How to get simplicity AND purity in our music, using clever math
If you like microtones, you’ve probably encountered the two camps. On one side are the just intonation purists, who insist that only exact small-integer frequency ratios — 3:2 for the perfect fifth, 5:4 for the major third, and so on — will do. On the other side are equal temperament enthusiasts, who point out the elegant simplicity of dividing the octave into some number of perfectly equal steps.
What if both camps are missing the bigger picture? That’s exactly the argument Paul Erlich makes in his landmark paper A Middle Path Between Just Intonation and the Equal Temperaments (originally published in Xenharmonikôn vol. 18, 2006). His paper introduces musicians and theorists to a rich universe of regular temperaments — a middle ground that has been central to Western music for centuries, yet has gone largely unnoticed as a concept, even by the musicians who practice it every day.
The False Dichotomy
Most discussions of tuning present the choice as binary: either you tune your intervals pure (just intonation), or you accept the compromises of equal temperament. But Erlich points out that this framing leaves out the most historically important tuning system in Western music: meantone temperament.
Meantone isn’t just intonation — its fifths are slightly flattened from the pure 3:2. But it isn’t equal temperament either — its steps are not all the same size. It’s something in between, and understanding why it exists and what it does unlocks an entire class of musical systems that composers and theorists have only recently begun to systematically map out.
Just Intonation: Beautiful but Complicated
Let’s start with just intonation (JI). The ancient insight is simple: certain frequency ratios between notes sound especially pure and resonant. The octave is 2:1. The perfect fifth is 3:2. The major third is 5:4. The further you go up the chain of primes — ratios involving 7, 11, 13 — the more exotic and complex the harmonies become.
The problem is that JI is inherently multi-dimensional. If you build a scale from pure fifths (the 3:2 ratio), you get Pythagorean tuning. If you also want pure major thirds (5:4), you need to add another dimension — the prime 5 — and now your scale requires at least a two-dimensional grid of pitches to describe it. Add the 7th harmonic, and you need three dimensions.
This means that in strict JI, the note D reached by going up a fifth from G is slightly different from the D reached by going up two pure major thirds from C. These two D’s differ by a tiny interval called the syntonic comma, about 21.5 cents (roughly a fifth of a semitone). In a real performance, this means you either need multiple versions of the same note — a tremendous practical complication — or you have to let some chords drift slightly out of tune.
Equal Temperament: Simple but Coarse
Equal temperament solves this by collapsing everything into one dimension. In 12-tone equal temperament (12-TET), you simply divide the octave into 12 equal steps of exactly 100 cents each. Every note has one and only one pitch. The perfect fifth becomes 700 cents instead of the pure 701.96 cents — just two cents flat. The major third becomes 400 cents instead of the pure 386.31 cents — about 14 cents sharp, which is actually a much bigger compromise, but one Western ears have mostly learned to accept.
The elegance is undeniable. You can transpose freely, modulate to any key, and build any instrument with a fixed set of keys. But you’ve paid a price: the pure intervals are gone, replaced by approximations of varying quality.
And crucially — equal temperament is just one choice. There’s nothing special about 12. You could divide the octave into 19 equal steps, or 22, or 31, or 53. Each of these gives a different set of compromises and a different sonic character.
The Middle Path: Tempering Out a Comma
Here’s the key insight that Erlich’s paper builds on. Instead of thinking about tuning systems as either “pure JI” or “equal steps,” think about them in terms of which small intervals are allowed to vanish.
Remember the syntonic comma — that 21.5-cent gap between two different versions of D in 5-limit JI? Meantone temperament is what you get when you declare that gap to be zero. You adjust the tuning of the perfect fifth very slightly (making it about 696–697 cents instead of 702), so that four pure fifths minus two octaves now lands exactly on a pure major third. The two versions of D collapse into one. The syntonic comma has been tempered out, or made to vanish.
The result is a 2-dimensional system — you can reach any note by combining octaves and fifths (or equivalently, octaves and major thirds). That’s exactly what the circle of fifths represents. Meantone is the theoretical foundation of most Western tonal music from roughly 1500 to 1900.
When one comma vanishes, dimensionality drops by one. JI with primes 2, 3, and 5 is three-dimensional. Temper out the syntonic comma, and you get a two-dimensional system: meantone. Temper out another independent comma on top of that, and you get a one-dimensional system — an equal temperament.
This is the unifying framework. Equal temperaments are just the extreme case where enough commas have been tempered out to collapse everything to a single dimension. Regular temperaments are the more general family where you stop before that final collapse.
Generators and Periods
Every regular 2-dimensional temperament can be described by two generating intervals:
- A period: usually the octave (1200 cents), though sometimes a fraction of an octave like a half-octave or third-octave.
