A musician's guide to 33-tone equal temperament: where your familiar scales become something stranger, more colourful, and entirely new.
Theory & Practice · For the Curious Musician · All Skill Levels
You already know what it feels like to play a major chord and have it sound right — that satisfying ring, the way the notes lock together. You probably also know that Western music has been using the same twelve-note octave for centuries. But what if that twelve-note grid is not the only way to divide musical space? What if there were a parallel musical universe — one with 33 notes per octave instead of twelve — where familiar chords take on a new colour, new scales emerge that no Western tradition has ever named, and the very feel of harmony shifts beneath your fingers?
That universe exists. It's called 33 equal divisions of the octave, or 33edo (also written 33tet or 33et). This article is a guide to what it is, why it sounds the way it does, and how you — a musician fluent in standard theory — can start to make sense of it.
📖 A QUICK PRIMER: WHAT IS AN "EDO"?
The tuning system you already use is called 12edo: twelve equal divisions of the octave. Every semitone is exactly 100 cents (there are 1,200 cents in an octave). This system is a convenience — it lets you transpose freely between all keys — but it's a compromise. Your major third (400 cents) is actually about 14 cents wider than the pure, acoustically "perfect" major third (386 cents). You've probably never noticed, because you've heard nothing else.
An EDO simply means: divide the octave into N equal slices. 24edo gives you quarter-tones. 19edo gives you a system beloved by Renaissance theorists. 31edo closely resembles Renaissance meantone temperament. And 33edo gives you slices of about 36.4 cents each — roughly one third of a standard semitone.
The Basic Architecture of 33edo
In 33edo, every step of the scale is approximately 36.4 cents. To put that in perspective: a standard semitone in 12edo is 100 cents, so each 33edo step is a little over a third of a semitone. The octave is reached after 33 of these steps.
When you compare this to the twelve notes you already know, you immediately see both familiarity and strangeness. Some 33edo intervals line up quite well with intervals you already use. Others land in between your familiar notes, in territory that 12edo simply cannot reach. Here are some of 33edo's key intervals alongside their closest familiar equivalents:
| Steps | Cents | Closest familiar interval | Difference | Character |
|---|---|---|---|---|
| 4 | 145¢ | Minor 2nd (100¢) | +45¢ | Wide, tense semitone |
| 5 | 182¢ | Major 2nd (200¢) | −18¢ | Very close to pure 10/9 |
| 8 | 291¢ | Minor 3rd (300¢) | −9¢ | Narrow minor 3rd, close to 13/11 |
| 9 | 327¢ | Minor 3rd (300¢) | +27¢ | Wide minor 3rd, close to 6/5 |
| 11 | 400¢ | Major 3rd (400¢) | 0¢ | Identical to 12edo! |
| 14 | 509¢ | Perfect 4th (498¢) | +11¢ | Slightly wide fourth |
| 19 | 691¢ | Perfect 5th (702¢) | −11¢ | Noticeably flat fifth |
| 22 | 800¢ | Minor 6th (814¢) | −14¢ | Same as 12edo augmented 5th |
You'll notice something immediately striking: step 11 gives you exactly 400 cents — the same major third as 12edo. That's not a coincidence; it arises because 33 is a multiple of 11 (3 × 11), which means 33edo shares the augmented chord (three equal major thirds dividing the octave into thirds) with 12edo. But most everything else diverges.
The Thing About the Fifth
If there's one interval that defines tonal music, it's the perfect fifth. It underpins harmony, voice leading, the circle of fifths — everything. In 12edo, the fifth is 700 cents, just 2 cents flat of the pure acoustic fifth (702 cents). It's close enough that you never notice.
In 33edo, the fifth lands at 691 cents — about 11 cents flat. That's a meaningful difference. It means 33edo fifths have a slightly "melting" quality, a gentle sag. Stacked fifths don't circle back to a clean chromatic scale the way they do in 12edo; instead, the system drifts in its own direction.
