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Nature has not finished when she has given us the Diatonic Scale of notes as first generated. In the diatonic scale, in ascending from C, the root of the tonic, the first step is an interval of 9 commas, supposing that we adopt the common division of the octave into 53 commas, which is the nearest practical measuring rule; the second step has 8 commas; the third has 5; the fourth has 9; the fifth has 8; the sixth has 9; and the seventh and last has 5 commas. So we have three steps of 9 commas, two steps of 8, and two of 5. The order of the steps in the major is 9, 8, 5, 9, 8, 9, 5. In the minor the magnitudes are the same, but the order is 9, 5, 8, 9, 5, 9, 8. So there are three magnitudes.¹ But Nature has an equalizing process in the course of her musical marshallings, in which these greater ones get cut down, and have to change places with the lesser, when her purpose requires them so to do.
     It should not be supposed that this division of the notes into semitones, as we call them, is something invented by man; it is only something observed by him. The cutting of the notes into twelve semitones is Nature's own doing. She guides us to it in passing from one scale to another as she builds them up. When we pass, for example, from the key of C to the key of G, Nature divides one of the intervals into two nearly equal parts. This operation we mark by putting a # to F. We do not put the # to F to make it sharp, but to show [Scientific Basis and Build of Music, page 47]

1 See Plate XIII. - The Vertebral Column.