Means, Meaning, and Music: Pythagoras, Archytas, and Plato

Scott Makeig

 

Introduction

 

Modern studies of Greek music have tried to discover how far the game of music theory actually went in the times of Plato and his Academy. However, the earliest Pythagoreans, possibly the most important and surely the most intriguing of Greek musical philosophers, kept largely to themselves the teaching they had been given and which in due course they modified and developed.

 

The most powerful and influential precept of Pythagoreanism was that all is ordered according to the order inherent in whole numbers. Recent scholarship finds no ground to doubt that this doctrine came through Pythagoras himself, though he may well, as his biographers claimed, have gotten it from Babylonia and/or Egypt.1

 

The simplest application of number philosophy to music involved the 6 : 8 : 9 : 1 2 ratios which formed a basis for Greek tuning theory as they in fact do for the modern Western scale system. Today these ratios are understood as frequency ratios. To the Greeks they probably represented ratios of string lengths or of pipe lengths. But to the Pythagoreans, the meaning of the musical ratios was primarily esoteric, meaning hidden from ordinary understanding. Although Plato, for instance, discusses their importance to both music and esoteric philosophy, he concentrates on the latter. Thus his discourses do not provide physical demonstrations of the ratios, for they were not meant as textbooks, but as spiritual philosophy. As Socrates explains:

 

We must endeavor to persuade those who are to be the principal men of our state to go and learn arithmetic, not as amateurs, but they must carry on the study until they see the nature of numbers with the mind . . . for the sake of the soul.2

 

Lasserre, in his 1964 book on the mathematics of the Academy, traces the transformation of the esoteric number philosophy of Pythagoras through philosophical applications by Plato and his contemporaries, into the more and more scientific mathematics and astronomy of succeeding generations.3 In this paper will attempt a parallel description of the development of certain elements of numerical tuning theory from Pythagoras through Archytas and Plato. I have written this paper not so much as an historian, but rather as a student of basic music theory. The off-hand references to Pythagorean music theory one often finds in music theory writing are seldom based on any detailed understanding or analysis of the historical record. It was to dispel my own vagueness about Greek tuning theory that I wrote this paper; I hope it may prove similarly useful to others.

 

The Means

 

As Pythagorean music theory was centered around the concept of means, I would like first to discuss the system of means developed by Archytas, Plato and later writers. The dimensions of the doorway to the tomb of the pyramid of Gizeh (dated around 4000 B.C.) are in the proportion which is today called the "golden mean" -  after Pythagorean sources.4 Both Plato and Aristotle mentioned its esoteric significance. It was most probably already in mathematical use in Babylonia for constructing the star pentagon, which later became the symbol of the Pythagoreans.5

 

While Nichomachus states that Pythagoras brought his mathematical knowledge from Babylonia, there is no evidence of the calculation of the golden ratio in the time of Plato. However, its geometrical construction is comparatively simple (see Figure 1). This construction is shown with full proof in Euclid, Book II, which Van der Waerden thinks was drawn from books of the time of Plato and before.6 Certainly artisans had long known of a method for constructing the

golden mean; it is found exactly expressed in the dimensions of the facade of the Parthenon, in Greek urns, and in Egyptian designs. It is usually called "phi" today in honor of reference to its use by the sculptor Phideus. However as we shall see, the "golden mean" was not the foremost among the means in use in musical philosophy.

 

Figure 1: The Golden Mean

 

The philosophic appeal to the Greeks of the concept of a mean between two extremes is not hard to fathom. In both Babylonian and Greek philosophy, life was seen to be a matter of opposites.7 To the Pythagoreans, life was created and held in balance by mind through the play of opposites. A numerological philosophy would therefore give high importance to discovering an orderly and dynamically balanced finite mean between two opposites, and such a mean would naturally be thought to share the absolute qualities of its bounding and life-giving pair of extremes. When these opposite extremes are considered primal, then the mean itself becomes a direct descendant of their primal complementarity.

 

The "golden mean" has only very rarely been involved in tuning theory - and never in the ancient world, so far as is known -  for the simple reason that it cannot be expressed in terms of whole numbers. (Furthermore, it generates intervals which don't sound very good!) But at least as early as the time of Archytas, and reputedly well before, several other means did figure prominently in musical and numerological philosophy. In particular, they are found at the heart of the musical systems of Archytas and the Platonic Epinomis.

 

Nichomachus, writing in the second century A.D., lists ten means.8 Of these ten, he says three were given by Pythagoras and Archytas, and others by later writers. The extant fragments of Archytas, Pythagorean and friend of Plato, give prime importance to the first three - the arithmetic, geometric and harmonic. The second set of three are usually attributed to Eudoxus, a contemporary and acquaintance of Plato.9 Eudoxus did not give musical applications to this second trio of means, but together with the first three they form a complete system. As Lasserre points out, Eudoxus was not directly a Pythagorean, but was a key figure in the modification of the Pythagorean number teaching into rigorous and self-contained mathematics.

