the major, the root of the major chord is the middle of the relative minor, and the middle of the major chord is the top of the relative minor; and as the note which has a semitonic progression downward to the top of the major has a semitonic progression upward to the root of the relative minor, so the same three notes which move in semitonic progression to the top, root, and middle of the major chord, likewise move by the semitonic progression to the root, top, and middle of the relative minor. In both cases the progressions are upward to the roots and downward to the tops; but in the major the movement is downward to the middle, while in the minor it is upward. So each one of these three of the four notes of the chromatic chord has two various movements.1
In the laws of quantities and motions the three primary ratios, 1:2, 1:3, 1:5, with the three different units, F1, C3, and G9, the roots of the chords of the subdominant, tonic, and dominant, produce the three chords of the musical system major, the one not interfering with the other; and by an inverse process are produced, from B720, E240, and A80, its generating notes, the three chords of the musical system minor; the one chord not interfering with the other. In a similar way the chromatic chords can be produced from three different units, without the one interfering with the other; and, like the subdominant, tonic, and dominant chords of the diatonic scale, they are fifths apart. So we may call them the subdominant, tonic, and dominant chromatic chords. Each of the three chromatic chords has also kinship with the major and minor modes, from the way in which the diatonic minor triad is constituted a chromatic chord by its supplement coming in the one side from the minor, and on the other side from the major system.
In going round the circuit of the common chords of the major and its relative minor, and beginning our circuit with the minor, we encounter a triplet,2 differing from all the rest in its constituents, thus -
Here we have passed through minor and major from D to D, and seem to have come to the point from which we started; but we know that these two D's are
1 See Plates XVII, and XVIII.
1 Triplet means in this usage simply three notes without regard to time. [Scientific Basis and Build of Music, page 53]
"There are three chromatic chords, and each of these three is related to eight particular tonic chords. When one the these chromatic chords goes to any one of its eight tonic chords, three of its notes move in semitonic progression, and the other note moves by the small tone, the ratio of 9:10. There is exception to this rule, whether the key be major or minor. But when the chromatic chord which should resolve to the tonic of C is followed by the subdominant, or the tonic of F (the example in Mr. Green's book), only two of its notes move in semitonic progress. Your friend describes the chord as if it had gone to the tonic of B; and what he said about it, and about D going to C, is what is supposed to be [Scientific Basis and Build of Music, page 94]
Here on the keyboard we see, nearest to the front, the great 3-times-3 chord of the full genesis of the scale from F1 to F64. When this chord is struck by the notched lath represented in front of the keyboard, the whole harmony of the key is heard at once. Behind this great chord are placed, to the left the subdominant, tonic, and dominant chords of the minor. D F A, A E C, E G B; and to the right the subdominant, tonic, and dominant chords of the major, F A C, C E G, G B D. When the notched lath is shifted from F to D, the minor third below F, and the 3-times-3 minor is struck down in the same way as the major, the whole harmony is heard; and the complete contrast of effect between major and minor harmony can be fully pronounced to the ear by this means. Behind these six diatonic chords, major and minor, on the part of the keyboard nearest to the black keys, are the three chromatic chords in their four-foldness, in both major and minor form. The center one shows the diatonic germ of the chromatic chord, with its supplement of G# on the one hand completing its minor form, and its supplement of A? on the other hand completing its major form. A great deal of teaching may be illustrated by this plate.