the constellation · instruments / tempered
Pure octaves, pure fifths, a circle that closes — pick two. Drag one slider and hear which one you gave up.
A whole tuning grows from one interval, stacked twelve times. Make it the pure fifth (3:2) and the fifth rests — but the thirds turn sour and the circle won't close. Narrow it and the thirds sweeten while the fifths begin to beat. There is no setting that makes everything pure. Drag and listen.
One octave, tuned live by the slider above. Tap a key, or strike a chord and switch temperaments under it — a just triad rests; an equal one shimmers; a Pythagorean one grinds. Drone the root and play against it to feel the beats.
Every pitch you hear is a real frequency ratio off the root. The only aesthetic choices are the timbre (a six-partial organ tone, so the beats between partials are audible) and the root pitch — both held fixed across temperaments, so any change you hear is a change of tuning, nothing else.
Two intervals anchor almost every tuning on earth: the octave (a frequency doubled, 2:1) and the perfect fifth (3:2, the simplest agreement after the octave). The trouble is they don't fit together. Stack twelve pure fifths and you should land seven octaves up, back where you began. You don't — you overshoot by the Pythagorean comma:
So the circle of fifths is not a circle. Something has to give, and a temperament is a choice of what. The slider is that choice, made continuous.
There is a second comma hiding here. Four pure fifths, folded down, give a major third of 81:64 ≈ 407.82¢ — but the third the ear actually wants is 5:4 ≈ 386.31¢. The gap between them is the syntonic comma, 81/80 ≈ 21.51¢. Pure fifths therefore guarantee sharp, restless thirds. That is why the Pythagorean setting rests its fifths but grinds its thirds.
Meantone made the opposite bargain. Narrow every fifth by exactly a quarter of the syntonic comma and four of them land on a pure 5:4 third — beatless, golden — at the price of fifths that now beat. This is the ¼-comma preset, the sound of the Renaissance and early-Baroque keyboard. Equal temperament refuses to choose: it smears the Pythagorean comma evenly across all twelve fifths (each 1.955¢ flat), so the circle closes exactly, no key is pure, and none is unusable. Every fifth beats a little; every third beats more.
And the leftover, the one fifth that has to absorb whatever the other eleven didn't — the wolf. In Pythagorean it comes out narrow (678.5¢); in meantone wide (737.6¢); in equal temperament there is no wolf at all, because all twelve fifths are already the same. The ring above draws exactly where the chain fails to close.
— A fixed twelve-note keyboard cannot be in "just intonation" in every key at once; that is the whole predicament. The keyboard here is a regular temperament — twelve notes generated by one repeated fifth — which is exactly the family Pythagorean, equal, and meantone all belong to. The ¼-comma preset is what gives genuinely pure major thirds on the home keys.
— It is often said Bach's Well-Tempered Clavier was written to prove equal temperament. That is most likely a myth: well-tempered named a family of unequal tunings in which every key is playable but each keeps its own colour. Which one Bach meant is still argued. The distinction is set out in the companion stratum, The Comma.
— Why it can never be fixed: for the circle to close you would need (3/2)m = 2k, i.e. 3m = 2m+k — a power of three equal to a power of two. One is always odd, the other always even. The fifth and the octave are incommensurable, in the same exact sense the diagonal of a square is incommensurable with its side. The comma is the size of the gap that proves it — one refusal, heard here as beating.