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. [Scientific Basis and Build of Music, page 53]
common, to mingle with more chord-society. So those added thirds which constitute compound chords are like accomplishments acquired for this end, and they make such chords exceedingly interesting. The dominant assumes the root of the subdominant, and so becomes the dominant seventh that it may be affiliated with the subdominant chords. Inversely, the subdominant assumes the top of the dominant chord that it may be affiliated with the dominant. The major tonic may exceptionally be compounded with the top of the minor subdominant when it comes between that chord and its own dominant; and the minor tonic may in the same way assume the root of the major dominant when it comes between that chord and its subdominant. The minor subdominant D F A, and the major dominant G B D, are too great strangers to affiliate without some chord to introduce them; they seem to have one note in common, indeed, but we know that even these two D's are a comma apart, although one piano-key plays them both, and the F G and the A B are as foreign to each other as two seconds can be, each pair being 9 commas apart, and G A are 8 commas apart. In this case, as a matter of musical courtesy, the tonic chord comes in between; and when it is the minor subdominant that is to be introduced, the major tonic assumes the top of that chord, and then turns to its own major dominant and suavely gives the two to enter into fellowship; for the tonic received the minor subdominant through its semitonic E F, and carries it to the major dominant through its semitonic B C, along with C in common on the one side and G in common on the other. When it is the major dominant that is to be introduced to the minor subdominant the minor tonic fulfills the function, only the details are all reversed; it assumes the root of dominant, and by this note in common, and its A in common with its own subdominant, along with the semitonic second B C on the one hand and the semitonic E F on the other, all is made smooth and continuous. The whole of this mediatorial intervention on the part of the tonic is under the wondrous law of assimilation, which is the law of laws all through creation; but when the tonic chord has fulfilled this graceful action, it immediately drops the assumed note, and closes the cadence in its own simple form.1 [Scientific Basis and Build of Music, page 71]
In the first six chords of the scale the tonic is the first of each two. The tonic chord alternating with the other two produces an order of twos, as - tonic dominant, tonic subdominant, tonic subdominant. The first three notes of the octave scale are derived from the root, the top, and the middle of the tonic dominant and tonic; the second three are derived from the root, top, and middle of the subdominant, tonic, and subdominant. The roots, tops, and middles of the chords occurring as they do produce an order of threes, as - root, top, middle; root, top, middle. The first, third, fifth, and eighth of the scale are from the tonic chord; the second and seventh from the dominant; and the fourth and sixth from the subdominant. In the first two chords of the scale the tonic precedes the dominant; in the second two, the subdominant; and in the third two the tonic again precedes the subdominant; and as the top of the subdominant chord is the root of the tonic, and the top of the tonic the root of the dominant, this links these chords together by their roots and tops. The second chord has the top of the first, the third has the root of the second, the fourth has the root of the third, the fifth has the top of the fourth, and the sixth has the root of the fifth; and in this way these successive chords are woven together. The only place of the octave scale where there are two middles of chords beside each other is at the sixth and seventh. The seventh note of the octave scale is the middle of the dominant, and the sixth is the middle of the subdominant. These two chords, though both united to the tonic, which stands between them, are not united to each other by having a note in common, inasmuch as they stand at the extremities of the system; and since they must be enabled to succeed each other in musical progression, Nature has a beautiful way of giving them a note in common by which to do so - adding the root of the subdominant to the top of the dominant, or the top of the dominant to the root of the subdominant, and this gives natural origin to compound chords. The tonic chord, being the center one of the three chords, is connected with the other two, and may follow the dominant and sub- [Scientific Basis and Build of Music, page 97]
