The evil twin of quarter tones: your intro to 23-equal temperament (23edo)

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Imagine 24edo, but instead of half the notes being microtones — all of them are!

23edo is usually best played on percussive metallophone instruments, like those of an Indonesian gamelan ensemble

You’ve probably heard of microtonal music — tuning systems that use intervals smaller than the semitones of standard Western music. Most people’s first stop is somewhere comfortable: 19-edo, which still has recognisable major and minor scales, or 31-edo, often described as a more expressive version of the tuning system we already use. Then there’s 23-edo — and 23-edo is not comfortable.

23-edo (also written 23EDO, 23ed2, or 23-TET) is a tuning system that divides the octave into 23 equal steps of about 52.2 cents each. For reference, a standard semitone in 12-tone equal temperament (12-edo) is exactly 100 cents, so each step of 23-edo is roughly half a semitone. That already sounds unusual. But the really strange thing about 23-edo isn’t how small its steps are — it’s what those steps refuse to do.

A Quick Primer: What Even Is an EDO?

If you’re comfortable with standard music theory but new to microtonality, here’s the one thing you need to understand: equal temperament is a compromise.

In standard 12-tone equal temperament, we divide the octave into 12 equal semitones. This makes it easy to transpose music into any key, but it means many intervals are slightly out of tune compared to their “pure” acoustic versions. A perfect fifth in 12-edo is 700 cents; the pure acoustic perfect fifth (the 3:2 frequency ratio) is about 702 cents. That 2-cent error is imperceptible to most listeners. A major third in 12-edo is 400 cents; the pure acoustic major third (the 5:4 ratio) is 386 cents — a 14-cent error that trained ears can actually hear.

Different EDOs (Equal Divisions of the Octave) make different compromises. 19-edo has a beautiful, nearly pure minor third. 31-edo has an extremely accurate major third. 72-edo is accurate enough to approximate almost everything. And 23-edo? 23-edo says: forget all of that.

The Fundamental Strangeness of 23-edo

Most musicians, when they first explore microtonality, look for EDOs that still sound somewhat familiar — ones that approximate a good perfect fifth and a usable major third, just with extra notes added in. 23-edo does not do this.

23-edo is the last EDO that fails to approximate the 3rd, 5th, 7th, and 11th harmonics within 20 cents — all at the same time. To put that in plain language: the building blocks of standard Western harmony (the perfect fifth, the major third, the dominant seventh, the 11th) are all significantly mistuned in 23-edo. Its “perfect fifth” is about 678 cents — a full 24 cents flat. That’s not a slightly out-of-tune fifth; that’s a different-sounding interval altogether. Its “major third” is about 417 cents, nearly a third of a semitone sharp of where it should be.

This is not a tuning system you can use to play your existing music in a subtly different flavour. 23-edo is a genuinely alien sound world.

Yet here’s the paradox that makes 23-edo so fascinating: while it fails to approximate those harmonics directly, it approximates the intervals between them remarkably well. The ratio 5/3 (a pure major sixth) is approximated within 7.4 cents. The ratio 11/7 (an interval not even present in standard Western music) is approximated to within 0.4 cents — essentially perfect. The ratio 13/8 is accurate to within 5.7 cents. 23-edo is like a musician who can’t play the root notes but nails every chord extension.

The Antidiatonic Scale: Where Major Becomes Minor

The most musically useful thing to understand about 23-edo is the antidiatonic scale — and to understand it, we need to briefly talk about how diatonic scales are built.

In standard music theory, a major scale is built by stacking perfect fifths (702 cents each) and folding them back into one octave. The result is the familiar pattern of whole steps and half steps: W-W-H-W-W-W-H (or, in semitones, 2–2–1–2–2–2–1).

The antidiatonic scale works the same way — except the “fifth” being stacked is the flat 678-cent fifth of 23-edo. When you stack seven of these flat fifths, something remarkable and eerie happens: major and minor get swapped.

In a normal diatonic major scale, stacking four fifths and reducing by an octave gives you a major third (roughly 386 cents). In the antidiatonic scale of 23-edo, that same process gives you something close to a minor third (about 313 cents, very close to the pure 6/5 ratio). What used to be a major third is now a minor third, and vice versa. The familiar emotional logic of major = bright, minor = dark is inverted, or at least scrambled.

The antidiatonic scale in 23-edo has the step pattern 3–3–4–3–3–3–4 (in steps of 23-edo), which you can compare to the standard major scale’s 2–2–1–2–2–2–1 (in steps of 12-edo). Notice that the antidiatonic scale has two “large” steps and five “small” ones, exactly like a diatonic major scale — but which size is which is inverted. Where the diatonic scale has a half step (the small interval), the antidiatonic scale has its large interval, and where the diatonic has a whole step, the antidiatonic has its small one.