This flat fifth is actually central to 33edo's most intriguing scale system: flattertone. Two stacked 33edo fifths (each 691¢) produce an interval of only 1,382 cents — just 0.6 cents shy of the pure 10/9 whole tone (182¢ above the octave). This means that in flattertone, the familiar major scale's whole tones are essentially tuned to the pure ratio 10/9, while the semitones expand to 145 cents. The result is a diatonic scale that feels simultaneously familiar and exotic — like a major scale heard through slightly foggy glass.
"Common practice minor and major chords become more supraminor and submajor in character, making everything sound almost neutral in quality." — Xenharmonic Wiki on 33edo
Major and Minor Chords: Familiar but Shifted
Here's where things get interesting for any musician steeped in tonal harmony. When you build what looks like a major or minor chord in 33edo using the standard flattertone diatonic scale, the result doesn't sound quite the same as what you're used to.
The major third in the flattertone diatonic is 11 steps (400 cents) — identical to 12edo. But the fifth is 19 steps (691 cents), about 11 cents flat. This gives major chords a slightly softer, less triumphant quality. Meanwhile, the minor third in the flattertone diatonic comes out at 8 steps (291 cents) — somewhat narrow compared to the 12edo minor third (300 cents) — adding a pressing, searching quality to minor chords.
Think of it this way: in 12edo, major sounds confident and minor sounds sad. In the flattertone system of 33edo, both become more ambiguous, neutral, questioning. Neither fully commits. Some listeners find this deeply expressive; others find it unsettling. Both reactions make sense.
Two "minor thirds" for the price of one
One of the more delightful quirks of 33edo is that it offers two distinct minor third-sized intervals where 12edo has only one. Step 8 (291¢) is a narrow minor third, very close to the pure ratio 13/11 — a bright, slightly tense sound. Step 9 (327¢) is a wide minor third, approximating the pure 6/5 ratio — darker and rounder. In 12edo, you get one minor third at 300 cents, splitting the difference between these two. In 33edo, you get to choose.
🎵 ON "PURE" INTERVALS AND WHY THEY MATTER
Intervals like the perfect fifth, major third, and minor third have "pure" versions defined by simple frequency ratios — 3:2 for the fifth, 5:4 for the major third, 6:5 for the minor third. When two notes form these ratios, the sound is acoustically stable and smooth; the waveforms align with minimal "beating." 12edo approximates these ratios reasonably well, but not perfectly.
One way to evaluate any alternative tuning system is to ask: how close does it get to these pure (just) ratios, and what new ones does it introduce? 33edo makes different trade-offs from 12edo. It's not especially well-suited to pure 5-limit harmony (the world of thirds and fifths), but it gets meaningfully closer to some higher harmonics — particularly the 11th, 13th, and even 17th — that 12edo barely touches at all.
Scales You've Never Seen Before
Because 33edo has 33 notes per octave, it supports a rich variety of scales with different step sizes. Here are a few of the most interesting ones — many with no equivalent in Western tradition:
Flattertone Diatonic (the "home base")
The most natural seven-note scale in 33edo: step pattern 5 5 4 5 5 5 4. The two types of steps are 182¢ (whole tone) and 145¢ (semitone). It looks like a major scale, but the semitones are wider and the whole tones narrower than in 12edo. Play it and you'll recognise the shape — but something is quietly different. This is an example of a MOS scale (moment of symmetry), the same underlying structure that generates familiar scales like the diatonic and pentatonic in 12edo.
Diasem
A striking nine-note scale with pattern 5 3 5 1 5 3 5 1 5. The tiny 1-step intervals (36¢) act almost like ornamental microtonal grace notes embedded within the scale itself. Diasem has no Western analogue. It's modal and otherworldly, and one of the scales where 33edo sounds most unlike anything in the standard repertoire.
The Iranian Calendar Scale
A quirky connection: 33 is also the number of years in Iran's traditional calendar leap cycle, where a leap year is added once every 4 or 5 years. This maps onto the step pattern 5 4 4 4 4 4 4 4 — a nearly equidistant eight-note scale with one slightly larger step. The connection between musical and calendrical mathematics is one of those unexpected places where number theory reveals hidden symmetry in nature.
