53-equal temperament (53edo): near-just intonation for fixed pitch instruments

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This article is a guided tour of 53-edo — what it is, why it sounds the way it does, and how you might actually use it

What Is 53-edo?

If you’ve spent any time playing with tuning systems beyond standard 12-tone equal temperament (12-TET), you’ve probably heard the name “53-edo” thrown around. It has a reputation in microtonal circles as something close to a holy grail: a tuning system large enough to be remarkably accurate, yet structured enough to feel familiar to any musician who already knows their way around a circle of fifths.

53-edo (also written 53-EDO, 53-et, or 53-TET) stands for 53 equal divisions of the octave. Just as 12-TET divides the octave into 12 equal semitones of exactly 100 cents each, 53-edo divides the octave into 53 equal steps of approximately 22.6 cents each.

To put that in perspective: 22.6 cents is less than a quarter of a standard semitone. Each step is smaller than what most listeners would consciously perceive as a “different note” in a melodic context, yet together these tiny steps give the system extraordinary harmonic flexibility — as we’ll see.

The full system contains 53 distinct pitches per octave, and since 53 is a prime number, the system has no smaller equal-tempered subsets (unlike 12-TET, which contains 6-TET, 4-TET, 3-TET, and 2-TET as subsets).

A Very Brief History

The remarkable properties of the number 53 in musical contexts were apparently first noted by Isaac Newton, who observed that 53 divisions of the octave provide an excellent approximation of the 5-limit just intervals (more on what that means below). The tuning has since attracted interest from theorists across centuries and cultures — it has found particular use in Arabic, Turkish, and Persian music, where microtonal inflections are not exotic curiosities but core features of the tradition.

The smallest step of 53-edo is sometimes called the Holdrian comma (after the Ottoman theorist Mikhail Mushaqah, sometimes called Holdrian), because it closely approximates both the Pythagorean comma (~23.46 cents) and the syntonic comma (~21.51 cents). This is not a coincidence — it is the key to understanding why 53-edo is so harmonically powerful.

The Core Idea: Just Intonation and Why It Matters

To understand why 53-edo is special, you first need to understand just intonation (JI).

When two notes sound together, their relationship is most consonant when their frequencies form simple whole-number ratios. A perfect fifth is a ratio of 3:2 — the higher note vibrates 3 times for every 2 vibrations of the lower. A major third is 5:4. A minor third is 6:5. These ratios produce minimal beating (the “wobble” you hear when two mistuned notes clash), resulting in chords that sound stable, pure, and resonant.

Our standard 12-TET is a compromise. It tunes the perfect fifth to 700 cents rather than the pure 701.955 cents — a tiny error of about 2 cents, barely noticeable. But the major third is tuned to 400 cents, when the pure just major third is 386.3 cents. That’s an error of nearly 14 cents — very noticeable to trained ears, and the reason pure 12-TET major chords have that slight tension compared to, say, a well-tuned barbershop quartet singing a cappella.

In just intonation, all intervals are tuned to exact simple ratios, and chords can be completely beatless and pure. The downside: JI is difficult to navigate across keys. Transpose a JI scale to a different root and many intervals shift slightly, making consistent keyboard or fretboard layouts nearly impossible.

53-edo solves this problem. It approximates just intervals so accurately that for most practical purposes it sounds like just intonation, while maintaining the equal-tempered consistency that makes transposition and modulation straightforward.

How Accurate Is 53-edo?

The accuracy figures are striking. Here’s how 53-edo approximates the core just intervals compared to 12-TET:

IntervalJust ratioJust value (cents)53-edo valueError12-TET errorPerfect fifth3/2701.955¢701.887¢−0.07¢−1.96¢Major third5/4386.314¢384.906¢−1.41¢+13.69¢Minor third6/5315.641¢316.981¢+1.34¢−15.64¢Major second (large)9/8203.910¢203.774¢−0.14¢−3.91¢Major second (small)10/9182.404¢181.132¢−1.27¢+17.60¢Minor second16/15111.731¢113.208¢+1.48¢−11.73¢

The perfect fifth in 53-edo is so accurate that the next equal temperament with a more accurate fifth is 200-edo — nearly four times as large. Because the fifth is so perfect, 53-edo is also essentially a seamless Pythagorean tuning, meaning stacked fifths produce intervals indistinguishable from pure Pythagorean ones in most musical contexts.

