A list of scales in 31edo, with descriptions

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Actual scales you can make music with in 31-equal temperament (31-tet)

So someone recommended 31edo to you. Maybe you watched a YouTube video, heard a piece of microtonal music that caught your ear, or read that 31-tone equal temperament has better major thirds than standard tuning. Now you're wondering: "Great, but what scales do I actually use to write music in this thing?"

This article is for you. You don't need a maths degree. You do need to already be comfortable with concepts like diatonic modes, pentatonic scales, and the idea of a "scale" as a selection of notes from a larger pitch set. We'll go through the most musically useful scale families in 31edo, explain what makes each one interesting, and link to the Xenharmonic Wiki so you can go deeper.

First: A Quick Orientation to 31edo

31edo divides the octave into 31 equal steps, each about 38.7 cents wide (a cent is 1/100th of a semitone, so each step is roughly 40% of a standard semitone). One step is called a diesis — you'll hear this word a lot.

Your familiar 12-note chromatic scale fits inside 31edo as a subset. The major scale, minor scale, and all the church modes still exist and work just as you'd expect. But now you have access to new intervals that don't exist in 12-tone tuning:

  • Neutral seconds and thirds — halfway between major and minor, with a characteristic "floating" or Middle Eastern quality
  • Subminor and supermajor thirds — a third that's noticeably flatter than minor (6:7 ratio, very resonant) or sharper than major (9:7 ratio, tense and bright)
  • Septimal intervals — intervals that lock in perfectly with the 7th harmonic of the overtone series, giving chords an unusually pure, ringing quality

The perfect fifth (3/2) in 31edo is just 5.2 cents flat of pure — barely noticeable. The major third is less than 1 cent sharp of the pure 5/4 ratio. This means 31edo is one of the best equal temperaments for playing music that sounds in tune while still being a closed, repeatable system you can write standard notation for.

Part 1: The Familiar Scales (Meantone Family)

These are the scales you already know. In 31edo they work exactly as in 12-tone, just with better tuning — especially those gorgeous major thirds.

The Diatonic Scale (Major/Minor and Modes)

Step pattern: 5 5 3 5 5 5 3

This is your standard major scale. In 31edo terms, it uses two step sizes: a "whole tone" of 5 dieses and a "semitone" of 3 dieses. All seven church modes (Dorian, Phrygian, Lydian, Mixolydian, etc.) are rotations of this pattern.

The diatonic scale in 31edo is strictly proper — a technical term meaning every interval class is unambiguously larger or smaller than adjacent classes. In practical terms this means the scale has excellent perceptual clarity: listeners can always tell where they are within it. By contrast, the same scale in 12-tone is only "proper" (not strictly proper), because the tritone is ambiguous. 31edo resolves this ambiguity.

Melodic Minor

Step pattern: 5 3 5 5 5 5 3

Identical in structure to the familiar jazz melodic minor scale (Ionian ♭3). All its modes — including the Lydian Dominant and the Altered scale — work as expected.

Harmonic Minor

Step pattern: 5 3 5 5 3 7 3

The characteristic augmented second between scale degrees 6 and 7 is now 7 dieses wide — noticeably larger than in 12-tone, giving it an even more dramatic, "Arabic" flavour.

Harmonic Major

Step pattern: 5 5 3 5 3 7 3

The harmonic major scale (major scale with a flat 6) works similarly, with the augmented second appearing between degrees 5 and 6.

Major Pentatonic / Minor Pentatonic

Step pattern: 5 5 8 5 8 (major pentatonic)

The familiar five-note scale. The "wide step" is 8 dieses (two whole tones), which is the same as in 12-tone relative to the octave. Works exactly as you'd expect for pentatonic melodies and improvisation.

Part 2: Scales With Neutral Intervals (Mohajira Family)

This is where 31edo starts to open up new territory. The neutral third — exactly halfway between major and minor — is one of 31edo's most expressive resources. It sits close to the pure 11:9 ratio (about 347 cents) and has a quality that many listeners describe as "floating," modal, or Middle Eastern.

The generator of this scale family is 9 dieses (about 348 cents), half of a perfect fifth. This family is called mohajira.

Mohajira / Neutral Dorian (the "home" mohajira mode)

Step pattern: 4 5 4 5 4 5 4

Seven notes, entirely built from alternating minor seconds (4 dieses) and neutral seconds (5 dieses... wait — here "neutral second" means the step of about 193 cents, close to 9/8). Actually, the step sizes here are: 4 dieses ≈ 155¢ (a neutral second interval smaller than a whole tone) and 5 dieses ≈ 194¢ (a whole tone). The result is a scale with no major or minor thirds — only neutral thirds everywhere.

