character of its own. And as Nature has constituted them, these various forces all converge to the Center of the Tonic Chord, and, with the exception of the interval of the octave itself, the notes of the tempered scale being a little nearer each other than the mathematically perfect notes, these converging forces and this tempering mutually assist each other, and give a greater decision to the resolution of chords.
The twelve semitones being the practical fulfillment of the ratios when the life-force of the notes is considered, the great masters had the ratios in their most workable form invested in their key-boards, and this along with the musical ear was the sure word of prophecy to them.1 In their great works they have thus been enabled to develop the science of music, and to express it in the language of its art.
Mathematicians have not recognized the life-power of the notes, and so they have misapplied their calculations, though these were perfect in themselves. Assuming the place of dictators, they say with an air of authority that, "strictly speaking, nothing could be more scientifically and musically untrue than the chromatic scale of twelve equal semitones as played on a tempered instrument; for in it, as in the diatonic scale, the same natural law prevails that no two tones of equal mathematical relations can melodically succeed each other." Saying that the same natural law prevails implies that they are reasoning from analogy; but in this assumption they are dictating to Nature. In a similar way they might assume that the interval of the octave, like the other intervals, should have a grave harmonic.2 But the fact that the octave interval has not a
1 "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
2 A grave harmonic is a secondary note which spontaneously arises when two different notes are sounded together. It is a note whose mathematical number is the difference of the two which awaken it; e.g., F2 and C3, the interval of the fifth, awakens a grave harmonic whose number is 1, which is the difference of 2 and 3; and this 1 strengthens F2, for it is its lower octave. But the interval of F2 and F4, or any other octave, does not awaken a grave harmonic, since there is no difference-number between the two. - Editor. [Scientific Basis and Build of Music, page 36]