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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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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


(3:2), octave plus fifth (3:1), fourth (4:3), and double
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-
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


that it was the perfect tuning, because both perfect
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

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

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


connection between consonance and ratio? He made
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


crystals, multiplying the same cell in successive and
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.


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
(Lyon, 1762); J. M. Barbour, Tuning and Tempera-
(East Lansing, Mich., 1953); E. Bottrigari, Il Desiderio
(Venice, 1594); M. R. Cohen and I. E. Drabkin, A Source
Book in Greek Science
(New York, 1948), pp. 286-310;
R. L. Crocker, “Pythagorean Mathematics and Music,” The
Journal of Aesthetics and Art Criticism,
22 (1963-64),
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
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
(New York, 1964); C. V. Palisca, 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
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,
Vorgemach der musikalischen Composition, Erster Theil
(Lobenstein, 1745), Ch. 5; B. Stillingfleet, Principles and
Power of Harmony
(London, 1771); G. Tartini, Trattato di
(Padua, 1754).


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