This scale is excellent for compositions that want to avoid the major/minor polarity of Western harmony entirely.

Maqam Rast (Sikah mode)

Step pattern: 4 5 5 4 4 5 4 (also listed as Neutral Diatonic Hypolydian)

Maqam Rast is one of the most foundational scales in Arabic music. In 31edo it's approximated very well. It has a neutral third above the tonic (giving it that characteristic "neither major nor minor" quality) and behaves somewhat like a major scale with a lowered third. This is one of the most useful scales for musicians interested in Middle Eastern or Mediterranean sounds.

Maqam Bayati

Step pattern: 4 4 5 5 3 5 5

Another foundational maqam. Bayati starts with a minor second followed by a neutral second, creating a distinctive downward pull at the opening of a phrase. 31edo approximates this well, and the scale has been used in compositions that blend Western and Arabic musical thinking.

Maqam Saba

Step pattern: roughly 4 3 6 5... (non-octave variants)

Saba is one of the most emotionally intense maqams, featuring a characteristic lowered fourth. In 31edo it can be approximated and used as a source of dramatic, tense melodic material.

Sheimanic

Step pattern: 4 4 4 4 5 5 5

A strictly proper mohajira-type scale with an even, gradual quality. Useful for composers wanting a neutral-flavored scale with a more "balanced" feel than Mohajira itself.

Thaiic

Step pattern: 4 5 5 4 4 4 5

Another strictly proper scale in the mohajira family, with a slightly different modal character — less symmetrical than Mohajira, giving it more of a directional, tonal feel.

For a complete list of strictly proper mohajira-type scales, see the Strictly Proper 7-Tone 31edo Scales page on the Xenharmonic Wiki.

Part 3: Scales With Septimal Intervals (Subminor/Supermajor Colors)

These scales use 31edo's excellent approximation of the 7th harmonic — the "barbershop seventh" or "blue note" that sits between the minor seventh and the major sixth. This interval (7:4 ratio) is only 1 cent flat in 31edo.

The subminor third (6:7 ratio, about 267 cents) and supermajor third (9:7 ratio, about 435 cents) give these scales a rustic, blues-inflected, or "just intonation" quality.

Septimal Natural Minor ("September Natural Minor")

Step pattern: 5 2 6 5 2 5 6

A variant of the natural minor scale where the minor third and minor seventh are pulled down to the septimal (7-limit) versions. The "2-step" semitones are very small — almost quarter-tones — giving this scale a gritty, bluesy edge.

Subminor Pentatonic

Step pattern: 7 6 5 7 6

A five-note scale built from subminor and supermajor intervals. Deep, resonant, and unusual — useful for slow, meditative music or anything that wants a strong "acoustic" or "just intonation" flavour.

Supermajor Pentatonic

Step pattern: 5 6 7 6 7

The "inversion" of the subminor pentatonic. Bright, stretched, and distinctive.

Part 4: The Orwell Scale Family

Orwell temperament is generated by the subminor third — 7 dieses (about 271 cents). Stack these, and you get a family of scales described by the musician Sevish as "bittersweet" — melodically flexible, with a quality somewhere between the familiarity of minor and something genuinely new.

Orwell[9] — the 9-note Orwell scale

Step pattern: 4 3 4 3 4 3 4 3 3

Nine notes generated by stacking subminor thirds. This is probably the most praised non-diatonic scale in 31edo among composers. It contains a mix of perfect fifths, subminor thirds, and neutral intervals that create a rich harmonic palette. The step sizes have a "soft" 4:3 ratio, which gives it a melodically flowing, ambiguous quality where you can hear both diatonic and microtonal logic at once. It contains the orwell tetrad chord (48:56:66:77) — a four-note chord unique to this temperament.

Orwell[5] — the 5-note Orwell scale

Step pattern: 7 7 7 7 3

A pentatonic subset of orwell. Very evenly spaced except for one "leftover" small step. Good for getting the flavour of orwell harmony with fewer notes to navigate.

Graham Orwell (MODMOS)

Step pattern: 3 4 3 4 4 3 4 3 3

A modified version of Orwell[9] (technically a MODMOS) named after microtonal theorist Graham Breed. It preserves the melodic character of Orwell[9] but adjusts the harmony to feature four consecutive perfect fifths and a major triad on the tonic that extends to a 4:5:6:7:11.

Part 5: The Squares Scale Family

Squares temperament is generated by the supermajor third — 11 dieses (about 426 cents, close to the 9:7 ratio). This is one of the more dramatic 31edo scale families.

