Musical Note Frequencies

The formula

In equal temperament, the octave is divided into twelve identical semitones. Each semitone multiplies the frequency by the twelfth root of 2, about 1.05946. Every note can therefore be calculated from a single reference, A4 at 440 Hz.

Numbering notes as MIDI does, where the tuning A is note 69, the frequency of note n is: f = 440 × 2^((n − 69) / 12).

Middle C is note 60. It lies 9 semitones below A4: f = 440 × 2^(−9/12) ≈ 261.63 Hz. The A an octave lower, note 57, vibrates at 220 Hz, and the A an octave higher, note 81, at 880 Hz.

Reference table

Calculating a frequency by hand

1. Count the semitones between your note and A4 at 440 Hz: positive above, negative below.
2. Divide that number by 12.
3. Raise 2 to that power.
4. Multiply by 440, or by your own reference if you play at 442 Hz, for example.
5. Round to two decimal places: the frequency chart gives the result for every note.

Scientific pitch notation and other octave numbering

In scientific pitch notation, used in English and by most software, middle C is C4 and the tuning A is A4. Some traditions number octaves differently, which is a common source of confusion: in French solfège, the same notes are called Do3 and La3, so the French octave number is one lower than the scientific one.

A standard 88-key piano therefore runs from A0 to C8 in scientific notation, from 27.50 Hz to 4,186.01 Hz.

The tuner lets you choose the display: letter names C D E with scientific octaves (A4), solfège Do Ré Mi with La3 = 440, or German notation with B and H and Helmholtz octaves (a′ for the tuning A).

• E2 = 82.41 Hz: the low string of the guitar.
• A2 = 110 Hz: the guitar's A string.
• C4 = 261.63 Hz: middle C.
• E4 = 329.63 Hz: the high string of the guitar.
• A4 = 440 Hz: the tuning A.

Cents: the unit for small differences

Hertz are handy for describing a note, but not for comparing two deviations: 1 Hz does not have the same effect in the bass as in the treble. That is why we use cents, hundredths of a semitone. An octave is 1,200 cents.

The difference in cents between two frequencies f1 and f2 is 1200 × log2(f2 / f1). Around A4, 1 Hz is about 3.9 cents; around A2 at 110 Hz, the same hertz is nearly 15.7 cents.

That is why the tuner displays cents: a needle at +5 cents means the same thing on a double bass as on a flute.

Beats: hearing a difference

Two sounds with close frequencies produce a regular pulsing, called beats, whose speed in beats per second equals the difference in frequency. An A at 441 Hz played with an A at 440 Hz beats once per second.

Beats also occur between harmonics. For the fifth A4 – E5, the 3rd harmonic of A (1,320 Hz) meets the 2nd harmonic of E. In equal temperament, E5 is 659.26 Hz and its 2nd harmonic 1,318.51 Hz, so the fifth beats about 1.5 times per second. A just fifth, with E at 660 Hz, would not beat.

The Sounds tab offers teaching beats so you can hear them, and intervals played in equal or just temperament for comparison.

Equal temperament and other tuning systems

The formula above describes equal temperament, today's standard. Its fifth is about 2 cents narrower than a just fifth, and its major third about 14 cents wider than a just third: an equal-tempered E4 is 329.63 Hz, versus 327.03 Hz for a just major third above C4.

Other temperaments, such as Pythagorean, Werckmeister III, Vallotti or meantone, change these values to favor certain keys. The tuner can apply them, with a chosen tonic and normalization to A.

If your reference is not 440 Hz

All frequencies are proportional to A. At 442 Hz, multiply every value in the chart by 442/440; at 415 Hz, by 415/440. In the app, change the reference in the settings: the tuner and the sounds follow automatically.