounting to only one cent in the case of only two intershy

vals are even smaller than those between the cycle of 31

and the -comma meantone temperament noted above

Table 18 gives in its five columns the names

of the intervals of Sauveurs system the values of his

intervals in cents the values of the corresponding just

intervals in cen ts the values of the correspondi ng intershy

vals 01 the system of ~-comma meantone temperament the

differences obtained by subtracting the third column fron

123

the second and finally the differences obtained by subshy

tracting the fourth column from the second

TABLE 18

1 2 3 4

SAUVEUHS JUST l-GOriI~ 5

INTEHVALS STiPS INTONA IION TEMPEdliENT G2-C~ G2-C4 IN CENTS IN CENTS II~ CENTS

VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl

It can be seen that the differences between Sauveurs

system and the just system are far ~reater than the differshy

1 ences between his system and the 5-comma mAantone temperashy

ment This wide discrepancy together with fact that when

in propounding his method of reCiprocal intervals in the

Memoire of 170121 he took C of 84 cents rather than the

Db of 112 cents of the just system and Gil (which he

labeled 6 or Ab but which is nevertheless the chromatic

semitone above G) of 781 cents rather than the Ab of 814

cents of just intonation sugpests that if Sauve~r waD both

utterly frank and scrupulously accurate when he stat that

the harpsichord tunings fell precisely on t1e meridional

21SalJVAur Systeme General pp 484-488 see vol II p 82 below

124

divisions of his monochord set for the system of 43 then

those harpsichords with which he performed his experiments

1were tuned in 5-comma meantone temperament This conclusion

would not be inconsonant with the conclusion of Barbour

that the suites of Frangois Couperin a contemnorary of

SU1JVfHlr were performed on an instrument set wt th a m0nnshy

22tone temperamnnt which could be vUYied from piece to pieco

Sauveur proposed his system then as one by which

musical instruments particularly the nroblematic keyboard

instruments could be tuned and it has been seen that his

intervals would have matched almost perfectly those of the

1 15-comma meantone temperament so that if the 5-comma system

of tuning was indeed popular among musicians of the ti~e

then his proposal was not at all unreasonable

It may have been this correspondence of the system

of 43 to one in popular use which along with its other

merits--the simplicity of its calculations based on 301

for example or the fact that within the limitations

Souveur imposed it approximated most closely to iust

intonation--which led Sauveur to accept it and not to con-

tinue his search for a cycle like that of 53 commas

which while not satisfying all of his re(1uirements for

the relatIonship between the slzes of the comma and the

minor semitone nevertheless expressed the just scale

more closely

22J3arbour Tuning and Temperament p 193

125

The sys t em of 43 as it is given by Sa11vcll is

not of course readily adaptihle as is thn system of

equal semi tones to the performance of h1 pJIJy chrorLi t ic

musIc or remote moduJntions wlthollt the conjtYneLlon or

an elahorate keyboard which wOlJld make avai] a hI e nIl of

1r2he htl rps ichoprl tun (~d ~ n r--c OliltU v

menntone temperament which has been shown to be prHcshy

43 meridians was slJbject to the same restrictions and

the oerformer found it necessary to make adjustments in

the tunlnp of his instrument when he vlshed to strike

in the piece he was about to perform a note which was

not avnilahle on his keyboard24 and thus Sallveurs system

was not less flexible encounterert on a keyboard than

the meantone temperaments or just intonation

An attempt to illustrate the chromatic ran~e of

the system of Sauveur when all ot the 43 meridians are

onployed appears in rrable 19 The prlnclples app] led in

()3( EXperimental keyhoard comprisinp vltldn (~eh

octave mOYe than 11 keys have boen descfibed in tne writings of many theorists with examples as earJy ~s Nicola Vicentino (1511-72) whose arcniceITlbalo o7fmiddot ((J the performer a Choice of 34 keys in the octave Di c t i 0 na ry s v tr Mu sic gl The 0 ry bull fI ExampIe s ( J-boards are illustrated in Ellis s Section F of t lle rrHnc1 ~x to j~elmholtzs work all of these keyhoarcls a-rc L~vt~r uc4apted to the system of just intonation 11i8 ~)penrjjx

XX to HelMholtz Sensations of Tone pp 466-483

24It has been m~ntionerl for exa71 e tha t JJ

Jt boar~ San vellr describ es had the notes C C-r D EO 1~

li rolf A Bb and B i h t aveusbull t1 Db middot1t GO gt av n eac oc IJ some of the frequently employed accidentals in crc middotti( t)nal music can not be struck mOYeoveP where (cLi)imiddotLcmiddot~~

are substi tuted as G for Ab dis sonances of va 40middotiJ s degrees will result

126

its construction are two the fifth of 7s + 4c where

s bull 3 and c = 1 is equal to 25 meridians and the accishy

dentals bearing sharps are obtained by an upward projection

by fifths from C while the accidentals bearing flats are

obtained by a downward proiection from C The first and

rIft) co1umns of rrahl e leo thus [1 ve the ace irienta1 s In

f i f t h nr 0 i c c t i ()n n bull I Ph e fir ~~ t c () 111mn h (1 r 1n s with G n t 1 t ~~

bUBo nnd onds with (161 while the fl fth column beg t ns wI Lh

C at its head and ends with F6b at its hase (the exponents

1 2 bullbullbull Ib 2b middotetc are used to abbreviate the notashy

tion of multiple sharps and flats) The second anrl fourth

columns show the number of fifths in the ~roioct1()n for tho

corresponding name as well as the number of octaves which

must be subtracted in the second column or added in the

fourth to reduce the intervals to the compass of one octave

Jlhe numbers in the tbi1d column M Vi ve the numbers of

meridians of the notes corresponding to the names given

in both the first and fifth columns 25 (Table 19)

It will thus be SAen that A is the equivalent of

D rpJint1Jplo flnt and that hoth arB equal to 32 mr-riclians

rphrOl1fhout t1 is series of proi ections it will be noted

25The third line from the top reads 1025 (4) -989 (23) 36 -50 (2) +GG (~)

The I~611 is thus obtained by 41 l1pwnrcl fIfth proshyiectlons [iving 1025 meridians from which 2~S octavns or SlF)9 meridlals are subtracted to reduce th6 intJlvll to i~h(~ conpass of one octave 36 meridIans rernEtjn nalorollf~j-r

Bb is obtained by 2 downward fifth projections givinr -50 meridians to which 2 octaves or 86 meridians are added to yield the interval in the compass of the appropriate octave 36 meridians remain

127

tV (in M) -VIII (in M) M -v (in ~) +vrIr (i~ M)

1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()

(33) -817 8 -250 (10) +25d () HuO (3~) -774 26 -)7) ( 11 ) +Hl1 (I) T~) (~ 1 ) -l1 (] ~) ) -OU ( 1) ) JUl Cl)

(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)

525 (21) -516 (12) 9 (22) +5) 1) ( Ymiddotmiddot ) (20) -473 (11) 27 -55 _J + C02 1 lt1 )

~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )

JYJ (l~) middot-2)L (6) 42 -775 (31) I- (j17 1 ) 2S (11) -53 (6) 17 -800 ( ) +j17

(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128

that the relationships between the intervals of one type

of accidental remain intact thus the numher of meridians

separating F(21) and F(24) are three as might have been

expected since 3 meridians are allotted to the minor

sernitone rIhe consistency extends to lonFer series of

accidcntals as well F(21) F(24) F2(28) F3(~O)

p41 (3)) 1l nd F5( 36 ) follow undevia tinply tho rulo thnt

li chrornitic scmltono ie formed hy addlnp ~gt morldHn1

The table illustrates the general principle that

the number of fIfth projections possihle befoTe closure

in a cyclical system like that of Sauveur is eQ11 al to

the number of steps in the system and that one of two

sets of fifth projections the sharps will he equivalent

to the other the flats In the system of equal temperashy

ment the projections do not extend the range of accidenshy

tals beyond one sharp or two flats befor~ closure--B is

equal to C and Dbb is egual to C

It wOl11d have been however futile to extend the

ranrre of the flats and sharps in Sauveurs system in this

way for it seems likely that al though he wi sbed to

devise a cycle which would be of use in performance while

also providinp a fairly accurate reflection of the just

scale fo~ purposes of measurement he was satisfied that

the system was adequate for performance on account of the

IYrJationship it bore to the 5-comma temperament Sauveur

was perhaps not aware of the difficulties involved in

more or less remote modulations--the keyhoard he presents

129

in the third plate subjoined to the M~moire of 170126 is

provided with the names of lfthe chromatic system of

musicians--names of the notes in B natural with their

sharps and flats tl2--and perhaps not even aware thnt the

range of sIlarps and flats of his keyboard was not ucleqUtlt)