August[12]: The Augmented Chromatic
A twelve-note scale built around the August temperament, with pattern 3 2 3 3 3 2 3 3 3 2 3 3. Built around 33edo's perfect augmented chord (three equal major thirds dividing the octave into thirds), this scale gives you twelve pitches organised around three "poles." It shares structural DNA with the augmented scale in 12edo, but with finer subdivisions between its notes.
33edo and the Bohlen–Pierce Scale
One of 33edo's most fascinating properties is its accurate approximation of the Bohlen–Pierce scale — a radical alternative tuning system based on the interval of a twelfth (the tritave, or 3:1 ratio) rather than the octave. The Bohlen–Pierce scale divides the tritave into 13 equal steps, and 33edo contains an excellent approximation: every 4 steps of 33edo (4 × 36.4¢ = 145.5¢) closely matches 1 step of 13edt (the Bohlen–Pierce scale). This means that composers interested in non-octave-based harmony will find 33edo a natural entry point.
Higher Harmonics: Where 33edo Genuinely Excels
While 33edo is admittedly imperfect at representing pure fifths and major thirds, it shows real talent in representing higher harmonics — the upper members of the overtone series that Western music rarely uses explicitly.
The 11th harmonic (which sits between a perfect fourth and tritone in pitch space) is represented with only about 6 cents of error. The 13th harmonic — a tone that sits between a minor and major sixth — is off by only 4 cents. The 17th, 19th, and 21st harmonics are all approximated within about 7 cents. These are the overtones that give instruments like the natural horn, the gamelan, and the human voice their rich, complex timbral textures.
In practical terms, this means that 33edo has a natural aptitude for timbre-like harmony: chords that evoke the inner resonance of a single rich tone, rather than the stacked simplicity of a triad. The full system of harmony provides what theorists call the optimal patent val for slurpee temperament in the 5-, 7-, 11-, and 13-limits.
🔢 THE HARMONIC SERIES IN BRIEF
Strike any note on a piano and you don't just hear one pitch — you hear a series of overtones above it. The first few are simple: the octave (2nd harmonic), the fifth above that (3rd), the double octave (4th), a major third up (5th). But from the 7th harmonic onward, the overtones start landing between the piano's keys — in the cracks of 12edo. A tuning system that approximates these higher harmonics well can produce chords that resonate with a kind of acoustic naturalness that standard triads alone cannot achieve. This is the core of what just intonation and high-limit temperaments are exploring.
33edo and Orgone Temperament
Because 33 is a multiple of 11edo, it inherits 11edo's approximations of the 7th and 11th harmonics through what's called orgone temperament. This means that certain chord structures available in 11edo carry over into 33edo with the same tuning, giving 33edo a connection to 11edo's distinctly modal, open sound world. The 33c and 33cd vals(the technical term for the way a tuning maps onto the harmonic series) also temper out 81/80 (the syntonic comma) and 49/48, placing 33edo in the family of meantone and godzilla temperaments.
How Would You Actually Play in 33edo?
This is the practical question. Most conventional instruments are locked into 12edo — pianos have twelve fixed pitches per octave, guitars have frets placed at 12edo positions. So how do you access 33edo?
Software synthesisers and DAWs are the easiest entry point. Most modern synths allow you to load custom tuning tables in the .tun or Scala format, instantly retuning every key on your MIDI controller to any EDO you like. Free tools like Scala or ODDSound MTS-ESP can send tuning data to compatible plugins.
Fretless instruments — violin, cello, fretless bass, trombone, voice — are physically capable of any tuning immediately. The challenge is purely one of ear training and muscle memory.
The Lumatone is a specialised isomorphic keyboard that can be mapped to any tuning system, including 33edo. The Xenharmonic Wiki lists specific Lumatone mappings for 33edo. It has become one of the most popular hardware options in the xenharmonic community.
For notation, 33edo can be written using standard staff notation with sharps and flats — because its chromatic semitone is exactly 1 step (36.4¢), the familiar accidentals have a consistent meaning. The catch: in distant "keys," you'll need double and triple sharps and flats, which can become visually busy. A specialised notation system called Sagittal (the same system used for 23edo and 28edo) is often preferred for its cleaner symbols.
