Either the one or the other must be at fault. Had the dictates of the mathematicians and the scale of mathematical intonation wholly ruled, the advent of the great masters would have been impossible. It was well said by one writing in The Choir - "Theory should be made from music, and not music from theory . . . the final judge of music is the Ear." The Great Masters are the exponent artists of what is true in the Science of Music, though it may differ from what has been taught by the merely mathematical-intonation advocates of music science. It should not be forgotten that the science of the mathematical theorists is one thing, and that of the composers is another. Schubert, Beethoven, Mozart, Haydin, Mendelssohn, and such inspired musicians, who walked in the liberty wherewith Nature made them free, are sufficient authority against the bondage of the one-law theorists who would tie us down to the mathematical command which comes from without, but who know nothing of the life within music which is the law unto itself.1
With twelve divisions in the Octave, each note is adapted to serve in any capacity, and does serve in every capacity by turns. It is quite clear that this cannot be said of the mathematically perfect notes. And this is where it is seen that what is perfect in mathematical ratios becomes imperfect in the Musical System. Indeed, the mathematical intonation does not give a boundary within which to constitute a System at all, but goes off into never-ending cycles.
In music, Nature begins by producing the Diatonic Octave of seven notes, derived by the mathematical ratios2; and when she has gone through all her
1 "Considerations such as those just alleged tend to show that while physical science is absolutely authoritative in all that relates to the constitution of musical sounds, and the smoothness of their combinations, the composer's direct perception of what is musically beautiful must mainly direct him in the employment of his materials. It would be a serious error to force upon him a number of rules planned on scientific principles to secure the maximum smoothness of effect; since mere smoothness is often a matter of extremely secondary importance compared with the grandeur of harmony and masterly movement of parts. The nature of the subject may sometimes call for a mode of treatment needing exceptional smoothness; in such a case the rules may become of considerable importance. It is well, therefore, that a composer should know and be able to handle them, but he should never allow them to fetter his freedom in wielding the higher and more spiritual weapons of his warfare." - Sedley Taylor.
2 See the Genesis of the Scale, Plate II.