- A generator: a single interval that you stack repeatedly to build the scale.
In meantone, the period is the octave and the generator is a slightly flattened fifth (around 696–697 cents). Stack five generators, fold back into the octave, and you get the pentatonic scale. Stack seven generators and you get the familiar diatonic scale. Stack twelve and you get the chromatic scale. This sequence — pentatonic → diatonic → chromatic — falls naturally out of the mathematics of meantone, and it’s exactly the sequence of scales Western music has historically used.
The horagrams in Erlich’s paper (circular diagrams similar to Ervin Wilson’s horagrams) show this process beautifully. Each concentric ring represents a new scale formed by adding more generator steps, and the numbers show the step sizes in cents. The rings where only two step sizes appear are the musically useful ones — these are called MOS scales (Moment of Symmetry scales), and they’re the microtonal generalization of the diatonic scale.
A Universe of Temperaments
Once you understand meantone as one specific comma-vanishing, it becomes clear that there are many other possible temperaments, each with its own musical character. Erlich’s paper catalogs dozens of them. Here are a few highlights:
Porcupine temperament — defined by the vanishing of the comma 250:243 (about 49 cents). Its generator is roughly 163 cents, a kind of large whole tone. It produces a natural 7-note scale and allows smooth chord progressions that would be impossible in 12-TET.
Magic temperament — defined by the vanishing of 3125:3072. Its generator is about 381 cents, close to a major third. Five generators stack up to a near-pure perfect fifth. It has a strikingly symmetrical structure and offers very accurate approximations of pure 5-limit harmony.
Miracle temperament — one of the most celebrated discoveries of the online tuning community, defined by two commas vanishing simultaneously (it’s a 7-limit temperament, meaning it involves the prime 7 as well as 2, 3, and 5). Its generator of about 117 cents is called the “secor.” It can approximate pure intervals involving the 7th harmonic — that bluesy, resonant sound of a 7:4 ratio — with remarkable accuracy.
Pajara temperament — the temperament underlying the 22-tone equal temperament that many microtonalists favor. Its period is a half-octave (600 cents) and its generator is about 492 cents. It supports a natural 10-note scale and handles both 5-limit and 7-limit harmonies well.
What Vanishes Tells You What’s Possible
Each temperament defines not just a set of pitches but an entire harmonic logic — a set of chord progressions that become possible, a set of scales that arise naturally, and a set of notational conventions that make sense within it.
Erlich illustrates this vividly with a chord progression by Graham Breed, most of whose chords approximate the frequency ratios 4:5:6:7 — a dominant seventh chord in just intonation, far purer and more resonant than its 12-TET equivalent. In 12-TET, repeating this progression causes a drift of a semitone. In strict JI, the drift is a tiny interval of 2401:2400, less than one cent — essentially negligible for a flexible ensemble. But conventional music notation is completely incapable of representing this progression accurately, because our notation is built around meantone logic.
In a temperament where 2401:2400 vanishes — which is true of Miracle temperament and several others — the progression closes into a perfect cycle, and a natural notation system falls into place.
This is Erlich’s deeper point: temperaments aren’t just tuning choices for instruments. They’re frameworks for thinking about and notating music. The diatonic scale and all of Western tonal harmony — the voice-leading conventions, the function of dominant and subdominant, the whole apparatus of Roman numeral analysis — are downstream of the choice to use meantone.
How to Tune a Temperament Optimally
Once you’ve decided which comma to temper out, you still have to decide how much to temper each prime interval. Should you tune the octave pure and spread all the damage across the fifths and thirds? Or should you stretch the octave slightly and share the load more evenly?
Erlich introduces an elegant answer he calls TOP tuning (Tenney OPtimal, also readable as “Tempered Octaves, Please”). The idea comes from James Tenney’s concept of harmonic distance — a measure of the musical complexity of an interval based on the sizes of the numbers in its frequency ratio.
The principle is this: distribute the tempering of a comma evenly across all the prime intervals it involves, proportional to the harmonic complexity (essentially the size) of each prime. This minimizes the maximum damage done to any interval in the entire harmonic lattice, weighted by how harmonically simple that interval is. Simple intervals like the octave (2:1) and fifth (3:2) are more sensitive to mistuning, so they need more accurate tuning; complex intervals are less sensitive and can tolerate larger deviations.
The result is that in TOP-tuned meantone, even the octave gets stretched very slightly (to about 1201.7 cents) — a departure from the “pure octaves” convention that most tuning systems assume. In practice this is inaudible in isolation but makes the overall system more internally consistent.
