Vortex-based Tuning

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.

Hexagonal Buttons

When creating the Musician’s Calculator iPhone app for ToruSound, I wanted the buttons to be more ergonomic and also wanted to be able to render them in patterns that made sense with regard to the underlying science, such as a “circle” of fifths surrounding “triangular” base-10 digit buttons, as well as piano-style buttons for producing actual musical pitches.

The source code for the hexagonal buttons (which is compatible with the Xcode interface designer) is up on GitHub:

https://github.com/DanielLewisRandall/iOpenCalculator/blob/master/OpenCalculator/OpenCalculator/UIHexButton.h

https://github.com/DanielLewisRandall/iOpenCalculator/blob/master/OpenCalculator/OpenCalculator/UIHexButton.m