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Dictionary of the History of Ideas | ||

#### MUSIC AND SCIENCE

Of all the arts, music has had most to rely upon a

scientific and mathematical analysis of its materials.

Both the relations between pitches and between dura-

tions are best defined by numbers and ratios. To con-

struct even the simplest instruments out of strings or

pipes, musicians had to derive as best they could the

laws of sound production. The most elementary fact,

generally accredited to the Pythagoreans but probably

known to the ancient Babylonians and Egyptians, is

that if a string is stopped in the middle, each of the

two halves sounds an octave higher than the whole;

if divided into three parts, two-thirds of the string will

sound a fifth above the whole; and so on.

Because it relies on precise measurement, music was

considered until fairly modern times, indeed until

around 1650, a branch of science. In late antiquity it

began to be included in the four mathematical disci-

plines of the quadrivium along with arithmetic, geom-

etry, and astronomy. But actually only theoretical

music was accorded this place. No singing or playing

was included in this curriculum. Practical music mak-

ing went its own way, maintaining only limited contact

with theoretical music, drifting farthest from it in the

Middle Ages and approaching nearer during the

Renaissance. The musical component of the mathe-

matical curriculum in the universities never went be-

yond the heritage of Greek music theory. Only the

Renaissance humanists succeeded in making this rele-

vant to Western musical art.

Because of this alliance with mathematics, music

figured prominently in cosmology, astrology, and num-

ber mysticism. Speculations about the harmony of the

universe were often inspired by musical facts, as in

Plato's *Timaeus* (31-39), or as in the theory that the

planets were governed in their motions by ratios of

the consonances and therefore produced an unheard

music (*Republic* X). These ideas were attacked by

Aristotle (*On the Heavens* II. 8-9), but the musical

world generally believed them until the end of the

fifteenth century, and Kepler much later was still seek-

ing to prove universal harmony when he discovered

the third law of planetary motion (the cube of any

planet's distance from the sun varies directly with the

square of the planet's period or time of revolution).

Greek writers credited Pythagoras (ca. 582-07 B.C.)

with the earliest acoustical observations. He is said to

have discovered the ratios of the octave (2:1), fifth

octave (4:1). These were the only consonances recog-

nized by Greek theory. Great metaphysical significance

was attached to the fact that the set of numbers from

1 to 4 was the source of all harmony.

It was assumed that these ratios produced the same

consonances whether the numbers applied to string

lengths, bore of pipes, weights stretching strings,

weights of disks, or volumes of air in vessels such as

bells or water-filled glasses. Theon of Smyrna (second

century A.D.) claimed that Pythagoras had verified these

ratios in all these circumstances. Boethius (fifth century

A.D.) reported that Pythagoras heard the consonances

also issuing from a blacksmith's hammers whose

weights were in the same ratios as the string lengths.

Actually, as Vincenzo Galilei, father of Galileo, was

to demonstrate in 1589 (Palisca, 1961), the ratios are

not the same in these cases as in the divison of the

string. Throughout the Middle Ages and early Renais-

sance, from Boethius to Gaffurio (Figure 1), almost

every author on music recounts the experiments of

Pythagoras without realizing their improbability.

The canonization of the octave, fifth, and fourth in

their natural ratios as the cornerstones of the harmonic

system had a deeper influence on music theory and

composition than on instrument building or playing.

Musicians tended to tune their instruments by ear,

tempering the ratios of the fourths and fifths, because

it was discovered early that if one tuned up a cycle

of twelve fifths from any note until that note was

reached again, this note was higher than that reached

through a cycle of seven octaves by

( 3/2 )12 : (2/1)7 or 531,441/524,288

approximately 24% of a semitone. According to the

theorists, the only acceptable tuning was that which

maintained the fifth and fourth at their proper ratios

of 3:2 and 4:3. This tuning was called by Ptolemy

Diatonic *ditoniaion,* and it is also known as the

Pythagorean tuning. In this scheme the tetrachord or

modular fourth is composed of two tones and a semi-

tone in the ratios 9:8, 9:8, 256:243. The fifth and

fourth, favored by this tuning, were the most promi-

nent consonances in written polyphony from the ninth

through the thirteenth centuries, particularly at points

of rhythmic or structural emphasis. The thirds and

sixths, which were not recognized as consonances by

Greek or orthodox medieval theory, were harsh-

sounding in the Pythagorean tuning. Nevertheless, they

were employed increasingly in polyphony, particularly

during the fourteenth and fifteenth centuries.

