IV
Like many of the ancient philosophers, Pythagoras (ca. 600 B. C.) is known only by his disciples and by their quotations from or commentaries on his teaching. Of these the most important are Archytas (400-365 B. C.) and the great mathematician Euclid (ca. 300 B. C), though there is some reason to suppose the part of Euclid’s work dealing with music to have been written by Cleonides (ca. 200 B. C.) and by the later Pythagorean Nichomachus (ca. 150 A. D.).
In Hierocles’ Commentaries on the ‘Symbols’ and ‘Golden Verses’ of Pythagoras, M. Dacier, the translator, amplifies the prefatory Life of Pythagoras found in Hierocles, and he recounts, as Gaudentius, Nichomachus, Macrobius, Boëtius, and others have recounted, the incident which drew the attention of the ancient founder of the great system of secret numbers to the numerical relations of Sound in Music. The quaint old story is as follows: ‘Pythagoras is honored with the Invention of Harmonical Measures; and ’tis related how it happened. They write, that one Day, after he had been meditating a long while on the Means of assisting the Hearing, as he had already found means of assisting the Sight, by the Rule, Compass, Astrolabe and other Instruments, and the Feeling, by the Balance and the Measures, he chanced to go by a Smith’s Shop, and heard several Hammers of different Sizes, beating Iron upon the Anvil. He was moved with the Justness of the Harmony, and going into the Shop, he examined the Hammers and their sound in regard to their Sizes; and, being returned home, he made an Instrument on the Wall of his Chamber, with Stakes that served for pegs and with strings of equal length, at the end of which he tied the different Weights, and by striking several of these strings at once he produced different Tones, and thereby learnt the Reasons of this different Harmony, and of the intervals that caused it.’
In general it may be pointed out that the Pythagorean system of harmonics was only incidental to philosophy. Thus Laloy, speaking of the musical system of Pythagoras, says: ‘One finds, amid their confused accounts and contradictory assertions, a body of rules and precepts which present a “Pythagoric life,” as there was an “Orphic life,” in which justice, order, friendship, abstinence, geometry, and music are an integral part ... even metempsychosis itself being merely the truth inherent in a number.’
The monochord, a single string stretched over a sliding bridge, was the basis of the acoustical experiments of Pythagoras. By shifting the position of the bridge he varied the pitch of this string. His great discovery, that which has rightly caused him to be regarded as the founder of a branch of acoustics, was that between the respective lengths of stretched strings which gave the three consonances of octave, fifth and fourth, there existed certain essentially simple relations, as follows: the octave was in the relation of a string of one half the length or double the length; in other words, the relation of 2/1; the fifth was in the relation of 3/2; and the fourth in the relation of 4/3. These intervals, apparently on account of the simplicity of their mathematical relationship, were henceforth regarded as consonant. All other intervals were dissonant, at any rate in theory. The essential difference between the mathematical theory of sound ratios as held by the Pythagoreans and that held in modern times lies in the conception of the Third. To the Greeks such an interval was entirely dissonant, not necessarily because it was displeasing to the ear, but because they either did not recognize its ratio as 4/5 or did not deem this ratio to fit in with the highly abstruse theology they had built up on other numerical ratios. The step of the Fifth was to the Pythagoreans not merely the fundamental, but also the only, basis for the determination of tone ratios, whereas to-day the Third and sometimes even the Seventh are taken into account.
As to the value that Pythagoras attached to these fundamentals, we may quote Hierocles: ‘Pythagoras,’ he says, ‘has a very particular Opinion concerning Musick, which nevertheless the Masters of that Science, after they have duly weigh’d it, will find Just and Reasonable. He condemned and rejected all judgment that was made of Musick by the ear: because he found the Sense of Hearing to be already so weaken’d and decay’d, that it was no longer able to judge aright. He would have Men therefore judge of it by the Understanding, and by the analogical and proportionable Harmony. This in my opinion was to show that the Beauty of Musick is independent of the Tune that strikes the Ear, and consists only in the Reason, in the confirmity and in the Proportion, of which the Understanding is the only Judge.’ And he adds this remark: ‘As to what he said, that the Sense of Hearing was become weak and impotent, it agrees with this other Assertion of his, that the reason why Men did not hear the Musick of the Universe was the weakness and imbecility of their Nature, which they had corrupted and suffered to degenerate.’
