The Octave
A Natural Phenomenon That Unites Virtually All Music
The octave is the fundamental interval that occurs between the 1st and 2nd harmonics in the harmonic series, with the higher pitch exactly twice the frequency of the lower.
In 12-tone equal temperament, the octave spans 12 semitones, or half-steps, encompassing the full cycle of pitch classes before returning to the starting note, albeit at a higher frequency. This interval is a universal feature in music across cultures and traditions, from Western classical and jazz to Indian ragas and African drumming, marking it as a near-universal structure in human music-making.
In Western music theory, notes an octave apart are treated as equivalents, sharing the same name and pitch class. For example, the note "C" at any octave represents the same pitch class, forming a kind of symmetry that musicians use to establish tonal centers and build harmony across the full range of their instruments. This equivalency simplifies musical structure and allows for complex, multi-octave compositions without losing harmonic identity.
Musicians and composers often avoid using octaves within chords, because doubling notes at the octave can increase the perceived loudness. In harmonic arrangements, especially within chords, composers generally include only the root octave, rarely doubling other intervals like the third, fifth, or seventh. The goal is to prevent any single note from dominating the harmony.
The Harken Music system equates unisons and octaves to the same numerical value. For example, the number "0" represents the same pitch class across all octaves, meaning the tonic is "0" whether it's in the bass, middle, or the treble. While octaves and unisons are technically intervals, they hold no distinct harmonic function beyond emphasizing the foundational tone, serving instead as structural points of return within the tonal space.
Like everything in the Harken Music system, we manage octaves moving in both directions (up and down) using only math. Here’s the formula:
x = (n + 12y) mod 12
Here’s how it works:
Moving up: Set y as a positive integer (e.g., y = 1 moves n up one octave)
Moving down: Set y as a negative integer (e.g., y = −1 moves n down one octave)
For example, if we assign MIDI pitch number 50 as the fixed tonic (0) and decide to shift that pitch up one octave (+12), we simply set variables n = 50 and y = 1 and apply the formula, which raises the pitch one octave from MIDI number 50 to 62.
Ultimately the octave remains one of the most unifying aspects of music, bridging traditions, instruments, and styles. Its ability to link disparate musical systems stems from its simplicity as a natural acoustic phenomenon and its ability to anchor tonal systems in virtually every culture.
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.
Please try our free open-source demo* at harkenmusic.com, where 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.
It cannot be overstated that the Harken Music system uses only mathematics and math related programming code. The programming does not include or use any databases, lookup tables, or other sources of data. Everything—nearly 1.2 billion musical combinations and permutations—is calculated on the fly.
This information and related software code are now available for free on HarkenMusic and Github as open-source software under the standard MIT License set forth below.
Copyright 2024 Mitchell Kahle and Holly J. Huber
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