Microtonal composer and theorist Sevish has described the overall character of mavila (the temperament that generates the antidiatonic scale) as “face swap, expectation and surprise” — a remarkably apt description. You hear something that has the architecture of familiar music, but with everything emotionally flipped.

Mavila Temperament: The Theory Behind the Strangeness

The antidiatonic scale in 23-edo is an example of mavila temperament. Like meantone temperament (which underlies standard Western music), mavila is based on a chain of fifths. But where meantone tempers out the syntonic comma (81/80), mavila tempers out the larger mavila comma (135/128).

The result is a family of tunings with very flat fifths — typically between 670 and 685 cents. Within this family, 16-edo and 23-edo are the most commonly used tunings. 23-edo’s fifth, at 678 cents, is on the sharper end of the mavila spectrum, which means it sounds somewhat less dissonant than 16-edo’s and is generally considered more musical.

One practical consequence of the flat mavila fifth: the perfect fourth (4/3) falls almost exactly between two scale degrees in 23-edo. There’s no single step that maps cleanly to the 498-cent perfect fourth; instead, there are two near-fourths, one slightly below and one slightly above. This splitting of the fourth is one of the most disorienting features of the system for trained ears — the one interval that medieval theorists considered most stable is suddenly ambiguous.

The antidiatonic scale extends naturally to a 9-note “superdiatonic” scale with the step pattern 3–3–3–1–3–3–3–3–1, which provides even more harmonic colour and has been explored extensively by composers in the xenharmonic community.

Erich von Hornbostel and the Blown Fifth

One of the more historically interesting facts about 23-edo is that it was proposed — at least approximately — by the German-Austrian ethnomusicologist Erich von Hornbostel in the early 20th century, as part of his theory of “blown fifths.”

Hornbostel observed that bamboo pipes, when overblown, produce a fifth that is slightly compressed compared to the standard acoustic 3:2 ratio. He argued that a circle of these flattened ~678-cent fifths could serve as the basis for a tuning system, and that this tuning might be the origin of certain non-Western scale systems, including some East Asian pentatonic scales. The 13th step of 23-edo — the one that functions as the “fifth” in mavila — sits at exactly 678.261 cents, and is sometimes called the Hornbostel generator in his honour.

While Hornbostel’s broader ethnomusicological thesis has been disputed, it’s a remarkable coincidence (or not) that his theoretical bamboo-blowing experiments led him to the same mathematical structure that modern xenharmonic composers explore today.

Harmony in 23-edo: Forget What You Know, Then Listen Again

So what chords can you actually build in this system?

Forget the standard major chord (4:5:6). The standard Western major triad doesn’t exist in 23-edo in any recognisable form — the major third is too wide and the fifth is too flat for the combination to produce those smooth harmonic ratios.

Instead, 23-edo excels at a completely different set of chords. The interval 11/7 — an “undecimal minor sixth” that doesn’t appear anywhere in standard Western music — is approximated in 23-edo to within less than half a cent (step 15, 782.6 cents). The interval 5/3 (the pure major sixth) is approximated at step 17 with about 7.4 cents error. The interval 13/8 appears at step 16.

Rather than thinking in the language of major and minor triads, composers working in 23-edo tend to think in terms of the 2.5/3.11/7.13.17 just intonation subgroup — a set of ratios that 23-edo approximates well. In practical terms, this means chord structures built on intervals like 11:7, 13:8, and 5:3 rather than 5:4 and 3:2.

One particularly interesting chord structure is the 16:18:21 triad, found at degrees 0–4–9 of the scale, which appears frequently in the 8-note mavila mos scale. This chord has a very different emotional quality from a Western major or minor triad — not quite either one, and not quite dissonant either.

Stretching the Octave: 23-edo Gets More Musical

One of the most fascinating practical discoveries in 23-edo is what happens when you slightly stretch the octave. In standard equal temperament, every octave is a perfect 1200 cents (a 2:1 frequency ratio). But there’s no law saying octaves have to be that size, and on some instruments — particularly pianos — the octave is already slightly stretched due to the physical properties of strings.

When you stretch the octave of 23-edo to approximately 1206 cents, all the approximations improve significantly. With this stretch, the system approximates the perfect fifth and various intervals involving the 5th, 7th, 11th, and 13th harmonics to within 18 cents — a level of error comparable to what 12-edo already asks us to accept for the major third.

The stretched version is sometimes described as not being an extension of or replacement for 12-edo, but rather a genuine alternative to it — with different strengths and weaknesses, and the two systems complementing rather than competing with each other. Stretched-23 is particularly well-suited to inharmonic instruments (instruments whose overtones don’t line up with the standard harmonic series), such as bells, marimbas, and metallophones. This makes it potentially very interesting for composers wanting to blend Western instruments with instruments from gamelan and other non-Western traditions.