The error of the Pythagoreans, it may be pointed out, did not lie in the misuse of experimental data, but in the philosophical deductions therefrom. To the followers of Pythagoras a harmonic consonance was not a perception, it was a thing the existence of which could be conceived independently, a thing as real as the string which had given it birth. Sound was to them, therefore, a distinct identity, possessing attributes pertaining only to itself, yet susceptible of impression from without; it was a number realized and concrete, a number simple and all-inclusive, but, above all, a series of numbers possessing a personality, the veiling power of which both illumined and obscured a myriad symbolisms. Strict Harmonic Consonance was the utmost of numerical potency, it was a divine thought, not embodied Being. How deeply this was felt to be a truth by the Pythagoreans is evidenced by the story told of the death of Pythagoras, when the great philosopher, turning to his disciples, gave as his last instruction “Always the monochord!”
As for the value of the Pythagorean school as a whole, it is manifest that it must be considered as a group of mystical speculators, professing to be students of music and claiming Pythagoras as their master, but, in actual verity, doing little more than reducing sounds to air vibrations and ascertaining the numerical relations of pitch. Lovers of music they were not, they were mathematical precisians, perceiving no beauty and hearing no inspiration in melodic sounds except in such wise as these fitted into an ordered sequence of arithmetical form.
The development of the Pythagorean school was rendered all the more self-centred by the vitality and strength of the Empiricists. This flourishing school of musical art was concerned with arbitrary regulations as to the most acceptable forms of composition. The Empiricists determined what melodies were suitable to certain instruments. They debarred the flute from certain festivals and admitted it to others, they decided upon the forms of construction of musical instruments, and, above all, they insisted upon the performance of certain compositions in the traditional style. While not avowedly hostile to the Pythagoreans, the Empirical school paid little heed to the mathematical speculations of the learned, and song and dance continued because music was an art. Great as was the symbolic majesty of the Monochord, the surging strain of the lyre meant infinitely more to the life of Ancient Greece.
The second great development of Greek musical philosophy is that of Aristoxenus (b. 354 B. C.), whose systemization of the transposition scales has already been mentioned. If Pythagoras established some of the fundamental rules of acoustics, Aristoxenus may be given the credit of establishing Musical Science; the former was a branch of a science, the second was the science of an art.
To put the essential principles of Aristoxenus in the simplest possible form it may be said that he established two principal rules: (1) that music accepts Sound as sounds heard by the ear, and that the science of music must be built upon the foundation of sounds that are heard; (2) that sound-functions exist, possessing properties of sonance not directly reducible to any simple or elemental numerical ratio. The work of Aristoxenus was a revolution in musical philosophy based upon the principle of music as an organic whole of sounds bearing a dynamic relation each to the other. Aristotle had not been able to break away from the old Pythagorean conception, but Aristoxenus brushed away the misty speculations of morality, the mathematical entanglements and the musty formalism that surround the music of his time and set himself to answer the one vital question: Why does Music employ certain sounds and reject certain others?
The third stage of development of Greek music may be represented by Claudius Ptolemy, who lived in the second century of the Christian era. He may, with considerable authority, be deemed the inventor of the first interpreter of the equal tempered scale. R. C. Phillips has thrown considerable light upon the disputed questions involved in this matter, and to his monograph on the ‘Harmonic Tetrachords of Claudius Ptolemy’ (1904) we may refer the reader desirous of detailed information. Leaving the question of theory, we now proceed to pick up the thread of mythical story and trace what we can of the history of Greek composition.