159-equal temperament (159edo) and the determined theorist who mapped it

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 DaffodilAura’s quest to tame a 159-note behemoth

This article is AI-generated

Before we dive into the deep end, let's establish some ground level.

The tuning system almost all Western music uses is called 12-EDO — short for 12 Equal Divisions of the Octave, also called 12-TET (twelve-tone equal temperament). It divides the octave into 12 equal semitones. Every piano, every guitar, every standard synthesizer lives inside this grid.

But 12-EDO is not the only way to slice an octave. Mathematically, you can divide an octave into any number of equal parts. 24-EDO gives you quarter-tones — you've probably heard these in Arabic or Turkish music. 19-EDO sounds warmer and more consonant than 12, popular with jazz and classical experimenters. 53-EDO is beloved by theorists for its extraordinarily accurate approximation of pure mathematical intervals.

And then there is 159-EDO.

So What Exactly Is 159-EDO?

159-EDO divides the octave into 159 equal steps. Each step is about 7.55 cents wide — for reference, a normal semitone in 12-EDO is 100 cents, so a single step of 159-EDO is less than a twelfth of a semitone. This is genuinely tiny: right around the threshold where most listeners begin to distinguish two pitches as separate at all (the so-called melodic just-noticeable difference, or JND).

The system isn't arbitrary. It has a beautifully logical structure: 159 = 3 × 53. This means 159-EDO contains 53-EDO as a subset — every note in 53-EDO exists in 159-EDO, plus 106 additional notes in between. It also contains 3-EDO (the three notes that divide the octave into equal thirds), which it shares with 12-EDO.

Think of it this way: if 53-EDO is a high-resolution photograph, 159-EDO is the same photograph with three times the pixel density. You get everything 53-EDO offers, with a new layer of fine detail on top.

Why Does 53-EDO Matter?

To appreciate 159-EDO, a quick word about its parent system.

Pure, untempered intervals — the kind that come directly from the physics of vibrating strings — are described by just intonation (JI), expressed as simple frequency ratios. A perfect fifth is 3/2. A major third is 5/4. These intervals sound perfectly smooth and resonant, with no beating or roughness.

The problem with JI is that it doesn't close into a neat equal-tempered scale — the math doesn't work out to let you freely modulate keys on a fixed-pitch instrument. 12-EDO solves this by slightly mistuning everything, which is why piano perfect fifths have a faint shimmer. 53-EDO solves it far more elegantly: its perfect fifth (at 701.9 cents) is almost indistinguishable from pure 3/2 (702.0 cents). Its major thirds are excellent too. For centuries, theorists have regarded 53-EDO as a near-perfect solution for 5-limit just intonation — the world of harmonics built from the numbers 2, 3, and 5.

159-EDO inherits all of this, and then goes considerably further.

What Makes 159-EDO Special?

1. Exceptional Accuracy Across a Wide Harmonic Range

Most EDOs do well in the lower prime limits — harmonics built from small primes like 2, 3, and 5. But as you climb into the territory of the 7th, 11th, 13th, and 17th harmonics, errors accumulate. 159-EDO is unusually well-behaved across this entire range.

Its most celebrated property is its near-perfect approximation of the 11th harmonic, also called the undecimal prime. The 11th harmonic creates an interval of roughly 551 cents — a slightly sharp tritone that sits in an eerie, ambiguous space between a perfect fourth and a tritone. In 12-EDO, this interval can only be clumsily approximated. In 159-EDO, the error is just 0.37 cents — essentially undetectable by human ears.

This matters because the 11th harmonic is the first "wild" harmonic in the overtone series: 2, 3, and 5 all produce familiar-sounding intervals (octave, fifth, major third), but 11 sounds genuinely otherworldly. Composers who want to use it in harmony — to create those strange, floating, just-slightly-wrong-in-a-beautiful-way chords — need a tuning system that handles it well. 159-EDO is one of the best in existence for this.

The system is consistent through the 17-odd-limit, meaning you can build complex chords using harmonics up to 17 without running into contradictions in how intervals add up. For context, 12-EDO is only consistent through the 3-odd-limit. You are working with a dramatically more harmonically capable system.

