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.’