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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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In the course of summarizing Pythagorean contri-
butions to Greek thought, Aristotle, having pointed out
the importance of mathematics to the Pythagoreans,
adds that “since... they saw that the modifications
and the ratios of the musical scales (༁ρυονιῶν) were
expressible in numbers;—since, then, all other things
seemed in their whole nature to be modelled in
numbers, and numbers seemed to be the first things
in the whole of nature, they supposed the elements
of numbers to be the elements of all things, and the
whole heaven to be a scale and a number
” (Meta-
A 5 986a, trans. W. D. Ross). Aristotle was
probably describing the views of fifth- and fourth-
century Pythagoreans such as Archytas of Tarentum,
under whom the doctrine of a universe ordered by the
same numerical proportions that govern musical
harmonies was developed.

How much the semimythical Pythagoras of Samos
(late sixth century B.C.) had to do with formulating the
laws of cosmic harmony is not known; he is credited
by Diogenes Laërtius with having discovered that the
principal musical consonances result from the sounding
of proportionate lengths of a stretched string, so that
within the series 1-4 (the sacred Pythagorean tetraktys)
simple ratios give forth the octave (2:1), the fifth (3:2),
and the fourth (4:3). In these same proportions and
their multiples, and particularly in the “means” found
within multiples of the duple proportion (arithme-
tic = 2:3:4; geometric = 1:2:4; harmonic = 3:4:6),
lay for subsequent Pythagorean thinkers the rela-
tionships between all sorts of natural phenomena. Since
numbers were for them not abstractions but quantities
with real, i.e., spatial, existence, the discovery of musi-
cal laws—more tangible than vague analogies—
governing the whole of creation, and especially the
starry universe, was an intoxicating one, and in its
precise and extended mathematical elaboration a
peculiarly Greek one.

Oriental and Near-Eastern cosmologies all show
some ordering principles at work, principles in many
instances exerting influence on terrestrial life. The
Greeks did not know where these ideas came from,
but Plutarch referred to the “Chaldeans” (De anim.
1028), and so did Philo Judaeus, who described
the Chaldeans (or Babylonians) as having “... set up
a harmony between things on earth and things on high,
between heavenly things and earthly. Following as it
were the laws of musical proportion (διἁ μουδικῆς
), they have exhibited the universe as a perfect
λὁγωνconcord or symphony produced by a sympathetic
affinity between its parts, separated indeed in space,
but housemates in kinship” (De migrat. Abrahami
XXXII, 177f., trans. F. H. Colson). It is possible that
not only Greek cosmologies but also Jewish beliefs in
an ordered universe hymning the praises of its
Maker—expressed in the Psalms, in the visions of Isaiah
and of Ezekiel, and especially in the Talmudic book
of Yoma—may have been influenced by Babylonian
lore. The same Philo who credited the Chaldeans with
discovering cosmic harmony wrote a lengthy commen-
tary on the six days of Creation (De opificio mundi)
with constant allusions to Pythagorean theories, thus
stressing what was for him the common parentage of
Greek and Jewish cosmology.

Among the fragments of pre-Socratic philosophy
there are a few references to symphonious order in
the heavens. Anaximander (b. 610 B.C.), for whom the
planets were wheels of fire visible through “breathing-
holes,” posited relative sizes of 27, 18, and (presuma-
bly) 9, thus 3-2-1, for the sun, moon, and “stars”
(planets?) with respect to the earth. This graduated
order was not accompanied by musical sound; but
Anaximander did compare the “breathing-holes” in his
fiery circles to the holes of musical pipes. In the Proem
of the fifth-century Parmenides' “Way of Truth” the
axle of a fast-moving heavenly chariot glows in its
socket and sings out like a pipe; a surviving portion
(frag. 12, Diels) of the second part, the “Way of Opin-
ion,” of the poem suggests further connection with
Anaximander's cosmology while at the same time
prefiguring one of the great statements of the music
of the spheres, the Myth of Er in Plato's Republic.
Finally, the mysterious “attunement of opposites” of
Heraclitus (frag. 51, Diels) was related both to the
cosmos (by Plutarch, De anim. procr. 1026 B) and to
music (by Plato, Symposium 187).


