How equal temperament produces the numbers
Twelve-tone equal temperament divides the octave into twelve equal ratios rather than twelve equal frequency steps. An octave is a doubling, so each semitone is the twelfth root of two, about 1.059463. The frequency of any note is therefore the reference frequency multiplied by 2 raised to the power of n over 12, where n is the number of semitones away from the reference, positive going up and negative going down.
With A4 at 440 Hz: A5 is 440 x 2 to the power 12/12, which is 880. C4 is nine semitones below A4, so 440 x 2 to the power minus 9/12, which is 261.6256 Hz. That is middle C, and it is not a round number in any tuning where A is. The two facts are inseparable, because the ratio between them is irrational by construction.
MIDI note numbers make the same arithmetic easier to carry around. A4 is MIDI 69, middle C is 60, and every semitone is one step. Convert a MIDI number to frequency with 440 x 2 to the power (m minus 69) over 12, and you never have to think about note names again.
Cents, and why they are the only sensible unit for pitch error
A cent is one hundredth of a semitone, defined as a ratio rather than a frequency difference: 1200 x the base-two logarithm of f2 divided by f1. Two frequencies an octave apart are 1200 cents apart whether they are 55 and 110 Hz or 3520 and 7040.
This is why hertz is useless for describing being out of tune. Being 3 Hz sharp at A2 (110 Hz) is roughly 47 cents, close to a quartertone and audible to anyone. Being 3 Hz sharp at A5 (880 Hz) is under 6 cents and almost nobody would flag it. The same absolute error, two completely different musical outcomes. Every tuner in existence reads in cents for exactly this reason.
| Deviation | What it corresponds to |
|---|---|
| 1 to 3 cents | Below the threshold most listeners detect on a sustained tone |
| 5 to 10 cents | Audible as beating between two sustained sources; the working tolerance of a careful tuning |
| 14 cents | The amount an equal-tempered major third sits above the pure 5:4 interval |
| Around 20 cents | Obviously out to most people on a held chord |
| 100 cents | A semitone. A different note. |
Beating, and using it to tune without a meter
When two tones are close in frequency, the sum rises and falls at a rate equal to the difference between them. Two strings at 440 and 442 Hz produce two beats per second, and the beating slows as they converge and stops when they match. That is the whole mechanism behind tuning by ear at the unison or the octave, and it is more accurate at the unison than any meter you can afford, because your ear is measuring a difference of a fraction of a hertz against a rate you can count.
The catch is that beating only works near a simple ratio. Well away from unison, octave or fifth, there is no slow beat to listen for and you are back to intervals and cents.
Reference pitch is a choice, not a constant
A440 is the modern convention and it is a convention, not a law of nature. Historical performance often works at 415, roughly a semitone below, and 430 turns up as well. Some orchestras habitually run at 442 or 443 to sound brighter, which is why a soloist arriving with a fixed-pitch instrument checks the reference before anything else.
The practical point for recording is only that everything in the session has to agree. A sampled instrument built at 440 and a synth set to 432 will never sit together, and no amount of processing fixes it. Set the reference on this page to whatever the session uses and read every number off that.
Wavelength, and why it matters in a room
Sound travels about 343 metres per second in air at ordinary room temperature, so a note at frequency f has a wavelength of 343 divided by f. A 40 Hz note is roughly 8.6 metres long, which is longer than most rooms, and that single fact explains why low-frequency response in a small room is dominated by the room rather than by the speakers. A 1 kHz note is 34 cm, comfortably shorter than any wall, so it behaves in ways that treatment can address directly.
If a room is fighting you at a particular pitch, look up its frequency here, take the wavelength, and compare it with the room dimensions. Anything close to a room dimension, or half or a third of one, is a standing wave and belongs to acoustic treatment rather than to mixing.
Where this connects
For the time-domain side of the same session, the BPM delay time calculator converts tempo to milliseconds. If you are working out what a tuning does to an instrument rather than to a track, the string tension calculator uses these frequencies directly, and the fret position calculator covers where the notes land on a neck. For a click to play along with, the metronome is the tool.
Questions people ask
What is the formula for the frequency of a note?
Frequency = reference x 2 to the power of n divided by 12, where n is the number of semitones between the note and the reference and the reference is normally A4 at 440 Hz. Semitones above the reference are positive, below are negative. Middle C is nine semitones below A4, so 440 x 2 to the power minus 0.75, which is 261.626 Hz. The same formula in MIDI terms is 440 x 2 to the power of (m minus 69) over 12, which avoids having to count note names.
How do I calculate cents between two frequencies?
Cents = 1200 x log base 2 of (f2 / f1). In a spreadsheet or a language without a log2 function, use 1200 x ln(f2 / f1) / ln(2). An octave gives exactly 1200, a semitone 100, and a unison 0. The result is signed: a negative figure means f2 is below f1. Because it is a ratio, the answer is independent of the register, which is the whole reason cents exist as a unit.
Is 432 Hz tuning better than 440?
There is no evidence for the claims made about it, and this page will happily calculate either. Changing the reference shifts every pitch by the same ratio, which is 31.8 cents in the case of 432 against 440, so the intervals between notes are identical and nothing about the harmony changes. What does change is that the whole recording sits about a third of a semitone lower, which some people prefer the sound of and which is a perfectly reasonable artistic choice. The only hard requirement is internal consistency: everything in a session has to share one reference or nothing will be in tune with anything.
Why does my tuner say the string is in tune but the chord sounds wrong?
Several reasons, and they compound. Equal temperament makes every third and sixth deliberately impure, so a major third on a correctly tuned guitar is 14 cents wider than a pure one and will beat noticeably. On top of that, fretting a note stretches the string and sharpens it, more on heavier strings and higher action, which is what saddle compensation is trying to correct. And a tuner reads an open string, whereas the chord is played with fingers pushing strings sideways. Guitars are tuned by compromise, which is why many players tune the open strings, then adjust by ear against the chords the song actually uses.
What does the MIDI note number tell me?
It is the integer index of a semitone, with middle C at 60 and A4 at 69, and it is what any MIDI device sends to specify a pitch. It is useful in three practical ways: it makes transposition into simple addition, it removes the octave-numbering confusion between different pieces of software, and it lets you convert to frequency with one formula and no note-name lookup. The one thing to watch is that not every manufacturer agrees on whether MIDI 60 is called C3, C4 or C5. The number itself is unambiguous; the name printed next to it is not.