Tuning Temperament Comparison

What this generator does

Takes a temperament's definition — which of the twelve fifths are pure and which are narrowed, and by what fraction of which comma — and walks that chain to derive all twelve pitches. It then measures the major third standing on each of them, which is where a temperament's character actually lives.

How to use this tool

1. Choose a temperament. Werckmeister III and Kirnberger III are the well temperaments Bach's era actually used.
2. Set the root — the C that everything is measured from.
3. Read the cents column for the tuning, and the last column for what it costs each key.
4. Switch to Equal and watch the last column go flat at +13.69 in all twelve keys.

Understanding the controls

Temperament

Equal, Pythagorean, quarter-comma meantone, Werckmeister III, Kirnberger III, or five-limit just. The first five are circles of fifths; just intonation is not, and the page says so.

Root frequency

The C everything is measured from, in hertz from 20 to 4,000, with decimals if you want them: 261.63 for middle C. Only the frequencies move with it — the cents and the thirds are properties of the temperament, not of the pitch you start at.

Common use cases

• Tuning a harpsichord or a chamber organ to a historical temperament
• Showing a class what “key colour” meant before equal temperament
• Comparing Werckmeister III against Kirnberger III before choosing one for a recording
• Finding which keys a given temperament makes unusable, and which it makes beautiful
• Explaining why quarter-comma meantone has eight perfect thirds and one unusable fifth

How this generator works

A temperament is not a table of twelve numbers; it is a rule about the twelve fifths of the circle. Pythagorean leaves eleven pure and dumps the whole Pythagorean comma into the twelfth, which is why the fifth from G♯ to E♭ is twenty-three cents flat and howls. Quarter-comma meantone narrows eleven fifths by a quarter of the syntonic comma each, which buys eight perfectly pure major thirds. Werckmeister III splits the Pythagorean comma four ways over C–G, G–D, D–A and B–F♯, so no key is unusable and no two are alike. Each is built here from that recipe rather than from a remembered table, and the check that matters follows from it: the twelve fifths must sum to exactly seven octaves, or the circle does not close. Every pitch is then re-derived by walking the chain again, every third recomputed from the pitches beside it, and the published cents tables for all six are asserted in the test suite.

Randomness and fairness

Nothing here is random. A temperament and a root give exactly one table, derived from commas and logarithms.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

• Six temperaments, which is a handful of the hundreds that have been proposed — Vallotti, Young, sixth-comma meantone and the rest are not here.
• Five-limit just is included for comparison but is a tuning rather than a temperament: its fifths are grossly uneven, it does not close into a circle, and you cannot modulate out of the home key with it.
• Everything is given from C. A temperament sounds different transposed, because its key colour is attached to note names rather than to scale degrees.
• Cents describe pitch, not consonance, which also depends on timbre and register — and real instruments are never exactly in tune with themselves anyway.
• For the beat rates you would actually count while tuning an interval by ear, the temperament cents calculator gives those; this page is about comparing whole systems.

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