Microtonality sounds like a specialist interest and it is better understood as the default that Western music opted out of. Twelve equal semitones is a compromise adopted for practical reasons, and knowing what it is a compromise with is the whole subject.
John Moriarty’s primer — short, clear, and the right place to start.
Ratios first
Pitch is frequency, and intervals are ratios. This is the one idea everything else follows from.
Double a frequency and you get an octave: 2:1. That is not a convention, it is physics — and it is why every musical culture that has a concept of octave equivalence arrived at the same one.
The next simplest ratios are the next most consonant intervals:
| Ratio | Interval | Why it sounds good |
|---|---|---|
| 2:1 | Octave | Every partial of the upper note coincides with one of the lower |
| 3:2 | Perfect fifth | Partials coincide every 2nd and 3rd |
| 4:3 | Perfect fourth | |
| 5:4 | Major third | |
| 6:5 | Minor third | |
| 7:4 | Harmonic seventh | Flatter than any seventh on a piano |
Consonance is coincidence of partials. Any harmonic sound contains a series of overtones at integer multiples of its fundamental. When two notes are in a small-integer ratio, large numbers of their partials land on exactly the same frequencies and there is nothing left to beat against. When the ratio is complex, partials land close-but-not-equal and you hear roughness.
Tuning by these ratios is just intonation, and it is what the Harmonic Eigenspace tool we wrote about this morning maps: its 100 consonance minima land on just intervals because that is where the physics puts them.
Cents, so you can talk about size
A cent is 1/1200 of an octave — 1/100 of an equal-tempered semitone. It is logarithmic, so cents add where ratios multiply, which makes it the practical unit.
cents = 1200 × log₂(ratio)
The numbers worth memorising:
| Just | 12-TET | Error | |
|---|---|---|---|
| Perfect fifth | 701.96¢ | 700¢ | −2¢ |
| Major third | 386.31¢ | 400¢ | +14¢ |
| Minor third | 315.64¢ | 300¢ | −16¢ |
| Harmonic 7th | 968.83¢ | 1000¢ | +31¢ |
The fifth is nearly perfect. The thirds are not. Equal temperament’s major third is 14 cents sharp, which is audibly bright and is the reason sustained chords on a piano have a slight restlessness that a barbershop quartet or a string quartet playing by ear does not.
About 5 cents is the threshold of noticing for most people on sustained tones; 14 is clearly audible if you know what to listen for.
Why equal temperament won
Because just intonation does not close.
Stack twelve pure fifths (3:2) and you should return to your starting note seven octaves up. You don’t — you overshoot by about 23.46 cents, the Pythagorean comma. The ratios 3:2 and 2:1 are incommensurable; no number of one equals any number of the other.
So a fixed-pitch instrument tuned in pure fifths is wrong somewhere, and historically that error was dumped into one interval — the “wolf fifth” — which was then avoided. That is workable if you only play in a few keys, and it is why Baroque keyboard music is key-specific in a way later music is not.
12-TET spreads the error evenly: every fifth is 2 cents flat and nothing is unusable. You lose pure intervals and gain unrestricted modulation, which is the trade that made nineteenth-century harmony possible.
The systems you will meet
Just intonation — tune to exact ratios. Pure consonances, key-dependent, needs either flexible-pitch instruments or retuning.
Equal divisions of the octave (n-TET / n-EDO) — divide the octave into n equal steps.
- 19-TET — better thirds than 12, usable fifths. A classic first step.
- 31-TET — excellent approximation of quarter-comma meantone, very good 5-limit and 7-limit intervals. Adriaan Fokker’s system; one of the most musically rewarding.
- 53-TET — nearly perfect fifths (0.07¢) and very good thirds. Underpins Turkish/Arabic maqam theory.
- 24-TET — quarter tones. The common ground with Arabic and Persian practice and the easiest to notate.
Meantone, well temperaments, Werckmeister — the historical compromises between just and equal, and what Bach’s keyboard music was actually written for.
Non-octave systems — the Bohlen-Pierce scale divides a 3:1 “tritave” into 13 steps. Genuinely alien and worth hearing once.
Three free tools to actually play this
1. Scala (huntsvillian-era but indispensable) — the reference application for tuning theory. It knows over five thousand historical and theoretical scales, calculates everything, and exports .scl files, the de facto standard format. Free, from Manuel Op de Coul.
2. MTS-ESP (ODDSound) — the thing that changed microtonal production. A plugin that broadcasts a tuning to every supported instrument in your DAW in real time, so you retune once and everything follows, including dynamically. MTS-ESP Mini is free. Supported by Surge XT, Vital, u-he, Arturia, Bitwig’s instruments and a growing list.
3. Surge XT — free, open-source, and the best microtonal synth available at any price. Loads .scl and .kbm files natively, supports MTS-ESP, and has a built-in tuning editor. If you want to hear 31-TET in the next ten minutes, this is the route.
Also worth knowing: Pianoteq (excellent native tuning support), Zynaddsubfx (free, deep), Sevish’s Scale Workshop (browser-based scale design, exports .scl), and on hardware, the Lumatone isomorphic keyboard — expensive, and the only mainstream controller designed for this.
Getting started in practice
Start with 31-TET, not just intonation. JI requires you to decide the tuning of every chord, which is a composition problem before you have any intuition. 31-TET is a fixed grid that sounds better than 12 on thirds and sevenths while still working like a keyboard. Load it in Surge XT and play familiar chord shapes — the harmony will sound cleaner and slightly unfamiliar, which is the right first experience.
Then try 7-limit harmony. The harmonic seventh (7:4) is 31 cents flat of an equal-tempered minor seventh and it is the single most striking thing in this territory: a dominant seventh chord in just 4:5:6:7 locks into a consonance that no piano can produce.
And remember timbre matters. Consonance depends on the spectrum, so a tuning optimised for harmonic sounds behaves differently on inharmonic ones. Sethares’ Tuning, Timbre, Spectrum, Scale is the book, and the practical upshot is that bells, metal and FM patches have their own consonances — which is an invitation rather than a problem.