The Circle That Refuses to Close
An octave represents a pure doubling of vibration frequency (a 2:1 ratio). A perfect fifth, the most consonant interval after the octave, increases frequency by half (a 3:2 ratio). If you start from a low bass note and step upward through 12 consecutive pure fifths, you touch all 12 pitch classes of the chromatic scale and should, in theory, land precisely on the original note, 7 octaves higher.
Nature’s prime numbers decree otherwise. The cumulative ratio of 12 pure fifths is (3/2)¹², or 531,441 / 4,096 — roughly 129.75 times the fundamental frequency. The ratio of 7 octaves is 2⁷, exactly 128. Powers of 3 can never equal powers of 2. Dividing these two quantities leaves a remainder of 531,441 / 524,288 (approximately 1.0136), known as the Pythagorean comma. In acoustic units where a semitone equals 100 cents, this excess measures exactly 23.46 cents — nearly a quarter of a semitone.
The Price of Tonal Freedom
For two millennia, instrument makers struggled to distribute this surplus. If you preserve 11 pure fifths, the entire 23.46-cent error accumulates on the 12th step. The resulting interval shrinks to just 678.5 cents — the infamous „wolf fifth”, named for its harsh howling dissonance. In Renaissance quarter-comma meantone tuning, major thirds were tuned purely, but the wolf fifth between G♯ and E♭ expanded to a monstrous 737.65 cents (+37.65 cents above equal temperament), rendering distant keys unusable.
The universal compromise adopted in the 19th century was 12-tone equal temperament (12-TET): the Pythagorean comma was divided evenly into 12 slivers of 1.955 cents, subtracted from each fifth on the wheel. Under this scheme, every fifth measures 700 cents rather than 702, and every semitone is identically 100 cents (a ratio of 2^(1/12)).
The burden of this mathematical symmetry fell squarely on the major thirds. A pure major third, derived from the 5th harmonic of a vibrating string (a 5:4 ratio), measures 386.31 cents. In equal temperament, that same third is forced to 400 cents — 13.69 cents sharper than physical reality. When a pianist strikes C4 and E4, the 5th harmonic of C (1,308.13 Hz) clashes with the 4th harmonic of E (1,318.51 Hz), generating a rapid, audible acoustic beat of 10.38 cycles per second.
Comparing the Four Major Systems
Every epoch in musical history chose where to place the inevitable mathematical error. The table below compares the four major historical tunings against the benchmark of pure harmonic intervals.
| Tuning System | Fifths (Cents) | Major Thirds (Cents) | Wolf Interval | Harmonic Modulation |
|---|---|---|---|---|
| Just Intonation (Pure) | 701.96 ¢ | 386.31 ¢ | Severe key-shift anomalies | Home key only (must retune) |
| Pythagorean (Antiquity) | 701.96 ¢ (11 pure) | 408.00 ¢ (+21.69 ¢) | 678.49 ¢ (G♯–E♭) | Limited to ancient modes |
| Quarter-comma Meantone | 696.58 ¢ (-5.38 ¢) | 386.31 ¢ (Pure) | 737.65 ¢ (+37.65 ¢) | Central keys only (≤ 3 accidentals) |
| Equal Temperament (12-TET) | 700.00 ¢ (-1.96 ¢) | 400.00 ¢ (+13.69 ¢) | None (Error shared equally) | Universal across all 24 keys |
Method Note & Acoustic Physics
Mathematical Interval Calculation
The logarithmic unit of acoustic cents was introduced by Alexander Ellis in 1884. The value in cents for any frequency ratio r = f₂ / f₁ is calculated via: C = 1,200 × log₂(r). An octave (r = 2) contains exactly 1,200 cents, and an equal-tempered semitone measures exactly 100 cents.
The Pythagorean comma ratio is (3/2)¹² / 2⁷ = 531,441 / 524,288 = 1.01364326477. Applying the formula yields: 1,200 × log₂(1.01364326477) = 23.460010 cents.
Physical beats (amplitude modulation) occur whenever two adjacent harmonics overlap, with a beat frequency equal to their absolute difference: f_beat = |n · f₁ - m · f₂|. Frequencies on this page are referenced to international concert pitch A4 = 440 Hz (C4 = 261.63 Hz).