The Complexity-Damage Tradeoff
Erlich organizes his catalog of temperaments using two key measures:
Complexity tells you how many notes you need before the temperament’s characteristic scales and harmonies start to appear. A low-complexity temperament like meantone gives you recognizable diatonic scales with just 5 or 7 notes. A high-complexity temperament like Miracle requires 10 or 21 notes before its most interesting scales emerge.
Damage (in the TOP sense) tells you how far the approximations of pure intervals stray from just intonation. A low-damage temperament like Helmholtz/Schismatic temperament approximates pure intervals with almost undetectable error — less than 0.1 cents on the octave — but at the cost of requiring many more notes to be useful.
The “main sequence” of temperaments in Erlich’s Table 1 represents the sweet spot of this tradeoff: systems that are both simple enough to be practical and accurate enough to be harmonically rich. Meantone sits near the center of this sequence, which is no accident — it’s exactly why Western music settled on it for so long.
Beyond 5-Limit: The World of 7
Most of Western tonal harmony operates in what theorists call the 5-limit: intervals built from the primes 2, 3, and 5. The major and minor triads, all the standard diatonic chords, the circle-of-fifths progressions — all of these live in this world.
But there’s a richer harmonic universe available if you include the prime 7. The interval 7:4 is about 969 cents — close to, but distinctly different from, the minor seventh of 12-TET (1000 cents). In just intonation, it’s the resonant, beatless seventh you hear in the best barbershop chords, or the natural seventh of a brass instrument playing without valves. It has a characteristic “sinking” quality that 12-TET’s minor seventh can’t quite capture.
Erlich’s paper includes a table of 7-limit temperaments — systems that approximate pure intervals involving the prime 7 as well as 2, 3, and 5. These require two independent commas to vanish (since 7-limit JI is 4-dimensional, and we want a 2-dimensional temperament). The catalog includes familiar names like Dominant temperament (essentially 12-TET with a 7-limit interpretation) and exciting systems like Miracle, Pajara, and Orwell temperament.
Why This Matters for Practical Musicians
You might reasonably ask: this is all very interesting theory, but what does it mean for someone who actually plays music?
A few things.
First, it reframes what “microtonal” means. The microtonal world isn’t just about equal divisions of the octave, and it isn’t just about pure just intonation. Regular temperaments offer a third option: tuning systems with real harmonic logic, scales with two (or a few) step sizes analogous to the diatonic scale, and chord progressions that close into satisfying cycles.
Second, it explains why certain equal temperaments are more interesting than others. 19-TET and 31-TET are good approximations of meantone — they sit on the same temperament “line” as quarter-comma meantone, just with different generator sizes. 22-TET is good for Pajara and Porcupine temperaments, which have their own characteristic harmonic logic. 72-TET supports Miracle temperament with exceptional accuracy. Knowing the underlying temperament tells you why a given equal temperament sounds good for certain kinds of music.
Third, it opens up new compositional possibilities. Imagine a scale that plays the same structural role as the diatonic scale — seven notes, two step sizes, a natural hierarchy of tensions and resolutions — but whose thirds are purer, whose seventh chords are more resonant, and whose harmonic progressions follow a different kind of logic. That’s what you get with temperaments like Magic or Porcupine. They’re not weird for weirdness’s sake; they have genuine musical coherence.
Finally, it reconnects tuning theory with notation. Each temperament implies a natural notation system, just as meantone implies our familiar staff notation with sharps and flats. If you wanted to write music in Miracle temperament, you’d use a staff based on a 10-note scale rather than the 7-note diatonic. The notation would be strange to Western eyes, but it would be logical — each symbol would correspond to a unique pitch class in the system, and the visual relationships between notes would mirror the harmonic relationships.
Getting Started
If this has sparked your curiosity, here are some places to explore further:
- The Xenharmonic Wiki is the community-maintained encyclopedia of microtonal music theory, with detailed articles on every temperament mentioned here and hundreds more.
- Erlich’s paper itself is freely available online and goes into much more mathematical detail, including the derivations of TOP tuning and extensive tables of temperament data.
- Software like Scala (free, cross-platform) lets you tune synthesizers and explore different temperaments. Many modern DAWs support microtuning via MIDI Tuning Standard or Scala files.
- The 22-tone equal temperament is a good first step beyond 12-TET: it supports familiar meantone-like harmonies alongside genuinely new structures, and a growing repertoire of music has been written in it.
The world beyond 12 is not a chaotic wilderness of arbitrary pitches. It’s a structured landscape of harmonic possibilities, waiting to be explored — and regular temperament theory is the map.
This article draws primarily on Paul Erlich, “A Middle Path Between Just Intonation and the Equal Temperaments,” Xenharmonikôn vol. 18 (Summer 2006), updated 2015. Erlich’s paper is essential reading for anyone who wants to go deeper into the mathematics.
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