The inner stave contains the chromatic scale of twelve notes as played on keyed instruments. The flat and sharp phase of the intermediate notes are both given to indicate their relation to each other; the sharpened note being always the higher one, although seemingly on the stave the lower one. The two notes are the apotome minor apart overlapping each other by so much; ♭D is the apotome lower than C#; ♭E the apotome lower than D#; F# the apotome higher than ♭G; G# the apotome higher than ♭A; and A# the apotome higher than ♭B. The figures for the chromatic scale are only given for the notes and their sharps; but in the mathematical series of notes the numbers are all given.
Fig. 1. - This figure shows the major and minor measured in commas and placed directly as they stand related, major and relative minor, the minor being set a minor third lower than the major. The interval between C and E in the minor is an 8-and-9-comma interval; between C and E in the major it is a 9-and-8-comma one. This difference arises from the minor D being a comma lower than the major D. In all the other intervals minor and major are the same.
Fig. 2. - In this figure the two modes are placed in their inverse relation, in order to show the notes standing opposite each other in their duality. Here the two D's come also opposite each other, inasmuch as in the two modes the C-D-E interval is inverted, becoming C D E in the one and E D C in the other. And so the 9-comma second between C and D in the major comes opposite the 9-comma second between D and E in the minor, and the two 8-comma seconds, of course, come opposite each other also.
Fig. 3. - This is a set of pendulum lengths for three octaves, given merely to assist any tyro who might wish to try them, but might find difficulty in calculating them.
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.
Fig. 3 illustrates the way Nature teaches us by example how to compound so as to enable chords that are separated by the intervention of others to pass to each other. In the middle of the chord scale Nature gives the root of the one chord to the top of the other, and the top of the one to the root of the other; in compounding we are taught by this example to do the same, and the top of the separated dominant is given to the root of the [Scientific Basis and Build of Music, page 120]
Ramsay - PLATE XXVII - The Mathematical Scale of Thirty two notes in Commas, Sharps and Flats