Twelve equal steps to the octave, so a keyboard can play in every key. Each semitone multiplies the frequency by 21/12 ≈ 1.059463; twelve of them land exactly on the octave, ×2. Down the center, data flows: an interval goes in, the engine computes the ratio and cents, the pitch comes out. The blue team builds and defends it; the red team tries to break it.
source Andreas Werckmeister, Musicalische Temperatur (Quedlinburg, 1691) — the treatise that made well-tempered keyboard tuning a standard; equal temperament as 21/12 was described earlier (Zhu Zaiyu 1584; Stevin c.1585; Mersenne 1636). No single canonical stable link (AMBER). Rendered, not quoted.
One rule, applied twelve times: to rise one semitone, multiply the frequency by 21/12. Not add Hz — multiply a ratio. So the interval of n semitones is 2n/12, and cents are just 100·n (1200 to the octave).
The full ladder from A (20/12) to A (212/12), computed live:
| n | note | ratio 2n/12 | cents |
|---|
Equal temperament is 21/12 per semitone: the compromise that plays every key. It is the-overtone-series smoothed into a grid — a uniform lattice laid over the ragged whole-number ratios nature actually emits.
That smoothing puts it at odds with the-just-intonation, where the fifth is exactly 3/2. ET's fifth is 27/12 — close, but not it. Each sphere is the next one's premise.
The blue team's live check: re-derive the octave, the semitone, the fifth, and transposition-invariance from the multiply rule and confirm them against known truth. If red switches to a linear step, this badge is where it shows.
Two anchors and one interval. The reference pitch is A4 = 440 Hz; the octave is fixed at exactly ×2; the octave is cut into 12 equal ratio-steps. Pick how many semitones n to rise (0 = unison, 12 = octave), and the machine returns the exact ratio, the cents, the frequency, and how far it sits from the pure just interval.
"Equal" means equal in ratio, not in Hz — every step is the same multiplier 21/12, so the Hz gap grows as you climb. That is what you feed the panel below.
Every step is the ratio 21/12. Only the octave is pure; the rest are the compromise.
Change n — the ratio is computed as 2n/12 on the spot, never looked up. Root key and interval key light on the octave below.
What the machine produces, proven: the octave doubles exactly (212/12 = 2 to 1e-12); the semitone is 1.0594630944; the fifth is 27/12 = 1.4983070769, about 1.955 cents flat of the pure 3/2; the major third is 13.686 cents sharp of 5/4. And the system is transposition-invariant — the same n gives the same ratio from any starting note.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
It is a grid laid over the overtone series, not derived from it. "How flat is too flat" is a perceptual judgement AMBER, not a theorem — the math here is exact; the verdict "acceptable" is a choice, which is why the panel lets you see the error rather than hide it.
"Werckmeister invented equal temperament in 1691." Cut. His 1691 book promoted unequal well-temperaments; he only accepted equal temperament in the 1700s. The 21/12 math predates him — Zhu Zaiyu 1584, Stevin c.1585, Mersenne 1636.
"Well-tempered means equal-tempered — Bach proved it." Cut. "Well-tempered" is a circulating tuning playable in all keys, not twelve equal steps. The WTC does not fix which temperament.
"Equal temperament has pure intervals." Kept, corrected. Exactly one is pure: the octave. Every fifth, third, and sixth is detuned by a fixed, computable amount — shown in the panel.
The red team's move: replace the multiply rule with a linear one — add a constant number of Hz per semitone instead of ×21/12. The blue team's witness (window 7) is watching.
Add equal Hz per step and the octave no longer doubles, and the same interval gives different ratios from different roots — transposition breaks. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.