Thomas Dechand
Humanities Center
The Johns Hopkins University
Music and Mathematics in Plato's Timaeus[1]
I would like to begin with two quotations. The first is from Samuel Coleridge's Biographia Literaria. As an illustration of his maxim, 'until you understand a writer's ignorance, presume yourself ignorant of his understanding,' Coleridge writes: "I have been re-perusing with the best energies of my mind the Timaeus of Plato. Whatever I comprehend, impresses me with a reverential sense of the author's genius; but there is a considerable portion of the work, to which I can attach no consistent meaning" (141). The other quote comes from the Timaeus itself, and I found it too apposite a description of what reading the dialogue often feels like not to use: "as you might imagine a person who is upside down and has his head leaning upon the ground and his feet up against something in the air, and when he is in such a position, both he and the spectator fancy that the right of either is his left, and the left right" (43e). The Timaeus is an extraordinarily rich dialogue. It is the story of Plato's cosmology, and in it one finds not only Plato's account of the creation of the universe, but a treatise on geometry disguised as a theory of the structure of matter, musings on the nature of time and space, a theory of vision, a chemistry of earth, air, fire and water, and most everything in between. Since this is obviously too much information to try to cover in an essay, after a few general remarks, I would like to focus on a few passages that seem particularly confusing the first time through.
Perhaps the most immediately striking feature of the dialogue, as Gregory Vlastos notes, is that the name Plato gives his creator literally means neither ruler nor king, but "craftsman" (Plato's Universe, 26). This is odd, since, as Vlastos tells us, "in Plato's Athens the craftsman is often a slave and as often a freeman working shoulder to shoulder with slaves in the same kind of work" (26). The contrast is to the Greek gods, who, as portrayed by the poets, lived lives best suited for daytime television. According to this portrait, an Athenian had to placate the gods with constant festivals and sacrifices, while simultaneously worrying about such things as being seduced by Zeus only to incur the jealously and wrath of Hera later on. Opposed to this image of the gods, Plato tells us that his creator is good, and as such, can "never have any jealousy of anything" (Timaeus, 29e). In the first of many similarities to the Biblical account of creation found in Genesis, Plato's Demiurge desires "that all things should be as like himself as they could be," or what comes to the same thing, "that all things should be good and nothing bad, so far as this was attainable" (30a).
As in Genesis, Plato needs two different creation stories to cover all the ground he wants to. The first story, describing what Plato calls "the works of intelligence," is juxtaposed with the second, which describes "the things which come into being through necessity" (47e). The "works of intelligence" are undertaken by the Demiurge; their telos ends in what is good. The works of necessity, as the name suggests, describes only those events logically entailed by other events. As Plato says: "the creation of this world is the combined work of necessity and mind. Mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection, and thus and after this manner in the beginning, through necessity made subject to reason, this universe was created" (47e-48a).
I will look closely at two passages, one from the works of intelligence dealing with Plato's construction of the world-soul, the other from the works of necessity dealing with the formation and interactions of the four elements.
Construction of the World-Soul
Plato's universe, as is to be expected, is created from a model (29b). It is based on the form of a perfect animal (30b, 32d). As such, it is self sufficient, implying that the universe is unique (30d-31a, 33d), and spherical, that being the most perfect shape (34a). Since it is based on the model of an animal, Plato's universe will also need a soul (30b, 34c). Having gathered up a sort of primordial ooze, Plato describes the Demiurge's creation of this world soul in the following passage, which I will quote once all the way through before attempting a more thorough line by line analysis:
And he proceeded to divide after this manner. First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty seven times the first [27]. After this he filled up the double intervals [that is, between 1, 2, 4, 8] and the triple [that is, between 1, 3, 9, 27], cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes [as, for example, 1, 4/3, 2, in which the mean, 4/3 is one third of 1 more than 1, and one third of 2 less than 2], the other being that kind of mean which exceeds and is exceeded by an equal number. Where there were intervals of 3/2and of 4/3 and of 9/8, made by the connecting terms in the former intervals, he filled up all the intervals of 4/3 with the interval of 9/8, leaving a fraction over, and the interval which this fraction expressed was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was exhausted by him. This entire compound he divided lengthwise into two parts which he joined to one another at the center like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting point (35b-36c).
So you can see that Coleridge was not exaggerating. To try and better understand what Plato is up to here, it is necessary to parse this passage one bit at a time.
After separating out the numbers 1, 2, 4, and 8, as well as 1, 3, 9, and 27, Plato's Demiurge fills up the spaces between each of these numbers by inserting two kinds of means. I will illustrate how Plato's Demiurges divides and fills the line by looking only at the numbers between 1 and 2. One kind of mean, that which "exceeds and is exceeded by an equal number" is more commonly known as the arithmetic mean. It is obtained by adding two numbers and then dividing their sum by two and given by the formula (a + b)/2. The arithmetic mean of 1 and 2 is 3/2.
1 3/2 2
The other kind of mean, what Plato describes as "exceeding and exceeded by equal parts of its extremes" is better known as the harmonic mean. It is given by the formula 2/(1/a + 1/b). The harmonic mean of 1 and 2 is 4/3.
