What the harmonic series is
A string clamped at both ends can only vibrate in patterns that fit whole numbers of half-wavelengths between the clamps. The lowest such pattern gives the fundamental; the next fits two half-wavelengths and vibrates twice as fast; the next three, and so on. The result is a series of frequencies at exact integer multiples of the fundamental. The same argument, with different boundary conditions, gives the same integer series for a cylindrical pipe open at both ends and for a conical bore.
Every sustained note from a string or wind instrument is that whole stack sounding at once. Your ear fuses it into a single pitch at the fundamental, and the relative loudness of the partials is what you perceive as timbre. Remove the odd partials and you get a hollow clarinet-like colour; strengthen the high ones and you get a bright, reedy tone. It is also why you can often still hear the pitch of a bass note through a small speaker that reproduces nothing near its fundamental: the partials are enough, and the brain supplies the missing fundamental.
The word choice matters and trips people up. Partial 1 is the fundamental. Partial 2 is the first overtone. A calculator, a physics text and a brass method book that all say "the seventh" may mean two different frequencies, so this page counts partials, and states it in every column heading.
Why the deviations from equal temperament are what they are
Two systems collide here. The harmonic series is arithmetic - partial n is n times the fundamental. Equal temperament is geometric - each semitone multiplies frequency by 21/12. An arithmetic sequence and a geometric one agree only at the octaves, where n is a power of two.
To see where any partial falls, convert its ratio to cents: 1200 x log2(n). Partial 2 gives exactly 1200, an octave. Partial 3 gives 1901.96, an octave plus 701.96 cents, so a pure fifth - only 1.96 cents from the tempered fifth, which is why fifths sound fine on a piano. Partial 5 gives 2786.31, two octaves plus 386.31 cents, a pure major third that sits 13.69 cents below the tempered 400. Partial 7 gives 3368.83, two octaves plus 968.83 cents, which is 31.17 cents below the tempered minor seventh. Partial 11 gives 4151.32, three octaves plus 551.32 cents, which is 48.68 cents below the tritone and 51.32 above the fourth - almost exactly in the crack.
These deviations are fixed properties of the integers. They do not depend on the fundamental, on the reference pitch, or on the instrument. Change A4 from 440 to 415 Hz and every partial moves, but partial 7 stays 31.17 cents flat of whatever tempered note is nearest. That is why brass players learn them as facts about the instrument: the third-space C on a trumpet is partial 4 and needs no help, while the written A above it is partial 7 and must be lipped up or fingered around.
Working the other way - from a measured frequency to a note name and a cents error - uses the same conversion the cents and interval calculator applies, and the tempered reference grid comes from the note to frequency calculator.
Worked example: the series on A2 at 110 Hz
Take a cello's open A string, A2, at 110 Hz with A4 = 440 Hz.
- Partial 2. 2 x 110 = 220 Hz. That is A3 exactly, because doubling is an octave in both systems. Deviation: 0 cents.
- Partial 3. 3 x 110 = 330 Hz. The tempered E4 is 329.6276 Hz. The deviation is 1200 x log2(330 / 329.6276) = 1200 x log2(1.00113) = +1.955 cents. Sharp, but inaudibly so.
- Partial 5. 5 x 110 = 550 Hz. Tempered C#5 is 554.3653 Hz. 1200 x log2(550 / 554.3653) = 1200 x log2(0.992126) = −13.686 cents. This is the pure major third, and the number is the same 13.69 that separates 5:4 from 400 cents.
- Partial 7. 7 x 110 = 770 Hz. Tempered G5 is 783.9909 Hz. 1200 x log2(770 / 783.9909) = −31.174 cents. Nearly a third of a semitone flat, which is unmistakably audible.
- Partial 9. 9 x 110 = 990 Hz. 1200 x log2(9) = 3803.91 cents, which is three octaves (3600) plus 203.91 - a pure 9:8 major second, +3.91 cents sharp of tempered.
- Partial 11. 11 x 110 = 1210 Hz. 1200 x log2(11) = 4151.32 cents, three octaves plus 551.32. The nearest tempered note is the tritone at 600, so the deviation is −48.68 cents.
Counting the partials within 15 cents of a tempered note across the first sixteen gives partials 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15 and 16 - twelve of them. Only partials 7, 11, 13 and 14 miss, by 31.17, 48.68, 40.53 and 31.17 cents respectively. That ratio is the practical summary of the whole subject: most of the low series is usable on a keyboard, and the notes that are not are the ones every brass method book warns about.
Reading the deviations as a player
A deviation within about 5 cents needs no action; within 15 cents it is correctable by embouchure on a wind instrument and by finger placement on a fretless string. Beyond 30 cents you are dealing with a note that a fixed-pitch instrument simply cannot produce, and the historical solutions are all workarounds: valves and slides that let a player pick a different, better-tuned partial of a different fundamental, or keys that shorten the tube.
