Ramsay
In just such a manner, only by more obvious leaps, the middle of the dominant in the advancing major scales is raised a sharp - i.e., four commas. When D27, the dominant top of the key of C, is multiplied by 5, it generates F#135; so, taking it one octave lower, F64 in C major is F#67 1/2 in the key of G. C96 in the key of G is C#101 1/4 in the key of D; G72 in the key of D is G#151 7/8 in the key of A. And this raising of the middle of the dominant goes on through all the twelve major keys.[Scientific Basis and Build of Music, page 62]
The interval F G A, which in the scale of A was a 9-8-comma interval, must take the place of C D E of the scale of A, which is an 8-9-comma interval; and in order to do this, G has been mathematically lowered a comma. As the middle of the dominant in the major is raised a comma, so the root of the subdominant is lowered a comma. The interval A B C, which in the scale of A was a 9-5-comma interval, is here to take the place of E F G in the scale of A, which is a 5-9-comma interval; and in order to do so, B is lowered 4 commas, and so becomes ?B; and this mathematical process makes the new scale exactly like the old one. This is the way of the minors when calculated as a descending series of scales, which is their natural way. [Scientific Basis and Build of Music, page 84]
top of the dominant; the third is the middle of the tonic; the fourth is the root of the subdominant; the fifth is the top of the tonic; the sixth is the middle of the subdominant; the seventh is the middle of the dominant; and the eighth, like the first, is the root of the tonic. [Scientific Basis and Build of Music, page 97]
subdominant also moves by semitonic progression to the middle of the dominant, and so, like the simple chords, they are brought into continuity. When the subdominant follows the dominant, the top of the dominant is lent to the root of the subdominant, and they come to have also a note in common; and the middle of the dominant moves by semitonic progression to the top of the subdominant; and thus resolving continuity is established between them. [Scientific Basis and Build of Music, page 112]
In Fig. 1 is shown the way in which duality arranges the new sharp in the majors to the middle of the dominant, and the new flat to the middle of the subdominant in the minors, all through the six scales done in flats and sharps. The flat goes to the root of the subdominant and the sharp to the top of the dominant in the other six, as in Fig. 2. This is the invariable way that the new sharps and flats are responsively added all through the system. [Scientific Basis and Build of Music, page 120]
See Also