Renaissance humanism had a somewhat delayed

effect on musical theory as compared to other disci-

plines. Only in the last quarter of the fifteenth century

did the ancient Greek treatises begin to be read first-

hand and translated; for example, Ptolemy's *Har-
monics,* Aristotle's

*Problems,*the short introductions of

Cleonides and Euclid, and eventually the

*Harmonics*

of Aristoxenus. At about the same time an antitheoret-

ical movement began among composer-theorists. The

first writer to break with the Boethian-Pythagorean

doctrine of consonances was Bartolomé Ramos de

Pareja. In his

*Musica practica*(Bologna, 1482) he pro-

posed a tuning system that allowed for sweetly tuned

thirds in the simple ratios of 5:4 and 6:5, as against

the Pythagorean 81:64 and 32:27. Ramos' innovations

met resistance among conservatives like Franchino

Gaffurio. But soon Lodovico Fogliano (1529), Gioseffo

Zarlino (1588), and Francisco de Salinas (1577) joined

Ramos in dethroning for all times the Pythagorean

tuning system. All three leaned upon the recently

rediscovered

*Harmonics*of Claudius Ptolemy in which

a tuning very similar to Ramos' was described under

the name Diatonic

*syntonon.*Zarlino was convinced

and imperfect consonances were in simple ratios of the

class

*n*+ 1/

*n,*known as superparticular. Zarlino mod-

ernized the pre-Ramos number mysticism by replacing

the number four by the set of numbers from 1 to 6,

the

*numero senario.*

A growing use of thirds and sixths paralleled theo-

retical recognition of a sweet-third tuning. However,

if there is a causal connection, it is that the theorists

saw the anachronism of standing by a theoretical

harsh-third tuning.

The astronomer Ptolemy was known to the Renais-

sance as the theorist who took the middle road between

the rationalist position of the Pythagoreans and the

empiricist method of Aristoxenus. Zarlino was attracted

to Ptolemy because he too was inclined to worship

number while aiming to satisfy the ear. Aristoxenus,

on the other hand, rejected ratios as irrelevant to music.

He preferred to divide pitch-space directly, if some-

what subjectively. Of his tuning systems the one that

most appealed to sixteenth-century musicians was that

in which each whole tone contained 12 units of pitch-

space and each semitone 6 units. Sixteenth-century

interpreters assumed this meant the division of the

octave into an equal temperament of twelve equal

semitones, as in the tuning of the lute. Such a tuning

would permit a melody to sound equally well in any

key, something that could not be accomplished with

any other tuning. Aristoxenus began to find apostles

in the last quarter of the sixteenth century, notably

Vincenzo Galilei and Ercole Bottrigari.

Meanwhile no scientific discovery had yet deprived

the simple ratios of the consonances of their priority.

But in 1589-90 Vincenzo Galilei drafted a treatise that

reported some new experiments with sounding bodies.

He discovered that the ratios usually associated with

the consonances are obtained only when they represent

pipe or string lengths, other factors being equal. When

weights were attached to strings, the ratio had to be

4:1, not 2:1 to produce an octave. The volumes of

concave bodies had to be in the ratio of 8:1 to produce

the octave. Since, in terms of weights, the fifth and

fourth were 9:4 and 16:9 respectively, Galilei saw no

reason to prefer simple ratios within the numbers of

the *senario.*

The bastion of the simple ratios was besieged also

by another line of research. In a letter to the composer

Cipriano de Rore of around 1563 the scientist Giam-

battista Benedetti proposed a new theory of the cause

of consonance. Benedetti argued that since sound con-

sists of air waves or vibrations, in the more consonant

intervals the shorter more frequent waves concurred

with the longer less frequent waves at regular intervals.

In the less consonant intervals, on the other hand,

concurrence was infrequent and the two sounds did

not blend in the ear pleasantly. He showed that in a

fifth, for example, the two vibrations will meet every

two cycles of the lower note and every three of the

higher. He went on to show that in terms of frequency

of concurrence the hierarchy of ratios within the

octave would be 2:1, 3:2, 4:3, 5:3, 5:4, 6:5, 7:5,

8:5, which challenges both the superiority of super-

particular ratios and the sanctity of the *senario.* There

could be no abrupt break from consonance to disso-

nance but only a continuum of intervals, some more,

some less consonant. Benedetti's theory was espoused

in the next century by Isaac Beeckman and Marin

Mersenne, who sought René Descartes' opinion of it.

Descartes declined to judge the goodness of con-

sonances by such a rational method, protesting that

the ear prefers one or another according to the musical

context rather than because of any concordance of

vibrations.

