Why tuning is measured in cents rather than hertz
Pitch perception is ratio-based. Doubling a frequency always sounds like an octave, whether you go from 55 Hz to 110 Hz or from 1760 Hz to 3520 Hz, even though one of those spans 55 Hz and the other 1760 Hz. Any unit that measures musical distance must therefore be logarithmic, and the cent is the one the world settled on: 1200 cents to the octave, so 100 cents to an equal-tempered semitone.
Alexander Ellis introduced the unit in the 1880s, in his appendices to the English edition of Helmholtz's work on tone sensation, specifically so that tuning systems from different cultures could be compared on one scale. It works because logarithms turn multiplication into addition. Two intervals stacked on top of each other multiply their frequency ratios, and adding their cent values gives the same answer. A perfect fourth of 498.045 cents plus a perfect fifth of 701.955 cents is 1200 cents, and 4/3 x 3/2 is exactly 2.
The practical payoff is that a single number describes tuning error everywhere in the range. A tuner reading −6 cents means the same amount of flatness on a bass string as on a piccolo, which a reading in hertz never does. Five cents is the usual working threshold for an audible error against another instrument; twenty cents is unmistakable.
The formula, and how to reverse it
Divide the upper frequency by the lower one, take the base-2 logarithm to get the interval in octaves, and multiply by 1200 to get cents. If your calculator has no log base 2, use ln(x) / ln(2) or log(x) / log(2); both give the same answer.
Reversing it is just as important. Given a deviation in cents, the frequency ratio is 2 raised to cents over 1200. So a source 10 cents sharp is running at 210/1200 = 1.005793 times its nominal frequency, and A440 measured 10 cents sharp reads 442.55 Hz. A source 1 cent sharp is a factor of 1.000578, which is 0.0578% - a useful figure to remember when you need to convert between cents and the parts-per-million language that clock and sample-rate specifications use.
Signs matter and are easy to lose. A ratio greater than one gives a positive cent value and means the second frequency is higher. A ratio below one gives a negative value. Halving a frequency gives log2(0.5) = −1, so −1200 cents, one octave down. The calculator keeps that sign rather than reporting a magnitude, because in tuning work the direction of an error is exactly what you need to know.
To name an interval, the calculator reduces the cent value into a single octave by subtracting whole multiples of 1200, then rounds to the nearest hundred to find the equal-tempered semitone count. What is left over is the deviation from equal temperament. It then looks up that same interval in the comparison tuning system and reports the deviation from it separately. Those two deviations answer different questions: the first says how far you are from a keyboard, the second says how far you are from a pure ratio.
Worked example: how sharp is the equal-tempered major third
Take middle C at 261.6256 Hz and the E above it as a piano plays it, at 329.6276 Hz, and compare it against the pure 5:4 third a singer or a brass section would gravitate toward.
- Form the ratio. 329.6276 / 261.6256 = 1.259921. That is the cube root of two, which is what four equal-tempered semitones must be.
- Take the logarithm. log2(1.259921) = 0.333333 octaves - exactly a third of an octave.
- Convert to cents. 0.333333 x 1200 = 400.00 cents. Four semitones of 100 cents each, as equal temperament defines them.
- Size the just third. The pure major third is the ratio 5:4 = 1.25. log2(1.25) = 0.3219281, and 0.3219281 x 1200 = 386.314 cents.
- Take the difference. 400.000 − 386.314 = 13.686 cents. The equal-tempered third is that much wider than the pure one.
- Hear it as beating. The fifth harmonic of C4 sits at 5 x 261.6256 = 1308.128 Hz. The fourth harmonic of the tempered E4 sits at 4 x 329.6276 = 1318.510 Hz. Those two differ by 10.38 Hz, so a sustained tempered major third in this octave beats about ten times a second - the roughness you hear on a piano and not in a barbershop quartet.
Run the same arithmetic on the fifth and the picture reverses. The equal-tempered fifth is 700 cents against the pure 701.955, so it is only 1.955 cents narrow. Its third harmonic against the second harmonic of the upper note beats slowly enough that most listeners never notice. That asymmetry - fifths nearly pure, thirds noticeably wide - is the defining sound of equal temperament.
What a given number of cents means
Below about five cents, a discrepancy is inaudible in isolation and produces beating too slow to notice on most sustained notes. This is the region tuners work in when they claim an instrument is in tune. From five to fifteen cents you hear beating between simultaneous notes long before you hear either note as wrong on its own, which is why unisons and octaves are tuned by counting beats rather than by ear-matching pitch. Above twenty cents most listeners describe a note as out of tune, and near fifty cents the note has no clear identity at all, sitting halfway between two semitones.
Some named quantities are worth memorising. The syntonic comma is 21.51 cents, the gap between four stacked pure fifths and a pure major third; it is the amount by which just intonation and Pythagorean tuning disagree about a third. The Pythagorean comma is 23.46 cents, the amount by which twelve pure fifths overshoot seven octaves - the error that every keyboard temperament exists to hide. A schisma is about 1.95 cents, which happens to be the amount an equal-tempered fifth is narrowed. These are the constants the whole history of temperament is built on.
