Microtonality has a navigation problem rather than a theory problem. The theory is centuries deep. What it lacks is an instrument — a layout where moving your hand a known distance produces a known harmonic change. Twelve-tone equal temperament has a keyboard, and the keyboard is why most Western musicians can think harmonically at all.
Harmonic Eigenspace: A Web-based Application for Navigating and Composing Microtonal Harmony, posted 2 October 2026 by David Dalmazzo and Ken Déguernel, is an attempt at a map instead of a keyboard.
And there is a live app: dazzid.github.io/ANIMA_Harmonic_Eigenspace.
What it is
A four-dimensional psychoacoustic space in which chord types are positioned by their spectral dissonance characteristics, using Sethares’ roughness/dissonance model.
The coordinate system is the clever part:
- It is transposition-invariant
- Three coordinates locate the upper notes relative to a root
So a chord’s position encodes its internal structure, not its absolute pitch. A major triad on C and a major triad on F♯ occupy the same point. That is the right abstraction, because harmonic quality is a property of intervals, not of pitches.
Over that space, a continuous dissonance volume containing 100 local minima that align with just-intonation intervals — and those minima organise the space into regions around the most consonant tetrads. Three equal temperaments are integrated, including 53-TET.
The app includes a 3D visualisation of the dissonance volume, a Modal Studio for composing progressions, and MIDI controller output compatible with a DAW.
Sethares’ model, and why it gives you a landscape
William Sethares’ Tuning, Timbre, Spectrum, Scale (1998) is the foundational text here and the model is worth understanding, because it explains why the space has valleys at all.
The core idea: dissonance is not a property of intervals, it is a property of spectra. Two tones sound rough when their partials fall close enough in frequency to beat against each other — within roughly a critical bandwidth. Plomp and Levelt measured this curve experimentally in 1965: roughness rises from zero as two partials approach, peaks at about a quarter of a critical bandwidth apart, and falls back toward zero as they converge to unison.
So to compute the dissonance of a chord: take every pair of partials across all its notes, evaluate the roughness curve for each pair, weight by amplitude, and sum.
The consequence is the important bit. For a harmonic spectrum — the overtone series of most acoustic instruments — this sum has minima exactly where partials coincide, which is at small-integer frequency ratios: 2:1, 3:2, 5:4, 7:4. Which is to say, just intonation falls out of the psychoacoustics rather than being imposed by theory.
The paper’s 100 local minima aligning with just-intonation intervals is that prediction, computed across a four-dimensional chord space and confirmed. The landscape’s valleys are the consonant chords, and they are there for a measurable physical reason.
And Sethares’ deeper point — the one that makes this more than analysis — is that the relationship runs both ways. Change the timbre and the minima move. An inharmonic spectrum has its consonances somewhere else entirely, which means a tuning system and a timbre are a matched pair. This is the theoretical basis for the whole tradition of designing scales for specific synthesised spectra.
Why 53-TET
Including 53-tone equal temperament is a specific and well-motivated choice rather than an arbitrary large number.
53-TET is remarkable because it approximates just intonation extremely well across the board: its fifth is within about 0.07 cents of a pure 3:2 — effectively perfect — and its major third is within 1.4 cents of 5:4. It has been known since antiquity (Ching Fang in the first century BCE, Mercator in the 17th) and is used in practice in Turkish and Arabic music theory, where the 53-comma division underpins the maqam system.
So 53-TET is where a map built on just-intonation minima becomes playable on a fixed-pitch instrument. It is the practical bridge between the continuous space and a keyboard you could actually build — which is why Lumatone and similar isomorphic controllers keep appearing in this territory.
What to do with it
Use it as a compositional tool, not a theory lesson. The Modal Studio plus MIDI output means progressions you discover in the space can be driven into whatever synth you already use. That is the difference between a visualisation and an instrument.
Pay attention to where the valleys are not. The interesting microtonal writing is frequently about the slopes — a progression that moves deliberately away from a consonance minimum and back has a tension profile you cannot get in 12-TET, because 12-TET’s chords are all sitting at roughly similar roughness.
And remember the timbre dependency. The space is computed for a given spectrum. Drive it into a synth with a wildly inharmonic patch and the map no longer describes what you hear. If you are working with FM, ring modulation or physical-modelled metal, the consonances are elsewhere — which is an opportunity, not a bug, and it is the thing Sethares’ book is actually about.
It is also the latest in a run of serious browser-based audio tools we have covered in a fortnight — Ian Dixon’s Web MIDI synth editors, the Convolution Playground — and the same argument applies: a URL you can hand someone beats an install, and Web MIDI means it talks to real hardware.