 

 

The Geometry  of Numbers

 

Problems of equal areas or volumes were the more difficult and important mathematical problems of the ancient world. It was Archytas himself who found a complex geometric solution to the problem of doubling the cube, which was equivalent to the problem of finding two successive geometric means between any two lengths. Archytas is also credited by Boethius with the arithmetic proof that given two whole numbers differing by one, their geometric mean cannot be expressed in terms of the ratio between two integers. For this reason, he justified the use of the arithmetic and harmonic means to fill out his harmonic system, not because of any practical problem in laying out a string length for the geometric mean, but because of the ontological importance of whole numbers.

 

 

 

Figure 2: The Geometric Mean

 

The geometric construction of the geometric mean (Figure 2) was well-known but the Pythagoreans argued that geometry arose from numbers. Nichomachus interprets this doctrine by pointing out that we could not conceive of triangles or squares unless the numbers three and four already existed. The Platonic Epinomis reasons more abstractly:

 

It is essential to learn mathematics, whose primary and most important discipline is the science of numbers, considered in their own right and apart from bodies. The aim of this science is the generation of odd and even, and their relation to the nature of other things.

Whoever has studied this will tackle next the discipline which is, properly speaking, the science by which numbers not in themselves comparable are made comparable by relating them to the category of surfaces.10

 

This passage opens the Epinomis, a dialogue published under Plato's name posthumously though possibly finished by Plato's disciple Philip.11 The passage later refers to the word "geometry"  - literally "geometry" or earth-measuring"  - as "absurd." For Plato, as we have seen, geometry was much more than a surveying methodology. Philip of Opus, who was apparently a fervent younger disciple of Plato, later drew Aristotle's criticism for his bald statements that nothing actually exists but numbers.12

 

It is important to understand that in the absence of modern algebraic notation (which was developed only very recently in Western history), each algebraic statement tended to seem independent of any other related statement. This is noticeably true for Nichomachus, a novice as a mathematician, but a follower of earlier mathematical texts. He gives algebraic statements about the means which today appear as simple restatements of each other, but which Nichomachus states as separate and independently meaningful results. He even states an algebraic truism as a meaningful property of first one, and then separately another of the means, showing how awkward it was for him to make simple algebraic generalizations in the absence of abstract notation.13

 

Nichomachus instead phrases his algebraic statements in terms of ratios of quantities which can be imagined geometrically. This shows that even several hundred years after Plato and Archytas, abstract thinking was tied either to

geometrical imagination, or to metaphysical representation of quantities. Furthermore, as we shall see, the metaphysical representations of numbers were also understood in geometrical terms.

 

Many scholars have argued that even in the philosophical teaching of arithmetic theory, a sand-table and pebbles were used to make diagrams.14 This was known as psephoi-arithmetic, (literally, "pebble-arithmetic") and is alluded to in Plato and Aristotle, as well as in many later writers. Aristoxenus makes apparent reference to it in referring to the unspecified but presumably well-known "close-packed diagrams of the harmonists".15

 

For the Pythagoreans, number was the begetter of geometry, and the true significance of whole numbers could be studied through geometric representation.  Nichomachus quotes Plato:

 

Every diagram, system of numbers, every scheme of harmony, and every law of the movement of the stars ought to appear one to him who studies rightly.16 (Laws, ix 1016)

 

The number one was at once the finite point, and the infinite Unity or God. Addition of successive units generated the line - represented in psephoiarithmetic by a line of 2, 3, or more pebbles. Multiplication produced plane surfaces - squares and rectangles of pebbles. Lines pertained to quantity; surfaces pertained to quality -  perhaps just as a real surface has perceptual qualities of texture, color. etc.

 

Multiplying a plane figure made of pebbles by another number was considered to produce a three dimensional figure. However, successive numbers could also be conceived of in successively higher dimensions. The "decade," for example was considered to be the largest simple number since it was the sum of the first four numbers -  1 plus 2 plus 3 plus 4 equals 10 - and was thus also thought of as the sum of the properties of a point, a line, a triangle, and a tetrahedron (0, 1, 2, and 3 dimensions respectively). The number 10, in this geometric and metaphysical sense, was called the tetractys, and had a unique place in number mysticism.

 

The number 5 was the "human" number since it was the sum of the firstmost even and odd numbers, 2 and 3. Two and three were thought of as the primal opposites among the numbers, three being masculine and two being feminine. Here we can find a geometrical analogy in genital symbolism (Figure 4).

 

 

 

 

Figure 3: The Tetractys

 

 

 

Figure 4: Geometric Construal of "2" and "3"

 

Six was considered to be a "perfect" number, since it is the sum of 1, 2, and 3 - representing the union of the divine, the masculine, and the feminine natures, respectively. Furthermore, it is a multiple of, (that is to say, "qualitatively projective from") each of these factors or natures. That is, 6 is a product of 1, of 2, and of 3.

 

 

The Babylonian Number System

 

The Babylonian system of counting was based on the earlier Sumerian system. The Sumerian system used multiples of both 1 0 and 6 times 10 or 60, as well as 10 times 60 and 60 times 60. The Babylonians later used 60 and 60 times 60 exclusively, and from their system we still have our sixty minutes, three hundred and sixty degrees, etc.17 Scholars have not found a convincing physical explanation of the origin of this system. But if in fact Pythagorean number philosophy came from Babylonia, the counting system may indeed have come from philosophical considerations in Babylonia and even earlier, in ancient Sumeria, from which clay tablets are known dating from at least three thousand B.C.18

 

The earliest Greek counting system was based not on 60 but on 5, 10, and 100.19 The use of 60 as number base was introduced into Greece, Van der Waerden notes, through the influence of Babylonian astronomy on later Greek astronomy. This does not contradict, however, its possible earlier introduction by Pythagoras into Greek number mysticism.'

