Background:
Cross-referencing the above three concepts yields a rather elementary description of the purity of Just Intonation.
Starting with the root note of A (432 Hz), which converts to 9 on the Vortex-based Math pattern (because 4+3+2 = 1+8 = 9), generate a Pure Fifth (3:2 ratio) with the formula 432*3/2 = 648. Since 648 converts to 9 on the Vortex-based Math pattern (because 6+8+4 = 1+8 = 9), a pattern becomes apparent: All of the following Just-intoned Diatonic musical frequencies convert to 9 on the Vortex-based Math pattern (the Just-intoned A Major Diatonic Scale):
Unison ( 1/1): 432 = 4+3+2 = 9 Fifth ( 3/2): 432 * 3/2 = 648 = 6+4+8 = 18 = 1+8 = 9 Fourth ( 4/3): 432 * 4/3 = 576 = 5+7+6 = 18 = 1+8 = 9 Sixth ( 5/3): 432 * 5/3 = 720 = 7+2+0 = 9 Third ( 5/4): 432 * 5/4 = 540 = 5+4+0 = 9 Second ( 9/8): 432 * 9/8 = 486 = 4+8+6 = 18 = 1+8 = 9 Seventh (15/8): 432 * 15/8 = 810 = 8+1+0 = 9 Octave ( 2/1): 432 * 2/1 = 864 = 8+6+4 = 18 = 1+8 = 9
Pythagorean Tuning produced some irrational numbers, so I experimented with other 12-tone systems:
Wendy Carlos 12-tone Just Intonation Scale (Key of A Major):
Note Ratio Formula Hz A (1:1) 432 * 1 / 1 = 432 = 4 + 3 + 2 = 9 A# (17:16) 432 * 17 / 16 = 459 = 4 + 5 + 9 = 18 = 1 + 8 = 9 B (9:8) 432 * 9 / 8 = 486 = 4 + 8 + 6 = 18 = 1 + 8 = 9 C (19:16) 432 * 19 / 16 = 513 = 5 + 1 + 3 = 9 C# (5:4) 432 * 5 / 4 = 540 = 5 + 4 + 0 = 9 D (21:16) 432 * 21 / 16 = 567 = 5 + 6 + 7 = 18 = 1 + 8 = 9 D# (11:8) 432 * 11 / 8 = 594 = 5 + 9 + 4 = 18 = 1 + 8 = 9 E (3:2) 432 * 3 / 2 = 648 = 6 + 4 + 8 = 18 = 1 + 8 = 9 F (13:8) 432 * 13 / 8 = 702 = 7 + 0 + 2 = 9 F# (27:16) 432 * 27 / 16 = 729 = 7 + 2 + 9 = 18 = 1 + 8 = 9 G (7:4) 432 * 7 / 4 = 756 = 7 + 5 + 6 = 18 = 1 + 8 = 9 G# (15:8) 432 * 15 / 8 = 810 = 8 + 1 + 0 = 9 A (2:1) 432 * 2 / 1 = 864 = 8 + 6 + 4 = 18 = 1 + 8 = 9
Diatonic Scale (Key of A Major):
Note Interval Ratio Formula Hz A Unison (1:1) 432 * 1 / 1 = 432 = 4 + 3 + 2 = 9 B Second (9:8) 432 * 9 / 8 = 486 = 4 + 8 + 6 = 18 = 1 + 8 = 9 C# Third (5:4) 432 * 5 / 4 = 540 = 5 + 4 + 0 = 9 D Fourth (4:3) 432 * 4 / 3 = 576 = 5 + 7 + 6 = 18 = 1 + 8 = 9 E Fifth (3:2) 432 * 3 / 2 = 648 = 6 + 4 + 8 = 18 = 1 + 8 = 9 F# Sixth (5:3) 432 * 5 / 3 = 720 = 7 + 2 + 0 = 9 G# Seventh (15:8) 432 * 15 / 8 = 810 = 8 + 1 + 0 = 9 A Octave (2:1) 432 * 2 / 1 = 864 = 8 + 6 + 4 = 18 = 1 + 8 = 9
Notice how the diatonic scale has all but two of the corresponding note frequencies in common; the exceptions being a difference of 9 Hz for the notes F# and D.
