Harmony built from whole numbers. The perfect fifth is 3/2, the major third 5/4, the fourth 4/3 — intervals so pure they beat almost silent. Stack four of those fifths and the ladder overshoots the third by a stubborn crumb: the syntonic comma, 81/80. That crumb is why pure tuning cannot freely change key. Rendered, not quoted.
source Ptolemy, Harmonics (Ἁρμονικά), c. 150 CE — the syntonic diatonic tetrachord (9:8 · 10:9 · 16:15) admitting 5-limit ratios; Pythagorean 3-limit tuning as foil. No stable canonical URL for the primary text — syntonic comma 81/80, Ptolemy's intense diatonic. AMBER · primary undated online
An interval is a frequency ratio. Just intonation restricts those ratios to small whole numbers, whose overtones line up and stop beating:
A major triad is root · third · fifth. Multiply out: 1 · 5/4 · 3/2 clears to the integer chord 4 : 5 : 6 — the simplest consonant harmony there is.
Harmony from whole numbers is the pure 3/2 and 5/4 that the-overtone-series sings on its own. Just intonation writes those ratios down and builds chords from them — until the syntonic comma breaks the ladder, the flaw the-equal-temperament later paved over by bending every fifth 2¢ flat so any key could be reached.
Re-runs the engine's checks against the live constants and flips red if the panel is tampered.
Watches the fifth ratio and the 4:5:6 triad. If window 6 swaps in the equal-tempered fifth, the integer-ratio check fails and this badge trips.
A reference pitch and the just ratios. Default root 264 Hz (just C), interval set { 3/2, 5/4, 4/3, 2/1 }, and the stack depth for the comma (four fifths, less two octaves).
Proven result:
Every number above is computed live from the ratios — no stored constants.
"Whole-number ratios are just a nice story — any tuning is arbitrary." No. The consonance is physical: when two tones share a ratio of small integers their overtones coincide, so the beat rate falls to zero. 3/2 beats at |2·f₃ − 3·f₂| = 0. Detune the fifth by 2¢ and the beating returns — audible, measurable. The ratios are not chosen, they are found.
"Twelve pure fifths return to the starting note." They miss by the Pythagorean comma 3¹²/2¹⁹ ≈ 1.0136. ✓ The circle of fifths does not close in pure tuning.
"Four just fifths equal a just major third plus two octaves." They overshoot by 81/80. ✓ That gap is the syntonic comma — the reason a chord tuned pure in one key drifts sharp in another.
"The just fifth equals the piano's fifth." The piano's is 2^(7/12) ≈ 1.4983, about 2¢ narrower than 3/2 = 1.5000. ✓ Equal temperament trades purity for free transposition.
Swap the pure fifth 3/2 for the equal-tempered 2^(7/12). The 4:5:6 triad stops being exact and the syntonic comma miscomputes — window 7 catches it.