There are only 2 permutations for each of the 11 possible intervals in 12-tone music, making intervals the easiest way for anyone to start learning 12-tone music. Watch this short video and then try it yourself using the free demo at harkenmusic.com.
Intervals relative to the fixed tonic [0] are traditionally named as follows:
[0] is the fixed “tonic” or [0, 0] is a “unison” or “octave”
[0, 1] is the “minor 2nd” or “diminished 2nd”
[0, 2] is the “major 2nd”
[0, 3] is the “minor 3rd” or “augmented 2nd”
[0, 4] is the “major 3rd” or “diminished 4th” (rare)
[0, 5] is the “perfect 4th”
[0, 6] is the “tritone” or “augmented 4th” or “diminished 5th”
[0, 7] is the “perfect 5th”
[0, 8] is the “minor 6th” or “augmented 5th”
[0, 9] is the “major 6th” or “diminished 7th”
[0, 10] is the “minor 7th” or “augmented 13th” (rare)
[0, 11] is the “major 7th”
Intervals are the basic building blocks of 12-tone harmony. A major triad, for example, is constructed by combining two intervals that are based on the same root [0]: a major 3rd [0, 4] and a perfect 5th [0, 7]. Thus a major triad is [0, 4, 7].
See my article “The Grand Order of Intervals” for a full explanation of each interval-tonic relationship and how each relates to others in the hierarchy of harmony.
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.