 

It is interesting that the Pythagorean geometrical view of whole numbers can be said to have been revived qualitatively in some of the abstract mathematics of our own century, in which objects of mathematical attention are structured collections of elements (cf. pebble-arrays), which have both ordinal size and also geometrical structure. I am thinking particularly of modern mathematical group theory, in which, for example, the "four-ness" of a square is represented as a symmetry group of four elements, consisting of the four ninety-degree rotations of the square as in Figure 5.

 

 

 

Figure 5: The "Rotational" Symmetry of the Number 4

 

 

Summary

 

In the Pythagorean number mysticism, the number was highly structured, and highly geometrical. The geometry of numbers was both highly abstract and at the same time symbolically malleable, and, on the evidence of counting systems, it may have been inherited from Babylonia or even from Sumeria.

 

The Tetractus of Means

 

Figure 6 summarizes the system of six means of Eudoxus in terms of the ratios between the differences among two given quantities, a and c, and their mean, b. The table shows that the six means systematically exhaust all such possible ratios consisting of combinations of the mean and the two extremes - that is, the ratio of (a-b) to (b-c) equals successively, a/a, a/b, a/c, b/a, b/b, b/c, etc.20

 

 

 

 

Figure 6

 

The set of six means also shows a remarkable structural similarity to the Pythagorean view of the number six itself. The primary mean is the geometric mean, begetter of three-dimensional life and all its qualities. It is also the mean of two opposing means, the arithmetic and harmonic. The geometric, harmonic, and arithmetic means form a triad. These are the three means of Archytas. Eudoxus introduced his three means as the conceptual complements to the means of Archytas. Thus the six-fold structure of the set of means was built up of first one, then two and then three more means. Not surprisingly, the next extension of the system of means, by still later writers, was to ten means, by addition of four more conceptually related means giving a tetractys, or perfect set of ten altogether.

 

The Sub-Contrary Mean

 

The mean mentioned foremost by Archytas in the construction of his tuning system is the harmonic mean. Archytas says that this was formerly called the sub-contrary mean. Why was it so called? And why was its name changed?

 

Answers to these questions should take into account the Greek method of calculating fractions. Fractions were very well handled in Babylonia, and Plato mentions their being useful for traders, though not for philosophers!21 The Greek method for calculating fractions, according to Van der Waerden, was to use ratios of whole numbers in terms of a least common multiple. This allowed problems with fractions to be thought of geometrically in terms of whole numbers of units. For example:

 

1:3+ 1:4= 4:12+ 3:12= 7:12

 

These units could be then thought of as lengths, and in particular could be lengths of musical strings or pipes. For instance Archytas mentions measuring half or other "parts" of a musical pipe. The arithmetic mean of a length and half of that length, (i.e. musically, of a tone and its octave), would have been thought of in terms of numbers of equal parts - the whole length would be thought of as four units, and the mean would be seen to be three of the four units in length.

 

The harmonic mean would be thought of in the same way. This time the whole length would have to be thought of as composed of six equal parts, and the harmonic mean could be seen to be equal to four of these parts. Furthermore, both means could be measured and compared with one another by thinking of the whole length as composed of twelve parts. This was how the 6 : 8 : 1 2 proportion was understood arithmetically. That the perfect musical intervals could be heard when a string or pipe was so divided would have been most pleasing to the Greek mind.

 

The two means were thus thought of as follows:

 

                                        `        The arithmetic mean          a - b = b - c

                                                 The harmonic mean         (a - b) / a     =  (b - c) / c

                                                                                                                                

The only difference in the above two statements is the addition of a (the longer fixed length) and c (the smaller fixed length) as denominators.22

 

The denominator represented the underlying subdivision of the larger length into units by which the extremes and means could be compared through simple counting. In the expression above for the harmonic mean, the "contrary" extremes a and c are inserted below the expression for the arithmetic mean. The harmonic mean might therefore fittingly be called the "sub-contrary" mean.

 

An even clearer modern demonstration of the complementary relation between the two means is the set of equations below. The form of these equations would not have been beyond Babylonian arithmetic.23 Perhaps the name, "sub-contrary", came to the Pythagoreans without the associated Babylonian methods of calculation. In any case, successive application of the arithmetic and harmonic means to the unity was the method of construction of Archytas' tuning system, the arithemetic mean being equal to [(a/1) + (c/1 )]/2 and the harmonic mean being equal to 2/[(1 /a) + (1/c)].

 

A third possible explanation is suggested by Heath.24 Nichomachus identifies the geometric proportionality of the first three means as known to Pythagoras. If arithmetic mean divided by geometric mean is equal to geometric mean divided by harmonic mean, then the harmonic mean is seemingly "sub-contrary" to the arithmetic.

 

Figure 7 shows some of the mean relationships among tones in Archytas' tuning system. From the figure, we can see yet another sense in which the harmonic mean may rightly have been called "sub-contrary" to the arithmetic mean.

