Microtonal Systems and Harmonic Implications

Core Idea

Microtonal music divides the octave into intervals smaller than the semitone. Each tuning system (19-ET, 31-ET, Bohlen-Pierce, spectral tunings) has distinct harmonic properties and suggest different harmonic vocabularies. Analyzing microtonality requires understanding both the mathematical structure of tuning systems and their perceptual effects.

Explainer

You already understand just intonation: tuning intervals to small-integer ratios (perfect fifth = 3:2, major third = 5:4) to achieve acoustically pure intervals with minimal beating. The problem is that pure intervals in one key create wolf intervals in others, which is why 12-tone equal temperament (12-TET) divides the octave into 12 equal logarithmic steps and accepts small deviations from just ratios in exchange for full transposability. Microtonal systems ask a deeper question: what if we treat the tuning system itself as a compositional resource, choosing the octave division for its specific harmonic affordances rather than accepting 12-TET as a neutral default?

In equal temperament systems (n-ET), the octave is divided into n equal steps of 1200/n cents each. 19-ET uses steps of ~63 cents and produces a major third (6 steps = 378 cents) closer to just (386 cents) than 12-TET's 400 cents, along with a very narrow chromatic semitone that gives it a distinctive leading-tone pull. 31-ET (steps of ~39 cents) approximates just intervals still more accurately — its perfect fifth, major third, and harmonic seventh all fall within 5 cents of just values — and enharmonic equivalents like C# and Db become genuinely distinct pitches separated by a diesis of ~41 cents. Each system supports different harmonic vocabularies: 31-ET opens up extended just-intonation chord structures within an equal-tempered framework, while 19-ET has a natural grammar for chromatic voice-leading.

Systems like Bohlen-Pierce break from the octave entirely. Instead of dividing the 2:1 ratio, Bohlen-Pierce divides the tritave (3:1) into 13 equal steps. This produces a scale with no conventional octave equivalence but strong consonances built from 3:5:7 ratios — the same odd-integer ratios that dominate the seventh partial of the harmonic series. Spectral tunings derive pitch collections directly from the overtone series of a single fundamental, so the intervals match the actual partials produced by a given instrument's timbre. The mathematical tools you need to work with these systems are logarithms (to convert frequency ratios into cents: cents = 1200 × log₂(ratio)) and modular arithmetic (to understand how n-ET step classes wrap around the octave or tritave and define equivalence classes).

Analyzing a piece in a microtonal system requires starting from scratch with interval tables. In 12-TET, a tritone (600 cents) is its own inversion and creates tritone-substitution equivalences. In 31-ET, the tritone and its complement are distinct, and the system contains approximations of 7-limit intervals (7:4 ≈ 969 cents, 7:6 ≈ 267 cents) absent from 12-TET. Voice-leading, harmonic roots, and set-class equivalences all depend on the step size of the system you are in. The theoretical apparatus — interval classes, transposition, inversion, prime forms — must be rebuilt relative to the system's modular structure. This is what makes microtonal analysis genuinely different from retuning 12-TET music: the harmonic logic is not just stretched or compressed, it is structurally reorganized.