On brass, the low partials are widely spaced and the high ones crowd together, which is why bugle calls use only partials 2 to 6 - those are the notes a valveless instrument can reach reliably and hit accurately. Above partial 8 the partials are less than a whole tone apart, so hitting the intended one demands precise embouchure control. The natural horn's much-admired ability to play scale passages above partial 8 comes from exactly this crowding, and hand-stopping supplies the notes still missing.
On strings, the flageolet or natural harmonic is produced by touching the string lightly at a node. Touching at the halfway point suppresses the fundamental and leaves partial 2; touching a third of the way along gives partial 3; a quarter gives partial 4. The node positions are exactly the 1/n divisions of the string, which is also why the twelfth fret sits at half the scale length - see the fret spacing calculator for the full geometry.
One caution about real instruments: this is the ideal series. Piano strings are stiff, and stiffness raises the frequency of every partial above the ideal multiple, increasingly with partial number. That inharmonicity is why piano tuners stretch octaves, and why a low piano note's partial 16 can sit 30 cents or more above 16 times the fundamental. Thin, long strings under high tension are less inharmonic than short thick ones, which is one reason a concert grand sounds cleaner than a spinet.
The first sixteen partials and their deviation from equal temperament
| Partial | Frequency (Hz) | Nearest tempered note | Cents above fundamental | Cents off tempered |
|---|---|---|---|---|
| 1 | 110.0 | A2 | 0.00 | 0.00 |
| 2 | 220.0 | A3 | 1200.00 | 0.00 |
| 3 | 330.0 | E4 | 1901.96 | +1.96 |
| 4 | 440.0 | A4 | 2400.00 | 0.00 |
| 5 | 550.0 | C#5 | 2786.31 | −13.69 |
| 6 | 660.0 | E5 | 3101.96 | +1.96 |
| 7 | 770.0 | G5 | 3368.83 | −31.17 |
| 8 | 880.0 | A5 | 3600.00 | 0.00 |
| 9 | 990.0 | B5 | 3803.91 | +3.91 |
| 10 | 1100.0 | C#6 | 3986.31 | −13.69 |
| 11 | 1210.0 | D#6 | 4151.32 | −48.68 |
| 12 | 1320.0 | E6 | 4301.96 | +1.96 |
| 13 | 1430.0 | F6 | 4440.53 | +40.53 |
| 14 | 1540.0 | G6 | 4568.83 | −31.17 |
| 15 | 1650.0 | G#6 | 4688.27 | −11.73 |
| 16 | 1760.0 | A6 | 4800.00 | 0.00 |
The cents-off column repeats for partials that share a prime factorisation pattern: 3, 6 and 12 all give +1.96 because each is the previous doubled, and doubling adds exactly 1200 cents. Partial 13 sits +40.53 cents above F6 and partial 11 sits 48.68 below D#6, the two most out-of-tune members of the low series.
Assumptions and limits
- This is the ideal series. Real strings have bending stiffness, which raises high partials above the ideal multiples. On a piano the effect is large enough to require stretched tuning.
- Cylindrical pipes closed at one end skip the even partials. A clarinet's air column produces roughly partials 1, 3, 5, 7 and so on, which is why it overblows at a twelfth rather than an octave.
- Partial numbering is not overtone numbering. Partial 2 is the first overtone. Sources that number overtones are consistently one behind sources that number partials.
- Deviations do not depend on the fundamental. Changing the fundamental moves every partial, but the cents-above-fundamental column and the deviations from tempered notes are properties of the integers only.
- Strong partials are not the same as audible ones. Partials above about the sixteenth are usually too close together to hear as separate pitches and merge into a formant-like colour instead.
- The tolerance figure is a convention, not a physical fact. A 15-cent window is a reasonable working threshold for correctable pitch, but it is a choice, and shifting it changes the count of usable partials.
Where the series turns up outside acoustics
The harmonic series explains far more than instrument tuning. Just intonation is built directly out of it: the pure major third is the ratio between partials 5 and 4, the pure fifth is 3 to 2, and the pure minor third is 6 to 5. When singers or brass sections lock a chord and the beating vanishes, they have tuned to those ratios rather than to the tempered ones, which is precisely the difference the cents and interval calculator quantifies.
In synthesis, additive methods build tone by summing partials with chosen amplitudes, and subtractive methods start from a harmonically rich waveform and filter it. A sawtooth contains every partial with amplitude falling as 1/n; a square wave contains only the odd partials, also falling as 1/n, which is why it sounds hollow like a clarinet. Any periodic waveform, however complex, is a sum of exactly these frequencies - that is Fourier's theorem, and it is the reason the series is unavoidable.
The series also constrains instrument design. Brass instruments are shaped so their resonances line up with the integer series despite the bell and mouthpiece disturbing it; the flare of the bell and the volume of the mouthpiece cup are corrective, not decorative. Historic natural trumpets and horns had access only to these pitches, which is why baroque trumpet parts live in the high register where the partials are close enough together to form a scale, and why horn parts before valves use so many crooks. Playing that repertoire on a modern instrument is a transposition problem as much as a technical one - the transposition calculator handles the arithmetic.