The philologist and student of ancient music

Girolamo Mei summed up this emancipation of music

from scientific determinism in a letter to Vincenzo

Galilei of 1572:

The true end of science is altogether different from that

of art.... The science of music goes about diligently

investigating and considering all the qualities and properties

of the existing constitution and ordering of musical tones,

whether these are simple qualities or comparative, like the

consonances, and this for no other aim than to come to

know the truth itself, the perfect goal of all speculation,

and as a by-product the false. It then lets art exploit as

it sees fit, without any limitation, those tones about which

science has learned the truth

(Palisca [1960], p. 65).

The revolution in musical thought encouraged ex-

perimentation in composition, in which a search for

new musical resources had already spontaneously

begun. Composers found a new harmonic richness; and

even in the old tunings they braved modulations to

distant keys and ventured melodic motion by semitone.

The Aristoxenian “equal temperament,” which would

have made these things easier, was demanded even by

some conservative musicians like Giovanni Maria

Artusi, a loyal disciple of Zarlino. Dissonances—

seconds, sevenths, and diminished and augmented

intervals—were introduced more and more freely into

compositions.

If scientific discovery stimulated musical change, the

opposite is also true: musical problems stimulated

scientific investigation. Benedetti and Vincenzo Galilei

were moved by musical problems to inquire into the

mechanics of sound production. The most notable case

is that of Galileo, who was disturbed by the very

problem that stumped his father: Is there a stable

perhaps the most fundamental discovery in acoustics

when he proved that there is: ratios between the fre-

quencies of vibration are the inverse of the ratios of

string lengths.

The most difficult challenge that music presented

to science in the seventeenth century was to explain

the multiple pitches that could be heard when a single

string vibrated. Aristotle noted that he could hear the

octave above (*Problems* 919b 24; 921b 42) and observed

the related phenomenon of hearing a string respond

sympathetically to one tuned an octave higher. It took

Mersenne's acute musical ear to hear from a single

vibrating string not only the upper octave but the

octave plus fifth, double octave, double octave plus

major third, and the double octave plus major sixth.

Neither her nor Descartes, nor any of the other scientists

of their circle could explain why this happened. In

1673 two Oxford scientists, William Noble and Thomas

Pigot, showed that strings tuned to the octave, octave

plus fifth, and octave plus major third below a plucked

string sounded sympathetically at the unison to the

plucked string by vibrating in aliquot parts. They

demonstrated this by placing paper riders on the sym-

pathetic strings at the points where, if the string were

stopped, unisons would be produced. In 1677 John

Wallis reported that multiple sounds would occur in

a vibrating string only if it was not plucked at the

points that marked off the aliquot parts. This showed

that a single string simultaneously vibrated as a whole

and in its aliquot parts. Thus harmonic vibration, which

was important also for mechanics and optics, was es-

tablished as a fact.

Of all the laws of acoustics that of harmonic vibra-

tion exercised most the imagination of theorists of

music. The first to utilize the information was Jean-

Philippe Rameau. In his *Traité de l'harmonie* (1722),

he had constructed a new system of harmony on the

ratios of the divisions of the string. When he learned

rather belatedly of harmonic vibration from an exhaus-

tive paper by Joseph Sauveur (1701), Rameau decided

that this was the original principle he had been looking

for. In his opinion it firmly established his theory of

fundamental bass, as he called the bottom note of a

triad whose notes are arranged in thirds, for the first

six notes of the harmonic series arranged as a chord

is equivalent to a major triad over a fundamental bass.

Thus his system was a copy of nature. The fundamental

bass, in his view, determines the progress of the har-

mony, as when it leaps down a fifth from the dominant

(fifth note) to the tonic (first note) of a key. But Rameau

did not stop at this. He made of the harmonic series

a Cartesian first principle from which he built up, by

manipulating the numbers of its ratios, a system of

theory that embraced every aspect of music. Unfor-

tunately, his numerical operations were often faulty

and drew severe criticism from the geometer Jean

d'Alembert and the mathematician Leonhard Euler.

The concept of fundamental bass received further

support when the celebrated violinist and composer

Giuseppe Tartini in 1754 announced his discovery of

“the third sound.” This is a subjective sensation now

known as “difference tone” that is believed to occur

because of the presence of nonlinear resonance in the

ear. When two pitches are sounded, a third lower one

seems to resound. Tartini found it by listening carefully

to double-stops played on a violin. Actually, unknown

to Tartini, Georg Andreas Sorge had noted the same

phenomenon nine years previously (1745). The “third

sound” usually reinforced the note of a chord that

Rameau identified as the fundamental bass, which

Tartini too accepted as a keystone of his system. Like

Rameau, he indulged in sweeping mathematical and

geometric speculations, which, however, did not with-

stand the scrutiny of mathematicians such as Benjamin

Stillingfleet and Antonio Eximeno.