The 36edt tuning (36 equal divisions of the tritave, where the octave is about 1215 cents) represents an even more stretched version, and approximates the harmonics 3, 5, 7, and 13 well while making 2 less pure — an unusual trade-off that opens up its own compositional possibilities.

The Sephiroth Modes: An Advanced Scale System

For those interested in going deeper, theorist Kosmorsky has proposed that the most musically significant modes of 23-edo are not the antidiatonic scale but a 10-note scale based on the pattern 2–2–2–3–2–2–3–2–2–3 (the 3L 7s mos or “fair mosh” scale). This scale arises from the 13th harmonic: two steps of the 13th harmonic generator land approximately on the 21st harmonic, and three steps land approximately on the 17th harmonic. Even more intriguingly, the chord 8:13:21:34 built within this system is a fragment of the Fibonacci sequence.

Kosmorsky named the ten modes of this scale after the ten Sephiroth of the Kabbalistic Tree of Life:

  • Keter (Crown): 2–2–2–3–2–2–3–2–2–3
  • Chesed (Loving-kindness): 2–2–3–2–2–3–2–2–3–2
  • Netzach (Eternity): 2–3–2–2–3–2–2–3–2–2
  • Malkuth (Kingdom): 3–2–2–3–2–2–3–2–2–2

…and so on for all ten modes. Whether or not you find the Kabbalistic framing meaningful, the underlying musical observation is real: this scale has a distinctive internal logic tied to the higher harmonics that 23-edo approximates well.

Notable Music in 23-edo

Despite (or because of) its difficulty, 23-edo has attracted a committed body of composers and performers. Here are some notable works to explore:

Easley Blackwood — one of the pioneering composers of microtonal music — included a study in 23-edo in his 12 Microtonal Etudes for Electronic Music Media, Op. 28 (1980). Blackwood systematically explored a range of EDOs, and his etude in 23 is among the most unsettling in the series — exactly as you’d expect.

Cryptic Ruse (Igliashon Jones) released UNFERTILE (2020), a 13-track album entirely in 23-edo that sits at the intersection of drone, doom metal, and extreme xenharmony. The album title tracks include compositions such as “Laying Fallow” and “The Bleak Majesty of Fire-Ravaged Lands” — titles that give a sense of the emotional territory the tuning evokes.

NullPointerException Music has released “Desertification” from the album Edolian (2020), and the follow-up Resurgence (2020), both in 23-edo.

Francium released From Melancholie (2023), a more melodic and accessible take on the tuning, with tracks “Elise Ain’t Right” (a nod to the famous Für Elise) and “Moonlight Party” available on Spotify and Bandcamp.

Carlo Serafini has produced multiple pieces over many years, including the notably approachable Desert Winds(2011)and a reworking of Barber’s Adagio for Strings in 23-edo — a fascinating exercise in hearing a deeply familiar piece rendered in an alien tuning.

Chris Vaisvil famously built a DIY 23-edo electric guitar from under $50 of materials, documenting the process and playing the finished instrument on YouTube — a reminder that you don’t need expensive custom instruments to explore xenharmony.

How to Explore 23-edo Yourself

If you want to experiment with 23-edo without building a custom guitar, here are some practical starting points:

Scale Workshop (https://scaleworkshop.luminus.app) — a free browser-based tool where you can build and play any EDO scale using your computer keyboard or a MIDI controller. Simply enter 23 as your division number and start playing.

Reaper, Bitwig Studio, and several other DAWs support microtuning via .scl or .tun files. You can find 23-edo tuning files pre-made in various xenharmonic archives online.

Surge XT is a free, open-source synthesiser with native microtonal support — one of the best free tools for xenharmonic exploration.

For deeper theory, the Xenharmonic Wiki page on 23-edo and the page on mavila temperament are thorough if somewhat technical. The 23-edo and octave stretching page is particularly useful if you want to try the more musical stretched-octave variant.

Why Bother?

It’s a fair question. If 23-edo fails to approximate the harmonic building blocks of the music we already know and love, why would anyone spend time with it?

The answer, I think, is the same reason anyone explores unfamiliar music from other cultures: not to replace what you know, but to discover that the emotional vocabulary of music is larger than you thought. The antidiatonic scale doesn’t just sound “wrong” — it sounds differently right. The swapped major and minor, the split fourth, the precise approximation of intervals you’ve never heard in a Western context — these aren’t bugs. They’re a genuinely different way of organising musical emotion.

Sevish put it well when he described 23-edo as “bright colour, unstable.” That instability isn’t necessarily a flaw. Some of the most interesting music exists precisely at the edge of what feels settled and resolved. 23-edo lives there permanently — a tuning system that refuses to come home to the consonances you were raised on, and invites you to find a new home instead.

Further reading:

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