2. It Can Sound Like Other Tuning Systems

One of 159-EDO's quirky strengths is its ability to impersonate other EDOs. Its step size is small enough that you can select subsets of its notes that closely resemble 10-EDO, 13-EDO, 17-EDO, 19-EDO, 22-EDO, 24-EDO, 31-EDO, and others. This makes it useful as a kind of universal translator for xenharmonic music — a composer can write in 159-EDO and approximate the flavor of many other systems.

3. Multiple Flavors of Diatonic Scale

In 12-EDO, there is essentially one perfect fifth (700 cents), and it generates one style of diatonic scale. In 159-EDO, there are five different possible generators for a diatonic scale, each with a distinct character:

  • 93\159 (≈ 701.9¢) — the classic Pythagorean fifth, generating the same diatonic sound found in 53-EDO
  • 92\159 (≈ 694.3¢) — a slightly flat "meantone-like" fifth, great for compositions in the Western classical tradition
  • 94\159 (≈ 709.4¢) — a sharp "superpyth" fifth with a harder, brighter sound
  • 91\159 (≈ 686.8¢) — an extremely soft, nearly equalized diatonic scale
  • 95\159 (≈ 716.9¢) — a nearly collapsed diatonic scale, more exotic

Each of these leads to a different family of extended scales and a different harmonic personality. You are not locked into one kind of "major scale." 159-EDO gives you a menu.

4. Essentially Tempered Chords

159-EDO supports a host of exotic chord types that arise from tempering out specific small intervals called commas. Without getting too deep into the math, this means certain chord progressions that are theoretically impossible in just intonation (because small discrepancies would accumulate) become possible and beautiful here. These include:

These represent genuinely new harmonic resources — chord types that simply do not exist in 12-EDO.

The History: Where Did 159-EDO Come From?

Despite being the "tripled superset" of the well-known 53-EDO, 159-EDO was surprisingly late to attract serious attention.

The first person to work with it was Ozan Yarman, a Turkish musicologist and composer who, in late 2005 and early 2006, was searching for a tuning system capable of accurately representing the intricate pitch world of maqam — the modal system underlying Turkish, Arabic, and Persian classical music. Maqam music uses intervals that fall between the notes of the standard Western scale, and no standard Western tuning captures them properly.

Yarman's approach was elegant: he divided the perfect fourth (4/3) into 33 equal parts and extended the resulting comma upward. He arrived at a 79-note scale that he called the Yarman-79 tuning, which he later had physically built into a Qanun (a Middle Eastern zither-like instrument) by affixing special "mandal" bridges. The theorist Gene Ward Smith later recognized that Yarman's scheme was actually a mode of MOS (moment of symmetry) derived from 159-EDO, and that it had an 80-note twin.

Yarman defended his doctoral thesis at Istanbul Technical University's Turkish Music State Conservatory in June 2008, with the Qanun featuring his 79-tone system. This work gave 159-EDO its first practical, physical realization.

Enter DaffodilAura: The Theorist Who Mapped the Territory

If Yarman opened the door to 159-EDO, it was a theorist writing under the name DaffodilAura — real name Dawson Berry, known on the Xenharmonic Wiki as User:Aura — who mapped what lay beyond it.

Aura describes their introduction to microtonal music as coming through fascination with the harmonic series, particularly the 11th harmonic. In 2014, they wrote their first microtonal piece, Folly of a Drunk, in 24-EDO — featuring a modulation built explicitly on both the 11th harmonic and the 11th subharmonic. It was the beginning of a deep theoretical journey.

The Scope of the Work

Aura's contributions to the Xenharmonic Wiki are extensive, and they have focused considerable energy on 159-EDO specifically. Their stated reason is telling: they were drawn to the system because of its near-perfect approximation of the 2.3.11 subgroup — that is, the world of harmonics built from the primes 2, 3, and 11. This is precisely where the 11th harmonic lives, and 159-EDO handles it better than almost any comparable system.