The major third — the interval most brutally mistuned in 12-TET — is off by only 1.41 cents in 53-edo. For reference, most listeners can’t detect pitch differences below 5–6 cents in a melodic context, and pure JI major thirds beating against 53-edo major thirds would produce only about 0.5 beats per second at concert pitch. Chords in 53-edo can sound remarkably close to beatless.

The Syntonic Comma: 53-edo’s Secret Weapon

Here is the detail that makes 53-edo feel surprisingly manageable for musicians who know standard theory: the syntonic comma is mapped to a single step.

The syntonic comma (ratio 81/80, approximately 21.5 cents) is the interval that describes the difference between the Pythagorean major third (built from four pure fifths) and the pure just major third. In 12-TET, both are mapped to the same pitch: the syntonic comma is “tempered out” entirely and disappears. This is why 12-TET has only one kind of major third.

In 53-edo, the syntonic comma is not tempered out — it’s mapped to one step of 22.6 cents. This means 53-edo distinguishes between a Pythagorean major third (18 steps, ~407¢) and a pure just major third (17 steps, ~385¢). These are genuinely different notes.

This single-step comma has a beautiful practical consequence: you can think of 53-edo as a standard 12-note chromatic scale with a small arrow accidental (one step up or down) that moves any note to its harmonically sweeter just-intonation version. The Pythagorean scale you already know is embedded in 53-edo; the extra steps let you fine-tune each note toward pure resonance.

This is exactly what the Xen Wiki describes as the “spiral of fifths”: the standard circle of fifths “bent” into a 12-spoked spiral, with each note having neighbors one comma away. For a musician already fluent in functional harmony, this is a remarkably intuitive way to navigate the system.

Beyond 5-Limit: Seventh Chords and Further

So far we’ve discussed “5-limit” harmony — intervals whose ratios involve only the primes 2, 3, and 5 (unisons, octaves, fifths/fourths, and thirds). 53-edo also does useful work in the 7-limit, which adds ratios involving the prime 7, giving access to the harmonic seventh (7/4 ≈ 969¢), the septimal minor third (7/6 ≈ 267¢), and others.

The harmonic seventh (7/4) is the interval that gives blues and barbershop their characteristic sound — slightly flat of a standard minor seventh, producing an extremely stable, non-dissonant quality that 12-TET’s 1000-cent minor seventh can only approximate. In 53-edo, 7/4 is represented by step 43 (973.6¢), an error of just +4.76 cents — usable, though less accurate than the 5-limit intervals.

Crucially, 53-edo tempers out the marvel comma (225/224 ≈ 7.7 cents), which means that the small discrepancy between the Pythagorean and just versions of certain 7-limit intervals is resolved, connecting 5-limit and 7-limit harmony into a consistent framework. This is what allowed composer and theorist John O’Sullivan to design his 12-note “Eagle 53” subset of 53-edo — a scale with both a pure just major third and a usable harmonic seventh, with every chord in the scale close to beatless.

The 11-limit and 13-limit (ratios involving primes 11 and 13) are where things get more contested. The 53-edo representation of the 11th harmonic (11/8, the “acoustic fourth”) carries an error of about −7.9 cents, which is right at the edge of what most listeners find tolerable. Some theorists consider 53-edo a fully adequate 13-limit tuning; others prefer 41-edo or 46-edo for 11-limit harmony. What’s clear is that 53-edo offers more harmonic territory than any equal temperament smaller than it, and far more than 12-TET.

The Familiar Inside the Unfamiliar: Diatonic Scales

One of the most reassuring things about 53-edo for a classically trained musician: your existing scales are still there.The diatonic major scale still exists in 53-edo; it just has more than one form depending on which just intervals you prioritize.

The Ptolemy–Zarlino just major scale (the one that was the theoretical ideal for Renaissance polyphony) maps to 53-edo steps as: 9 8 5 9 8 9 5 — adding up to 53. Compare to the 12-TET major scale pattern of 2 2 1 2 2 2 1 (in semitones): the structure is analogous. The step sizes 9, 8, and 5 correspond to the large whole tone (9/8), the small whole tone (10/9), and the diatonic semitone (16/15) of just intonation — the very intervals that 12-TET collapses into two sizes (tone and semitone).

