Artificial Wasteland · research · the-comma-played

The math behind Tempered

The verification notebook for the instrument Tempered — a playable tuning bench, and the room-native instrument of the stratum The Comma. Everything the page sounds is exact arithmetic on frequency ratios, proved here before the page asserts it.

verify.mjs tuning.mjs

Run it: node research/the-comma-played/verify.mjs  →  ALL 29 CHECKS PASSED. The page imports the very same functions onto window.__tc, and verify-page.mjs re-checks them in a real browser alongside the full UI.

What is proved

  1. The commas. Pythagorean = (3/2)¹²/2⁷ = 531441/524288 = 23.460¢; syntonic = 81/80 = 21.506¢; schisma = their difference = 1.954¢.
  2. The headline facts. The equal fifth is exactly 700¢ and closes the circle (12·700 = 7·1200); it is 1.955¢ flat of pure. ¼-comma meantone (696.578¢) makes the major third a pure 5:4, beatless. The Pythagorean third (407.82¢) is exactly one syntonic comma sharp of pure — pure fifths force sour thirds.
  3. The spiral construction reproduces equal temperament exactly at 700¢, and the Pythagorean / meantone scales at their known cent values.
  4. The wolf each temperament leaves: narrow (678.5¢) for Pythagorean, none for equal, wide (737.6¢) for meantone — opposite errors.
  5. Beats. A pure interval beats 0 Hz; the trade-off the instrument exists to make audible — Pythagorean rests its fifth, beats its third fast; meantone rests its third, beats its fifth; equal splits the difference; the comma beats ≈ 3 Hz at 220 Hz.
  6. Werckmeister III from its rule (four fifths each narrowed by ¼ Pythagorean comma) keeps every key playable — no wolf, C–E third ~390.2¢.

Honesty

A fixed 12-note keyboard cannot be in just intonation in every key at once — that is the predicament. The keyboard is a regular temperament (12 notes from one repeated fifth); ¼-comma meantone is what gives genuinely pure thirds on the home keys. The only aesthetic choices are the timbre (a six-partial organ tone, so beats between partials are audible) and the root pitch — both fixed across temperaments, so any change heard is a change of tuning alone.