To return to Pythagoras himself, it is impossible to
sort out historical truth from the welter of myth sur-
rounding this figure. But according to the doxographer
Hippolytus, Pythagoras is said to have taught that the
universe is put together by means of harmonic laws
and so produces, through the motion of the seven
planets, rhythm and melody (see Diels, Doxographi
[1879], p. 555). The very enthusiastic Neo-
Pythagorean Iamblichus went so far as to claim that
Pythagoras could actually hear the cosmic music
inaudible to other mortals. And since all discoveries
about the Pythagorean cosmos were dependent on the
numerical ratios sounded by the stretched string or
monochord, it was reported by the Neo-Platonic musi-
cal theorist Aristides Quintilianus (third century A.D.)
that Pythagoras' dying injunction to his students was
μανοχορδἱζειν (“work the monochord”).

It is not from Pythagoras himself, nor yet from any
of his direct followers, that we get a full and circum-
stantial account of the formation of the universe by
the laws of harmony; the first such account—and
certainly the most important—is that given in Plato's
Timaeus. In this dialogue Timaeus the Locrian is
spokesman for Plato's version of Pythagorean cosmic
doctrine. (On the basis of a spurious Alexandrian dia-
logue paraphrasing Plato, “Timaeus of Locris” was
long thought to be the source of Plato's Pythagorizing
views.) The wondrous tale of the Demiurge fashioning
the World-Soul is told (Timaeus 35-36); after this
psychogony has been completed it serves as model
(παρἀδειγμα) for the creation of the corporeal world.
Out of a material blended of Sameness and Difference,
ideal and bodily Existence, the Demiurge constructs
a model for the universe. The psychic material is cut
or marked into proportionate lengths before being split
and bent into circles illustrating the makeup of the
planetary system. It is these proportions, yielding the
series 1-2-3-4-9-8-27, a compound of two geometric
series (1-3-9-27 and 2-4-8), that outline the Pythagorean
harmonic world. Plato forms a scale that “sounds” his
ideal celestial distances. Within the two geometric
series are placed arithmetic and harmonic means,
creating proportions of 3:2, 4:3, and (their difference)
9:8. The proportion 4:3 (in musical terms an interval
of a fourth) is filled in, or marked off, with intervals
of the size 9:8 (in music, a whole tone), leaving in
each fourth a small difference (λεῖμμα), or semitone,
of 256:243. The result is a musical scale, based on a
tuning of intervals that has ever since been termed
Pythagorean, of nearly five octaves—truly universal,
since it is by far greater in compass than any scale
given by Greek musical theorists. Out of material
marked with this scale, then, the Demiurge forms the
World-Model, and thus it is that one could suppose
“the whole heaven to be a scale and a number,” its
paradigm made out of a kind of celestial monochord.

No one knows whether Plato in the Timaeus was
himself thinking as a Pythagorean or was reporting
current theories not originated, perhaps not even fully
believed, by him. But the Timaeus is for us the main
source of Pythagorean cosmology; and so it was for
the later ancient world as well. A large surviving body
of commentary by Neo-Platonic and Neo-Pythagorean
writers (the fullest and best is that of Proclus in the
fifth century A.D.) shows what fascination Plato's work
had—as well as how unclear his meaning was. Through
the Latin commentary of Chalcidius the Timaeus was
known in the Middle Ages, and Renaissance Neo-
Platonists, such as Marsilio Ficino, added to the body
of work seeking to amplify and explicate Plato's ac-
count. The tuning system outlined in the Timaeus
became a regular part of Greek musical theory, given
full statement in the κατατομὴ κανὀνος (ca. 300 B.C.)
attributed to Euclid, and included in the musical
treatise of the great astronomer Ptolemy (second cen-
tury A.D.), whose celestial harmonies are an elaborate
scientific restatement of Plato's cosmic sketch.

Plato does not describe his harmoniously conceived
universe as sonorous in the Timaeus—the musical the-
ory outlined there belongs to the Greek discipline of
harmonics, the tuning of intervals, rather than to music
itself. What is known to us as the “music of the
spheres” comes from another source in Plato, the Myth
of Er at the end of the Republic. Er the Pamphylian,
a hero slain in battle, was given the privilege of seeing
the next world and then returning to life to describe
what he had seen. The vision of Er includes once again
a model of the universe, a set of concentric rings or
whorls—the planets—hung on the spindle of Necessity.
The rims of these whorls are of different sizes and
colors, and they revolve at different speeds—all the
inner ones in opposition to the movement of the outer
rim, the firmament. The Pythagorean proportions of
the Timaeus are lacking here; but present is actual
music, for as the spindle turns, “on the upper surface
of each circle is a siren, who goes round with them,
hymning a single tone or note. The eight together form
one harmony (ἁρμονἰαν)” (Republic X. 617, trans. B.