to perform the music of for example Couperin of whose

suites for c1avecin only 6 have no more than 12 different

scale c1egrees 1I28 Throughout his fJlemoires howeve-r

Sauveur makes very few references to music as it is pershy

formed and virtually none to its harmonic or melodic

characteristics and so it is not surprising that he makes

no comment on the appropriateness of any of the systems

of tuning or temperament that come under his scrutiny to

the performance of any particular type of music whatsoever

The convenience of the method he nrovirled for findshy

inr tho number of heptamorldians of an interval by direct

computation without tbe use of tables of logarithms is

just one of many indications throughout the M~moires that

Sauveur did design his system for use by musicians as well

as by methemRticians Ellis who as has been noted exshy

panded the method of bimodular computat ion of logari thms 29

credited to Sauveurs Memoire of 1701 the first instance

I26Sauveur tlSysteme General p 498 see vol II p 97 below

~ I27Sauvel1r ffSyst~me General rt p 450 see vol

II p 47 b ow

28Barbol1r Tuning and Temperament p 193

29Ellls Improved Method

130

of its use Nonetheless Ellis who may be considerect a

sort of heir of an unpublicized part of Sauveus lep-acy

did not read the will carefully he reports tha t Sallv0ur

Ugives a rule for findln~ the number of hoptamerides in

any interval under 67 = 267 cents ~SO while it is clear

from tho cnlculntions performed earlier in thIs stllOY

which determined the limit implied by Sauveurs directions

that intervals under 57 or 583 cents may be found by his

bimodular method and Ellis need not have done mo~e than

read Sauveurs first example in which the number of

heptameridians of the fourth with a ratio of 43 and a

31value of 498 cents is calculated as 125 heptameridians

to discover that he had erred in fixing the limits of the

32efficacy of Sauveur1s method at 67 or 267 cents

If Sauveur had among his followers none who were

willing to champion as ho hud tho system of 4~gt mcridians-shy

although as has been seen that of 301 heptameridians

was reintroduced by Savart as a scale of musical

30Ellis Appendix XX to Helmholtz Sensations of Tone p 437

31 nJal1veur middotJYS t I p 4 see ]H CI erne nlenera 21 vobull II p lB below

32~or does it seem likely that Ellis Nas simply notinr that althouph SauveuJ-1 considered his method accurate for all intervals under 57 or 583 cents it cOI1d be Srlown that in fa ct the int erval s over 6 7 or 267 cents as found by Sauveurs bimodular comput8tion were less accurate than those under 67 or 267 cents for as tIle calculamptions of this stl1dy have shown the fourth with a ratio of 43 am of 498 cents is the interval most accurately determined by Sauveurls metnoa

131

measurement--there were nonetheless those who followed

his theory of the correct formation of cycles 33

The investigations of multiple division of the

octave undertaken by Snuveur were accordin to Barbour ~)4

the inspiration for a similar study in which Homieu proshy

posed Uto perfect the theory and practlce of temporunent

on which the systems of music and the division of instrushy

ments with keys depends35 and the plan of which is

strikingly similar to that followed by Sauveur in his

of 1707 announcin~ thatMemolre Romieu

After having shovm the defects in their scale [the scale of the ins trument s wi th keys] when it ts tuned by just intervals I shall establish the two means of re~edying it renderin~ its harmony reciprocal and little altered I shall then examine the various degrees of temperament of which one can make use finally I shall determine which is the best of all 36

Aft0r sumwarizing the method employed by Sauveur--the

division of the tone into two minor semitones and a

comma which Ro~ieu calls a quarter tone37 and the

33Barbou r Ttlning and Temperame nt p 128

~j4Blrhollr ttHlstorytI p 21lB

~S5Romieu rrr7emoire Theorioue et Pra tlolJe sur 1es I

SYJtrmon Tnrnp()res de Mus CJ1Je II Memoi res de l lc~~L~~~_~o_L~)yll n ~~~_~i (r (~ e finn 0 ~7 ~)4 p 4H~-) ~ n bull bull de pen j1 (~ ~ onn Of

la tleortr amp la pYgtat1llUe du temperament dou depennent leD systomes rie hlusiaue amp In partition des Instruments a t011cheR

36Ihid bull Apres avoir montre le3 defauts de 1eur gamme lorsquel1e est accord6e par interval1es iustes j~tablirai les deux moyens aui y ~em~dient en rendant son harm1nie reciproque amp peu alteree Jexaminerai ensui te s dlfferens degr6s de temperament d~nt on pellt fai re usare enfin iu determinera quel est 1e meil1enr de tons

3Ibld p 488 bull quart de ton

132

determination of the ratio between them--Romieu obiects

that the necessity is not demonstrated of makinr an

equal distribution to correct the sCale of the just

nY1 tnm n~)8

11e prosents nevortheless a formuJt1 for tile cllvlshy

sions of the octave permissible within the restrictions

set by Sauveur lIit is always eoual to the number 6

multiplied by the number of parts dividing the tone plus Lg

unitytl O which gives the series 1 7 13 bull bull bull incJuding

19 31 43 and 55 which were the numbers of parts of

systems examined by Sauveur The correctness of Romieus

formula is easy to demonstrate the octave is expressed

by Sauveur as 12s ~ 7c and the tone is expressed by S ~ s

or (s ~ c) + s or 2s ~ c Dividing 12 + 7c by 2s + c the

quotient 6 gives the number of tones in the octave while

c remalns Thus if c is an aliquot paTt of the octave

then 6 mult-tplied by the numher of commas in the tone

plus 1 will pive the numher of parts in the octave

Romieu dec1ines to follow Sauveur however and

examines instead a series of meantone tempernments in which

the fifth is decreased by ~ ~ ~ ~ ~ ~ ~ l~ and 111 comma--precisely those in fact for which Barbol1r

38 Tb i d bull It bull

bull on d emons t re pOln t 1 aJ bull bull mal s ne n6cessita qUil y a de faire un pareille distribution pour corriger la gamme du systeme juste

39Ibid p 490 Ie nombra des oart i es divisant loctave est toujours ega1 au nombra 6 multiplie par le~nombre des parties divisant Ie ton plus lunite

133

gives the means of finding cyclical equivamplents 40 ~he 2system of 31 parts of lhlygens is equated to the 9 temperashy

ment to which howeve~ it is not so close as to the

1 414-conma temperament Romieu expresses a preference for

1 t 1 2 t t h tthe 5 -comina temperamen to 4 or 9 comma emperamen s u

recommends the ~-comma temperament which is e~uiv31ent

to division into 55 parts--a division which Sauveur had

10 iec ted 42

40Barbour Tuning and Temperament n 126

41mh1 e values in cents of the system of Huygens

of 1 4-comma temperament as given by Barbour and of

2 gcomma as also given by Barbour are shown below

rJd~~S CHjl

D Eb E F F G Gft A Bb B

Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9

The total of the devia tions in cent s of the system of 1Huygens and that of 4-coInt~a temperament is thus 9 and

the anaJogous total for the system of Huygens and that

of ~-comma temperament is 13 cents Barbour Tuninrr and Temperament p 37

42Nevertheless it was the system of 55 co~~as waich Sauveur had attributed to musicians throu~hout his ~emoires and from its clos6 correspondence to the 16-comma temperament which (as has been noted by Barbour in Tuning and 1Jemperament p 9) represents the more conservative practice during the tL~e of Bach and Handel

134

The system of 43 was discussed by Robert Smlth43

according to Barbour44 and Sauveurs method of dividing

the octave tone was included in Bosanquets more compreshy

hensive discussion which took account of positive systems-shy

those that is which form their thirds by the downward

projection of 8 fifths--and classified the systems accord-

Ing to tile order of difference between the minor and

major semi tones

In a ~eg1llar Cyclical System of order plusmnr the difshyference between the seven-flfths semitone and t~5 five-fifths semitone is plusmnr units of the system

According to this definition Sauveurs cycles of 31 43

and 55 parts are primary nepatlve systems that of

Benfling with its s of 3 its S of 5 and its c of 2

is a secondary ne~ative system while for example the

system of 53 with as perhaps was heyond vlhat Sauveur

would have considered rational an s of 5 an S of 4 and

a c of _146 is a primary negative system It may be

noted that j[lUVe1Jr did consider the system of 53 as well

as the system of 17 which Bosanquet gives as examples

of primary positive systems but only in the M~moire of

1711 in which c is no longer represented as an element

43 Hobert Smith Harmcnics J or the Phi1osonhy of J1usical Sounds (Cambridge 1749 reprint ed New York Da Capo Press 1966)

44i3arbour His to ry II p 242 bull Smith however anparert1y fa vored the construction of a keyboard makinp availahle all 43 degrees

45BosanquetTemperamentrr p 10

46The application of the formula l2s bull 70 in this case gives 12(5) + 7(-1) or 53

