(Prelude to the Song itself)
By Ernest G. McClain.
New York: Nicolas Hays, 1978. x, 192 pp., 57 charts and tables.
Review by Siegmund Levarie in The Musical Quarterly Vol. 64, No. 3 (Jul., 1978), pp. 402-407
There is something very special about this book as regards both method and contents. Scholars in various fields - mathematics, philology, political science, education, philosophy, music - can profit from it significantly and in due time are likely to recognize it as essential to further thought in their respective disciplines.
The area is defined, but in no way constricted, by Plato's late dialogues. In dealing ostensibly with moral education and behavior, Plato writes in analogies and metaphors whenever they serve to convey abstract ideas in a concrete way. While thus utilizing images of all kinds (one remembers the dramatic cave), Plato repeatedly declares his preference for musical similes. Music provides him with a perfect model. It operates with both natural forces (the immutable interval relationships) and human artifacts (the conditioned scale selections). It involves the senses (the direct experience of sound) but also the mind (the empirical investigation of proportion). It is a quality which nonetheless can be approached mathematically. It is one field of human behavior in which man has succeeded, for better or for worse, in setting up and enjoying an artificial order within a potentially unlimited chaos.
As a consequence, Plato's dialogues abound in musical examples and their corresponding mathematical equations which have driven philologists and philosophers of the last two millennia to open despair. What is one to do with Plato's statement that "the tyrant's life is 729 times more painful" than that of the good man (Republic 587e)?1 Or that "the root four-three mated with the five, thrice increased, produces two harmonies sovereign of better and worse begettings" (Republic 546cd)?
Plato scholars have coped with such issues in various ways, none of them satisfactory to either them or their readers. The most radical solution was found by the philologist Francis Cornford who in his famous English translation earlier this century "simplified the text" by omitting such troublesome numbers. Others, like A. E. Taylor, have valiantly struggled with them and occasionally solved an equation with out having any strength left for a meaningful explanation of the solution. A few exceptional minds, like those of Albert von Thimus in the nineteenth century and Robert Brumbaugh in the twentieth, have come very close to understanding Plato's intentions. Both sensed the musical implications of all of Plato's difficult numbers but, not being professionally trained musicians, stopped short of a full exploration of the issues.
Ernest G. McClain, a music professor at the City University of New York, has succeeded where angels either feared to tread or tripped. His method is simple enough: he started by taking Plato literally. He believed Plato's statements concerning the supremacy of music: "Education in music is most sovereign" (Republic 401d). "Argument mixed with music alone, when it is present, dwells within the one possessing it as a savior of virtue throughout life" (Republic 549b). "Moderation" will stretch through the whole city as it does "from top to bottom of the en tire scale, making the weaker, the stronger, and those in the middle ... sing the same chant together" (Republic 432a). "Our songs have be come laws" (Laws 799d).
1 The musical reader of this review knows, or is reminded, that 729 is the tritone number, 93. He also rightly suspects that McClain's book contains the answers to all such questions.
All these statements, and many more, are intimately connected with the famous inscription above the door leading to Plato's Academy which barred the entrance to anyone not trained in mathematics (ageometretos). In modern jargon, music and mathematics are prerequisites for an under standing of Plato's thoughts.
Greek mathematics poses a problem of its own, but it is far less weighty than that of Greek music. The reader who learns a few basic facts will quickly lose his apprehension of Plato's ample wordage and huge numbers. First, Plato transcribes all mathematical formulas into literary sentences. Thus he defines a number as being "of equal length in one way but an oblong: on one side, of one hundred rational diameters of the five, lacking one for each; or, if of irrational diameters, lacking two for each; on the other side, of one hundred cubes of the three" (Republic 546c). James Adams correctly disentangled this wording to read 4800 x 2700, a modern formula all of us readily understand.·To appreciate this particular difficulty of reading Plato, a twentieth-century reader need only try to express in words any algebraic formula with which he is familiar.
Second, in line with his time,Plato avoids fractions. Hence any number may also stand for its inverse: the concept of 3, for example, always contains that of 1 / 3. This kind of inversion bothers modern mathematicians more than it does musicians who readilydefine pitch of any tone by either frequency or equally well by its inverse, wavelength.
Third, the avoidance of fractions generates large numbers when even simple ratios have to be adjusted to a common denominator. While we customarily express the ratios of the fixed tetrachordal tones within the octave, as 1 : 3/4 : 2/3 : 2, Plato necessarily reads the same proportion as 6:8:9:12, or 12:9:8:6; and both rows correspond to the same pitch relations. In the definition of a chromatic scale of eleven tones, the octave frame has "waxed" to 360:720 - different numbers, just toavoid fractions, for a constant musical interval. The octave that accommodates integer tones (a figure Plato experiments with in the Republic 546a-d and throughout Critias) can be no smaller than 6,480,000 : 12,960,000.
McClain's method, in short, consists in fully respecting Plato's in sistence on musical interpretation of numbers, and in accepting and translating the somewhat clumsy Greek mathematical formulations. 2
2 An earlier book by McClain, The Myth of Imvariance (New York, 1976) demonstrates the age-old tradition of such an approach. Plato appears as a late link in a chain stretching to him from India across Babylon, Palestine, and Egypt.
