A.Harmonic Series Visualizer
| n | Hz | ratio | nearest 12-TET | ¢ off |
|---|
The "wrong" ones. The 5th harmonic sits −13.7¢ below the 12-TET major third. The 7th is −31.2¢ below the 12-TET minor seventh — used as the harmonic seventh in barbershop and blues. The 11th lands almost exactly halfway between the perfect 4th and the tritone: +51¢ above the 4th, −49¢ below the tritone — the famous "blue" 11th. The 13th is +41¢ above the minor 6th. These are perfectly in tune to the natural overtone series, but exiled by 12-tone equal temperament.
B.Tuning System Comparator
| step | cents | Hz | vs 12-TET |
|---|
C.Beating Demo
About this prototype — psychoacoustics, comma wars, & non-Western tunings
Modern psychoacoustics begins with Hermann von Helmholtz's On the Sensations of Tone as a Physiological Basis for the Theory of Music (Die Lehre von den Tonempfindungen, 1863). Helmholtz proposed that musical consonance and dissonance are grounded in the interactions of harmonic partials — that two tones sound rough when their nearby harmonics produce rapid beats inside a critical band. Module C of this lab demonstrates the underlying phenomenon: two sines at f₁ and f₂ produce amplitude modulation at |f₁ − f₂| Hz.
The harmonic series (Module A) is the seed of every tuning controversy. Pure fifths (3:2) stacked twelve times overshoot seven octaves by the Pythagorean comma (~23.46¢). Stacking four pure fifths and comparing to the pure major third (5:4) leaves the syntonic comma (~21.51¢). The semitone gap left over after stacking diatonic whole tones is the limma (~90¢). Western tuning history — Pythagorean → meantone → well-temperaments → equal temperament — is essentially a series of strategies for where to hide the comma.
1/4-comma meantone, dominant from the 16th to early 18th centuries, makes every major third pure (5:4) at the cost of dramatically narrow fifths and an unusable wolf fifth. Equal temperament distributes the Pythagorean comma evenly: every key is equally (slightly) out of tune, but every key is playable. ET won in Europe by ~1850 because composers wanted modulation freedom across all 24 keys.
A common misconception: Bach's Well-Tempered Clavier (Books I & II, 1722 / 1742) was not written for equal temperament. "Well-tempered" means an unequal but circulating temperament (Werckmeister III, Kirnberger III, and others have all been argued) — every key is usable, but each retains its own characteristic color. This is why the C-major and F♯-major preludes feel different in period performance.
Contemporary systematic musicology continues the Helmholtzian tradition through figures like Helga de la Motte-Haber (Berlin) and Richard Parncutt (Graz), whose work on pitch perception, harmony cognition, and cross-cultural psychoacoustics extends the empirical study of tuning into the present.
Outside Europe, tuning theory takes radically different paths. Marc Perlman's Unplayed Melodies (2004) documents how Javanese gamelan musicians theorize slendro and pelog — neither based on a fixed cycle of fifths nor confined to fixed cents. A. J. Racy's ethnomusicology of Arab maqam shows that microtonal inflection in the jins of Rast or Bayati is not a deviation but a structural feature. Slendro and pelog vary measurably from one gamelan to another; ensemble-specific tuning is not a defect of standardization but a defining aesthetic property. The cents values in Module B for these systems are labeled approximate for exactly this reason — a single number cannot speak for an entire tradition.