The 22 Temperaments

Equal temperament divides the octave into twelve mathematically identical semitones, each exactly 100 cents wide. No interval except the octave is pure: major thirds are roughly 14 cents wider than a pure 5/4, fifths are 2 cents narrower than a pure 3/2.

Its practical advantage is total symmetry: you can modulate to any key without re-tuning. That convenience made it the default on modern pianos, synths, and almost every piece of software since the early twentieth century. Its cost is that no interval has the beating-free purity you hear in just or meantone tunings — the compromise is evenly smeared across all of them.

Five-limit just intonation is built from three prime numbers: 2, 3, and 5. Octaves are 2/1, fifths are 3/2, major thirds are 5/4. Every interval is a pure beatless ratio — play a C major chord and it locks without any audible beating, because the partials of the three notes align perfectly.

The catch is that purity is only available in one key at a time. Modulate a fourth away and the new thirds become Pythagorean-wide and start to beat; go further and you hit the wolf fifth. This is why just intonation lived in vocal music (where singers can bend notes in real time) and never caught on for fixed-pitch instruments until the digital era gave us per-note retuning.

Seven-limit just adds the prime 7 to the mix, unlocking the "harmonic seventh" 7/4 — about 969 cents, noticeably flatter than the 1000-cent minor seventh you are used to. That interval sits in between a minor and a major seventh and has a particular consonant quality, almost resolved by itself.

Composers like Harry Partch and La Monte Young built entire sound worlds around 7-limit and higher ratios. On Vivaldi Keyboard, this tuning is the easiest way to hear what a "natural" dominant seventh chord actually sounds like — and why equal temperament had to approximate it.

The defining meantone of the Renaissance and early Baroque. Every fifth is flattened by exactly one quarter of a syntonic comma (about 5.4 cents), which has the magical consequence of making four consecutive fifths land on a mathematically pure 5/4 major third.

Eight keys — roughly E♭ to E major — sound glorious, with beatless thirds that ring like nothing on a modern piano. The four remote keys are the price: the accumulated tempering piles up on one fifth, the famous "wolf" (usually G♯–E♭), which howls if you try to use it. Composers from Frescobaldi to early Bach wrote around this limitation.

Stretches the meantone principle further: flatten each fifth by a third of a comma, and you get minor thirds that are pure instead of major ones. The trade-off is steeper — fifths are now audibly narrow and the wolf is even more dramatic.

In practice this temperament is a curiosity. Theoretical treatises describe it, but surviving instruments and manuscripts tuned this way are rare. Vivaldi Keyboard includes it so you can hear what "too much" purity costs.

A softer meantone: each fifth is flattened by a fifth of a comma instead of a quarter. The thirds are no longer beatless, but they are still noticeably sweeter than equal; the fifths are much closer to pure, and the wolf is smaller.

Think of it as "meantone with training wheels" — the cost of modulation is reduced, and a couple more keys become usable. Late Renaissance Spanish and Italian theorists experimented with it as organs started to be played in more keys.

The gentlest of the meantones. Fifths are only a sixth of a comma narrow — nearly pure. Thirds are wider and no longer beatless, but all twelve keys become playable, each with a slightly different color.

This is the kind of tuning associated with the Silbermann organs of central Germany in Bach's lifetime: Bach himself disliked Silbermann's tuning as "too pure," which tells you how spoiled he was by meantone thirds. Today it makes a great compromise tuning for early eighteenth-century music on keyboard.

Proposed by Zarlino in his Istitutioni harmoniche, 2/7 comma is a theoretical sweet spot: it distributes the syntonic comma so that major thirds are a tiny bit smaller than pure and fifths are a bit more tempered than 1/4 comma. The overall consonance is, by some measures, mathematically optimal.

In practice it sounds very close to 1/4 comma with slightly more "bite" on the fifths. A beautiful oddity — worth trying back-to-back with the others.

Werckmeister's "correct" temperament (Correctus) was the first widely circulated well-temperament. Four of the fifths are tempered by a quarter of a Pythagorean comma each, the rest are pure. The result: every key works, but C, G, and D have a nearly-meantone sweetness while F♯ and C♯ lean Pythagorean-bright.

Generations of scholars have proposed it as the tuning Bach had in mind for Das Wohltemperierte Klavier (1722). Whether or not that is true — and the debate is still open — it is one of the most musical ways to play Bach's preludes and fugues in all twenty-four keys.

Kirnberger was a Bach student, and this temperament was designed to preserve what he thought was his teacher's sensibility. Most fifths are pure; the syntonic comma is split between just four fifths (C–G–D–A–E), giving C major a near-pure major third.

The side effect is that remote keys retain a strongly Pythagorean character — bright, restless, and a little wild. Many organists consider it the most authentic tuning for late Bach organ works.

Vallotti simplified the well-temperament idea: take six adjacent fifths (F–C–G–D–A–E–B) and narrow each by a sixth of a Pythagorean comma; leave the other six pure. The result is a perfectly symmetric "good in C, wild in F♯" curve with no surprises.