Squares[5]

Step pattern: 9 9 2 9 2

A five-note scale with a very high contrast between large steps (9 dieses) and tiny steps (2 dieses, about 77 cents — smaller than any semitone in 12-tone). These small steps create strong directional pull, making the scale feel harmonically intense and "purposeful." The hardness ratio is 9:2, giving it a very angular, resolved quality.

Squares[8]

Step pattern: 2 7 2 7 2 2 7 2

Eight notes, still with those same dramatic contrasts. This scale has 4 perfect fifths and is rich in neutral triads, subminor triads, and supermajor triads. The Xenharmonic Wiki notes it has a hardness of 7:2, "leading to the scale feeling dramatic and intense."

Oneirotonic / A-Team[8] (related to Squares)

Step pattern: 5 5 2 5 5 2 5 2

An 8-note scale generated by 12 dieses (about 464 cents). This "oneirotonic" scale has been called dreamlike — it contains good approximations of 9/8, 5/4, and 7/6, and the step pattern creates a sense of flowing between different tonal centers. It was described as "similar to the 5L 3s scale in 13edo but with better-tuned intervals."

Part 6: Mothra / Mosura Scale Family

Mothra temperament is generated by 6 dieses (about 232 cents — a supermajor second, close to 8:7). Three of these generators make a perfect fifth.

Mothra[6]

Step pattern: 6 6 6 6 1 6

Six notes, nearly equal, with one tiny 1-diesis "leftover" step. The near-equal spacing gives this scale an almost whole-tone quality while the single tiny step provides direction. The Mothra hexad chord (5 Parent Chord 84:98:128:147:192:224) is a six-note chord unique to this scale family.

Mothra[11] — de Vries 11-tone scale

Step pattern: 5 1 5 1 5 1 5 1 5 1 1

Eleven notes, with a chromatic feel but with the characteristic supermajor seconds giving it a different flavour from a standard 12-tone chromatic scale. Named after Dutch theorist de Vries.

Part 7: Valentine and Miracle (Chromatic-like Scales)

For composers who want more notes per octave — more chromatic density — 31edo has two notable scale families that provide 15–21 notes while remaining structured and coherent.

Valentine[16]

Step pattern: 2 2 2 2 2 2 2 2 2 2 1 2 2 2 2 2 (called "Semi-Equalized Armodue")

Valentine temperament is generated by 2 dieses (about 77 cents — close to a minor semitone). The 16-note scale is nearly equal, giving it a chromatic feel with some quarter-tone flexibility. Useful for composers who want to write densely chromatic music with occasional microtonal colour.

Blackjack / Miracle[21]

Step pattern: 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

Miracle temperament is generated by 3 dieses (about 116 cents — a minor second or "secor"). The 21-note Blackjack scale is a near-equal division of the octave into 21 steps, with every note close to a pure just intonation interval. This is an advanced scale for composers who want near-just intonation with full chromatic flexibility. Blackjack is one of the most-discussed scales in all of microtonal theory.

Miracle[10]

Step pattern: 3 4 3 4 4 4 4 3 4 4 (approximately)

A 10-note subset of Miracle. More manageable than the 21-note Blackjack, and containing the key harmonic resources of miracle temperament.

Part 8: MOS Scales for Specific Moods

The following are some individual scale recommendations from the literature, each with a distinct character:

Würschmidt[10]

Step pattern: 8 1 1 8 1 1 8 1 1 1

Generated by the major third (10 dieses). Has a hardness ratio of 8:1 — the largest contrast of any common 31edo MOS. The single-diesis steps act as strong leading tones, making resolutions very dramatic. Described as "superhard."

Myna[11]

Step pattern: 1 1 6 1 1 6 1 1 6 1 6

Generated by 8 dieses (a minor third). A mostly equal scale with small "diesis" steps scattered throughout. The dieses add colour without disrupting the near-equal melodic flow. Named after the myna bird.

Neutral[7]

Step pattern: 4 5 4 5 4 5 4

Wait — this is the same as Mohajira above, but the 7-note MOS of the neutral third generator specifically. Pure alternation of two similar step sizes gives it an almost pentatonic spaciousness despite having seven notes.

Part 9: Non-Octave and Double Modes

Erose–McClain Double Modes

These are a family of scales unique to 31edo, created by starting with a diatonic mode and flattening the octave by one diesis (from 31 steps to 30 steps). The result is a scale with a "pseudo-octave" — it repeats at a slightly compressed interval. Zhea Erose discovered Double Lydian; Levi McClain worked out the full set of seven.