Both Tartini and Rameau were pioneers in musical

composition, at the forefront in technique, style, and

structure. But as thinkers they were anachronistic. In

the midst of the Enlightenment, which was skeptical

of systems and the deductive process, they spun webs

of numbers in blissful isolation, only to become hope-

lessly entangled in them.

Musicians have continued to search for a natural

basis for music theory. In the twentieth century, Paul

Hindemith extended the idea of fundamental bass to

all kinds of dissonant chords. Like his predecessors he

based the theory on the harmonic series and on differ-

ence tones. He believed, like Rameau, that all harmonic

movement depends on the progress of roots of chords,

but he freed this process from the simple cadence-like

successions of Rameau. Hindemith's theory has been

challenged because its premisses are fully valid only

in a system of just intonation, and because, while ex-

ploiting difference tones, he ignored combination tones

that are sometimes more audible.

Lately, scientific facts and theories outside the realm

of acoustics have inspired philosophies of music. From

thermodynamics the concepts of entropy and indeter-

minacy have been seized upon as a justification for

music to copy nature by following laws of chance

rather than willful combinations. From information

theory and physics composers have borrowed the con-

cept of the stochastic process, in which events are

interconnected through a succession of probabilities.

It is also possible by analogy to defend as expressive

of the contemporary view of reality the modular struc-

tures of serial compositions, which are constructed like

simultaneous juxtapositions. Meanwhile the entire

corpus of acoustical science is called into play by

electronic and computer music, in which art merges

indissolubly with engineering. Music may be on the

road to becoming once again a branch of science, or

at least of technology.

## *BIBLIOGRAPHY*

For references concerning Beeckman, Benedetti,

Descartes, Fogliano, V. Galilei, G. Galilei, Gaffurio, Kepler,

Mersenne, Mei, B. R. de Pareja, Rameau, Salinas, and

Sauveur, see C. V. Palisca, “Scientific Empiricism in Musical

Thought,” in H. H. Rhys, ed., *Seventeenth-Century Science
and the Arts* (Princeton, 1961), pp. 91-137. See also Jean

Le Rond d'Alembert,

*Élémens de musique, théorique et*

pratique(Lyon, 1762); J. M. Barbour,

pratique

*Tuning and Tempera-*

ment(East Lansing, Mich., 1953); E. Bottrigari,

ment

*Il Desiderio*

(Venice, 1594); M. R. Cohen and I. E. Drabkin,

*A Source*

Book in Greek Science(New York, 1948), pp. 286-310;

Book in Greek Science

R. L. Crocker, “Pythagorean Mathematics and Music,”

*The*

Journal of Aesthetics and Art Criticism,22 (1963-64),

Journal of Aesthetics and Art Criticism,

189-98, 325-35; Signalia Dostrovsky, “The Origins of Vi-

bration Theory: The Scientific Revolution and the Nature

of Music” (Ph.D. dissertation, Princeton University, 1969,

unpubl.); Stillman Drake, “Renaissance Music and Experi-

mental Science,”

*Journal of the History of Ideas,*30 (1969),

483-500; L. Euler,

*Tentamen novae theoriae musicae*

(Petropoli [Saint Petersburg], 1739); A. Eximeno,

*Dell'origine*

delle regole(Rome, 1774); E. E. Helm, “The Vibrating

delle regole

String of the Pythagoreans,”

*Scientific American,*217 (1967),

92-103; H. Helmholtz,

*On the Sensations of Tone*(New

York, 1964); P. Hindemith,

*Craft of Musical Composition*

(New York, 1942); E. Lippman,

*Musical Thought in Ancient*

Greece(New York, 1964); C. V. Palisca,

Greece

*Girolamo Mei*

(Rome, 1960); idem, “The Interaction of the Sciences and

the Arts: A Historical View,”

*Proceedings of the Fourth*

National Conference on the Arts in Education(Philadelphia,

National Conference on the Arts in Education

1965), pp. 19-25; idem, “Fogliano,” “Galilei,” “Gogava,”

“Mei,” “Ramos,” “Salinas,” “Valla,” “Zarlino,”

*Die Musik*

in Geschichte und Gegenwart(Kassel, 1955-68); G. A. Sorge,

in Geschichte und Gegenwart

*Vorgemach der musikalischen Composition,*Erster Theil

(Lobenstein, 1745), Ch. 5; B. Stillingfleet,

*Principles and*

Power of Harmony(London, 1771); G. Tartini,

Power of Harmony

*Trattato di*

musica(Padua, 1754).

musica

CLAUDE V. PALISCA

[See also Astrology; Cosmology; Number; Pythagorean Har-mony; Renaissance Humanism.]

Dictionary of the History of Ideas | ||