It became the de facto Italian tuning of the eighteenth century and is arguably the most usable well-temperament for modern ears: the key colors are there, but none of them are abrasive.

Thomas Young — yes, the same Young of the double-slit experiment and the Young's modulus — proposed this tuning in 1799. It shifts Vallotti's tempered fifths one step along the circle, giving a marginally different key-color profile.

In blind A/B tests most listeners cannot tell Young from Vallotti. Scholars use it for specific late-eighteenth-century English repertoire where that tiny difference might matter.

Neidhardt proposed three versions of his temperament, graded by how much color they allowed: "Village," "Small City," and "Big City." The Big City version, included here, is the one closest to equal temperament — a mild well-temperament for a sophisticated urban audience that expected to hear modulations to any key.

It is the quietest of the well-temperaments: the colors are there if you listen for them, but they never get in the way. Some musicologists argue Bach was thinking of this version, not Werckmeister, for the WTC.

In 1979 the English harpsichordist John Barnes published a study analysing the frequency of thirds and fifths in every prelude and fugue of Bach's WTC, and proposed a temperament that would serve them all best.

The result is a modern well-temperament: gentle colors, every key comfortable, a tiny bit of extra consonance where Bach used the most thirds. A scholarly choice, not a historical one — but an informed one.

In 2005 the harpsichordist Bradley Lehman noticed that the loops and curls drawn at the top of Bach's WTC autograph might be a cipher describing how to temper the instrument. He decoded them into a specific distribution of fifths and published the result.

The musicological community split. Some accepted the proposal as plausible, others dismissed it as pattern-matching on a doodle. Whatever its historical status, the tuning itself sounds very good — every key is usable, the colors are distinct, and Bach's more harmonically adventurous pieces shine in it.

Rameau — better known as a composer and theorist than as a tuner — described this temperament in his Nouveau système de musique théorique (1726). It is a modified meantone: eight of the thirds are preserved nearly pure, but the wolf fifth is distributed more gently than in 1/4 comma.

The French Baroque school used it for organ and harpsichord repertoire until the late eighteenth century. It feels warmer and less bright than a German well-temperament.

Kellner proposed his temperament in 1975 as an answer to the same question Lehman would re-open thirty years later: what did Bach really mean by "wohltemperiert"? His answer is simple — split the Pythagorean comma across exactly five fifths, keep the other seven pure.

The resulting sound is distinctly Kirnberger-like, but not identical: Kellner argued that his version fits the WTC more symmetrically. A pre-Lehman standard for musicologists who wanted something beyond Werckmeister.

Build a keyboard by stacking pure fifths — 3/2, 3/2, 3/2 — twelve times, and the cycle almost closes. Almost. The accumulated discrepancy (the Pythagorean comma, about 23.5 cents) has to go somewhere, and in the Pythagorean tuning it lands entirely on one fifth, the famous "wolf" between G♯ and E♭, which is a quarter-tone narrow and unusable for harmony.

This was medieval Europe's tuning of choice. It sounds spectacular for melody and for the open-fifth "organum" polyphony of the twelfth century, because every fifth is pure. The problem is the thirds: Pythagorean major thirds are 22 cents sharper than pure 5/4, enough to feel restless to modern ears. When Renaissance composers started using thirds as consonances, Pythagorean tuning had to go.

Twenty-four equal quarter-tones per octave is the lingua franca used by modern Arabic music theory textbooks to notate maqam intervals on a Western staff. In practice, performers bend the "neutral" intervals by ear — real maqamat use slightly larger or smaller quarter-tones depending on the mode, the region, and the player.

Vivaldi Keyboard's 24-TET is the textbook approximation. Use it to get a feel for the neutral second and neutral third, or to play along with recordings that already use the equal-tempered version.

Divide the octave into nineteen equal steps and something remarkable happens: the major third lands at about 379 cents — much closer to pure (386 cents) than 12-TET's 400 cents. Fifths are a touch narrow but still usable, and every interval is enharmonically consistent (C♯ and D♭ are literally different pitches).

The Dutch composer Joel Mandelbaum and the American Easley Blackwood both wrote extensively in 19-TET. It feels like a parallel universe where the Renaissance won.

Huygens noticed that if you divide the octave into thirty-one equal parts, you get something almost indistinguishable from 1/4 comma meantone — except that the wolf disappears. Every key becomes playable with meantone-quality thirds.

Two and a half centuries later, the Dutch physicist Adriaan Fokker built an organ in 31-TET and founded the Huygens-Fokker Foundation, still active today in Amsterdam. This is the tuning if you love meantone colors but want to modulate freely.

Divide the octave into fifty-three equal parts and you get a ridiculously good approximation of almost every interval the ear cares about: pure fifths (within 0.07 cents), pure major thirds (within 1.4 cents), and the Pythagorean comma sits neatly on one step.

Turkish classical music theory formalised itself around 53-TET in the nineteenth century, and it remains the standard notation for maqamat today. For ear training it is also a revelation: you can finally hear the difference between a Pythagorean third and a just third without retuning.