"When Nature, who weighs the mountains in scales and the hills in a balance, and number and names the stars, takes hold of this vibration-material, she subjects it to the most rigorous measurement in marshalling that series of notes which constitute The Musical System. She is in no hurry in giving us all her gifts in this matter. She first of all gives us the interval of an Octave at one stroke, and in generating the seven notes which compose the three chords of the Diatonic Scale she carries the process up through six octaves..." [Scientific Basis and Build of Music, page 19]
"This definition of a vibration answers to all the requirements of the case; and in exhibiting the ratio 1:2 with two strings, it brings the two strings into the same position at every second vibration of the higher one. And so with every ratio of the musical system."[Scientific Basis and Build of Music, page 23]
she is found to have produced the Chromatic scale of twelve semitones, derived from her own vital operations; so that there are no anomalies. It is a degradation of the mathematical primes to apply them to the getting of the semitones of the chromatic scale, as even Euler himself mistakenly does. The mathematical ratios lead the way in getting the notes of the diatonic scale, and that is all that is required of them. The true praise to the ratios is that they have constituted an organic structure with form and life-powers adapted for self-development. It would be little credit to the mother if the child required to be all its life-long pinned to her apron-strings. As the bird when developed so far leaves the shell, and is afterwards fully developed in new conditions; so the System of music when developed so far leaves the law of ratios, its mathematical shell, and is afterwards fully developed by other laws. Music has an inspirational as well as a mathematical basis, and when mathematicians do not recognize this they reckon without their host. [Scientific Basis and Build of Music, page 35]
"Mathematicians would demonstrate the System of Music to little purpose if we did not find in ourselves an unlearned evidence of the fact . . . Mathematical demonstration is obliged at last to appeal to the Natural Ear." - The Olio [Scientific Basis and Build of Music, page 36]
Some of the elements of the Chromatic System were known 200 years ago. The Diatonic scale, being called the "Natural scale," implied that the chromatic chords were consider to be artificial; but the notes of the chromatic chords, from their PROXIMITY to the notes of the tonic chord, fit to them like hand and glove. Nothing in music is more sweetly natural and pleasingly effective than such resolutions; and hence their extensive use in the hands of the Masters. The chromatic chords have close relations to the whole system of music, making the progressions of its harmonies easy and delectable, and producing effects often enchanting and elevating, as well as often subtle and profound; and while they are ever at hand at the call of the Composer, they are ever in loyal obedience to the laws of their own structure and system. When a diatonic chord precedes another diatonic chord belonging to the same scale, it has one note moving in semitonic progression;1 but when a chromatic chord precedes a diatonic chord, it may have three semitonic progressions.2 The primary chromatic chord resolves into 8 of the 24 diatonic tonic chords, with 3 semitonic progressions. These identical notes of the chromatic chord, with only some changes of names, resolve into another 8 of the 24 tonic chords, with 2 semitonic progressions and one note in common; and when they resolve into the third and last 8 of the 24 tonic chords, they move with one semitonic progression and 2 notes in common. So to the chromatic chord there are no foreign keys.3 And as it is with the first chromatic chord, so with the other two. [Scientific Basis and Build of Music, page 51]
See Also