1 4/3 2
Placing these two means in the same diagram, Plato notes there are intervals of 4/3, 3/2, and 9/8, the latter obtained by dividing 3/2 by 4/3 (36a-b).
1 4/3 3/2 2
(x 9/8)
Plato then fills up the entire segment with such 9/8 intervals, noting that this does not quite work out because it leaves remainder 256/243 intervals (36b). Plato's construction, reading off the numbers between 1 and 2 from left to right, thus gives us the sequence 1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, and 2. Each term in this sequence is obtained by multiplying the prior term with either the interval 9/8 or the interval 256/243.
1 9/8 81/64 4/3 3/2 27/16 243/128 2
(x 9/8) (x 9/8) (x 256/243) (x 9/8) (x 9/8) (x 9/8) (x 256/243)
Having thus filled up all the intervals between 1, 2, 3, 4, 8, 9, and 27, Plato's Demiurge stops dividing, telling us only that the "whole mixture out of which he cut these portions was exhausted" (36b). And without telling us what, if anything, these numbers mean, Plato is off and running in the very next sentence, cutting this strip into two and tying it back together so as to create two circles which he uses to recreate the motions of the fixed stars in relation to the motion of the planets, sun, and moon about the earth.
So what in the world is going on here? What significance does Plato see in the arithmetic and harmonic means, and why does he stop dividing the intervals at 9/8 and 256/243? Having sought in vain for a mathematical interpretation of this sequence, I eventually happened to hit upon a musical interpretation.
If one considers the ratios as the relative frequencies of various notes, or as was more likely in Plato's time, the relative lengths of string plucked to produce pleasing sounds, a scheme quickly emerges for making sense of Plato's construction of the world-soul. One can verify the following ratios: an octave corresponds to a 2/1 ratio; a perfect fifth to 3/2; a fourth to 4/3; a tone to 9/8; and a near half-tone to 256/243. Looking back at the division between 1 and 2 of the world-soul, one notices that Plato has constructed a musical scale! This scale is nothing but Plato's version of a sort of 'Do Re Me Fa So La Ti Do.'
1 9/8 81/64 4/3 3/2 27/16 243/128 2
(x 9/8) (x 9/8) (x 256/243) (x 9/8) (x 9/8) (x 9/8) (x 256/243)
C D E F G A B C
Had Plato built a piano, it would sound somewhat along the lines of what one would hear, if, starting with middle C, Plato played only the white keys until he hit another C. This scale is known as the Pythagorean scale, and it is Pythagoras who is credited with first discovering the rations governing the octave, fourth, fifth, and tone.
This interpretation may also help explain the nature of the stuff that Plato's Demiurge used to create the world soul. This mixture, as Gregory Vlastos notes, "contains no physical matter, and none of the properties of physical matter except one: it can move" (Plato's Universe, 31). Its movement is special both because soul is the only material capable of moving itself, and also because its motion is confined to revolving in a circle (31). Plato describes circular motion as "most appropriate to mind and intelligence" (Timaeus, 34a). The musical ascent of one octave, since it begins and ends on the same note, could be thought of as describing a circle.
As mentioned earlier, Plato uses this scale to construct the motions of the fixed stars, planets, sun, and moon around the earth. An interesting application of this idea, one I have had absolutely no luck with, would be to apply this scale to Plato's description of the motion of the heavenly bodies.
Plato's Particle Physics
In a celebrated line from his television series Cosmos, astronomer Carl Sagan said, "If you want to make an apple pie from scratch, you must first invent the universe." Plato's version, at least according to the Timaeus, would likely read: "If you wish to make goat cheese from scratch, you must first invent the triangle." Remembering that Plato marked the entrance to his academy with the words: "Let no one who is destitute of geometry pass beyond these walls," let us proceed to examine what may be termed 'Plato's Particle Physics.'
Plato chooses the two triangles he believes have "the most beautiful forms" as the fundamental particles of his universe (54a). The first is "that triangle which has its hypotenuse twice the lesser side," or a 30-60-90 degree triangle (54d). The second is an isosceles triangle, which has angles equal to 45, 45, and 90 degrees (55b).
The 30-60-90 degree triangle (left) and the isosceles triangle (right) constitute Plato's fundamental particles.
The four elements in Plato's universe, earth, air, fire, and water, are made by combinations of these two triangles. Plato envisions this system on a similar part to whole relation as he uses for letters and words in the Cratylus.
Three of the four elements, those made from 30-60-90 degree triangles, have equilateral triangles as their faces. Plato describes the construction of a face as follows: "When two such triangles are joined at their diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a center, one equilateral triangle is produced from six of these" (54d-e). The following diagram shows such a face.
The first object Plato makes with these equilateral triangles is a tetrahedron. In his words: "four equilateral triangles, if put together, make out of every three plane angles one solid angle. And out of the combination of these four angles arises the first solid" (54e-55a).
The second object constructed in this manner is a octahedron: "the second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed" (55a).
The third object is an icosahedron: "the third body is made up of one hundred and twenty triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle" (55a-b).