Instrument context changes what counts as acceptable. Fretted instruments are locked to equal temperament by their geometry, so a guitar cannot play a pure third even in principle; the fret spacing calculator shows exactly where those frets have to sit. Wind and brass players bend pitch continuously and unconsciously tune thirds toward just when sustaining chords. Piano tuners deliberately stretch octaves well beyond 1200 cents at the extremes, because string stiffness pushes the overtones sharp - the harmonic series calculator shows where those partials land.
Interval sizes in three tuning systems
| Interval | Equal | Just ratio | Just cents | Just − equal | Pythagorean ratio | Pyth. cents |
|---|---|---|---|---|---|---|
| Unison | 0 | 1:1 | 0.00 | 0.00 | 1:1 | 0.00 |
| Minor 2nd | 100 | 16:15 | 111.73 | +11.73 | 256:243 | 90.22 |
| Major 2nd | 200 | 9:8 | 203.91 | +3.91 | 9:8 | 203.91 |
| Minor 3rd | 300 | 6:5 | 315.64 | +15.64 | 32:27 | 294.13 |
| Major 3rd | 400 | 5:4 | 386.31 | −13.69 | 81:64 | 407.82 |
| Perfect 4th | 500 | 4:3 | 498.04 | −1.96 | 4:3 | 498.04 |
| Tritone | 600 | 45:32 | 590.22 | −9.78 | 729:512 | 611.73 |
| Perfect 5th | 700 | 3:2 | 701.96 | +1.96 | 3:2 | 701.96 |
| Minor 6th | 800 | 8:5 | 813.69 | +13.69 | 128:81 | 792.18 |
| Major 6th | 900 | 5:3 | 884.36 | −15.64 | 27:16 | 905.87 |
| Minor 7th | 1000 | 16:9 | 996.09 | −3.91 | 16:9 | 996.09 |
| Major 7th | 1100 | 15:8 | 1088.27 | −11.73 | 243:128 | 1109.78 |
| Octave | 1200 | 2:1 | 1200.00 | 0.00 | 2:1 | 1200.00 |
The tritone and the minor seventh have no single agreed just ratio; 45:32 and 16:9 are used here, while 7:5 (582.51) and 7:4 (968.83) are common septimal alternatives. Note that just and Pythagorean thirds sit on opposite sides of equal temperament, differing from each other by the syntonic comma of 21.51 cents.
Pitfalls when working in cents
- Cents add, ratios multiply. Stacking two intervals means multiplying their ratios, never adding them. 3/2 stacked on 3/2 is 9/4, not 3.
- Percent is not cents. One cent is a factor of 1.000578, so it is 0.0578%, not 1%. A 1% frequency error is 17.3 cents.
- A deviation near 50 cents has no reliable name. The nearest-interval label becomes arbitrary, and the usual cause is that a tuner has locked onto an overtone rather than the fundamental.
- Just intonation has no single answer for every interval. The tritone and the sevenths depend on which prime limit you accept, and the choices differ by tens of cents. State which ratio you mean.
- A pure ratio in one key is not pure in another. Fixing a keyboard in just intonation makes one key beautiful and remote keys unusable, which is the entire reason equal temperament exists.
- Beat rate is not proportional to cents alone. The same cent error produces faster beating at higher frequencies, because beat rate depends on the absolute frequency difference between the coinciding partials.
Temperament: what the cent numbers are really measuring
The central problem of tuning is that pure intervals do not close. Stack twelve pure 3:2 fifths and you land 23.46 cents - the Pythagorean comma - above seven pure octaves. There is no way to keep every fifth pure and still return to the note you started on, because no power of 3/2 is ever a power of 2. Every tuning system is a decision about where to hide that discrepancy.
Pythagorean tuning keeps eleven fifths pure and dumps the whole comma into the twelfth, producing the notorious wolf fifth. Its thirds come out at 407.82 cents, wider than equal temperament, which suited medieval music where thirds were not treated as consonances. Just intonation instead prioritises thirds at 5:4 and sixths at 5:3, giving beatless triads in one key at the cost of unusable ones elsewhere. Quarter-comma meantone narrows each fifth by a quarter of the syntonic comma to make eight major thirds exactly pure, and dominated European keyboard music for two centuries. Equal temperament spreads the error perfectly evenly: every fifth 1.955 cents narrow, every third 13.686 cents wide, every key equally usable and equally impure.
That last figure is the price of modulation. It is also why choral and brass ensembles sound different from a piano on the same chord - free-pitch ensembles drift toward just ratios automatically, because the beating disappears there, while the piano cannot. When you measure a real performance and find thirds sitting 10 cents below the tempered value, you are not finding an error; you are finding musicians tuning by ear.
To go from cents back to actual pitches, the note to frequency calculator converts any note and detune into hertz, and the transposition calculator handles the key arithmetic that only equal temperament makes safe.