peregrinations, 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.
     The number 3 is the creative power in music, producing fifths, but it is under the control of the Octave prime - the number 2. It is the supreme octave which forms a boundary by making twelve fifths and seven octaves unite in one note. Within this horizon lies the musical system in its threefoldness - major, minor, and chromatic.
     Twelve divisions in the Octave serve all the purposes of music. This is the master-stroke of Nature in putting so much into seemingly so little. Twelve is the most genial of all numbers; it is nature's representative of the social order in music. It is the manifold divisibility of twelve which makes the chromatic system in music possible. The equalized scale of twelve semitones in the octave and the chromatic system of music are indissolubly connected. With this the scale of mathematical intonation has neither part nor lot.
     The life-force of the notes from the law of position gives them a versatility which they could never have had from fixed ratios, however numerous. If the interval of the octave be excepted, there are no two notes together in a chord, nor succeeding each other in the octave scale, having the same amount of specific levity or gravity; consequently each note has an expression and [Scientific Basis and Build of Music, page 35]