Similarly, the just minor scale maps to: 9 5 8 9 5 9 8.

This means that if you’re playing or writing in a single key, 53-edo essentially gives you the just-intonated version of the scale you already know, with the added benefit that — unlike strict JI — you can modulate to any key without retuning.

A 12-Note Practical Subset: Eagle 53

For musicians who want to explore 53-edo without redesigning their instruments or workflow from scratch, composer John O’Sullivan developed Eagle 53: a carefully chosen 12-note subset of 53-edo that maximises harmonic purity while remaining playable on standard 12-note interfaces (with appropriate retuning).

Eagle 53 is essentially a just-intonated chromatic scale built around the ratios 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, and 15/8 — nearly identical to the historical “Euler’s Genus Diatonico-chromaticum” and “Ellis’ Duodene”. Every one of these notes is within 1.5 cents of its just target (except the tritone 7/5, which is ~6.2 cents sharp, still well within most listeners’ tolerance).

In Eagle 53, a major chord (2:3:4:5:6:8) can be played on the roots E, F, G, A, and B; minor chords (10:15:20:24:30:40) on E, G#, A, B, C#, and D#. Every single chord in the subset is within 1.4 cents of pure just intonation. For a guitarist or pianist interested in just intonation without the full complexity of a 53-note per octave instrument, Eagle 53 is a practical entry point.

Notation: How Do You Write 53-edo Music?

Notation is one of the practical challenges of any microtonal system. Several approaches exist for 53-edo:

Ups and downs notation is perhaps the most intuitive for musicians who already read standard staff notation. In this system, the existing note names (C, D, E, F, G, A, B and their sharps/flats) are kept, but small arrows (↑ for “up” and ↓ for “down”) indicate one-step adjustments. A note written as “↓E” (a “down E”) is one step lower than a standard E — the difference between the Pythagorean and just major third. Two arrows (↑↑ or ↓↓) indicate two steps. This maps directly onto the spiral-of-fifths structure described above, keeping the system legible to classically trained readers.

Sagittal notation is a more comprehensive microtonal notation system that uses a variety of arrow-like accidentals to specify exact intervals across many different tuning systems. 53-edo can be fully notated using the Spartan subset of Sagittal symbols, where the apotome (chromatic semitone) equals 5 steps and the limma (diatonic semitone) equals 4 steps.

Temperament Theory: What 53-edo “Tempers Out”

For those interested in going deeper into the theory: a regular temperament is defined by the commas it tempers out — small just intervals that the system treats as zero. The commas a tuning ignores determine its harmonic character.

53-edo’s most important tempered commas include:

  • The schisma (32805/32768, ~2 cents): The difference between a Pythagorean and a just major third, reached by stacking 8 fifths versus 5 octaves + a major third. Tempering this out makes 53-edo a schismatic temperament, meaning that in Hanson/catakleismic mapping, the best approximation to a pure major third is actually a diminished fourth (e.g., E–Ab rather than E–G#). This is musically counterintuitive but theoretically elegant.
  • The kleisma (15625/15552, ~8.1 cents): This comma links stacked minor thirds to a pure major sixth. Tempering it out means 53-edo supports the Hanson temperament (also called catakleismic), where a chain of 6 minor thirds approximates a pure octave.
  • The marvel comma (225/224, ~7.7 cents): Tempered out in 53-edo’s 7-limit mapping, this links 5-limit and 7-limit harmony into a coherent framework.
  • Mercator’s comma (353/284, ~3.6 cents): 53-edo is in fact named after this comma — 53 perfect fifths nearly exactly span 31 octaves, with the tiny remainder being Mercator’s comma, which 53-edo sets to zero.

Linear Temperaments Within 53-edo

Because 53-edo is a relatively large system, it contains many interesting rank-2 temperaments — tuning systems that can be derived from it by choosing a generator other than the single step.

