Job says concerning the planet on which we live, the earth-
"He stretcheth out the north over empty space;
He hangeth the earth upon nothing." - Job xxvi. 7.
So the dual system of Music revolves round a mathematical point which is in none of its notes, but in the empty space between the two D's. Like the earth, it is hung upon nothing. This is an exceedingly interesting musical phenomenon. In that comma of vacant space is music's center of gravity.
1 3 9 27
F C G D
D A E B
26 2/3 80 240 720
A LESSON ON THE DEVELOPMENT OF KEYS FOR THE BEGINNER.
To develop the two new notes for a new key a fifth higher, you multiply the vibration-number of the top of the dominant of the key you have by 3 and by 5, thus-
C 9 D 8 E 5 F 9 G 8 A 9 B 5 C
24 27 30 32 36 40 45 48
48 54 60 64 72 80 90 96
There are two octaves in the key of C, as it is called. Now for the scale of a fifth higher than C, that is G, multiply the top of the dominant, that is the highest note of the chord-scale, by 3 and by 5, and the two new notes for the scale of G will be found; the rest of the notes are the same mathematically as those of C.
The chords of the scale are S F A C, T C E G, D G B D. Now D is the top of the dominant. Well, take it as D27 or D54 it is all the same, higher or lower octave.
2) 81 A, 2)185 F#
2) 40 1/2 A, 2) 67 1/2 F#
20 1/4 A, 33 3/4 F#
N.B. - The sharp comes here by the prime 5, and the comma by the prime 3. Now we have the key of G provided for;-
These are two octaves of the scale of G. G A B, which in the scale of C was an 8-9-comma third, must now take the place of C D E, which in C was a 9-6-comma