135

as it was in the Memoire of 1707 but is merely piven the

47algebraic definition 2s - t Sauveur gave as his reason

for including them that they ha ve th eir partisans 11 48

he did not however as has already been seen form the

intervals of these systems in the way which has come to

be customary but rather proiected four fifths upward

in fact as Pytharorean thirds It may also he noted that

Romieus formula 6P - 1 where P represents the number of

parts into which the tone is divided is not applicable

to systems other than the primary negative for it is only

in these that c = 1 it can however be easily adapted

6P + c where P represents the number of parts in a tone

and 0 the value of the comma gives the number of parts

in the octave 49

It has been seen that the system of 43 as it was

applied to the keyboard by Sauveur rendered some remote

modulat~ons difficl1l t and some impossible His discussions

of the system of equal temperament throughout the Memoires

show him to be as Barbour has noted a reactionary50

47Sauveur nTab1 e G tr J 416 1 IT Ienea e p bull see vo p 158 below

48Sauvellr Table Geneale1r 416middot vol IIl p see

p 159 below

49Thus with values for S s and c of 2 3 and -1 respectively the number of parts in the octave will be 6( 2 + 3) (-1) or 29 and the system will be prima~ly and positive

50Barbour History n p 247

12

136

In this cycle S = sand c = 0 and it thus in a sense

falls outside BosanqlJet s system of classification In

the Memoire of 1707 SauveuT recognized that the cycle of

has its use among the least capable instrumentalists because of its simnlicity and its easiness enablin~ them to transpose the notes ut re mi fa sol la si on whatever keys they wish wi thout any change in the intervals 51

He objected however that the differences between the

intervals of equal temperament and those of the diatonic

system were t00 g-rea t and tha t the capabl e instr1Jmentshy

alists have rejected it52 In the Memolre of 1711 he

reiterated that besides the fact that the system of 12

lay outside the limits he had prescribed--that the ratio

of the minor semi tone to the comma fall between 1~ and

4~ to l--it was defective because the differences of its

intervals were much too unequal some being greater than

a half-corrJ11a bull 53 Sauveurs judgment that the system of

equal temperament has its use among the least capable

instrumentalists seems harsh in view of the fact that

Bach only a generation younger than Sauveur included

in his works for organ ua host of examples of triads in

remote keys that would have been dreadfully dissonant in

any sort of tuning except equal temperament54

51Sauveur Methode Generale p 272 see vo] II p 140 below

52 Ibid bull

53Sauveur Table Gen~rale1I p 414 see vo~ II~ p 163 below


137

If Sauveur was not the first to discuss the phenshy

55 omenon of beats he was the first to make use of them

in determining the number of vibrations of a resonant body

in a second The methon which for long was recorrni7ed us

6the surest method of nssessinp vibratory freqlonc 10 ~l )

wnn importnnt as well for the Jiht it shed on tho nntlH()

of bents SauvelJr s mo st extensi ve 01 SC1l s sion of wh ich

is available only in Fontenelles report of 1700 57 The

limits established by Sauveur according to Fontenelle

for the perception of beats have not been generally

accepte~ for while Sauveur had rema~ked that when the

vibrations dve to beats ape encountered only 6 times in

a second they are easily di stinguished and that in

harmonies in which the vibrations are encountered more

than six times per second the beats are not perceived

at tl1lu5B it hn3 heen l1Ypnod hy Tlnlmholt7 thnt fl[l mHny

as 132 beats in a second aTe audihle--an assertion which

he supposed would appear very strange and incredible to

acol1sticians59 Nevertheless Helmholtz insisted that

55Eersenne for example employed beats in his method of setting the equally tempered octave Ibid p 7


57 If IfFontenelle Determination

58 Ibid p 193 ItQuand bullbullbull leurs vibratior ne se recontroient que 6 fols en une Seconde on dist~rshyguol t ces battemens avec assez de facili te Lone d81S to-l~3 les accords at les vibrations se recontreront nlus de 6 fois par Seconde on ne sentira point de battemens 1I

59Helmholtz Sensations of Tone p 171

138

his claim could be verified experimentally

bull bull bull if on an instrument which gi ves Rns tainec] tones as an organ or harmonium we strike series[l

of intervals of a Semitone each heginnin~ low rtown and proceeding hipher and hi~her we shall heap in the lower pnrts veray slow heats (B C pives 4h Bc

~ives a bc gives l6~ beats in a sAcond) and as we ascend the ranidity wi11 increase but the chnrnctnr of tho nnnnntion pnnn1n nnnl tfrod Inn thl~l w() cnn pflJS FParlually frOtll 4 to Jmiddotmiddotlt~ b()nL~11n a second and convince ourgelves that though we beshycomo incapable of counting them thei r character as a series of pulses of tone producl ng an intctml tshytent sensation remains unaltered 60

If as seems likely Sauveur intended his limit to be

understood as one beyond which beats could not be pershy

ceived rather than simply as one beyond which they could

not be counted then Helmholtzs findings contradict his

conjecture61 but the verdict on his estimate of the

number of beats perceivable in one second will hardly

affect the apnlicability of his method andmoreovAr

the liMit of six beats in one second seems to have heen

e~tahJ iRhed despite the way in which it was descrlheo

a~ n ronl tn the reductIon of tlle numhe-r of boats by ] ownrshy

ing the pitCh of the pipes or strings emJ)loyed by octavos

Thus pipes which made 400 and 384 vibrations or 16 beats

in one second would make two octaves lower 100 and V6

vtbrations or 4 heats in one second and those four beats

woulrl be if not actually more clearly perceptible than

middot ~60lb lO

61 Ile1mcoltz it will be noted did give six beas in a secord as the maximum number that could be easily c01~nted elmholtz Sensatlons of Tone p 168

139

the 16 beats of the pipes at a higher octave certainly

more easily countable

Fontenelle predicted that the beats described by

Sauveur could be incorporated into a theory of consonance

and dissonance which would lay bare the true source of

the rules of composition unknown at the present to

Philosophy which relies almost entirely on the judgment

of the ear62 The envisioned theory from which so much

was to be expected was to be based upon the observation

that

the harmonies of which you can not he~r the heats are iust those th1t musicians receive as consonanceR and those of which the beats are perceptible aIS dissoshynances amp when a harmony is a dissonance in a certain octave and a consonance in another~ it is that it beats in one and not in the other o3

Iontenelles prediction was fulfilled in the theory

of consonance propounded by Helmholtz in which he proposed

that the degree of consonance or dissonance could be preshy

cis ely determined by an ascertainment of the number of

beats between the partials of two tones

When two musical tones are sounded at the same time their united sound is generally disturbed by

62Fl ontenell e Determina tion pp 194-195 bull bull el1 e deco1vrira ] a veritable source des Hegles de la Composition inconnue jusquA pr6sent ~ la Philosophie nui sen re~ettoit presque entierement au jugement de lOreille

63 Ihid p 194 Ir bullbullbull les accords dont on ne peut entendre les battemens sont justement ceux que les VtJsiciens traitent de Consonances amp que cellX dont les battemens se font sentir sont les Dissonances amp que quand un accord est Dissonance dans une certaine octave amp Consonance dans une autre cest quil bat dans lune amp qulil ne bat pas dans lautre

140

the beate of the upper partials so that a ~re3teI

or less part of the whole mass of sound is broken up into pulses of tone m d the iolnt effoct i8 rOll(rh rrhis rel-ltion is c311ed n[n~~~

But there are certain determinate ratIos betwcf1n pitch numbers for which this rule sllffers an excepshytion and either no beats at all are formed or at loas t only 311Ch as have so little lntonsi ty tha t they produce no unpleasa11t dIsturbance of the united sound These exceptional cases are called Consona nces 64

Fontenelle or perhaps Sauvellr had also it soema

n()tteod Inntnnces of whnt hns come to be accepted n8 a

general rule that beats sound unpleasant when the

number of heats Del second is comparable with the freshy65

quencyof the main tonerr and that thus an interval may

beat more unpleasantly in a lower octave in which the freshy

quency of the main tone is itself lower than in a hirher

octave The phenomenon subsumed under this general rule

constitutes a disadvantape to the kind of theory Helmholtz

proposed only if an attenpt is made to establish the

absolute consonance or dissonance of a type of interval

and presents no problem if it is conceded that the degree

of consonance of a type of interval vuries with the octave

in which it is found

If ~ontenelle and Sauveur we~e of the opinion howshy

ever that beats more frequent than six per second become

actually imperceptible rather than uncountable then they

cannot be deemed to have approached so closely to Helmholtzs

theory Indeed the maximum of unpleasantness is


65 Sir James Jeans Science and Music (Cambridge at the University Press 1953) p 49

141

reached according to various accounts at about 25 beats

par second 66

Perhaps the most influential theorist to hase his

worl on that of SaUVC1Jr mflY nevArthele~l~ he fH~()n n0t to

have heen in an important sense his follower nt nll

tloHn-Phl11ppc Hamenu (J6B)-164) had pllhlished a (Prnit)