Some notable ones supported by 53-edo:

  • Garibaldi/Pontiac (generator: 22 steps ≈ 498¢, a near-perfect fourth): Essentially a meantone-adjacent system emphasising 5-limit accuracy.
  • Hanson/Catakleismic (generator: 14 steps ≈ 317¢, a minor third): Six minor thirds in a row approximate a pure major sixth. Produces rich, interleaved harmonies.
  • Orwell (generator: 12 steps ≈ 272¢): A generator slightly larger than a 7/6. 53-edo is the optimal patent val for 11-limit Orwell temperament.
  • Amity (generator: 15 steps ≈ 340¢): Named after the amity comma it tempers out.
  • Vulture/Buzzard (generator: 21 steps ≈ 475¢): A generator close to 21/16. Produces very complex but harmonically rich structures.

Each of these is a different way of “zooming in” on 53-edo — choosing a particular interval as your primary building block and generating scales from it, rather than working with all 53 notes at once.

MOS Scales in 53-edo

A MOS scale (moment of symmetry) is a scale generated by stacking a single interval and reducing to within an octave, so that the resulting scale has exactly two step sizes. The diatonic scale is the most familiar MOS: it’s generated by stacking fifths, and it has two step sizes (tones and semitones).

53-edo contains the diatonic MOS (7 notes, generated by the near-perfect fifth of 31 steps), but also supports many other MOS scales with different numbers of notes and different characteristic sounds. The 1953 scale (with steps defined by its generator pattern) is one notable larger MOS in 53-edo that has attracted interest from composers.

Because the system is so large and its fifth so accurate, it supports an unusually wide variety of well-formed MOS scales compared to smaller equal temperaments.

53-edo in Practice: Instruments and Software

Playing or composing in 53-edo requires either specially designed instruments or software retuning. Some practical options:

  • The Lumatone keyboard (an isomorphic hexagonal MIDI controller) has official 53-edo mappings, including “Preset 9–53-EDO Bosanquet,” which uses the Bosanquet layout aligning the system with standard Pythagorean structure. There are also Hanson, Semaja, and other generator-based layouts.
  • Microtonal guitar and luthiery: Guitars and other fretted instruments can be refretted for 53-edo, though the small step size (22.6 cents) means some frets are very close together, which is physically challenging on smaller instruments.
  • Software: DAWs with microtuning support (via MTS-ESP, Scala, or similar tools) can retune virtual instruments to 53-edo. Free tools like Scala have 53-edo built in.

Music in 53-edo: What Does It Sound Like?

Modern composers and arrangers have explored 53-edo across a wide stylistic range. Some notable examples:

Renderings of classical music: Several composers have reimagined baroque and classical repertoire in 53-edo. Bach’s “Jesus bleibet meine Freude” (BWV 147), various pieces from the Well-Tempered Clavier, works from the Art of Fugue, Handel harpsichord suites, Chopin preludes and études, and even Scott Joplin’s Maple Leaf Rag have all been arranged and rendered in 53-edo — with the goal of approximating the harmonic purity these composers may have been striving toward with meantone or Pythagorean temperaments of their era.

Contemporary composition: Artists working in 53-edo today include Bryan Deister (microtonal improvisations and original works), Francium (whose album releases include music in Hanson/53-edo tuning available on streaming platforms), and others in the wider xenharmonic music community.

The general sonic character of 53-edo is often described as “lush” — chords are noticeably more stable and resonant than their 12-TET equivalents, and the availability of multiple shades of each interval (Pythagorean, just, and septimal versions of thirds, sixths, and sevenths) gives composers unusual expressive latitude within a single key.

Who Should Explore 53-edo?

53-edo is particularly well-suited for:

  • Musicians interested in just intonation but frustrated by the practical difficulties of strict JI tuning (keyboard layouts, pitch drift in modulation, etc.)
  • Composers working in meantone or historical temperaments who want to explore what a more accurate version of those systems sounds like
  • Players of Middle Eastern, Turkish, or Persian music or those interested in incorporating those traditions, since 53-edo is already used in those contexts
  • Theorists and students of harmony who want to hear the difference between Pythagorean, just, and septimal versions of familiar intervals — 53-edo makes all of them audible and distinct
  • Anyone curious about why 12-TET major thirds sound the way they do — once you hear a 53-edo major chord, the roughness of the 12-TET version becomes impossible to unhear

53-edo is not a beginner’s microtonal system in terms of instrument design or full mastery. But conceptually, it is one of the most accessible large equal temperaments, because so much of what you already know maps directly onto it. The circle of fifths is still there. The diatonic scale is still there. The familiar chord qualities are still there — just noticeably sweeter.

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