Thus for Plato the universe was designed according
to harmonious proportions, and this intellectual har-
mony could be described, in the metaphoric language
of a dream-vision, as sounding music. Whether or not
the cosmic myths of the Timaeus and the Republic
were meant by their author to be related, most people
in the ancient world took them to be so. Since the
term harmonia could mean, among other things, the
interval of the octave, some commentators made of


the Sirens' music a single octave of the Timaeus scale—
sounding simultaneously but audible successively to
anyone privileged to move through the planetary
realms (such a voyage is described in a long didactic
poem by the encyclopedist Martianus Capella in the
fifth century A.D.). Whether the scale went up or down
from outer to inner planets, whether the motionless
earth “sounded” in this celestial scale—these and simi-
lar questions were treated in detail by Neo-
Pythagorean writers. And despite Aristotle's rejection
of sounding planetary rims (in favor of his own silent,
frictionless spheres; see De caelo II. 9. 290-91) belief
continued strong in a literally musical universe, with
the harmonious gradation of sound produced by the
differing planetary speeds. These speeds in turn are
regulated by the distances of the planets from earth,
the center of the system, or from the firmament, its
outer rim.

Plato's Pythagorean universe was studied, com-
mented upon, and imitated; one of the most popular
and long-lived imitations was Cicero's Somnium
a dream-vision placed at the end of his De
in direct imitation of Plato. For Cicero it
is the motion of the spheres that produces the “great
and pleasing sound” of the universe. This sound is a
concord of “carefully proportioned intervals,” there
being seven tones in all; these seven planetary tones
were equated (by Macrobius and other later commen-
tators) with the seven numbers of Plato's geometric
series in the Timaeus. Mortal beings, accustomed from
birth to the sound of the cosmos, cannot ordinarily hear
it; only in a vision, or after death, does its sublime
harmony, of which terrestrial music is an imitation,
reveal itself.

The core of Pythagorean belief in universal harmony
is the music—heard or inaudible—of the celestial ele-
ments. But the sublunary world also partook of this
harmony: the elements of fire, air, water, and earth;
the seasons; the days of the week; the flow of rivers
and the tides of the sea; the direction of winds; the
growth of plants. These and many other earthly
phenomena were viewed as directly related to the
heavens, and so governed by the same principles of
harmonics or musical mathematics. An elaborate set
of these correspondences is given by the late Greek
theorist Aristides Quintilianus (Περἱ Μονσικῆς, Book
III). Man, the microcosm, shares in this harmony:
everything from the gestation period of the human
embryo and bodily proportions to the smallest details
of human behavior is governed by analogy with, or
dependence upon, celestial harmony. Even such
apparently prosy dicta as ἀρχὴ δἑ τοι ᾕμισμ παντός
(freely rendered as “once begun is half done”) could
be related to the proportion 1:2, to the interval of
the octave, to the midpoint of a monochord (see M.
Vogler in Festschrift J. Schmidt-Görg [1957], pp.
377-82). Where human actions are concerned the line
between cosmic harmony and astrology is a fine one;
indeed there was no real distinction between the two
for ancient writers, although the more vulgar aspects
of astrological belief were scorned by serious thinkers.

The third book of Ptolemy's Harmonics, devoted to
cosmic analogies of all kinds, shows a distinction be-
tween ἁρμονία κίσμον and ἁρμονία ψνχῆς; these
categories, rendered in the Latin of Boethius' De
(early sixth century) as musica mundana, and
musica humana, are joined with the music sung and
played by men (musica instrumentalis) to form a
tripartite division of the science and art of music that
was to be canonical for the next thousand years in
academic circles. In fact, the place of music in the
curriculum, as a part of the quadrivium (along with
geometry, astronomy, and arithmetic) is really due to
the central importance of Pythagorean views of the
subject in late antiquity. It should be remembered that
academic study of music was primarily the study of
harmonics—of tuning systems and of musical arithme-
tic, the properties of the numerus sonorus.