67de 1 Iarmonie in which he had attempted to make music

a deductive science hased on natural postu1ates mvch

in the same way that Newton approaches the physical

sci ences in hi s Prineipia rr 68 before he l)ecame famll iar

with Sauveurs discoveries concerning the overtone series

Girdlestone Hameaus biographer69 notes that Sauveur had

demonstrated the existence of harmonics in nature but had

failed to explain how and why they passed into us70

66Jeans gives 23 as the number of beats in a second corresponding to the maximum unpleasantnefls while the Harvard Dictionary gives 33 and the Harvard Dlctionarl even adds the ohservAtion that the beats are least nisshyturbing wnen eneQunteted at freouenc s of le88 than six beats per second--prec1sely the number below which Sauveur anrl Fontonel e had considered thorn to be just percepshytible (Jeans ScIence and Muslcp 153 Barvnrd Die tionar-r 8 v Consonance Dissonance

67Jean-Philippe Rameau Trait6 de lHarmonie Rediute a ses Principes naturels (Paris Jean-BaptisteshyChristophe Ballard 1722)

68Gossett Ramea1J Trentise p xxii

6gUuthbe~t Girdlestone Jean-Philippe Rameau ~jsLife and Wo~k (London Cassell and Company Ltd 19b7)

70Ibid p 516

11-2

It was in this respect Girdlestone concludes that

Rameau began bullbullbull where Sauveur left off71

The two claims which are implied in these remarks

and which may be consider-ed separa tely are that Hamenn

was influenced by Sauveur and tho t Rameau s work somehow

constitutes a continuation of that of Sauveur The first

that Hamonus work was influenced by Sauvollr is cOTtalnly

t ru e l non C [ P nseat 1 c n s t - - b Y the t1n e he Vir 0 t e the

Nouveau systeme of 1726 Hameau had begun to appreciate

the importance of a physical justification for his matheshy

rna tical manipulations he had read and begun to understand

72SauvelJr n and in the Gen-rntion hurmoniolle of 17~~7

he had 1Idiscllssed in detail the relatlonship between his

73rules and strictly physical phenomena Nonetheless

accordinv to Gossett the main tenets of his musical theory

did n0t lAndergo a change complementary to that whtch had

been effected in the basis of their justification

But tte rules rem11in bas]cally the same rrhe derivations of sounds from the overtone series proshyduces essentially the same sounds as the ~ivistons of

the strinp which Rameau proposes in the rrraite Only the natural tr iustlfication is changed Here the natural principle ll 1s the undivided strinE there it is tne sonorous body Here the chords and notes nre derlved by division of the string there througn the perception of overtones 74

If Gossetts estimation is correct as it seems to be

71 Ibid bull

72Gossett Ramerul Trait~ p xxi

73 Ibid bull

74 Ibi d

143

then Sauveurs influence on Rameau while important WHS

not sO ~reat that it disturbed any of his conc]usions

nor so beneficial that it offered him a means by which

he could rid himself of all the problems which bGset them

Gossett observes that in fact Rameaus difficulty in

oxplHininr~ the minor third was duo at loast partly to his

uttempt to force into a natural framework principles of

comnosition which although not unrelated to acoustlcs

are not wholly dependent on it75 Since the inadequacies

of these attempts to found his conclusions on principles

e1ther dlscoverable by teason or observabJe in nature does

not of conrse militate against the acceptance of his

theories or even their truth and since the importance

of Sauveurs di scoveries to Rameau s work 1ay as has been

noted mere1y in the basis they provided for the iustifi shy

cation of the theories rather than in any direct influence

they exerted in the formulation of the theories themse1ves

then it follows that the influence of Sauveur on Rameau

is more important from a philosophical than from a practi shy

cal point of view

lhe second cIa im that Rameau was SOl-11 ehow a

continuator of the work of Sauvel~ can be assessed in the

light of the findings concerning the imnortance of

Sauveurs discoveries to Hameaus work It has been seen

that the chief use to which Rameau put Sauveurs discovershy

ies was that of justifying his theory of harmony and

75 Ibid p xxii

144

while it is true that Fontenelle in his report on Sauveur1s

M~moire of 1702 had judged that the discovery of the harshy

monics and their integral ratios to unity had exposed the

only music that nature has piven us without the help of

artG and that Hamenu us hHs boen seen had taken up

the discussion of the prinCiples of nature it is nevershy

theless not clear that Sauveur had any inclination whatevor

to infer from his discoveries principles of nature llpon

which a theory of harmony could be constructed If an

analogy can be drawn between acoustics as that science

was envisioned by Sauve1rr and Optics--and it has been

noted that Sauveur himself often discussed the similarities

of the two sciences--then perhaps another analogy can be

drawn between theories of harmony and theories of painting

As a painter thus might profit from a study of the prinshy

ciples of the diffusion of light so might a composer

profit from a study of the overtone series But the

painter qua painter is not a SCientist and neither is

the musical theorist or composer qua musical theorist

or composer an acoustician Rameau built an edifioe

on the foundations Sauveur hampd laid but he neither

broadened nor deepened those foundations his adaptation

of Sauveurs work belonged not to acoustics nor pe~haps

even to musical theory but constituted an attempt judged

by posterity not entirely successful to base the one upon

the other Soherchens claims that Sauveur pointed out

76Fontenelle Application p 120

145

the reciprocal powers 01 inverted interva1su77 and that

Sauveur and Hameau together introduced ideas of the

fundamental flas a tonic centerU the major chord as a

natural phenomenon the inversion lias a variant of a

chordU and constrllcti0n by thiTds as the law of chord

formationff78 are thus seAn to be exaggerations of

~)a1Jveur s influence on Hameau--prolepses resul tinp- pershy

hnps from an overestim1 t on of the extent of Snuvcllr s

interest in harmony and the theories that explain its

origin

Phe importance of Sauveurs theories to acol1stics

in general must not however be minimized It has been

seen that much of his terminology was adopted--the terms

nodes ftharmonics1I and IIftJndamental for example are

fonnd both in his M~moire of 1701 and in common use today

and his observation that a vibratinp string could produce

the sounds corresponding to several harmonics at the same

time 79 provided the subiect for the investigations of

1)aniel darnoulli who in 1755 provided a dynamical exshy

planation of the phenomenon showing that

it is possible for a strinp to vlbrate in such a way that a multitude of simple harmonic oscillatjons re present at the same time and that each contrihutes independently to the resultant vibration the disshyplacement at any point of the string at any instant

77Scherchen Nature of llusic p b2

8Ib1d bull J p 53

9Lindsay Introduction to Raleigh Sound p xv

146

being the algebraic sum of the displacements for each simple harmonic node SO

This is the fa1jloUS principle of the coexistence of small

OSCillations also referred to as the superposition

prlnclple ll which has Tlproved of the utmost lmportnnce in

tho development of the theory 0 f oscillations u81

In Sauveurs apolication of the system of harmonIcs

to the cornpo)ition of orrHl stops he lnld down prtnc1plos

that were to be reiterated more than a century und a half

later by Helmholtz who held as had Sauveur that every

key of compound stops is connected with a larger or

smaller seles of pipes which it opens simultaneously

and which give the nrime tone and a certain number of the

lower upper partials of the compound tone of the note in

question 82

Charles Culver observes that the establishment of

philosophical pitch with G having numbers of vibrations

per second corresponding to powers of 2 in the work of

the aconstician Koenig vvas probably based on a suggestion

said to have been originally made by the acoustician

Sauveuy tf 83 This pi tch which as has been seen was

nronosed by Sauvel1r in the 1iemoire of 1713 as a mnthematishy

cally simple approximation of the pitch then in use-shy

Culver notes that it would flgive to A a value of 4266

80Ibid bull

81 Ibid bull

L82LTeJ- mh 0 ItZ Stmiddotensa lons 0 f T one p 57 bull

83Charles A Culver MusicalAcoustics 4th ed (New York McGraw-Hill Book Company Inc 19b6) p 86