Their wiki contributions include:

  • Aura's Introduction to 159-EDO — a detailed guide to intervals, notation, and diatonic scale construction in the system
  • 159-EDO Interval Names and Harmonies — a comprehensive catalog of all 159 intervals, their JI approximations, their harmonic and melodic compatibility ratings, and notes on how to use each one in composition
  • Multiple sections of the main 159-EDO wiki page itself
  • A broader body of music theory work including guides to diatonic functional harmony, EDO impressions, and ideas on consonance and tonality

One of Aura's notable theoretical contributions is helping develop Syntonic-Rastmic Subchroma (SRS) notation — a notation system capable of writing 159-EDO music. Notating a 159-note-per-octave system is genuinely hard: standard sharps, flats, and even existing microtonal accidentals are insufficient. SRS notation was designed to handle this with clarity, using a layered system of accidentals that remains legible even at high EDOs.

The "Dinner Party Rules" — A Framework for Harmony

One of the more charming concepts Aura has contributed is an adaptation of what the xenharmonic community calls the "Dinner Party Rules" — a metaphor for which intervals "get along" harmonically, like guests at a dinner party.

In 12-EDO, the rule of thumb is simple: consonance comes from intervals near pure JI ratios, dissonance from those far away. In 159-EDO, the sheer number of available intervals makes this more complex. Aura's interval table assigns each of the 159 intervals both a Harmonic Compatibility Rating (how well it blends when sounded simultaneously) and a Melodic Compatibility Rating (how well it works in a melodic line), on a scale from −10 to +10. The perfect unison and octave score 10/10 on both. A step of 7.5 cents scores 0/0 — technically usable as a kind of microtonal color, but not something you'd build a chord on.

This kind of systematic rating is genuinely useful for composers approaching 159-EDO without wanting to rediscover all its harmonic rules from scratch.

The Diatonic Modes — A 5-Limit Overhaul

One of the more practically useful sections of Aura's introduction to 159-EDO is their treatment of 5-limit diatonic music — that is, the familiar major and minor modes, but optimized for the purer intervals available in 159-EDO.

The key insight is this: in standard 12-EDO meantone tuning, every diatonic mode uses the same set of intervals. But in a non-meantone system like 159-EDO, each mode can and should have its own optimized scale. The major third used in Ionian (major) mode is not the same major third that sounds best in Dorian or Phrygian.

Aura works through four modes in detail:

  • Ionian (Major): Optimized using the Didymean scale — a Pythagorean major second (9/8) for the supertonic, a pure 5/4 major third, and a pure 27/16 major sixth. This yields two types of major triads on different scale degrees, each with their own character.
  • Dorian: Not symmetrical, as one might expect — Rothenberg propriety requires an asymmetric layout. Features a pure 6/5 minor third and a Pythagorean 27/16 major sixth.
  • Phrygian: Optimized using the left-handed Zarlino scale. Features a pure 16/15 minor second and 8/5 minor sixth.
  • Lydian: Optimized using the right-handed Zarlino scale. Features the sharp #IV (a pure 45/32) as its defining character note.

This is music theory that any classically trained musician can follow — the concepts of major thirds, perfect fourths, and modal scales are all still here, just rendered with finer precision.

Aura's Music

Aura is not only a theorist but an active composer, and their music directly demonstrates the possibilities of 159-EDO. Some highlights:

Space Tour (2020) is a long-form instrumental piece and possibly their most ambitious work. It tells the story of a space tourism voyage gone wrong — from liftoff to an emergency landing and rescue in a Canadian forest. Musically, it functions as a tour of EDOs, moving through approximations of 1-EDO, 12-EDO, 14-EDO, 17-EDO, 19-EDO, 22-EDO, 24-EDO, 27-EDO, 31-EDO, 35-EDO, 41-EDO, and 53-EDO before arriving at full 159-EDO in its final act. Each transition represents a new location in the story and a new harmonic world.

Welcome to Dystopia (2021) began as a 12-EDO piece and was later remade in 159-EDO, adding xenharmonic modulations upward by 11/8 and downward by 16/11 — the very intervals that drew Aura into microtonality in the first place.