 

 

Figure 7

 

Note first that C- is the arithmetic mean of A and D'. We can write the intervals in the following manner:

Figure 7.

 

 

               Figure 8


Now note that F is the harmonic mean of D and A; we can write the intervals:

In the above expressions. the numbers 8, 7, and 6 descend in order, (8 : 7, then 7 : 6), but the numbers 6, 5, and 4 descend `contrarily,' (as 5 : 4, and then 6 : 5). These 'orderly' and 'contrary' arrangements of numbers are peculiar to the arithmetic and harmonic means of epimoric intervals (intervals of the form n+ 1 : n). But, as epimoric intervals form the heart of Archytas' system, this might have been another, more primitive reason for calling the harmonic mean, "sub-contrary."

 

Eventually, Pappus, following Eratosthenes showed how to derive ten means from unity by means of geometric and arithmetic proportions, and justified his effort by mentioning Plato's philosophy that all things are derived from unity by proportion.25 But the most sophisticated use of means in musical tuning theory was that of Archytas and Plato.

 

 

The Pythagorean Origins of the Tuning System of Archytas

 

Archytas of Tarentum was apparently both a student in a direct line of teaching from Pythagoras, and at the same time a respected and highly inventive public leader, thinker, and inventor. Ptolemy called him the most important Pythagorean music theorist,26 and several scholars connect Plato's praise for mathematics throughout his later discourses with his two visits to Archytas.

 

In fact, Archytas' tuning system is apparently presented in detail with philosophical interpretation in the posthumous Platonic Epinomis, which is regarded as having been written (or finished) by a loyal student of Plato, after Plato's death. The relevant text is as follows:

 

In fact what is divine and marvelous for those who understand it and reflect upon it is this - that through that power which is constantly whirling about the duplication and through its opposite, according to the different proportions [i.e. through the different kinds of means], the pattern and type of all nature receives its mark.

 

[The text then speaks of successive duplications of unity, from 1 to 2, from 2 to 4, and finally from 4 to 8, which is called, "the spatial and tangible

 

Finally, [by] the [power which is the opposite] of duplication, which tends towards the middle, . . [in particular, by the arithmetic and harmonic means] , . . . in the middle of 6 to 12, the ratios 2 : 3 and 3 : 4 come together.

By starting from these, and moving from the center to both sides, [the power which is the opposite] of duplication, [that is, the means] presented to mankind melodious consonance and measured charm of play, rhythm and harmony, abandoned to the blessed dance of the muses.27

 

Plato calls the third doubling of unity, the 8, "spatial and tangible," in reference to the cube of 2. The arithmetic and harmonic means are defined quite explicitly, using the same definitions which Archytas gives them. (I have omitted the exact wording above for easier reading). Van der Waerden quite reasonably interprets this passage as referring to the construction of Archytas' tuning system.28

 

Figure 9 shows the construction of Archytas' system, beginning from the octave "duplication," DD'. Opposed to the outward-directed duplication (DD') is the inward-directed generation of the harmonic and arithmetic means, G and A. Finally, moving outward again from these central tones (G and A), the arithmetic and harmonic means of the intervals D/A, A/D', and D/G generate three more tones of Archytas' system, F, C-, and F-.29

 

 

 

Figure 9

 

 

The harmonic mean of G/D' generates a Bb which Erickson feels was a transposition tone used by Archytas in representing some of the actual scales of his time using his system. Archytas limited his presentation of his system to one model tetrachord for each genera - (diatonic, chromatic, and enharmonic) - but as Erickson also notes, this may be formalism, and the Bb could easily have been used in the scales representable only by transposition within the system. The process above of taking successive means in fact generates all the basic intervals, which for the Pythagoreans, were the epimoric (Latin, superparticular) ratios, that is, ratios of the form, (n+ 1) : n.

 

As Figure 10 (pg. 51, left) shows, all the epimoric ratios 1:2, 2:3, 3:4, 4:5, 5:6, 6:7, and 7:8, are present in the basic tone set. The same figure (right) shows that between three of these same basic tones, the epimores 9:8 and 10:9 have also been generated. In fact, the complete set of epimoric ratios present in Archytas' system amount to these same ratios 2:1 through 10:9, with the addition of 36:35 between F and F-, and C and C-, and 28:27 between E and F-, and between B and C-. It is the 28:27 interval which figures prominently in Ptolemy's presentation of Archytas' system - for Archytas uses it in his model tetrachords for each genera. But it is the 'hidden' ratios,  1:2, 2:3, 3:4, . . . 9:10 , according to the testimony of the Epinomis,

 

       through that power . . . [of] duplication and through its opposite, according to the different (means), the pattern and type of all nature receives its mark. which formed the philosophical heart of the system. In fact, to the Pythagoreans, the tetractys (encompassing the first ten numbers) did underlie, "the pattern and type of all nature."


Figure 10: Intervals Generated by epimoric Ratios

 

 

 Now recall the famous acusma of Pythagoras, to the effect:

 

Q: What is the Oracle of Delphi?