147

which is close to the A of Handel84_- came into widespread

use in scientific laboratories as the highly accurate forks

made by Koenig were accepted as standards although the A

of 440 is now lIin common use throughout the musical world 1I 85

If Sauveur 1 s calcu]ation by a somewhat (lllhious

method of lithe frequency of a given stretched strlnf from

the measl~red sag of the coo tra1 l)oint 86 was eclipsed by

the publication in 1713 of the first dynamical solution

of the problem of the vibrating string in which from the

equation of an assumed curve for the shape of the string

of such a character that every point would reach the recti shy

linear position in the same timeft and the Newtonian equashy

tion of motion Brook Taylor (1685-1731) was able to

derive a formula for the frequency of vibration agreeing

87with the experimental law of Galileo and Mersenne

it must be remembered not only that Sauveur was described

by Fontenelle as having little use for what he called

IIInfinitaires88 but also that the Memoire of 1713 in

which these calculations appeared was printed after the

death of MY Sauveur and that the reader is requested

to excuse the errors whlch may be found in it flag

84 Ibid bull

85 Ibid pp 86-87 86Lindsay Introduction Ray~igh Theory of

Sound p xiv

87 Ibid bull

88Font enell e 1tEloge II p 104

89Sauveur Rapport It p 469 see vol II p201 below

148

Sauveurs system of notes and names which was not

of course adopted by the musicians of his time was nevershy

theless carefully designed to represent intervals as minute

- as decameridians accurately and 8ystemnticalJy In this

hLLcrnp1~ to plovldo 0 comprnhonsl 1( nytltorn of nrlInn lind

notes to represent all conceivable musical sounds rather

than simply to facilitate the solmization of a meJody

Sauveur transcended in his work the systems of Hubert

Waelrant (c 1517-95) father of Bocedization (bo ce di

ga 10 rna nil Daniel Hitzler (1575-1635) father of Bebishy

zation (la be ce de me fe gel and Karl Heinrich

Graun (1704-59) father of Damenization (da me ni po

tu la be) 90 to which his own bore a superfici al resemshy

blance The Tonwort system devised by KaYl A Eitz (1848shy

1924) for Bosanquets 53-tone scale91 is perhaps the

closest nineteenth-centl1ry equivalent of Sauveur t s system

In conclusion it may be stated that although both

Mersenne and Sauveur have been descrihed as the father of

acoustics92 the claims of each are not di fficul t to arbishy

trate Sauveurs work was based in part upon observashy

tions of Mersenne whose Harmonie Universelle he cites

here and there but the difference between their works is

90Harvard Dictionary 2nd ed sv Solmization 1I

9l Ibid bull

92Mersenne TI Culver ohselves is someti nlA~ reshyfeTred to as the Father of AconsticstI (Culver Mll~jcqJ

COllStics p 85) while Scherchen asserts that Sallveur lair1 the foundqtions of acoustics U (Scherchen Na ture of MusiC p 28)

149

more striking than their similarities Versenne had

attempted to make a more or less comprehensive survey of

music and included an informative and comprehensive antholshy

ogy embracing all the most important mllsical theoreticians

93from Euclid and Glarean to the treatise of Cerone

and if his treatment can tlU1S be described as extensive

Sa1lvellrs method can be described as intensive--he attempted

to rllncove~ the ln~icnl order inhnrent in the rolntlvoly

smaller number of phenomena he investiFated as well as

to establish systems of meRsurement nomAnclature and

symbols which Would make accurate observnt1on of acoustical

phenomena describable In what would virtually be a universal

language of sounds

Fontenelle noted that Sauveur in his analysis of

basset and other games of chance converted them to

algebraic equations where the players did not recognize

94them any more 11 and sirrLilarly that the new system of

musical intervals proposed by Sauveur in 1701 would

proh[tbJ y appBar astonishing to performers

It is something which may appear astonishing-shyseeing all of music reduced to tabl es of numhers and loparithms like sines or tan~ents and secants of a ciYc]e 95

llatl1Ye of Music p 18

94 p + 11 771 ] orL ere e L orre p bull02 bull bullbull pour les transformer en equations algebraiaues o~ les ioueurs ne les reconnoissoient plus

95Fontenelle ollve~u Systeme fI p 159 IIC t e8t une chose Clu~ peut paroitYe etonante oue de volr toute 1n Ulusirrue reduite en Tahles de Nombres amp en Logarithmes comme les Sinus au les Tangentes amp Secantes dun Cercle

150

These two instances of Sauveurs method however illustrate

his general Pythagorean approach--to determine by means

of numhers the logical structure 0 f t he phenomenon under

investi~ation and to give it the simplest expression

consistent with precision

rlg1d methods of research and tlprecisj_on in confining

himself to a few important subiects96 from Rouhault but

it can be seen from a list of the topics he considered

tha t the ranf1~e of his acoustical interests i~ practically

coterminous with those of modern acoustical texts (with

the elimination from the modern texts of course of those

subjects which Sauveur could not have considered such

as for example electronic music) a glance at the table

of contents of Music Physics Rnd Engineering by Harry

f Olson reveals that the sl1b5ects covered in the ten

chapters are 1 Sound Vvaves 2 Musical rerminology

3 Music)l Scales 4 Resonators and RanlatoYs

t) Ml)sicnl Instruments 6 Characteri sties of Musical

Instruments 7 Properties of Music 8 Thenter Studio

and Room Acoustics 9 Sound-reproduclng Systems

10 Electronic Music 97

Of these Sauveur treated tho first or tho pro~ai~a-

tion of sound waves only in passing the second through

96Scherchen Nature of ~lsic p 26

97l1arry F Olson Music Physics and Enrrinenrinrr Second ed (New York Dover Publications Inc 1967) p xi

151

the seventh in great detail and the ninth and tenth

not at all rrhe eighth topic--theater studio and room

acoustic s vIas perhaps based too much on the first to

attract his attention

Most striking perh8ps is the exclusion of topics

relatinr to musical aesthetics and the foundations of sysshy

t ems of harr-aony Sauveur as has been seen took pains to

show that the system of musical nomenclature he employed

could be easily applied to all existing systems of music-shy

to the ordinary systems of musicians to the exot 1c systems

of the East and to the ancient systems of the Greeks-shy

without providing a basis for selecting from among them the

one which is best Only those syster1s are reiectec1 which

he considers proposals fo~ temperaments apnroximating the

iust system of intervals ana which he shows do not come

so close to that ideal as the ODe he himself Dut forward

a~ an a] terflR ti ve to them But these systems are after

all not ~)sical systems in the strictest sense Only

occasionally then is an aesthetic judgment given weight

in t~le deliberations which lead to the acceptance 0( reshy

jection of some corollary of the system

rrho rl ifference between the lnnges of the wHlu1 0 t

jiersenne and Sauveur suggests a dIs tinction which will be

of assistance in determining the paternity of aCollstics

Wr1i Ie Mersenne--as well as others DescHrtes J Clal1de

Perrau1t pJnllis and riohertsuo-mnde illlJminatlnrr dl~~covshy

eries concernin~ the phenomena which were later to be

s tlJdied by Sauveur and while among these T~ersenne had

152

attempted to present a compendium of all the information

avniJable to scholars of his generation Sauveur hnd in

contrast peeled away the layers of spectl1a tion which enshy

crusted the study of sound brourht to that core of facts

a systematic order which would lay bare tleir 10gicHI reshy

In tions and invented for further in-estir-uti ons systoms

of nomenclutufte and instruments of measurement Tlnlike

Rameau he was not a musical theorist and his system

general by design could express with equal ease the

occidental harraonies of Hameau or the exotic harmonies of

tho Far East It was in the generality of his system

that hIs ~ystem conld c]aLrn an extensIon equal to that of

Mersenne If then Mersennes labors preceded his

Sauveur nonetheless restricted the field of acoustics to

the study of roughly the same phenomena as a~e now studied

by acoustic~ans Whether the fat~erhood of a scIence

should be a ttrihllted to a seminal thinker or to an

organizer vvho gave form to its inquiries is not one

however vlhich Can be settled in the course of such a

study as this one

It must be pointed out that however scrllpulo1)sly

Sauveur avoided aesthetic judgments and however stal shy

wurtly hn re8isted the temptation to rronnd the theory of

haytrlony in hIs study of the laws of nature he n()nethelt~ss

ho-)ed that his system vlOuld be deemed useflll not only to

scholfjrs htJt to musicians as well and it i~ -pprhftnD one

of the most remarkahle cha~actAristics of h~ sv~tem that

an obvionsly great effort has been made to hrinp it into

153

har-mony wi th practice The ingenious bimodllJ ar method

of computing musical lo~~rtthms for example is at once

a we] come addition to the theorists repertoire of

tochniquQs and an emInent] y oractical means of fl n(1J nEr

heptameridians which could be employed by anyone with the

ability to perform simple aritbmeticHl operations

Had 0auveur lived longer he might have pursued

further the investigations of resonatinG bodies for which

- he had already provided a basis Indeed in th e 1e10 1 re

of 1713 Sauveur proposed that having established the

principal foundations of Acoustics in the Histoire de

J~cnd~mie of 1700 and in the M6moi~e8 of 1701 1702

107 and 1711 he had chosen to examine each resonant

body in particu1aru98 the first fruits of which lnbor

he was then offering to the reader

As it was he left hebind a great number of imporshy

tlnt eli acovcnios and devlces--prlncipnlly thG fixr-d pi tch

tne overtone series the echometer and the formulas for

tne constrvctlon and classificatlon of terperarnents--as

well as a language of sovnd which if not finally accepted

was nevertheless as Fontenelle described it a

philosophical languare in vk1ich each word carries its

srngo vvi th it 99 But here where Sauvenr fai] ed it may

b ( not ed 0 ther s hav e no t s u c c e e ded bull

98~r 1 veLid 1 111 Rapport p 430 see vol II p 168 be10w

99 p lt on t ene11_e UNOlJVeaU Sys tenG p 179 bull

Sauvel1r a it pour les Sons un esnece de Ianpue philosonhioue bull bull bull o~ bull bull bull chaque mot porte son sens avec soi 1T

iVORKS CITED

Backus tJohn rrhe Acollstical Founnatlons of Music New York W W Norton amp Company 1969

