a minor third. So by adding the middle of the minor dominant, G, but made G#, that the third so produced may be a minor third, according to the nature of the chromatic chords, we have on this minor side of the chord G#, B, D, F, which we may call its minor form, inasmuch as the semitone of its second minor third is the one, B-C, which genetically arises in the minor genesis; and inasmuch as it has also received its supplemental G# from the minor dominant. How shall we find its complement on the other side? We have seen that D, the Janus-faced center of this triad, B, D, F, looks, as D27, toward the major also; it has already F in common with the major subdominant. The very next step is to the middle of this chord, A. Middles, we have just seen, are ever ready to accommodate themselves; and this minor third triad claims that A be flattened, for on this side also, though its major side, it must have a minor third; so by adding the middle of the major subdominant, A, but made A♭, according to the nature of chromatic intervals, that this F-A♭ also may be a minor third; and now we have it as B, D, F, A♭, which we may call its major form, inasmuch as the semitone of its minor third, E-F, is the one which genetically arises in the major genesis, and inasmuch as it has now received its supplemental A♭ from the major subdominant. This, then, is the chromatic chord in its native place, and in its native constitution; a 4-note chord, wholly of the minor thirds. It will be observed that it has now, in its two forms, divided the octave into minor thirds - 4 minor thirds, so it is very much at home anywhere in the octave; indeed it is at home everywhere - G#, B, D, F, A♭. And as every diatonic common chord in music is constituted of materials found in the octave of notes, it cannot be far from a chromatic chord in some one of its forms. [Scientific Basis and Build of Music, page 55]
Moreover, it is only from one to five, that is from C to G in ascending, which is its proper direction in the genesis, that the major in being harmonized does not admit of minor chords, but if we descend this same natural major scale of the fifth from five to one, that is from G to C, the first chord is C E G; the next chord is F A C; if this is succeeded by the minor chord A C E, there are two notes in common and one semitonic progression, as very facile step in harmony; and the following two notes are most naturally harmonized as minor chords. So modulation into the minor, even in this major scale, is very easy in descending, which is the proper direction of the minor genesis.2 In a similar way, it is only from five to one, that is from E to A in descending, which is its proper genetic direction, that the minor in being harmonized does not admit of major chords; but if we ascend this same minor scale of the fifth from one to five, the first chord is A C E, the next is E G B, and if this chord be followed by the major C E G, there are here again two notes in common and one semitonic progression; and the two notes following are then most naturally harmonized as major chords. So modulation into the major, even in this minor scale, is very natural and easy in ascending, which is the proper direction of the major genesis.3 The dominant minor and the tonic major are, like the subdominant major and the tonic minor, very intimately related in having two notes in common and one semitonic progression. [Scientific Basis and Build of Music, page 65]
Now we come to a remarkable arrangement of Nature. The minor does not grow in the same way out of this third chord's top. Two features come before us: first the minor chord grows out of the major, but it is taken not from the top but from the middle, from a rib out of his side. B, the middle of the major dominant chord; B, the last-born of the major genesis; B is the point of departure in the outgrowth of the minor mode. The feminine is a lateral growth from the masculine. Another feature: it grows downward, like a drooping ash or willow. Its first generated chord is its dominant, and its last is its subdominant. Its middle chord, like the middle one of the major, is its tonic. Still further, it is generated by division, not multiplication; B45 is divided by 3 and by 5 for the root and middle of this highest chord, E and G. E15 is divided by 3 and 5 for the root and middle of the tonic chord, A and C. A5 is divided by 3 and 5 for the root and middle of the lowest chord, D and F. Thus we have the whole generation of the elements of music, six generations of harmony, like the six days of creation. Up to this point the whole process and aspect is inverse; growing from a middle; growing downward; growing by division;- while the major is growing from the top; growing upward; growing by multiplication. But here the inverse aspect ends. The generating primes of the major are 3 and 5; 3 and 5 are also the generating primes of the minor. In this essential phase of their creation their comparison is direct, not inverse. [Scientific Basis and Build of Music, page 67]
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 [Scientific Basis and Build of Music, page 77]
The scales march on following each other methodically, whether they be written with sharps or flats, and
"Not a step is out of tune, as the tides obey the moon."
The most natural, because the genetic, way to write the scales is to make the major scales all in sharps, after C, because the major genesis is upward in ratios ascending; and to make the minor scales all in flats, after A, because the minor genesis is downward in ratios descending. Let the young student, however, always keep in mind that the sharps and flats are simply marks to show how Nature, at whatever pitch we are taking the scales, is securely keeping them in the same form as when they are first generated; and in their birthplace no sharps or flats are needed. [Scientific Basis and Build of Music, page 90]