A: The tetractys - the harrnonia the Sirens sing!30

 

Burkert interprets this to mean that the numbers 1, 2, 3, and 4 generate the basic musical ratios 2/1 (the octave), 2/2 (the fifth), and 4/3 (the fourth).31 But by the tetractys, the Pythagoreans meant the generation of the numbers 1 through 10, out of the numbers 1, 2, 3, and 432 As these are the numbers which are in musical (i.e. epimoric) ratio in Archytas' system, Archytas' system actually does "present to mankind melodious consonance and measured charm of play, rhythm, and harmony", in" the form of the epimoric ratios of the tetractys. Archytas' system is, then, in effect a practical working out in musical terms of the original acusma of Pythagoras! The philosophical base of Archytas' system is confirmed by the Epinomis, which, as we have seen, calls it the music of the "blessed dance of the Muses!" The Muses' musike - which encompasses rhythm, melody, and dance simultaneously, is thus ordered by the tetractys through the opposing 'powers' of doubling and taking means.

 

Now, Archytas is credited with a proof of the impossibility of taking the geometric mean of 1 : 2 or of any other epimoric ratio in terms of a ratio of whole numbers.33 If, in fact, a teaching of Pythagoras expanding the meaning of the above acusma was known to Archytas, a teaching which contained the essence, at least, of the material presented in the quote from the Epinomis, then the key point would be how to take the mean - i.e. just which mean was meant by, "the opposite power of duplication"?

 

This would in turn give Archytas a reason to show in detail that it could not be the geometric, but the arithmetic and "sub-contrary" (i.e. harmonic) means which generate the epimore ratios of the tetractys, to which the acusma tic teaching refers. And this in turn would be ample justification for changing the name of the 'sub-contrary' mean to the 'harmonic' mean.

 

Archytas in fact attributes this change of name to Hippasus, an early Pythagorean and possibly a student of Pythagoras himself. That Archytas specifically gives credit to Hippasus for this re-naming shows that he was referring to an already developed theory which we might reasonably imagine to have come at least in part from Pythagoras. The power and verve of the above acusma does not strike me as a fabrication of a loyal disciple, but as rather the spontaneous jesting remark of a confident teacher. Of course, it is not possible to prove how detailed was the knowledge which Pythagoras may have given Hippasus.

 

Archytas' system, then, appears as the invention of a practical man who was interested in translating the fascinating Pythagorean mystical teaching into understandable and practical musical form.

 

Scholars have often assumed that Pythagoras gave only a more "primitive" form of the Platonic numerology. But in view of the high state of the mathematics of the Babylonian tablets, we may perhaps better imagine Pythagoras as having selected from Babylonian wisdom a teaching which fit his own spiritual purposes. How much numerological detail he may have communicated directly to his disciples cannot be judged. On the evidence of the acusma, it is not impossible that the method of construction of the system came from Pythagoras himself, though we do not know this.

 

Such a practical interest is in character with some of Archytas' other reputed achievements, as inventor, and as discoverer of a geometric method of 'doubling the cube' - a problem in fact equivalent to finding two geometric means in the octave! However, this geometric solution had, for Archytas, nothing to do with music or musical tuning. The acusma tells us that Pythagoras taught that music is generated by the tetractyc relationships among whole numbers. The physical geometry of string or pipe lengths could only be seen as a passive correlate, but not an active principle of music.

 

Van der Waerden notes a characteristic fuzzy logic in those propositions of Euclid which he suspects came from Archytas.34 Ptolemy's evident perplexity over Archytas' system is in keeping with this supposed mental style of Archytas, for whatever text of Archytas Ptolemy was working from contained what for Ptolemy seems to have been a puzzling statement, to the effect that between the F# of Archytas' chromatic, and the G of his diatonic scales, there occurs the ratio 256 : 243.

 

Ptolemy finds fault with this ratio for not being an epimore - but evidently he doesn't understand which epimores are the ones which define Archytas' system, (i.e. the epimores composed of the numbers one through ten, generated by the tetractys, and not primarily those which appear between neighboring notes of his scales.)

 

 

The  Completion of Archytas'  System

 

We have seen how Archytas generated his basic set of tones. But how did he complete his tone-system? The answer is simple: he used the scale of whole tones (9 : 8) and limmas35, constructed from string of consecutive fourths and fifths. This method of scale construction was known in Babylonia since at least 1800 B.C., and today is called the "Pythagorean" diatonic scale, after Boethius' description.36 The simple method of actually tuning this scale is explained in detail on tablets dating from 1800 B.C.37 In 150 A.D. Ptolemy referred to the practical use of this tuning by musicians. It is also the scale generated by Plato's explanation of creation and cosmic order in the earlier dialogue Timaeus.38

 

 

Plato: The Scale of the Timaeus

 

In the Timaeus passage, Plato constructed so-called "Pythagorean" tetra-chords:

Figure 11. The Pythagorean Tetrachord and this is evidently just what Archytas did also, starting from his first tone, D, and going up two 9 : 8 tones to E and his chromatic F#, which then stands in 243 : 256 ratio to his diatonic G, which was the tone already produced as the harmonic mean of D and D'.