are always when they have returned to the side from which they were started. The Pendulographer, also, when writing the beautiful pictures which the musical ratios make when a pen is placed under the control of the pendulums, always finds his figure to begin again when the pendulums have finished their period, and have come for a fresh start to the side from which the period began. This confirms our author's definition of an oscillation of a pendulum. Fig. 3 is an illustration of the correct definition of a Musical Vibration, as also given in this work. Although the definition of an oscillation is not identical with that of a vibration, yet on account of their movement in the same ratios the one can be employed in illustration of the other as we have here done. Fig. 4 is a uniform rod suspended from the end as a pendulum; it will oscillate, of course, at a certain speed according to its length. In such a pendulum there are three centers related in an interesting way to the subject of Music in its three chords - subdominant, tonic, and dominant, which roots are F, C, and G. The center of gravity in the middle of the rod at 2, suspended at which the rod has no motion, corresponds to F, the root of the subdominant, in which there is the maximum of musical gravity. The center of oscillation at 3, which is one-third of the length of the rod from the end, is like the root of the tonic whose number is 3 in the genesis of the scale from F1. In this point of suspension the oscillations are the same as when suspended from the end at 1. The point at 9 is at a ninth from the center of oscillation. Our author discovered that, if suspended at this point, the pendulum had its highest rate of speed. Approaching the end, or approaching the center of oscillation from this point, the rate of speed decreases. Exactly at one-ninth from the center of oscillation, or two-ninths from the end, is this center of velocity, as Ramsay designated it; and it corresponds in some sort also to the root of the dominant G, which is 9 in the genesis of the scale from F1; its rate of vibration is nine times that of F1. The dominant chord is the one in which is the maximum of levity and motion in music. [Scientific Basis and Build of Music, page 105]
In the major system, when the tonic chord follows the subdominant one, there is one semitonic progression to the middle of the tonic, and one note in common with the root, so these two chords are linked together in different ways. When the tonic chord follows the dominant one, there is one semitonic progression to the root of the tonic, and one note in common with its top, so these two chords also are linked together in two different ways. When the tonic chord follows the compound dominant, i.e., the dominant seventh, there are two semitonic progressions, one to the middle and one to the root, and one note in common with its top, so these two are linked together in the same two ways; but the semitonic progression being double gives this resolution great urgency. And now we come to the two chords, the subdominant and dominant, which have no note in common, and must, when they succeed each other, be helped to come together. Nature teaches us how this is to be done by a process of borrowing and lending which will establish between them a similar relationship to that which keeps the continuity of the other chords in succession. We have seen that the top of the subdominant and the root of the tonic are a note in common to these chords, and so the top of the tonic and the root of the dominant also are a note possessed in common by these two chords. In like manner in this disjunct part, when the dominant follows the subdominant, the root of the subdominant is lent to the top of the dominant, and thus they come to have a note in common. The top of the [Scientific Basis and Build of Music, page 111]
With perfect duality of response does resolution of chords go on in the minors. When the tonic chord follows the subdominant one, they have for their note in common A, i.e., in the key of A; and the middle of the subdominant moves by semitonic progression to the top of the tonic. When the tonic chord follows the dominant one, the top of the tonic and the root of dominant E is a note in common, and the top of the dominant goes by semitonic progression to the middle of the tonic. These simple chords are thus linked together exactly with the same degree of continuity as the simple chords of the major. When the tonic chord follows the compound subdominant, this compound chord, like the compound dominant in the major, has two semitonic progressions - one to the top and one to the middle of the tonic - and they have one note in common. When the compound dominant follows the subdominant, the root of the subdominant is lent to the top of the dominant, and thus a note in common is created, and the middle of the subdominant moves by semitonic progression to the root of the dominant. When the compound subdominant follows the dominant, the top is lent to the root of the subdominant, creating a note in common between them, and the root of the dominant goes to the middle of the subdominant in semitonic progression. This is the way of Nature. The unbroken continuity of her ways is perfectly illustrated in the linked sweetness and kinship of chords in a key; or when one key passes by modulation to another key; and that through all the chords and all the keys. We shall see wondrously more of this when we come to the study and contemplation of the Chromatic System of Chords. [Scientific Basis and Build of Music, page 112]
See Also
Dominant
Dominant Chord
middle of the dominant
Root
top of the dominant