Harmonic Series Deviation from Equal Temperament
How each partial in the harmonic series from 1 through 20 compares in pitch to its nearest equal temperament equivalent, and what determines the magnitude of each deviation.
The harmonic series produces pitches through integer frequency ratios. Equal temperament divides the octave into twelve semitones, each a frequency ratio of 2^(1/12). Because equal temperament is built from an irrational number and the harmonic series from integers, the two systems agree only at the octave. Every other partial in the series deviates from its nearest equal temperament pitch by an amount that is entirely predictable from the harmonic number.
The position of the nth harmonic above the fundamental, in cents, is:
c = 1200 × log₂(n)
The deviation from the nearest equal temperament pitch is then:
deviation = 1200 × log₂(n) − 100 × round(12 × log₂(n))
where the round function selects the nearest integer, identifying the nearest semitone.
The table below shows this calculation for harmonics 1 through 20, using C as the fundamental. Positive values indicate sharpness relative to equal temperament; negative values indicate flatness.
Harmonic Series — Deviation from Equal Temperament (C fundamental, cents)
| Harmonic | Nearest ET Pitch | Deviation |
|---|---|---|
| 1 | C | 0 |
| 2 | C | 0 |
| 3 | G | +2.0 |
| 4 | C | 0 |
| 5 | E | −13.7 |
| 6 | G | +2.0 |
| 7 | Bb | −31.2 |
| 8 | C | 0 |
| 9 | D | +3.9 |
| 10 | E | −13.7 |
| 11 | F# | −48.7 |
| 12 | G | +2.0 |
| 13 | Ab | +40.5 |
| 14 | Bb | −31.2 |
| 15 | B | −11.7 |
| 16 | C | 0 |
| 17 | C# | +5.0 |
| 18 | D | +3.9 |
| 19 | Eb | −2.5 |
| 20 | E | −13.7 |
Powers of two. Harmonics 1, 2, 4, 8, and 16 are octaves of the fundamental and have zero deviation. Equal temperament reproduces the octave exactly by definition.
Harmonics involving the prime factor 3. The ratio 3:2 — the just perfect fifth — falls 2.0 cents sharp of the equal temperament fifth. Harmonics 3, 6, and 12 all carry this deviation, since the additional factors of 2 are exact octave shifts. Harmonic 9 (3²) compounds it: two factors of 3 produce approximately double the sharpness, landing at +3.9 cents above D. Harmonic 18 (2×3²) carries the same deviation.
Harmonics involving the prime factor 5. The ratio 5:4 — the just major third — falls 13.7 cents flat of the equal temperament major third. Equal temperament achieves good approximations of the fifth at the cost of a significantly worse major third. Harmonics 10 (2×5) and 20 (4×5) carry the same 13.7 cent flatness as harmonic 5. Harmonic 15 (3×5) combines the factor-5 flatness with a small factor-3 sharpening: −13.7 + 2.0 = −11.7 cents from B natural. The intrinsic flatness of the fifth harmonic — independent of fingering or valve physics — is the same tendency documented in valve combination tuning.
Harmonic 7 — the harmonic seventh. The seventh harmonic falls 31.2 cents below Bb, the nearest equal temperament pitch. This interval — sometimes called the harmonic seventh or natural seventh — does not approximate any equal temperament pitch closely. The same 31.2 cent flatness appears one octave higher at harmonic 14 (2×7).
Harmonics 11 and 13. These introduce prime factors for which equal temperament has no close approximation. The eleventh harmonic falls 48.7 cents below F#, placing it 51.3 cents above F — nearly equidistant between the two. The thirteenth harmonic falls 40.5 cents above Ab, well short of A. Neither partial maps usefully to an equal temperament pitch.
Harmonics 17 and 19. The seventeenth harmonic falls 5.0 cents sharp of C# and the nineteenth falls 2.5 cents flat of Eb — two of the closer approximations among the upper harmonics. These results follow from where those primes happen to fall within the equal temperament grid rather than from any structural relationship between them and the twelve-semitone division.
Practical notes for Bb trumpet
The fifth harmonic produces a written E that runs 13.7 cents flat, workable enough that open is the standard fingering. The tenth harmonic is that same E an octave higher, carrying the same prime-5 flatness. With the first valve down, the lengthened tube drops the tenth harmonic to a written D — slightly flatter still, as shown in the tuning tendencies article. The ninth harmonic open gives that same D at only 3.9 cents sharp.
The density of workable open harmonics increases further up the series. Harmonics 15 through 20 produce a consecutive chromatic run — B, C, C#, D, Eb, E — all available without valves. Harmonics 16 through 19 fall within 5 cents of equal temperament; harmonics 15 and 20 reflect the prime-factor tendencies established earlier in the table, landing at −11.7 and −13.7 cents respectively.