For the Church Fathers Pythagorean beliefs were
acceptable as long as biblical parallels could be found
for them; and for notions of cosmic harmony there
were, as scholars like Philo discovered, plenty of paral-
lels. The second-century Alexandrian philosopher
Numenius went so far as to say “For what is Plato,
but Moses speaking in Attic Greek?”—words quoted
by Clement of Alexandria and other Fathers; and ac-
cording to Josephus (Contra Apionem I, xxii), Pythag-
oras himself was an admirer and imitator of Jewish
beliefs. Acceptance of the literal reality of the music
of the spheres varied from writer to writer, but re-
ceived a great boost when Saint Jerome translated from
the Book of Job a passage (38:37) dealing with rain-
clouds as concentum coeli quis dormire faciet, “who
can make the harmony of heaven to sleep?” (Douay

Jerome, in making this translation, drew upon the
Greek version of Symmachus, a member of an early
Judeo-Christian sect with strong Gnostic tendencies.
Pythagorean cosmic ideas as developed by Gnostic
writers took on a much more obvious astrological cast,
with the planets becoming deities invoked by mystic
hymns using music “proper” to each planet. This sort
of thing was strongly opposed by orthodox Christian
writers, but, as in the case of Jerome's translation, it
may occasionally have exerted some influence. Jewish
belief in the angelic habitation of the universe, colored
by Gnostic angelology and given orthodox standing by
the sixth century (when the nine angelic hierarchies


of Dionysius the Areopagite became accepted),
ultimately led to belief in a musica coelestis, angel-
music in or above the starry heavens. This form of
cosmic harmony persisted even in the later Middle
Ages, when Pythagorean thinking was rather dis-
credited because of the scholastic adherence to the
anti-Pythagorean Aristotle in all things; and in Dante's
Paradiso one finds musica mundana and musica
combined in a blazing vision of light and

In general, Pythagorean ideas were repeated and
elaborated whenever currents of Neo-Platonism were
strong: in the sixth-century commentaries of Boethius
and Cassiodorus; in the ninth-century Carolingian re-
vival (John Scotus Erigena, Regino of Prüm); in the
writings of the Chartres school of the twelfth century
(Guillaume de Conches, Alain de Lille). Cultivation of
Neo-Platonic thought in the medieval Arab world was
marked by preoccupation with musica humana—the
harmonious makeup and workings of the human body—
resulting in theories about the curative powers of music
that were taken literally enough to cause music to be
played as a therapeutic agent in hospitals. The great
revival of Neo-Platonism among fifteenth-century
humanists led to some imaginative restatements of
Pythagorean cosmic belief by such men as Giorgio
Anselmi of Parma (Dialoghi, 1434) and Marsilio Ficino
(in a number of works, but most fully in a commentary
on the Timaeus), as well as to encyclopedic compi-
lations of everything the ancients said on the subject,
the fullest being that of the theorist-composer Fran-
chino Gafori (Theorica musicae, 2nd ed. [1492], I, i,
“De musica mundana”).

Although the literal acceptance of the Pythagorean
cosmos in the early Renaissance was tempered with
a certain sophisticated skepticism in the sixteenth cen-
tury, enthusiastic restatements of the old beliefs con-
tinued to appear; a work like Pontus de Tyard's Soli-
taire second
(1555) contained a 200-page exposition of
musique mondaine & musique humaine. Universal
harmony was described in poetry: Italian, French, and
especially English. And it was depicted in fêtes and
intermedi; Leonardo da Vinci designed the planetary
sets and celestial mechanism for a Festa del Paradiso
given at Milan in 1490. No music for this survives;
but a score showing how a Renaissance musician
thought of cosmic harmony does exist for a tableau
staged as part of the festivities at a Medici wedding
in Florence in 1589. This tableau, designed by
Giovanni de' Bardi, was called “L'Armonia delle sfere,
and contemporary accounts make it clear that Bardi
was trying to depict on the stage Plato's Myth of Er.

Other aspects of Renaissance culture were touched
by Pythagorean doctrine. The use of “harmonious”
proportions in architecture, perhaps practiced in
antiquity, was revived in the building of the Gothic
cathedrals, and became a preoccupation with archi-
tects from the time of Alberti (mid-fifteenth century).
The Venetian monk Francesco di Giorgio, author of
an enormous, relentlessly Pythagorean treatise called
De harmonia totius mundi (1525), wrote a memoran-
dum recommending the use of the Timaeus series in
the building of a church. A generation later, when
Andrea Palladio completed the facade of this church
(S. Francesco della Vigna), he used a scheme of 27
moduli, thus Plato's outer limit, for its width.

Pythagorean ideas seem to have a less vivid appeal
after the Renaissance. But before retreating, driven by
seventeenth-century rationalism into poetic metaphor,
the Platonic-Pythagorean cosmos received a splendid,
consummatory restatement in Kepler's Harmonices
(1619), which Kepler himself describes as a work
picking up where Ptolemy left off. It separates the
Copernican spheres by intervals defined by the five
regular solids of Greek geometry, and finds harmonic
proportions expressible in musical terms—a seven-
teenth-century chordal complex rather than a Greek
scale—in the relationships between the movements of
planets and their respective medium distances from the
sun. Kepler, like Ptolemy, includes a whole treatise on
musical theory to lay the groundwork for his theories
of cosmic harmony.

