Calculate the Circle of 5ths/4ths
Only two simple formulas are needed to calculate the Circle of 5ths/4ths, ascending and descending.
Calculating the Circle of 5ths/4ths (aka the “Cycle Order”)
Ascending Cycle Order
(n * 7) % 12 for n = 0 to 11
(0 * 7) % 12 = 0
(1 * 7) % 12 = 7
(2 * 7) % 12 = 2
(3 * 7) % 12 = 9
(4 * 7) % 12 = 4
(5 * 7) % 12 = 11
(6 * 7) % 12 = 6
(7 * 7) % 12 = 1
(8 * 7) % 12 = 8
(9 * 7) % 12 = 3
(10 * 7) % 12 = 10
(11 * 7) % 12 = 5
[0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5]
Descending Cycle Order
(n * 5) % 12 for n = 0 to 11
(0 * 5) % 12 = 0
(1 * 5) % 12 = 5
(2 * 5) % 12 = 10
(3 * 5) % 12 = 3
(4 * 5) % 12 = 8
(5 * 5) % 12 = 1
(6 * 5) % 12 = 6
(7 * 5) % 12 = 11
(8 * 5) % 12 = 4
(9 * 5) % 12 = 8
(10 * 5) % 12 = 2
(11 * 5) % 12 = 7
[0, 5, 10, 3, 8, 1, 6, 11, 4, 9, 2, 7]
Cycle Order (Bi-directional)
The cycle order is bi-directional, so ascending and descending cycles may be combined: where the tonic [0] is located in the middle of the sequence, with the tritone [6] at each end.
The tritone interval [6] appears twice to preserve left-right symmetry, but is only counted once, so the cycle order remains entirely within base-12.
Please read the article “The Future of Music is Math” and then try our simple HTML/Javascript demo* at harkenmusic.com to see and hear for yourself. Using this demo, you may choose any combination number (from 1 to 12 notes); then select any combination to view and listen to all possible permutations, reflections (inversions), and rotations (transpositions).
*Note: a desktop or laptop computer and current web browser are required for using the demo.