The results, all of them intelligible to thinking musicians, are highly persuasive. The contents emerge as various attempts at coordinating disparate elements to function in an orderly manner within a system. Plato's primary concern is politics, but he pursues the problem on three levels: astronomy, music, and the res publica. "In heaven," Socrates says, "a pattern is laid up for the man who wants to see and find a city within himself" (Republic 592b). What can we learn from the multitude of stars of which the movements are incompatible with each other in relation to a fixed observer and which yet function in an apparently orderly manner? What has man done as an imitator of the Supreme Artist to build musical systems all of which have an inevitable internal conflict between the "perfect" octave and the powers of any other interval, and which nevertheless permit him to proceed in well-regulated compositions? And what can man do to set up a political state in which the many mutually irreconcilable members may "get along" with each other and flourish in spite of inescapable frictions?
The answer on all three levels is the need for compromise. All calendars at all times are compromise approximations of apparently irrational heavenly forces. In music, the compromise is called temperament. Every musical system of more than three tones, in all places and at all times, whether practically applied or theoretically conceived, is subject to some kind of temperament. Proof for this generalization is simple: any musical division of the octave corresponds to a root of 2, and any such root will be irrational. Whatever intervals we pile up, we can never reach the octave. We can only approximate it. There will be some left over that will cause a disturbance unless "attuned" to the prevailing system by a compromise. The particular form of the compromise defines the particular temperament. Many temperaments are theoretically possible. Each is characterized by the preservation of some principle at the expense of others. The system depends on the choice one has made, on style.
Plato's late dialogues form a continuity extending from the Republic through Timaeus and Critias to Laws. In the face of Plato scholars who generally hesitate to elucidate one dialogue by another, McClain has had the courage to trace the musical connections. The political connections are obvious, for all these dialogues address themselves to the possibilities of the good life under different forms of government. The Republic postulates two different city-states, one of them trying to function on an "indispensable minimum" (369d), and the other "luxurious" and surfeited. In Timaeus and Critias, these two cities are subsequently identified as, respectively, good Old Athens and doomed Atlantis. In Laws, finally, his last work, Plato describes a government so different from the earlier two that some scholars have doubted the common authorship. The detailed description of all three governments is full of musical allusions to number sets.
McClain has had the ingenious idea of interpreting Plato's three different relevant number sets as different kinds of musical temperament. His solution is self-consistent, considerate of every detail, and completely persuasive. Everything falls into place, apparently for the first time since early medieval theorists, separated from classical Athens by many generations and misunderstandings, began to worry a text that had originally been very clear to Plato's friends and disciples.
McClain's analysis could not be more specific. He supports every step in his logical chain by direct references to Plato's own words. Ancient Athens emerges in analogy to a tuning system based on the number 3, that is, Pythagorean tuning in perfect fifths. For the surfeited city Atlantis, doomed to destruction because of internal frictions, Plato builds a tuning system based on the numbers 3 and 5, or what was later called Just tuning, with perfect major triads on the fixed tetrachordal tones of the octave scale. In Laws, finally, Plato, under the influence of his friend Archytas, experiments with the introduction into a music system of the number 7. This city he calls Magnesia, and life in it seems particularly dreary to us.
Although McClain does not say so, I consider the identification of Pythagorean tuning with Ancient Athens, and of Just tuning with sinful Atlantis, revealing and appropriate. Tuning in pure fifths sufficed for the monodic style of Greek classical music. It was the "indispensable minimum" for adequate functioning. By comparison, a temperament based on pure major triads could properly be condemned as "luxurious" and "superfluous" when viewed against the prevalent musical style of Plato's contemporaries. Just tuning found its deserved recognition about fifty generations later when the triad as a unit became the style criterion of the late Renaissance. By that time, Pythagorean tuning had necessarily been abandoned together with monodic composition.
McClain's most startling but (to me) irrefutable insight concerns Plato's "heavenly" speculation with equal temperament. The Republic culminates in a famous dream, the journey to heaven of a man named Er. The complex mathematics determining stages of the journey are as explicitly defined by Plato as the modern concepts of frequency or wave length might be by a precise physicist; and McClain's book elucidates the subtlest relationships and meanings. By properly interpreting the specific revolutions of the eight heavenly circles and the adjustments of these revolutions by the Daughters of Necessity, "seated round about at equal distances," McClain has here discovered the first theoretic exploration of equal temperament in history, necessarily bound up with that of irrationals.
The culmination of the Republic in a dream of equal temperament is particularly apt when understood in relation to the central concern of the dialogue, namely, justice. Earlier in the dialogue, Socrates forcefully rejects the suggestion "that it is just to give to each what is owed." Rather, true justice to him means doing "what is best for the city." This solution becomes possible only when every member of the group – be it a tone in a musical scale or a citizen in a political community – voluntarily abandons a small fraction of his natural rights. Equal temperament, unlike other tuning systems, precisely does not give to any interval within the octave "what is owed" but demands the small compromise sacrifice of each that "is best" for the whole system.
To follow Plato's and McClain's detailed reasoning, one must study their complete texts. The result will be satisfactory and stimulating, particularly to musicians, who will have a headstart on philologists, philosophers, and even mathematicians. McClain's accomplishment strikes me as a major intellectual breakthrough in man's perennial struggle with his own spiritual history. The purpose of this review is to alert the musical community among scholars to the data at hand. Other disciplines are likely to follow once the musical primacy of the explanations has been accepted.
(This review is an abbreviated version of a lecture to the Greater New York Chapter of the American Musicological Society on December 15, 1977).