 

 

 

Figure 11

 

Figure 12 below shows Archytas' system in modern notation as a tone-lattice. The seven tones related by successive fifths are shown connected to each other left to right. Relations involving the factor 5 are shown under the line of fifths (using double lines), and relations involving the prime factor 7 are shown above the line of fifths (using wiggly lines). Thus all adjacent tones are connected by epimoric ratios 3:2, 4:3, 5:4, 6:5, 7:6, or 8:7. Note the compactness and simplicity of the whole system, shown here in terms of its prime factor relations.


Figure 12

 

 

WHY 28 : 27?

 

It is now possible to suggest why Archytas used the 28:27 ratio in all three genera. This is the unique occurrence of 28 : 27 in Greek tuning literature. For most writers, the pyknon (the sum of the lowest two intervals of the scale), is of different size in each genus, therefore the lowest step (subdividing the pyknon) should also be of various sizes (depending on the size of the pyknon).

 

Archytas, however, uses F-/E (28/27) exclusively. We have seen how Archytas generated the F and F- in his system, using means. It would seem possible for Archytas to have used for the diatonic genus the tone F, giving the following tetrachord:

            Figure 13

 

However, the F is generated from the octave DD' by taking first the arithmetic and then the harmonic mean, whereas the F- is generated by first taking the harmonic mean, and then the arithmetic. Perhaps Archytas chose his diatonic genus as the following:

 

Figure 14

 

because of the prominence of the "harmonic" mean in the generation of the A (harmonic mean of E'/E), the G (harmonic mean of D'/D), and the F- (the mean of the harmonic means G and D).

 

It is also possible that Archytas wished to relate his system to the system of the unknown "Harmonists", aulos scale theorists whom Aristoxenus criticizes in his Harmonics.39 Aristoxenus mentions their "close-packed diagrams" of twenty-eight tones. Schlesinger believed that the system of the "Harmonists" was a well-developed tuning system based not on ratios but on equal divisions of the aulos by its finger-holes.40  I have not the space to review her evidence here, but if indeed such a system was as widely known as was the aulos itself, then Archytas might have wished to relate his system to that of the"Harmonists" twenty-eight toms. Of course this possibility rests on a string of "ifs!" But, as we have seen, Archytas seems to have intended his system to relate to both the practical musical world as well as to the philosophy of Pythagoras, so the idea remains an intriguing possibility.

 

In the Timaeus, Plato gives a numerical construction which he applies to cosmology rather than explicitly to music. However, in the ancient world these subjects were highly interrelated; in building his cosmology, Plato uses musical terminology, and in fact seems to construct a "Pythagorean" diatonic scale. He first successively doubles the unity three times to obtain the numbers 1, 2, 4, and 8. Then he successively triples the unity, giving the numbers 1, 3, 9, and 27. He is using the principle of the primal tetrad, elaborated through a geometric series based on 2 and 3, the only two prime numbers among 1, 2, 3, and 4. The numbers 2 and 3 were thought of as the first two numbers (not unity), the primal Even-Odd pair, which produce all duality (2), and dialectic (3). (Plato notes that in order to synthesize the three elements of the dialectic into one whole, "force" is required,41 which makes the number 3 the representative of primal force).

 

Evidently, Plato then arranges these numbers in a triangular form, and takes what are in effect geometric means,-6, 1 2, and 18 - between them (see Figure 15) where 2 : 3 = 4 : 6 = 6 : 9 = 8 : 12  -  12 : 18 = 18 : 27. (Note that these 3 : 2 ratios can be thought of musically as perfect fifths or fourths).

 

 

 


Figure 15

 

Plato then mentions that, "there are two means in every interval,"42 that is, both the harmonic and arithmetic means of 6 : 12 and of 6 : 18 are already present in the number system. For the arithmetic mean of 6 and 12 is 9, and of 6 and 18 is 12, while the harmonic mean of 6 and 12 is 8, and of 6 and 18 is 9.43

 

He then speaks of filling in the 4:3 ratios with 9:8 (tones) and limmas, which he says correctly, "in terms of number, subsist between 256 and 243," i.e. the limma = 256:243. In order to see what Plato may have been describing in terms of whole numbers, we must first multiply all the tetractys of numbers by 64, then fill in the 4:3 intervals with two whole tones. The limmas 512:486 and 768:729 are two and three times the limma in lowest terms, 256:243. The numbers 384 through 768 define a diatonic scale in lowest terms44 (see Figure 16).

 

Note that the modern prejudice which equates "ancient thought" with "primitive thinking", especially tends to color our attitudes towards ancient abstract thought, (although in abstract thought, human beings seem forever to remain beginners!). For example, Plato did emphasize the simple ratios 6:8:9:12 above all others in his references to music. But another indication of the true level of complexity of his numerical philosophy can be glimpsed in the following fragment from the Republic. Plato says:

 

The root 4:3 mated with the 5 thrice increased produces two harmonies. One of them is equal an equal number of times, taken one hundred times over. The other is of equal length in one way but is an oblong: on one side, of one hundred rational diameters of the 5, lacking one for each, or if of irrational diameters, lacking two for each; on the other side, of one hundred cubes of the three. This whole geometrical number is sovereign of better and worse begettings.45

 

 

 

Figure 16

James Adams has succeeded at least in reconstructing the numbers Plato refers to.46 The least common denominator of (4x5)4 and (3)4 can be calculated as (360)2 x 1 00, or ((60)292 or as (62 x 100)2 which equals 12,960,000.

