13-equal temperament (13edo): an intro to the strangest tuning you’ll ever love

Have you ever wondered what music would sound like if the octave were sliced into 13 equal pieces instead of 12 — welcome to 13edo: the tuning that sounds like it came from a parallel universe, and is secretly one of the most fascinating musical systems ever explored

What Even Is an “EDO”?

Before we go further, a quick primer. EDO stands for Equal Divisions of the Octave. The tuning system you’ve used your whole life — the piano, the guitar, virtually all Western music since the 18th century — is 12edo: twelve equal divisions of the octave, each step being exactly 100 cents. A semitone is 100¢, a whole tone is 200¢, a perfect fifth is 700¢, and so on.

13edo simply divides that same octave into 13 equal steps instead. Each step is about 92.3 cents — just slightly narrower than a semitone. That doesn’t sound like much of a difference, but it cascades into something completely alien from standard Western harmony.

This field of exploring non-12 tuning systems is called xenharmonics (from the Greek xenos, meaning foreign or strange), and 13edo is one of its most beloved and challenging playgrounds.

Why Would Anyone Do This?

Fair question. Here’s the honest answer: not because it’s easy or familiar, but because it opens up sounds that simply cannot exist in 12edo.

Your standard tuning system approximates the natural harmonic series — the overtones that ring out whenever any acoustic instrument plays a note — reasonably well for the 3rd, 5th, and 7th harmonics. This is why major triads and dominant seventh chords feel consonant: they echo nature. 13edo makes a completely different trade. It gives up good approximations of the 3rd harmonic (the perfect fifth) almost entirely, but in exchange it gains remarkably close approximations of the 11th, 13th, 17th, 19th, and 21st harmonics — intervals that are totally absent from 12edo and virtually never heard in Western music.

The result is a tuning with a dreamy, suspended, otherworldly quality. Familiar melodic shapes morph into something uncanny. Chord progressions resolve in ways you’ve never felt before. As one composer put it, 13edo sounds like music from a parallel universe.

The Elephant in the Room: The Fifth Is Broken

If you try to use 13edo like 12edo — stacking thirds, building triads, chasing dominant-to-tonic resolutions — you’ll be disappointed. The “fifth” in 13edo lands at 738 cents, which is about 38 cents sharp of the perfect fifth you know. That’s nearly half a semitone off. It doesn’t function as a perfect fifth in any familiar sense.

Likewise, the “fourth” sits at 646 cents (the dual flat fifth) or 554 cents (closer to a tritone). The major triad shapes 0–4–7 and 0–3–7 (in scale degrees) both sound rough and unsatisfying. If you go into 13edo expecting those intervals to do what they do in 12edo, you’ll just hear a mess.

The key insight is this: 13edo isn’t a broken version of 12edo. It’s a completely different instrument that requires a completely different harmonic language.

The New Consonances: Forget the Fifth, Hello the Ninth

So what does work in 13edo? The answer starts with the major second (184.6¢, approximately 9/8 or 10/9 — a ratio very close to what you’d call a whole tone in 12edo). This interval, and especially its octave equivalent the major ninth, takes on the structural role that the perfect fifth plays in standard harmony. It’s the most stable, open-sounding interval after the octave itself.

The major third (369.2¢, approximating 5/4) and minor third (276.9¢) also work well and feel surprisingly familiar — they’re only about 17 cents flat and sharp of their just counterparts respectively, which is well within the range of normal musical expression.

The most characteristic consonant chord in 13edo isn’t a triad at all, but a three-note chord approximating the ratio 4:5:9 — a root, a major third, and a major ninth. This is the “major triad” equivalent in 13edo. You can think of it as a major third plus a whole tone stacked on top, with no fifth at all. It sounds open, bright, and strangely satisfying once your ears have adjusted.

The corresponding “minor” version (4:9:21, or root + minor third + major ninth) and various four-note extensions involving the 11th and 13th harmonics fill out the harmonic palette considerably.