I~ull i w f~ouse A ~hort Acc011nt of the lIiltory of ~athemrltics New York Dovor Publications l~)GO

Barbour tTames Murray Equal rPemperament Its History from Ramis (1482) to Hameau (1737)ff PhD dissertashytion Cornell University 1932

Tuning and Temnerament ERst Lansing Michigan State College Press 1951

Bosanquet H H M 1f11 emperament or the division of the octave iiusical Association Froceedings 1874-75 pp 4-1

Cajori 1lorian ttHistory of the Iogarithmic Slide Rule and Allied In[-)trllm8nts II in Strinr Fipures rtnd Other F[onorrnphs pre 1-121 Edited by lrtf N OUSG I~all

5rd ed iew York Ch01 sea Publ i shinp Company 1 ~)G9

Culver Charles A Nusical Acoustics 4th edt NeJI York McGraw-hill Book Comnany Inc 1956

Des-Cartes Hene COr1pendium Musicae Rhenum 1650

Dtctionaryof Physics Edited hy II bullbullT Gray New York John lJiley and Sons Inc 1958 Sv Pendulum 1t

Dl0 MllSlk In Gcnchichte lJnd Goponwnrt Sv Snuvellr J os e ph II by llri t z Iiinc ke] bull

Ellis Alexander J liOn an Improved Bimodular Method of compnting Natural and Tahular Loparlthms and AntishyLogarithms to Twelve or Sixteen Places vi th very brief Tables II ProceedinFs of the Royal Society of London XXXI (February 188ry-391-39S

~~tis F-J Uiographie universelle des musiciens at b i bl i ographi e generale de la musi que deuxi erne ed it -j on bull Paris Culture et Civilisation 185 Sv IISauveur Joseph II

154

155

Fontenelle Bernard Ie Bovier de Elove de M Sallveur

Histoire de J Acad~mie des Sciences Ann~e 1716 Amsterdam Chez Pierre Mortier 1736 pre 97-107

bull Sur la Dete-rmination d un Son Fixe T11stoire ---d~e--1~1 Academia Hoyale des Sciences Annee 1700

Arns terdnm Chez Pierre fvlortler 17)6 pp Id~--lDb

bull Sur l Applica tion des Sons liarmonique s alJX II cux ---(1--O---rgues Histol ro de l Academia Roya1e des Sc ienc os

Ann~e 1702 Amsterdam Chez Pierre Mortier l7~6 pp 118-122

bull Sur un Nouveau Systeme de Musique Histoi 10 ---(~l(-)~~llclo(rnlo lioynle des ~)clonces AJneo 1701

Amsterdam Chez Pierre Nlortier 1706 pp 158-180

Galilei Galileo Dialogo sopra i dlJe Tv1assimiSistemi del Mondo n in A Treasury of 1Vorld Science fJP 349shy350 Edited by Dagobe-rt Hunes London Peter Owen 1962

Glr(l1 o~~tone Cuthbert tTenn-Philippe Ram(~lll His Life and Work London Cassell and Company Ltd 1957

Groves Dictionary of Music and usiciA1s 5th ed S v If Sound by Ll S Lloyd

Harding Hosamond The Oririns of li1usical Time and Expression London Oxford l~iversity Press 1938

Harvard Dictionary of Music 2nd ed Sv AcoustiCS Consonance Dissonance If uGreece U flIntervals iftusic rrheory Savart1f Solmization

Helmholtz Hermann On the Sensations of Tone as a Physiologic amp1 Basis for the Theory of hriusic Translnted by Alexander J Ellis 6th ed New York Patnl ~)nllth 191JB

Henflinrr Konrad Specimen de novo suo systemnte musieo fI

1iseel1anea Rerolinensla 1710 XXVIII

Huygens Christian Oeuvres completes The Ha[ue Nyhoff 1940 Le Cycle lIarmoniauetI Vol 20 pp 141-173

Novus Cyelns Tlarmonicus fI Onera I

varia Leyden 1724 pp 747-754

Jeans Sir tTames Science and Music Cambridge at the University Press 1953

156

L1Affilard Michel Principes tres-faciles ponr bien apprendre la Musigue 6th ed Paris Christophe Ba11ard 170b

Lindsay Hobert Bruce Historical Introduction to The Theory of Sound by John William strutt Baron ioy1elgh 1877 reprint ed Iltew York Dover Publications 1945

Lou118 (~tienne) flomrgtnts 011 nrgtlncinns ne mllfll(1uo Tn1 d~ln n()llv(I-~r~Tmiddot(~ (~middoti (r~----~iv-(~- ~ ~ _ __ lIn _bull__ _____bull_ __ __ bull 1_ __ trtbull__________ _____ __ ______

1( tlJfmiddottH In d(H~TipLlon ut lutlrn rin cllYonornt)tr-e ou l instrulTlfnt (10 nOll vcl10 jnvpnt ion Q~p e ml)yon (11]lt11101 lefi c()mmiddotn()nltol)yoS_~5~l11r~~ ~nrorlt dOEiopmals lTIaYql1cr 10 v(ritnr)-I ( mOllvf~lont do InDYs compos 1 tlons et lelJTS Qllvrarr8s ma Tq11C7

flY rqnno-rt ft cet instrlJment se n01Jttont execllter en leur absence comme s1i1s en bRttaient eux-m mes le rnesure Paris C Ballard 1696

Mathematics Dictionary Edited by Glenn James and Robert C James New York D Van Nostrand Company Inc S bull v bull Harm on i c bull II

Maury L-F LAncienne AcadeTie des Sciences Les Academies dAutrefois Paris Libraire Acad~mique Didier et Cie Libraires-Editeurs 1864

ll[crsenne Jjarin Harmonie nniverselle contenant la theorie et la pratique de la musitlue Par-is 1636 reprint ed Paris Editions du Centre National de la ~echerche 1963

New ~nrrJ ish Jictionnry on EiRtorical Principles Edited by Sir Ja~es A H ~urray 10 vol Oxford Clarenrion Press l88R-1933 reprint ed 1933 gtv ACOllsticD

Olson lJaPly ~1 fmiddot11lsic Physics anCi EnginerrgtiY)~ 2nd ed New York Dover Publications Inc 1~67

Pajot Louis-L~on Chevalier comte DOns-en-Dray Descr~ption et USRFe dun r~etrometre ou flacline )OlJr battrc les Tesures et les Temns ne tOlltes sorteD d irs ITemoires de l Acndbi1ie HOYH1e des Sciences Ann~e 1732 Paris 1735 pp 182-195

Rameau Jean-Philippe Treatise of Harmonv Trans1ated by Philip Gossett New York Dover Publications Inc 1971

-----

157

Roberts Fyincis IIA Discollrsc c()nc(~Ynin(r the ~11f1cal Ko te S 0 f the Trumpet and rr]11r1p n t n11 nc an rl of the De fee t S 0 f the same bull II Ph t J 0 SODh 1 c rl 1 sac t 5 0 n S 0 f the Royal ~ociety of London XVIi ri6~~12) pp 559-563

Romieu M Memoire TheoriqlJe et PYFltjone SlJT les Systemes rremoeres de 1lusi(]ue Iier1oires de ] r CDrH~rjie Rova1e desSciAnces Annee 1754 pp 4H3-bf9 0

Sauveur lJoseph Application des sons hnrrr~on~ ques a la composition des Jeux dOrgues tl l1cmoires de lAcadeurohnie RoyaJe des SciencAs Ann~e 1702 Amsterdam Chez Pierre Mortier 1736 pp 424-451

i bull ~e thone ren ernl e pOllr fo lr1(r (l e S s Vf) t e~ es tetperes de Tlusi(]ue At de c i cllJon dolt sllivre ~~moi~es de 11Aca~~mie Rovale des ~cicrces Ann~e 1707 Amsterdam Chez Pierre Mortier 1736 pp 259-282

bull RunDort des Sons des Cordes d r InstrulYllnt s de r~lJsfrue al]~ ~leches des Corjes et n011ve11e rletermination ries Sons xes TI ~~()~ rgtes (je 1 t fIc~(jer1ie ~ovale des Sciencesl Anne-e 17]~ rm~)r-erciI1 Chez Pi~rre ~ortjer 1736 pn 4~3-4~9

Srsteme General (jos 111tPlvnl s res Sons et son appllcatlon a tOllS les STst~nH~s et [ tOllS les Instrllmcnts de Ml1sique Tsect00i-rmiddot(s 1 1 ~~H1en1ie Roya1 e des Sc i en c eSt 1nnee--r-(h] m s t 8 10 am C h e z Pierre ~ortier 1736 pp 403-498

Table genera] e des Sys t es teV~C1P ~es de ~~usique liemoires de l Acarieie Hoynle ries SCiences Annee 1711 AmsterdHm Chez Pierre ~ortjer 1736 pp 406-417

Scherchen iiermann The jlttlture of jitJSjc f~~ransla ted by ~------~--~----lliam Mann London Dennis uoh~nn Ltc1 1950