 

Plato's meaning in calling this number, "sovereign of better and worse begettings", is more obscure. But certainly the passage does show that Plato had ability to deal with complex as well as simple numerical relationships, along with their philosophical corollary meanings. This passage relates music, geometry, arithmetic and metaphysics.47 Though its meaning seems unclear to us now, it was not Plato's style to speak vaguely. He surely had an exact meaning in mind, and assumed that at least some of his readers would understand it.

 

 

Conclusion

 

Plato may have restricted his written dialogues to discussion of simple musical ratios - but this is evidently, from the above example, no proof of how intricate his mathematical tuning theory may actually have been. Plato's use of small number ratios could therefore well have been only an appeal to first principles suitable for a widely read dialogue, and not the limit of his arithmetic or musical philosophy.

 

The Pythagorean esoteric teaching which Plato exposed as a combination of mysticism and philosophy eventually became the justifying basis for systematic mathematics. But the teaching was almost surely older than Pythagoras, and most probably from Babylonia, where it may have been integrated with the origins of the arithmetic system, and, one might imagine, into the whole of mental life. This is not to say that number mysticism was necessarily "common" knowledge in Babylonia, but that it may have been a tradition which referred to the number system and cultural knowledge of the Babylonians more than to that of the Greeks.

 

The system of Archytas is the combination of the systems of Plato's Timaeus and of the Epinomis. It may not be possible to trace the source of these interrelated conceptions, but Archytas seems to have been nearer to the Pythagorean source, which the acusma suggests played a determinative role.

 

At the heart of both systems are the tetractys and the means - the arithmetic and harmonic. In the Epinomis, the means are taken twice in succession, starting from the octave 2 : 1. In the Republic, the three-fold duplication (1 to 2 to 4 to 8) is accompanied by a three-fold triplication, (1 to 3 to 9 to 27), and the means are then taken only once - but between the geometric means of the two series! Finally, both systems are filled out by compounding by the ratio composed of the first two means (9 : 8), our most common musical whole step.

 

The Epinomis construction highlights epimoric ratios, while the system of the Republic highlights geometric construction through successive multiplications and divisions. The scale of the Republic was not discovered by Pythagoras. Recently deciphered Babylonian tablets, inscribed over a millenium before Pythagoras, give a practical method for tuning the scale. It seems probable that in the Republic, Plato was deriving an ancient scale, using numerological concepts which may have developed from Babylonian philosophy as transmitted by Pythagoras. It is possible that Plato learned of Archytas' system only after the Republic was completed following his visits to Archytas. Archytas' aim in building his system may have been, as Erickson has proposed, to provide a Pythagorean derivation of the scales actually in use at the time, a system which would also be compatible with the 28-tone diagrams of Aristoxenus' unknown "Harmonists."

 

Not knowing this for certain, we can, however at least conclude that both systems show a wonderful and interrelated economy of means(!), and speculate that the essential methods of construction of Archytas' tuning system, a system which we find interpreted as metaphysical philosophy in the Platonic Epinomis, came from a Pythagorean teaching which is condensed in the acusma:

 

Q: What is the Oracle of Delphi?

A: The tetractys - the harmonia the Sirens sing!


Bibliography

 

Barbera, Andre, "Arithmetic and Geometric Divisions of the Tetrachord," Journal of Music Theory 21,2(1977):294-323

Burkert, Walter, Lore and Science in Ancient Pythagoreanism, trans.

 Edward Minar, Jr., (Cambridge: Harvard University Press, 1972)
 Crocker, Richard, "Pythagorean Mathematics and Music," Journal of Aesthetics and Art Criticism, XXII (1963),189-198,325-333

Erickson, Robert, "The Musical System of Archytas," unpublished paper, 1965

Hamilton, Edith, The Greek Way (N.Y.: W.W. Norton, 1930,1942)

Heath, Sir Thomas, (1921), A History of Greek Mathematics, Vol. 1, (Oxford: Clarendon Press, 1921)

Heath, Sir Thomas, (1931), A Manual of Greek Mathematics, (Oxford: Clarendon Press, 1921)

Lasserre, Francois, The Birth of Mathematics in the Age of Plato (Larchmont, N.Y.: American Research Council, 1969)

Lippman, Edward A., Musical Thought in Ancient Greece, (N.Y.: Columbia University Press, 1964)

McClain, Ernest G., The Myth of Invariance, (N.Y.: Nicolas Hayes, 1976) Macran, Henry Stewart, The Harmonics of Aristoxenus, trans. Macran, (N.Y.: Georg Olms, 1974)

Nichomachus of Gerusa, Introduction to Arithmetic, trans. Martin D'Ooge, in Great Books of the Western World, Robert Hutchins, ed., Vol. 11, pp. 811-848 (Chicago: Encyclopedia Brittanica, 1952)

Philip, James A., Pythagoras and Early Pythagoreanism, (Toronto: University of Toronto, 1966), Phoenix Suppl. Vol. III

Plato, The Timaeus, trans. Thomas Taylor, (N.Y.: Pantheon, 1944), Bollingen Series,

Rogers, Michael, "The Golden Proportion in Musical Time: Speculations on Temporal Proportion", Ph.D dissertation, The University of Iowa, 1977 Schlesinger, Kathleen, The Greek Aulos, (London: Methuen, 1923)

Van der Waerden, B.L., Science Awakening, 3rd ed., trans. Arnold Dresden, (Groningen: Wolters Noordhoff,

 

 

See B.L. Van der Waerden, Science Awakening, trans. Arnold Dresden, 3rd ed.