The Scales: Two Diatonic Systems

One of the most remarkable things about 13edo is that it supports not one but two distinct “diatonic-like” scale systems, each with their own set of modes, chords, and characteristic moods. These are called the archaeotonic (a 7-note scale) and the oneirotonic (an 8-note scale), and they were named by the theorist/composer group Cryptic Ruse after elements of H.P. Lovecraft’s Dreamlands mythos — partly because 13edo itself has that dreamy, hypnagogic quality.

The Archaeotonic: 7 Notes, Lovecraftian Modes

The archaeotonic scale (also called 6L 1s in TAMNAMS notation) is generated by stacking major seconds (2\13, or 184.6¢). It has the step pattern L L L L L L s — six large steps and one small step. If you know the diatonic scale’s pattern of W W H W W W H (or 2 2 1 2 2 2 1 in semitones), you can think of archaeotonic as what happens when one of those half steps turns into a whole step. The result sounds a bit like the Lydian or whole-tone scale, but more tonal and grounded.

The archaeotonic is rich in 4:5:9 triads — there are five “major” versions and one “minor” version available across the scale’s seven degrees. Its seven modes are named after Lovecraftian figures (Nyogthian, Ryonian, Lobonian, etc.) and each mode feels like a slightly twisted cousin of a familiar diatonic mode.

The Oneirotonic: 8 Notes, Dream City Modes

The oneirotonic scale (5L 3s) is the one most 13edo composers gravitate toward first. It has the step pattern L L s L L s L s (in its brightest mode), which gives it 8 notes per octave. Think of it as the diatonic scale with an extra note squeezed in. That extra note doesn’t crowd things — it gives the scale a lush, octatonic ambiance while still being completely navigable.

The eight modes of the oneirotonic are named after Lovecraftian dream cities:

Mode PatternNameRough FeelL L s L L s L sDylathianBright, major-ishL L s L s L L sIllarnekianBright, slightly modalL s L L s L L sCelephaïsianMinor with leading toneL s L L s L s LUltharianDorian-like, no leading toneL s L s L L s LMnarianDorian-like, darkers L L s L L s LKadathianDarker minors L L s L s L LHlanithianDarks L s L L s L LSarnathianDarkest

Where “L” is a large step (184.6¢, roughly a whole tone) and “s” is a small step (92.3¢, roughly a semitone). The Dylathian mode — the brightest — serves as the “major scale” reference point for 13edo, and is used as the basis for the JKLMNOPQ notation system developed by composer Kentaku, which gives the eight scale degrees their own letter names to avoid confusion with standard notation.

The oneirotonic is particularly strong at suggesting modal harmony. The minor modes (Celephaïsian, Ultharian, Mnarian) each have their own characteristic cadential patterns — degrees that function like “dominants” — but they work through different mechanisms than the V–I motion you’re used to. Celephaïsian, for instance, uses its 3rd and 5th degrees as resolving points because of leading-tone motion, not fifth-relationship.

Harmony: Thinking in Whole Tones Instead of Fifths

Here’s a mental model that might help. In 12edo, the circle of fifths is the backbone of functional harmony — chords move by fifth (V to I, ii to V), and the tritone (the diminished fifth) creates the tension that drives resolution.

In 13edo, replace that backbone with chains of whole tones and minor thirds. The most characteristic chord motions are by major second (one whole tone up or down), by major third, and by minor fourth (461.5¢, approximating 21/16). One common functional pattern is:

Root (min) → 4th degree (min) → resolves back to root

…which feels a bit like a diatonic V–I in its tension/release profile, but with a completely different harmonic color.

Some important chord types to know:

Delta-rational major triad: 0–369–646¢ (root, major third, minor fifth) — the closest thing to a major chord with a stable, resonant quality
Delta-rational minor tetrad: 0–277–738–923¢ (root, minor third, major fifth, major sixth) — the “minor” sonority
The 4:5:9 triad: root, major third, major ninth — the most consonant basic harmony
The 4:5:9:11 tetrad: adds an approximation of the 11th harmonic (554¢) — buzzy, characteristic

A useful heuristic from theorist Inthar: the most consonant intervals in 13edo are the major second, major third, and minor third (the “basals”). The fourths, sixths, and sevenths have an intermediate “glittery” quality. The fifths — the intervals that anchor all of Western music — are the most dissonant intervals in 13edo. This is almost a perfect inversion of the standard consonance hierarchy.

