The middle octave will now be seen to be Phrygian (corresponding to No. 3 above) instead of Dorian as before. Now in their system of transposition scales (in reality transposed Complete Systems) the Greeks gave to every scale the name corresponding to the mode of its middle octave. Before the time of Aristoxenus only seven of these transposition scales, or keys ([Greek: tonoi]) were in use. That theoretician eventually rounded out the scheme to eighteen (of which six appear in modern notation as duplicates or octave transpositions). He did this systematically by taking the interval of the perfect fifth as a basis and building on each semi-tone degree a group of three scales (natural, hypo, and hyper). As there were not enough of the original modes to supply names for all of the new scales, it was, of course, necessary to invent arbitrary names for the superfluous ones. By this achievement it was possible to transpose a melody into any one of the eighteen (or really twelve) keys without changing its modal character. We may therefore assume with some justification that Aristoxenus’ system in a way did for the Greeks what our own equal temperament has done for modern music.
We end our brief sketch of Greek theory at this point, which may be assumed as the highest development of the system. Later systems were either based on Aristoxenus or were of reactionary nature. We must, however, for a moment retrace our steps to explain briefly the achievements of an earlier theoretician, the great philosopher Pythagoras, in the field of musical acoustics.
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!”