The Winterbright Symphony (2022) is a seven-movement work written entirely in 159-EDO. It is a programmatic symphony following a character named Felix from a tragic childhood through old age and reconciliation, with movements titled in Italian: Sonatina (Overture), Ponte, Minuet & Trio, Passacaglia, Double Fugue, Aspettativa, and Finale. It is a substantial undertaking — program music in the grand tradition, but sounding unlike any symphony you've heard.

The Fantasizer's Symphony (late 2024) is another seven-movement work in 159-EDO, similarly structured with an Overture, Passacaglia, Minuet & Trio, Fugue, and Finale.

A Mighty Fortress Is Our God (2021) is a hymn arrangement in 159-EDO, written around Reformation Day. Aura — a devout Lutheran who has been candid about keeping this part of their life quiet online — brings the 11th harmonic into a setting of Luther's famous chorale. Sacred music and xenharmonic theory, meeting in the same piece.

Commas: The Secret Architecture

One aspect of 159-EDO worth explaining for music-theory-literate readers is its comma structure — the list of small intervals it "tempers out," meaning treats as equivalent to zero.

A comma is a tiny interval that arises from the arithmetic of just intonation. The most famous is the syntonic comma(81/80, about 21.5 cents) — the difference between a Pythagorean major third (81/64) and a pure 5-limit major third (5/4 = 80/64). In 12-EDO, both are mapped to the same 4 semitones, so the syntonic comma is tempered out. This is what makes meantone tuning work.

159-EDO tempers out a long list of commas, including:

Each tempered comma enables a new kind of chord or progression. This is what theorists mean when they call 159-EDO "well-balanced" — it supports both precise JI approximations and a rich variety of essentially tempered chord types.

Is 159-EDO Playable?

The honest answer is: not easily on traditional instruments. But there are paths forward.

Software is the most accessible entry point. Programs like Surge XT, Vital, and ZynAddSubFX can be retuned to any EDO. MuseScore can be used for notation with microtonal plugins. Aura composes in approximations of 159-EDO using MuseScore and related tools.

The Neod is a physical instrument under development by Erik Natanael that directly uses 159-EDO. It is built primarily on 53-EDO keys, with additional buttons that modify pitch by a single 159-EDO step — a practical design that makes the system physically navigable by building on the familiar 53-EDO structure.

The Yarman Qanun remains the most historically significant physical realization — a traditional Turkish instrument rebuilt with the Yarman-79 tuning, demonstrating that 159-EDO-derived tuning has real-world acoustic roots in a living musical tradition.

Why Should a Classically Trained Musician Care?

Here's the honest pitch.

If you understand major and minor thirds, perfect fifths, dominant seventh chords, and modal theory, you already have most of the vocabulary needed to work in 159-EDO. The concepts carry over — the system just offers more options and higher precision.

The pure 5/4 major third (384.9 cents, step 51 in 159-EDO) is genuinely sweeter than the 12-EDO major third (400 cents). The pure 3/2 fifth (701.9 cents, step 93) is almost exactly just. The harmony, when built from these purer intervals, has a quality that players of fretless instruments and vocalists already know instinctively: it locks in in a way that equal-tempered harmony doesn't quite.

Beyond the familiar intervals, you gain access to the 11th harmonic — that beautiful, eerie interval that floats between the fourth and the tritone, which Aura calls the undecimal prime. You gain neutral thirds (halfway between major and minor), subminor thirds, supermajor sixths, and dozens of other intervals that have distinct harmonic personalities and are simply absent from 12-EDO.

And you gain the theoretical framework Aura has built: the interval tables, the compatibility ratings, the diatonic mode guides, the chord catalogs. You don't have to start from zero.

Where to Learn More

The best starting point is the Xenharmonic Wiki, which is a free, community-maintained resource. Key pages include:

For Aura's music specifically, their YouTube channel (linked from their wiki user page) hosts the compositions mentioned in this article.

A Final Thought

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