Notation: How Do You Even Write This Down?

This is genuinely tricky. Standard notation, built around 7 letter names and sharps/flats, doesn’t map cleanly onto 13 notes. Several solutions exist:

The JKLMNOPQ System (Kentaku/Fox-Raven): The 8 notes of the Dylathian oneirotonic mode get fresh letter names J through Q, avoiding any confusion with traditional note names. Chromatic alterations are added with sharps and flats between most adjacent letters (with no sharp between L/M, O/P, or Q/J — analogous to E/F and B/C in standard notation). This is probably the most beginner-friendly approach. See the full article here.

Ups and Downs Notation: Uses conventional note names but adds ^ (up) and v (down) symbols to represent the additional pitches. More backwards-compatible with standard notation but requires some mental gymnastics given that 13edo’s “best fifth” is 738¢, making the minor second a descending interval in this system.

Numbers: Simply referring to scale degrees as 0–12 works perfectly well for analysis and is common in xenharmonic writing.

For most composers starting out, the JKLMNOPQ system or plain scale-degree numbers are the most practical entry points.

The “Uncanny Valley” Effect: Why 13edo Is Uniquely Unsettling

One fascinating property of 13edo is that any standard 12edo scale can be converted into a 13edo scale either by adding one extra semitone or by replacing a semitone with a whole tone. This means melodic phrases in 13edo often begin sounding familiar — your brain starts to orient around recognizable patterns — and then subtly diverge into something just slightly wrong. This gives 13edo music a uniquely eerie, dreamlike, or uncanny quality that other microtonal systems don’t quite replicate. Phrases feel familiar but keep arriving somewhere unexpected.

This makes it particularly powerful for certain emotional registers: music that wants to feel like a half-remembered dream, something almost-but-not-quite right, or a world that mirrors ours but is subtly different. Several composers have noted that 13edo pairs naturally with surreal, fantastical, or dark atmospheric moods — hence the Dreamlands naming convention.

Actually Getting Your Ears Around It

The good news: you don’t need a custom instrument to start hearing 13edo. Here’s a practical roadmap.

Listen first. The 13edo music page on the Xenharmonic Wiki has an extensive list of compositions. Good starting points include Inthar’s Tonal Studies in various oneirotonic modes, Claudi Meneghin’s organ works, and Aaron Andrew Hunt’s keyboard pieces. The composer Sevish, though primarily known for other EDOs, offers a useful frame for what xenharmonic electronic music can sound like.

Use software. Tools like Scala, ODDSound MTS-ESP, or the Surge XT synthesizer (which has built-in microtuning support) let you retune any MIDI instrument to 13edo. Most modern DAWs support microtuning through MIDI Tuning Standard (MTS). Once you have a keyboard mapped to 13edo, spend time just improvising without trying to make it sound like 12edo music.

Start with the archaeotonic. The 7-note archaeotonic scale, with its wealth of 4:5:9 triads, is arguably the most immediately approachable entry point for someone coming from standard theory. It has cleaner harmonic structure and familiar interval categories, just with a slightly exotic color.

Then explore the oneirotonic. Once the archaeotonic feels somewhat natural, the oneirotonic opens up a much richer modal world. Start with Dylathian (the brightest mode) or Celephaïsian (a compelling minor mode with clear cadential motion).

Drop your expectations about fifths. This really bears repeating. Every time you feel the urge to resolve V–I or use a dominant seventh chord, consciously redirect. Ask instead: where does the 4th degree want to resolve? Where does the whole-tone motion want to go? Let the tuning guide you rather than fighting it.

The Microtone Fox

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