the excess of the vibrations of the one note over the other makes one or more sounds which are called "grave harmonics;" *e.g.*, in the interval of the fifth, in the ratio of 2:3, the excess of 3 over 2 is 1, so the grave harmonic is an octave below the lowest of the two notes, that is, the ratio of 1:2. This reinforces the lowest note, 2, and gives it a solid effect. In this way the octave is incorporated into the fifth, and unity with variety is combined with the law of continuity at the very threshold of harmony. In 32 of the 42 intervals the grave harmonics are notes which belong to the natural scale. In the 10 remaining intervals which have not the exact number of vibrations found anywhere in the natural scale, 6 of them are from the number 7, thus - 7, 7, 7, 21, 21, 35; the remaining 4 are from 11, 13, 13, and 19.

At the first, in the laws of quantities and motions adjusting musical vibrations, there is one chord of the three notes, F, A, C, the root, middle, and top of the five notes which compose the true natural scale; this one chord can be reproduced a fifth higher, C, E, G, in the same mathematical form, taking the top of the first for the root of the second chord. In like manner this second can be reproduced another fifth higher, G, B, D, still in the same mathematical form, and so fit to be a member of the chord-scale of a key. But the law does not admit of another reproduction without interfering with the first chord, so that a fourth fifth produces no new effect; but the whole key is simply a fifth higher, *i.e.*, if the fourth fifth has been properly produced by multiplying the top of the third fifth by 3 and by 5, the generating primes in music. That this carries us into a new scale is seen in that the F is no longer the F♮ but F#, and the A is no longer A♮ but A,. But if we suppose the fourth fifth to be simply the old notes with their own vibration numbers, then D, F, A would not be a fifth belonging either to the major or the minor mode, but a fifth a *comma less*. The letters of it would read like the minor subdominant, D, F, A; but the intervals, as found in the upward development of the major genesis, instead of being, when expressed in commas, 9, 5, 8, 9, which is the minor subdominant, would be 8, 5, 9, 8, which is not a fifth of the musical system; these having always, whether major or minor, two 9's, one

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