3c~ rst middoti~rich ilL I Affi 1nrd on tr-e I)Ygt(Cfl Go~n-t T)Einces fI

~i~_C lS~ en 1 )lJPYt Frly LIX 3 (tTu]r 1974) -400

1- oon A 0 t 1R ygtv () ric s 0 r t h 8 -ri 1 n c)~hr ~ -J s ~ en 1 ------~---

Car~jhrinpmiddoti 1749 renrin~ cd ~ew York apo Press 1966

Wallis (Tohn Letter to the Pub] ishor concernlnp- a new I~1)sical scovery f TJhlJ ()sODhic~iJ Prflnfiqct ()ns of the fioyal S00i Gt Y of London XII (1 G7 ) flD RS9-842

iA i 2

i

~ ) ~ )

vrrA

Robert 11nxham was born in Erie Pennsyl vnnla on

March 1 1947 and attended pnrochial schools there radushy

sting from Cathedral preparatory School with honors 1n

ItJGb liaving obtained his bachelors deproe with

distinction at the Eastman School of fIIusic in 1969 he

studied philosophy and theology at st Marks Seminary

in Erie bull Charles Horromeo Seminary in Philadelphia

where he won the Hanna Cusick Ryan Prize for General

Excellence in Bunrlamental Theology and St Mary 1 s

Seminary and University in Baltimore tie returned to the

stlJdy of musiC receiving the Master of Arts degree in

Musicology from the strnan School in 1975 ~he next

year was spent in preparation in absentia of the

present study

ii

PREFACE




















fast




iii




























1v













v

ABSTHACT
























vi



structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157














PREFACE




















fast




iii




























1v













v

ABSTHACT
























vi



structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157









































1v













v

ABSTHACT
























vi



structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157


























v

ABSTHACT
























vi



structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157














ABSTHACT
























vi



structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157
















structu-re












could be avoided













vii


















viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157































viii

bull bull bull

bull bull bull

CONTENTS





ix







x

LIST OF TABLES














b


xi

















1

2









3

























9 Ib i d p 99

4

























5















6






bishop ~~d judged




of his chosen field


































8
























9










frequenci es 30














10










regular usage











11



























12




























13







ori ontal s 1















I



vol II p 97 below



14







diatonic system n52



















15

























16













be made plain










17


























18
























vol II p 15~ below


19


















p 150 below








20

















how



21























22



epiglottis




















23






















ur 3Lli table 99






24




by a second lOO





100Ib1d p 107



CHAPTER I
















)-shy





p 26 below





25

26

























27





The rule was









28



























29






















13Ibid bull


30
















151bid bull





31

TABLE 1



I II III

feet

1 1 3-~ 2 4 14 3 9 31~ 4 16 56 dr 25 n7~ G 36 1 ~~ ( 7 49 171J

2

8 64 224 9 81 283~

10 100 350 11 121 483~ 12 144 504 13 169 551t 14 196 686 15 225 787-~ 16 256 896 17 289 lOll 18 314 1099 19 361 1263Bshy20 400 1400 21 441 1543 22 484 1694 23 529 1851~ 24 576 2016


26 676 ~~366 27 729 255lshy28 784 ~~744 29 841 ~1943i 30 900 2865





32













adjustments






33














respectively












21 Ibid bull

34

TABLE 2

T L L2


12 144~1~)2 3612 ~

11 121(1~)2 25 t 5i12 ~

10 100 12

(1~)2 ~

25

9 81(~) 2 16 + 412 4

8 64(~) 2 1612 4

7 (7)2 49 9 + 3t12 2 4

6 (~)2 36 912 4

5 (5)2 25 4 + 2-t12 2 4

4 16(~) 2 412 4

3 9(~) 2 1 Ii12 4 2 (~)2 4 I

12 4

1 1 + l(~) 2 0 412 4






35












second must thus be
















25Ibid bull

36









TABLE 3

T L Ll

70 4900(ig)260 155

71 5041(i~260 100

72 5184G)260 155

73 5329(ig)260 100

74 5476(ia)260 155

75 G~)2 5625 60 100






( ~ig1Jr e 1)


37

P P1 4l 3

I I- ~ 1

I I I

d K A M E rr

Fig 1

















- -

38

0 )

T-1--W I

cleT2

T deg1 0

00 rt-degIQ

shy

deg1degpound

CIOr0

01deg~

I J 1 CL l~

39




TABLE 4


tEW UmGTH)4~)OO

-f -100

P to 1 140 1 141 P to Y 5041 100 roo 100 100

P to 2 280 4 284 5184P to 100 100 100 100

P to 3 420 9 429 P to fA 5329 100 100 100 100

p to 4 560 16 576 p to y- 5476 100 100 roo 100











40

























41














CHAPTER II










of the first



two of these





42

43



























44





















24 27 30 32 36 40 45 48 54 60 64 72 80 90 98






7

45















7 Ibid bull




46




















12Ibid bull



47








second table

















48





















mantissa by seven



49










certain other ways














21 Ibid bull


50






















25 Ibid bull




51






of 1711


















35 35

52








to him





53






















5


54

















36Ib i d p xi





6

55























56

TABLE 5

JUST INTONATION



1 169 2498775 99601 996

VI 53 2218488 88429 804 6 85 2041200 81362 B14 V 32 1760913 7019 702 5 6445 1529675 60973 610

IV 4532 1480625 59018 590 4 43 1249387 49800 498

III 54 0969100 38628 386 3 65middot 0791812 31561 316

112 98 0511525 20389 204

III 109 0457575 18239 182

2 1615 0280287 11172 112












57



rABLE 6


VII 11 1100 +12 +12 72 71

10 1000 -IS + 4

-18 + 4

VI 9 900 +16 +16 6 8 800 -14 -14 V 7 700 - 2 - 2 5

JV 6 600 -10 +10

-10 flO

4 5 500 + 2 + 2 III 4 400 +14 +14

3 3 300 -16 -16 112 2 200 - 4 - 4 III +18 tIS

2 1 100 -12 -12






22 cents)




Table 7


58

TABLE 7

THE SYSTEM OF 31


VII 28 1084 - 4 - 4 72 71 26 1006

-12 +10

-11 +10

VI 6

23 21

890 813

--

6 1

- 6 - 1

V 18 697 - 5 - 5 5 16 619 + 9 10

IV 15 581 - 9 -10 4 13 503 + 5 + 5

III 10 387 + 1 + 1 3 8 310 - 6 - 6

112 III

5 194 -10 +12

-10 11

2 3 116 4 + 4





columns


Table 8 (Table 8)








59



TABLE 8

THE SYSTEM OF 43


VII 39 1088 0 0 -13 -1372 36 1005

71 + 9 + 8

VI 32 893 + 9 + 9 6 29 809 - 5 + 4 f-4]V 25 698 + 4 -4]- 4 5 22 614 + 4 + 4

IV 21 586 + 4 [-4J - 4 4 18 502 + 4 + 4

III 14 391 5 + 4 3 11 307 9 - 9-

112 - 9 -117 195 III +13 +13

2 4 112 0 0


Table 9 (Table 9)











60

TABLE 9

ThE SYSTEM OF 55


VII 50 1091 3 -+ 3 72

71 46 1004

-14 + 8

-14

+ 8

VI 41 895 -tIl +10 6 37 807 - 7 - 6 V 5

32 28

698 611

- 4 + 1

- 4 +shy 1

IV 27 589 - 1 - 1 4 23 502 +shy 4 + 4

III 18 393 + 7 + 6 3 14 305 -11 -10

112 III

9 196 - 8 +14

- 8 +14

2 5 109 - 3 - 3

TABLE 10

DIFFEHENCES

SYSTEMS


VII +12 4 0 + 3 72 -18 -12 [-11] -13 -14

71 + 4 +10 9 ~8] 8

VI +16 + 6 + 9 +11 L~10J 6 -14 - 1 - 5 f-4J - 7 L~ 6J V 5

IV 4

III

- 2 -10 +10 + 2 +14

- 5 + 9 [+101 - 9 F-10] 1shy 5 1

- 4 + 4 - 4+ 4 _ + 5 L+41

4 1 - 1 + 4 7 8shy 6]

3 -16 - 6 - 9 -11 t10J 1I2 - 4 -10 - 9 t111 - 8 III t18 +12 tr1JJ +13 +14

2 -12 4 0 - 3

61






















in which c = 1

His third principle




62




















the fa ct t1at




63




















64




be below 59













L J




67 below


65




























66



science bull 65
















II


66~3auveur




67
























68

TABLE 11


JUST RATIO IN CENTS

DIFFERENCE

COMPUTATION)

IV 4536 608 [5941 590 +18 [+4J 4 43 498 498 o

III 54 387 386 t 1 3 65 317 316 + 1

112 98 205 204 + 1

III 109 184 182 t 2 2 1615 113 112 + 1
















69













in his Chronometer










70























71







second







72






countable number



















73















in those units












84Ibid bull

v

74











D B

(86) (43) (0 )


