Notes and frequencies
Every musical note corresponds to a frequency of vibration in hertz (Hz). Western music uses twelve-tone equal temperament, where the octave is divided into twelve equal semitones. Each semitone multiplies the frequency by the twelfth root of 2, about 1.0595, so an octave exactly doubles it.
The reference is A4 = 440 Hz, the A above middle C, set as an international standard in 1939 and confirmed by ISO 16 in 1975.
The formulas
The calculator turns a frequency f into a fractional MIDI note number:
- m = 12 × log₂(f ÷ 440) + 69
- The nearest note is m rounded; its name comes from the remainder after dividing by 12.
- The octave is the note number divided by 12, rounded down, minus 1.
- The deviation in cents is (m − nearest note) × 100.
- The exact frequency of a note n is 440 × 2^((n − 69) ÷ 12).
Worked examples
| Frequency | Nearest note | Deviation |
|---|---|---|
| 440 Hz | A4 (MIDI 69) | 0 cents |
| 261.63 Hz | C4, middle C (MIDI 60) | 0 cents |
| 445 Hz | A4 | +19.6 cents, sharp |
At 440 Hz, the wavelength in air at 20 °C is about 78 cm: the speed of sound, 343 m/s, divided by the frequency.
Cents and tuning
A cent is one hundredth of a semitone. Most tuners show a note as in tune within about ±5 cents. Differences of 5 to 10 cents are often audible when two notes are played one after the other, and much smaller differences can be heard as beating when two notes sound together.
Other reference pitches
| Reference | Used by |
|---|---|
| 415 Hz | Baroque ensembles on period instruments |
| 432 Hz | Some alternative tuning advocates |
| 440 Hz | International standard |
| 442 to 443 Hz | Many European orchestras |
Changing the reference shifts every note by the same ratio. 415 Hz is almost exactly a semitone below 440.
Tips
- Octave numbers change at C, not at A: B3 is followed by C4.
- Some software numbers octaves differently and calls middle C “C3”, so compare MIDI numbers when in doubt.
- Equal temperament makes every key usable but slightly detunes pure intervals: an equal-tempered major third is about 14 cents wider than a pure 5:4 third.