Music That Has Been Made in 33edo
33edo is genuinely under-explored territory, but there is a growing body of work. Here is a selection of composers and pieces from the Xenharmonic Wiki's listings:
- Johann Sebastian Bach, "Contrapunctus 4 & 11" from The Art of Fugue — rendered by Claudi Meneghin (2024). A fascinating exercise in hearing Baroque counterpoint through a shifted harmonic lens.
- Claudi Meneghin, "Rising Canon on a Ground" for Baroque Oboe, Bassoon, Violone (2024) — an original 33edo composition in a Baroque style, demonstrating how idiomatic the system can be for contrapuntal writing.
- Claudi Meneghin, "Lytel Twyelyghte Musicke" for Brass and Timpani (2024)
- Bryan Deister — several freely improvised pieces (2023–2025): groove 33edo, 33edo jam, 33edo riff, 33edo improv.
- Budjarn Lambeth, "Enchanted Shopping Mall" (2024) — an evocative ambient/electronic piece making use of 33edo's neutral, ambiguous harmonic palette.
- Relyt R, "Nongenerate" and "Kolmekymmentäkolme" — available on Bandcamp and Spotify.
- Chris Vaisvil, "5 5 1 mode of 33 equal" — a study in one of 33edo's characteristic modal scales, with video.
- Xeno*n*, "Mysteries of Thirty-Three" (2024)
The Honest Assessment: What 33edo Is Good At (and What It Isn't)
No tuning system does everything well. 33edo's character comes directly from its trade-offs, so it's worth being honest about both sides.
Where 33edo struggles: Traditional tonal harmony — the world of I–IV–V–I progressions and standard triads — is genuinely compromised. The fifth is 11 cents flat, which is noticeable. The Xenharmonic Wiki bluntly notes that 33edo is "near-maximally bad for its size for tonal harmony" — meaning for a 33-note system, you'd have a hard time finding a worse choice if clean triads are your priority. Its triple, 99edo, is a much stronger system for 7-limitharmony.
Where 33edo excels: Higher-limit harmony (11th, 13th, 17th, 19th harmonics), neutral and ambiguous tonal colours, modal and scalar exploration, the Bohlen–Pierce sound world, and structurally interesting scales with unusual step ratios. It's a system that rewards composers willing to meet it on its own terms rather than trying to force it into a 12edo mould.
33edo's triple — the 99-note system — would be a very strong 7-limit system, and it indeed is. But 33edo alone carries its own strange charm: a system that sounds almost tonal, but isn't quite.
A Starting Point for Exploration
If you want to begin exploring 33edo, here is a practical path forward.
Start by loading a 33edo tuning table into your preferred software synth and spending time with the flattertone diatonic scale — the seven-note scale built from 33edo's flat fifths (step pattern 5 5 4 5 5 5 4). Play through the modes. Notice where major feels like it wants to be minor, and where minor feels like it wants to resolve somewhere unexpected. Let the neutral quality of the harmony wash over you before trying to analyse it.
Then explore the two different minor thirds (steps 8 and 9). Play them back to back. The difference between 291¢ and 327¢ is dramatic — one is bright and pressing, one is dark and settled. Neither is the 300¢ you're used to, but both are expressive in their own way.
From there, try building chords around the augmented triad (three stacked 400¢ major thirds) and use it as a pivot to modulate — since the augmented chord divides the octave symmetrically in 33edo just as it does in 12edo, it can act as a harmonic hinge between regions of the scale that have no equivalent in standard theory.
Above all: resist the urge to make 33edo sound like 12edo. Its power lies precisely in the places where it diverges. The flat fifth, the two minor thirds, the neutral chords, the wide semitone — these are not defects to be apologised for. They are the sound of a different musical logic, one that has been waiting patiently in the mathematics of the octave for someone to explore it.
33edo is, in the end, one answer to a question most musicians never think to ask: what if we had drawn the grid differently? The answer turns out to be: stranger, more ambiguous, and in its own way, beautiful.
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