The seven double modes are:

  • Double Lydian — 5 5 5 3 5 5 2
  • Double Ionian — 5 5 3 5 5 5 2
  • Double Mixolydian — 5 5 3 5 5 3 4
  • Double Dorian — 5 3 5 5 5 3 4
  • Double Aeolian — 5 3 5 5 3 5 5
  • Double Phrygian — 3 5 5 5 3 5 4
  • Double Locrian — 3 5 5 3 5 5 4

Each one sounds recognizably like its diatonic parent but with a slight "shimmer" or "drift" as the scale reaches its top — because the octave is compressed, melodies feel like they're curving slightly inward. This article from the Xenharmonic Wiki has more detail.

Part 10: Greek Tetrachord Modes

31edo is especially good at approximating the ancient Greek tetrachord system, which divides a perfect fourth into different patterns depending on the "genus" — diatonic, chromatic, or enharmonic.

Soft Diatonic (Ptolemy's Soft Diatonic)

Step pattern: 2 5 6 2 5 6 5 (Soft Diatonic Mixolydian)

Uses whole tones and "soft" half-tones that are smaller than standard but larger than the ancient enharmonic dieses. Has a gently exotic quality.

Hemiolic Chromatic

Step pattern: 2 2 9 2 2 9 5 (Hemiolic Chromatic Mixolydian)

A chromatic tetrachord genus with very large "augmented seconds" (9 dieses ≈ 348 cents — a neutral third!) and very small steps. Extremely dramatic, with a sound reminiscent of certain Balkan or Middle Eastern folk traditions.

Enharmonic

Step pattern: 1 2 10 1 2 10 5 (Enharmonic Mixolydian)

The ancient enharmonic genus, with tiny 1-diesis and 2-diesis steps and a massive “ditone” of 10 dieses (≈ 387 cents, a near-pure major third). Historically this genus was considered the most sophisticated and difficult to sing. In 31edo it becomes playable. The tiny dieses create an intense, focused sound.

Part 11: Harmonic Series Scales (Mode 8 etc.)

31edo can approximate scales built directly from the harmonic overtone series — something that regular 12-tone can’t do well.

Mode 8 of the Harmonic Series

Step pattern: 5 5 4 4 4 3 3 3

This scale uses the 8th through 16th harmonics as its pitches. In 31edo these land at:

  • 8: unison (tonic)
  • 9: major second (5 dieses)
  • 10: major third (10 dieses)
  • 11: superfourth (14 dieses)
  • 12: perfect fifth (18 dieses)
  • 13: neutral sixth (22 dieses)
  • 14: subminor seventh (25 dieses)
  • 15: major seventh (28 dieses)
  • 16: octave (31 dieses)

This is one of the most consonant scales available in any tuning system, since every note relates to the tonic by a simple ratio. The “superfourth” on the 11th harmonic is the most exotic note — not quite a tritone, not quite a fourth — but in context it sounds natural and pure.

Part 12: Euler–Fokker Genera

These are scales derived from products of prime numbers 3, 5, and 7 — a concept developed by the mathematician Leonhard Euler and championed in 31edo by Dutch physicist Adriaan Fokker, after whom the 31-tone pipe organ in Amsterdam is named.

Genus Primum

Step pattern: 5 8 5 13

Four notes. The simplest Euler–Fokker genus, a kind of tetrad scale. Very open and consonant, built on intervals derived from multiples of 3 and 5.

Genus Diatonicum

Step pattern: 5 5 3 5 5 3 2 3

Eight notes, extending the diatonic idea into septimal territory by adding a 7-limit note. Bridges familiar diatonic harmony with septimal colour.

For the full list of Euler–Fokker genera, see the 31edo Modes page on the Xenharmonic Wiki.

Practical Advice: Where to Start

If you’re just getting into 31edo, here’s a recommended path through the scales above:

Start here:

  1. The standard diatonic modes — they work just like in 12-tone, but the major thirds ring more purely and the minor thirds feel more resonant.
  2. The harmonic minor and harmonic major — the augmented second sounds more vivid and expressive than in 12-tone.

Then try: 3. Maqam Rast and Maqam Bayati — these use neutral intervals gently. They’re familiar from world music and provide an accessible entry into 31edo’s “new” territory. 4. Orwell[9] — this is the scale that often wins over composers who say “but I can’t hear where I am.” The bittersweet, flowing quality is immediately musical.

Once you’re comfortable: 5. The Erose–McClain Double Modes — a beautiful and very 31edo-specific sound. 6. Harmonic Series Mode 8 — deeply consonant, unlike anything in 12-tone. 7. Squares[8] or Oneirotonic — for those who want drama and a more radical departure.

Resources for Going Further

Step patterns in this article are given as sequences of dieses (single steps of 31edo, each ~38.7 cents).

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