Temperaments

A temperament is a system for tuning the twelve notes of the chromatic scale. Because no fixed-pitch instrument can sound every interval pure at once, every temperament is a compromise — a deliberate distribution of small impurities across the keyboard so that the music it serves can sing.

This page is a practical field guide. Each entry describes what the temperament is, who used it, and what it sounds like. Every system listed here ships in temperat.io and can be tuned to in your browser.

If you would rather skip the reading, open the tuner and pick a temperament from the menu.

Equal Temperament (12-TET)

Twelve-tone equal temperament divides the octave into twelve mathematically identical semitones of exactly 100 cents each. Every key sounds the same as every other key, transposed. This is the standard tuning for modern pianos, guitars, and most digital instruments built since the late 19th century.

The price of universal compatibility is that no interval except the octave is acoustically pure. The major third is sharp by about 14 cents; the perfect fifth is flat by 2. For most ears these compromises are inaudible, which is why 12-TET won. For a harpsichordist tuning Bach, they are audibly wrong.

Just Intonation

Just intonation tunes intervals to small whole-number frequency ratios — 2:1 for the octave, 3:2 for the fifth, 5:4 for the major third. The result is acoustically pure: triads ring without beating, and harmonies lock into place in a way that equal temperament cannot match.

The catch is that just intonation is key-specific. A keyboard tuned justly in C major will sound out of tune the moment you modulate to E-flat. This is why fixed-pitch instruments largely abandoned just intonation centuries ago, and why string ensembles, choirs, and barbershop quartets — all of whom can adjust pitch in real time — still gravitate toward it instinctively.

Pythagorean Tuning

The oldest Western tuning system. Pythagorean tuning stacks pure 3:2 fifths until twelve notes are produced, accepting whatever dissonance falls on the final, leftover fifth — the famous wolf. The result is brilliant, ringing fifths and perfect fourths, paid for with notably wide major thirds (about 22 cents sharp) that sound jangly to modern ears.

Pythagorean tuning suits medieval polyphony — Notre Dame organum, early Machaut — where fifths and fourths carry the harmony and thirds are passing dissonances. As thirds grew more important in Renaissance music, Pythagorean was abandoned in favour of meantone.

Meantone Temperament

Meantone trades the purity of the fifth for the purity of the third. Quarter-comma meantone tempers each fifth flat by a quarter of the syntonic comma, producing eight perfectly pure 5:4 major thirds — at the cost of an unusable wolf fifth somewhere on the keyboard, conventionally between G♯ and E♭.

Quarter-comma was the standard keyboard tuning of the Renaissance and early Baroque. Performers worked around the wolf by simply not modulating into the dark keys. Sixth-comma meantone is a milder compromise that softens the wolf and remains close to the pure third.

Well Temperaments

By the late 17th century, composers wanted to write in every key. Meantone could not survive the journey. The well temperaments are a family of irregular tunings — no two intervals of the same nominal size are exactly equal — designed so that every key is usable, while preserving distinct character from one key to the next. C major sounds bright and open; F-sharp major sounds dark and restless. This is what Bach meant by Das Wohltemperierte Klavier.

Werckmeister III (1691)

Andreas Werckmeister's third temperament tempers four fifths (C–G, G–D, D–A, B–F♯) by a quarter of the Pythagorean comma each and leaves the rest pure. The home keys ring sweetly; the remote keys retain Pythagorean intensity. A standard choice for German Baroque keyboard music.

Kirnberger III (1779)

Johann Philipp Kirnberger, a student of Bach, designed a temperament with one perfectly pure C–E major third and a single comma distributed across four fifths. Closer to just intonation in the central keys than Werckmeister, and a frequent choice for historically informed performance of Bach.

Vallotti (c. 1779)

Francesco Antonio Vallotti tempered six fifths (F through B) by a sixth of the Pythagorean comma each, leaving the other six pure. The result is symmetric and gentle — a popular modern choice for 18th-century repertoire from Vivaldi to early Mozart.

Young (1800)

Thomas Young proposed two temperaments around the turn of the 19th century. Young II (the more common modern choice) is similar to Vallotti but rotated, placing the pure fifths on the sharp side of the circle. Suits late Classical and early Romantic keyboard music.

Bach / Lehman (2005)

Bradley Lehman's reading of the squiggle drawn at the head of the autograph manuscript of The Well-Tempered Clavier as a tuning diagram. Whether or not Bach intended it, the resulting temperament is musically convincing for the ’48 and is now widely recorded.

Neidhardt, Kellner, Rameau, D'Alembert, Tartini-Vallotti, Sorge

temperat.io ships these and other lesser-known well temperaments for completeness. Each is a plausible candidate for unmarked repertoire of its period; the differences between them are subtle and largely a matter of taste.

Modern Reconstructions

John Barnes (1979) and Bernhard Billeter (1979 and later) each proposed reconstructions of the temperament Bach is most likely to have used, based on internal evidence in the music itself. These are conjectural but well-argued, and worth comparing side by side.

Thomas Young I is also listed alongside these for ease of comparison, but it is not a reconstruction — it is Young's own first temperament from 1800, included next to Young II so you can hear the difference between his two proposals.

Microtonal Tuning

Microtonal music uses pitches that fall outside the standard twelve-tone chromatic scale. Equal divisions of the octave (EDOs) beyond twelve — 19-EDO, 22-EDO, 31-EDO, 53-EDO — open up harmonic territory inaccessible to conventional Western tuning. Just-intonation lattices, xenharmonic scales, and historical enharmonic systems are the rest of the same world.

Real-time pitch detection makes microtonal practice possible outside a lab. The temperaments shipped here are 12-note systems, but the cents readout itself is microtonal-aware: any pitch between any two notes is shown to the cent, so you can tune deliberately to a quarter-tone, a sixth-tone, or any custom interval you can hear.

How to use temperat.io

Open the tuner, grant microphone access, and select a temperament from the menu. The cents readout shows how far each note deviates from its target in the chosen system. Reference pitch (A) is adjustable from 415 to 450 Hz — 415 for Baroque, 432 for alternative tuning, 440 for modern concert, 442 for most European orchestras, and 444 for orchestras at the sharper end of European practice.

For piano work, the optional Railsback-curve toggle compensates for inharmonicity in stretched piano tuning. Audio never leaves your device.