There are 6 variations for each of the 660 possible triads in 12-tone music. The Harken Music system proves this mathematically using combinatorics to calculate every combination and permutation possible in 12-tone equal temperament. With the Harken Music system we can calculate there are a total of 660 triads and 3,960 permutations in all.
Can you play all those triads and permutations? The Harken can. Watch the video. Check out the demo. Use the code. It’s all free and open-source.
For a detailed explanation of the high-school-level math used in creating the Harken Music system, please read, “The Future of Music is Math” by Mitch Kahle.
See also: “Harken Music: A New Mathematical Framework for 12-Tone Harmony”, by ChatGPT 4o, and for an even deeper dive into the mathematics, read Factorials and Fractions “Navigating the 12-tone matrix,” by Mitch Kahle.
Try the free open-source demo* at harkenmusic.com. You can choose any combination size (from 1 to 12 notes) from the pull-down menu; then select any particular combination from the numbered list, to see and hear all permutations, reflections (inversions), and rotations (transpositions).
*Note: a desktop or laptop computer and current web browser (Chrome, Safari, etc.) are required for using the demo; however there are no downloads, sign-up, or subscription requirements.