In the Utriusque cosmi (1617) of Robert Fludd, con-
temporary and archenemy of Kepler, the Ptolemaic
system is still the basis for an elaboration, lengthy and
formless, of Pythagorean ideas. In seventeenth-century
Italy Mario Bettini (Apiaria, 1641-42) and Giambat-
tista Riccioli (Almagestum novum, 1651) presented
traditional Pythagorean cosmologies with Keplerian
refinements tacked on. Polymaths such as Marin
Mersenne in France (Harmonie universelle, 1636) and
the German Jesuit Athanasius Kircher (Musurgia
1650) continued the process of summariz-
ing, and in a way bringing up to date, Pythagorean

Elements of Pythagorean thought have persisted
among philosophers, mathematicians, and astronomers.
Leibniz, for example, was fond of Pythagorean imagery
(see R. Haase, “Leibniz und die pythagoreisch-
harmonikale Tradition,” Antaios, 4 [1962], 368-76),
and his doctrine of “pre-established harmony” might
be seen as a new version of musica mundana/humana.
In the nineteenth century Platonic cosmology was
much studied, especially by German scholars; and
philosophers like Schopenhauer took up the old images
of world harmony once more. One of the most inter-
esting of nineteenth-century cosmological studies is A.
von Thimus' Die harmonikale Symbolik des Alterthums


(1868-76), in which the presence of esoteric numero-
logical lore is traced in the records of ancient cultures
around the world.

The establishment by the German astronomer J. E.
Bode (1747-1826) of a simple arithmetic series to rep-
resent planetary distances from the sun, and the later
inclusion of the newly-discovered planet Neptune in
the series (see Duhem, Système, II, 15-17) carried
forward Keplerian ideas of stellar harmony; in the
twentieth century a new system of harmoniously pro-
portionate planetary distances was worked out by the
cosmologist Wilhelm Kaiser (Geometrischen Vorstel-
lungen in der Astronomie, Kosmos und Menschen-
1930), while at the microscopic level V. Gold-
schmidt has found the proportions of the musical scale
in the relative dimensions of crystals (Ueber Harmonie
und Complication,
1931). The cosmos of the Timaeus
has found twentieth-century admirers such as Sir
Arthur Eddington and A. N. Whitehead; and one can
call twentieth-century scholars like Hans Kayser
(Lehrbuch der Harmonik, 1950) genuine Neo-
Pythagoreans, in that their aim is to reinterpret and
to reestablish, with the support of modern scientific
knowledge, the basic Pythagorean concepts of world


Still of fundamental importance is the first modern inves-
tigation of the subject, A. Boeckh, “Über die Bildung der
Weltseele im Timaeos des Platon,” Gesammelte kleine
(Leipzig, 1866), III, 109-80. See also: R. S.
Brumbaugh, Plato's Mathematical Imagination (Blooming-
ton, Ind., 1954); F. M. Cornford, Plato's Cosmology (New
York, 1937); R. Crocker, “Pythagorean Mathematics and
Music,” Journal of Aesthetics and Art Criticism, 22 (1963),
189-98, 325-35; P. Duhem, Le système du monde. Histoire
des doctrines cosmologiques de Platon à Copernic,
10 vols.
(Paris, 1913-59); E. Frank, Plato und die sogenannten
(Halle, 1923); M. Ghyka, Le nombre d'or. Rites
et rythmes pythagoriciens dans le développement de la
civilisation occidentale
(Paris, 1932); R. Hammerstein, Die
Musik der Engel
(Bern, 1962); J. Handschin, Der Toncharak-
(Zurich, 1948); J. Hutton, “Some English Poems in Praise
of Music,” English Miscellany, ed. M. Praz (Rome, 1951),
II, 1-63; H. Kayser, Lehrbuch der Harmonik (Zurich, 1950);
L. Spitzer, “Classical and Christian Ideas of World Har-
mony: Prolegomena to an Interpretation of the Word
'Stimmung,'” Traditio, 2 (1944), 409-64; 3 (1945), 307-64.


[See also Cosmic Images; Cosmology; Harmony or Rapture;
Macrocosm and Microcosm;
Neo-Platonism; 4">Pythagorean