  2 Book VII, sec. 525; in Jowett, The Dialogues of Plato, Clarendon Press, 1953, vol. 2. Quoted in Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972, p.44

  3 Lasserre, Francois, The Birth of Mathematics in the Age of Plato. (Larchmont, N.Y.: American Research Council, 1964)

  4 Rogers, Michael, "The Golden Section in Musical Time: Speculations in Temporal Proportion", Ph.D dissertation, University of Iowa, 1977, p. 13.

  5 Van der Waerden, p. 101.

  6 Van der Waerden, p. 164.

  7 See, for example, Lippman, Musical Thought in Ancient Greece, p. 11.

  8 Nichomachus, Introduction to Arithmetic, Bk. 2, XXVIII.

  9 However, Heath (1931), p. 86, notes that they were also attributed to Archytas himself.

10 Quoted in Lasserre, pp. 26-27. Taylor translates this as, ". . having regard to area-numbers," and interprets this as referring to irrational numbers whose products are rational". Taylor, p.243

11 Van der Waerden, p. 155.

12 J.A. Philip, Pythagorus and Early Pythagoreanism, (University of Toronto Press, 1966).

13 Nichomachus, Bk. 2, XXVIII, 3-4.

14 See Philip, p. 202-3.

15 Schlesinger points this out. See Macran, p. 190.

16 Nicomachus, p. 812.

17 Van der Waerden, pp. 40-41.

18 See Heath, pp. 52-53. 19Van der Waerden, p. 46.

19 Van der Waerden, p. 46.

20 See Heath, (1931), pp. 52-53. 21 See Van der Waerden, p. 49.

21 Van der Waerden, p. 49.

 22 Van der Waerden notes that the denominators of fractions were written under the numerators in some Greek texts dating back at least to the third century B.C. I do not know that Archytas actually wrote them this way, but these formulas follow Archytas' verbal definitions.

 23 Van der Waerden, pp. 37ff.

 24 Heath, (1921), p. 87.

 25 Van der Waerden, p. 232.

 26 Van der Waerden, p. 149.

 27 as interpreted by Becker; quoted by Van der Waerden, pp. 156-157.

 28 Van der Waerden, p. 157. The essential structure of his system is also pointed out by Erickson (1965).

 29 I am using Erickson's (1965) terminology.

 30 Burkert, pp. 170, 187, and 351 esp. Burkert cannot trace the meaning of the reference here to the Sirens. Later Pythagoreans said that the Muses sing the cosmic harmony, while the Sirens have to do with more earthly music. However the Sirens are also associated with the 'music of the spheres.' In any case, the reference to the oracle of Delphi confirms that the tetractys is to be associated here with a high spiritual plane.

 31Though he notes Eudemus reported that the Pythagoreans called nine (not ten) the musical number, because 2+ 3+ 4 = 9. (Burkert, p. 400).

 32 As emphasized by Crocker, "Pythagorean Mathematics and Music," The Journal of Aesthetics and Art Criticism XXII (1963), pp. 189-198.

 33 By Boethius. See Calvin Bower, "Boethius' The Principles of Music: an Introduction, Translation and Commentary," unpublished Ph.D. thesis, George Peabody College for Teachers, 1967. George Peabody College for Teachers, 1967.

 34 Van der Waerden, p. 155.

 35 The limma is the difference between two 9 : 8 tones and the 4 : 3 perfect fourth. Numerically, the limma is 256 : 243.

 36 Bower, p.198.

 37Kilmer, Crocker and Brown: "Sounds from Silence: Recent Discoveries in Ancient Near Eastern Music." Bit Enki Publications, Berkeley, 1976. Booklet and long playing record.

 38 See Timaeus, p. 123.

 39 Macran, p. 190. Nichomachus also refers to an Egyptian conception of the '28-sounding' universe.

 40 Kathleen Schlesinger, The Greek Aulos, (London, Methuen, 1923).

 41 Plato, Timaeus, p. 123.

 42 Timaeus, p. 123.

 43 Plato eventually uses these numbers to explain the movements of the planets. Taylor notes that the mean distance between the planets and the sun is actually very close to 2 : 3 : 5 : 8 : (14) : 27! Of course Plato does not use a heliocentric system, and would have had no access to observations on this order, in any case.

 44This interpretation of Plato's text is in accord with Thimus, Cornford, and MacClain. For an alternate interpretation, see Jacques Handschin, "The 'Timaeus' Scale," Musica Disciplina, Vol. IV, fasc. 1 (1950), p. 3-42. American Institute of Musicology, Rome.

 45 Plato, Republic, trans. Allen Bloom, p.545e, quoted in McClain, p. 164. For reference to other interpretations, see Heath (1921), pp. 305-308.

 46 see McClain, p. 164.

 47Although "harmonies" here may have been meant in an extra-musical sense.