(43) (215)

Fig 3





75















2













76























the scale





77





















p 33 below


78















into antilogaritr~s






92Ib1 d p 43 bull




79





fixed sound

TABLE 12

1 2 t U 16 32 601 128 256 512 1024 olte 1CY)( fH )~~ 160 gt1

2deg 21 22 23 24 25 26 27 28 29 210 211 212 213 214

32768 65536

215 216













of each octave


80









eluded that












81





1701









general system




24 27 30 32 36 40 ) 484








82


meridians























83






















84







SnUV0Ul adds that

















85

















of the voice








l12Ibid bull

II p

113Sauveur

90 below


86













classification

1 sem-GA( MI)



4 DO( S1)


6 BO(G RE SOL)







II pp 91-92 below

87


12 sub-LO(LA) tenor



15 sub-FA( NIl)




19 subbis-LO(LA)















p 196 below


88



























89



octave 122















I 2 II 3 III 4 IV V 6 VI VII 0 3 11 14 18 21 25 (28) 32 36 39






90





TABLE 13


Diatonic intervals

I 2 II 3 III 4 IV (5)

V 6 VI 7 VIr VIrI



UT PA 0 (3) 7 11 14 18 21 25 (28) 32 36 39 113

cJ)

r-i ro gtH OJ

+gt c middotrl

r-i co u 0 ~-I 0

-1 u (I)

H

Q)

J+l

d pi

HE RA

b go

MI GA

FA SO

d sa

0 4

0 4

0 (3)

a 4

0 (3)

0 4

(8) 11

7 11

7 (10)

7 11

7 (10)

7 11

(15)

14

14

14

14

( 15)

18

18

(17)

18

18

18

(22)

21

21

(22)

21

(22)

25

25

25

25

25

25

29

29

(28)

29

(28)

29

(33)

32

32

32

32

(33)

36

36

(35)

36

36

36

(40)

39

39

(40)

3()

(10 )

43

43

43

43

Il]

43

4-lt1 0

SOL BO 0 (3) 7 11 14 18 21 25 29 32 36 39 43

cJ) -t ro +gt C (1)

E~ ro T~ c J

u

d sa

LA LO

b de

5I DO

0 4

a 4

a (3)

0 4

(8) 11

7 11

7 (10)

7 11

(15)

14

14

(15)

18

18

18

18

(22)

(22)

21

(22)

(26)

25

25

25

29

29

(28)

29

(33)

32

32

32

36

36

(35)

36

(40)

3lt)

39

(40)

43

43

43

43




91







TABLE 14



(2) (3) 84 (C )

2 4 112 Db II 7 195 D

(II) (8 ) 223 (Ebb) (3 ) 3

(10) 11

279 3Q7

(DII) Eb

III 14 391 E (III)

(4 ) (15) (17 )

419 474

Fb (w)

4 18 502 F IV 21 586 FlI

(IV) V

(22) 25

614 698

(Gb) G

(V) (26) 725 (Abb) (6) (28) 781 (G)

6 29 809 Ab VI 32 893 A

(VI) (33) 921 (Bbb) ( ) (35 ) 977 (All) 7 36 1004 Bb

VII 39 1088 B (VII) (40) 1116 (Cb )




92





scrutinized









values of 0 112 223 304 419 502 614 725 809 921




in Table 15

TABLE 15



2 5 0 5 2 3 4 1 6 1 4 3




II p 84 below

1

93

c GD A E B F C G

1 2 3 4 3 25middot 6
















and their intervals

CHAPTER III

THE OVERTONE SERIES










stops










94

95






















6Ibid bull



96

























p 73 below


vol II p 73 below


97



moon 1113

























98

B

n

E

A c B












in small parts and

15Ibid bull

16Ibid

99




















p

18 Sauveur 77 below


19Ibid bull

100














are audible





observation that





101

























pp 77-78 below

102




number of parts

3o

3110 l~ IS 30 10

10
















26Ibid bull

10

103




















the 1 eaps




104



















29 Ibid bull




105


























106

TABLE 16


9 1 18 2 27 3 36 4 45 5 54 6 72 n

] on 12 144 16 216 24 288 32 4~)2 48 576 64 864 96






107




stops only imitated















second






108







his M~moire of 1701














109




























110


and proportions











CHAPTER IV














musical intervals




III

112


















founders






113





music in particular












114




















115












POlrr ~

connoi tre




lOIbid pp 193-195

116




























117

























12Ibid p 395

118























119



























120
















he noted that







121
























20Ibid p 36

9

122

TABLE 17

CYCLE OF 43 -COMMA


1)Vll lOuU 0 lOUU l

b~57 1005 0 1005 B _JloA ltjVI 893 0 893

V( ) 781 0 781 G-

_l V 698 0 698 G 5

F-~IV 586 0 586

F+~4 502 0 502

E-~III 391 +1 390

Eb~l0 53 307 307

1

II 195 0 195 D-~

C-~s 84 +1 83












123



TABLE 18

1 2 3 4



VII 1088 1038 1088 0 0 7 1005 1018 1005 -13 0

VI 893 884 893 + 9 0 vUI) 781 781 0 V

IV 698 586

702 590

698 586

--

4 4

0 0

4 502 498 502 + 4 0 III 391 386 390 + 5 tl

3 307 316 307 - 9 0 II 195 182 195 t13 0

s 84 83 tl














124

























more closely


125

























126

























127


1075 (43) -1075 o - 0 (0) +0 (0) c j 100 (42) -103 (

18 -25 (1) t-43 ) )1025 (Lrl) -989 36 -50 (2) tB6 J )

1000 (40) -080 11 -75 (3) +B6 () ()75 (30) -946 29 -100 (4) +12() (]) 950 (3g) - ()~ () 4 -125 (5 ) +129 ~J) 925 (3~t) -903 22 -150 (6) +1 72 (~)

- 0) -860 40 -175 (7) +215 (~))

G7S (3~) -8()O 15 (E) +1J (~

4 (31) -1317 33 ( I) t ) ~) ) (()


(~O ) -731 10 IC) -J2~ (13) +3i 14 )) 72~ (~9) (16) 37 -350 (14) +387 (0) 700 (28) (16) 12 -375 (15) +387 (()) 675 (27) -645 (15) 30 -400 (6) +430 (10)

(26) -6t15 (1 ) 5 (l7) +430 (10) )625 (25) -602 (11) ) (1) +473 11)

(2) -559 (13) 41 (l~ ) + ( 1~ ) (23) (13) 16 - SOO C~O) 11 ~ 1 )2))~50 ~lt- -510 (12) ~4 (21) + L)


~75 (19) -~73 (11) 2 -GOO (~~ ) t602 (11) (18) -430 (10) 20 (25) + (1J) lttb

(17) -337 (9) 38 -650 (26) t683 ) (16 ) -3n7 (J) 13 -675 (27) + ( 1 ())

Tl5 (1 ) -31+4 (3) 31 -700 (28) +731 ] ) 11) - 30l ( ) G -75 (gt ) +7 1 ~ 1 ) (L~ ) n (7) 211 -rJu (0 ) I I1 i )


(10) -215 (5) 35 -825 (33) + (3() I )

( () ) ( j-215 (5) 10 -850 (3middot+ ) t(3(j

200 (0) -172 (4) 28 -875 ( ) middott ()G) ri~ (7) -172 (-1) 3 -900 (36) +90gt I

(6) -129 (3) 21 -925 ( )7) + r1 tJ

- )

( ~~ (~) (6 (2) 3()

+( t( ) -

()_GU 14 -(y(~ ()) )

7) (3) ~~43 (~) 32 -1000 ( )-4- ()D 50 (2) 1 7 -1025 (11 )

G 25 (1) 0 25 -1050 (42)- (0) +1075 o (0) - 0 (0) o -1075 (43) +1075

128




























129

























II p 47 b ow



130






















131






















132




nY1 tnm n~)8





















133







10 iec ted 42





rJd~~S CHjl


Huygens 77 194 310 387 503 581 697 774 890 1006 1084

l-cofilma 76 193 310 386 503 579 697 773 (390 1007 lOt)3 4

~-coli1ma 79 194 308 389 503 582 697 777 892 1006 1085 9





134








major semi tones

















135














in the octave 49








p 159 below



12

136






















52 Ibid bull



137

























138




















middot ~60lb lO


139










that










140

























141


par second 66

















70Ibid p 516

11-2





















71 Ibid bull


73 Ibid bull

74 Ibi d

143




















cal point of view







75 Ibid p xxii

144




























145










origin













8Ib1d bull J p 53


146














question 82










80Ibid bull

81 Ibid bull



147






















84 Ibid bull


Sound p xiv

87 Ibid bull



148
























9l Ibid bull



149













language of sounds











150


























151





























152




















study as this one









153




























iVORKS CITED














154

155





















156













-----

157














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THE CONTRIBUTIONS OF