How to use the tuning system comparison calculator
- Enter the frequency of A4. 440 Hz is the usual reference pitch.
- Choose the tonic, the note your scale or key starts on, in the octave from C4 to B4.
- Read the table with each step in equal temperament, just intonation and Pythagorean tuning.
- Use the cents columns to see how far the other systems sit above or below equal temperament.
Formula
Equal temperament: frequency = tonic x 2^(step / 12), 100 cents per step
Just intonation: ratios 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8
Pythagorean: ratios built from stacked 3/2 fifths and folded into one octave
Cents = 1200 x log2(ratio)
A cent is 1/100 of an equal-tempered semitone. The deviation columns subtract the equal-temperament value, so positive means sharper.
C with A4 = 440 Hz
With A4 at 440 Hz the tonic C4 is 261.626 Hz. A perfect fifth above it is 391.995 Hz in equal temperament and 392.438 Hz at the pure ratio 3/2, which is 701.955 cents, or 1.955 cents sharper. A just major third (5/4) is 386.314 cents, 13.686 cents flatter than the equal-tempered 400, while the Pythagorean major third (81/64) is 407.820 cents, 7.820 cents sharper.
The difference between the two major thirds, 81/80, is 21.506 cents, called the syntonic comma. Twelve pure fifths overshoot seven octaves by 23.460 cents, the Pythagorean comma. With A4 = 432 Hz and a tonic of G, G4 is 384.868 Hz.
Why there are different tunings
Pure fifths and pure thirds cannot all be true at once, because twelve pure fifths do not return to the octave exactly. Pythagorean tuning keeps the fifths pure and lets the thirds be wide. Just intonation tunes the main intervals of one key to pure whole-number ratios, which sounds very smooth but only in that key. Equal temperament spreads the error evenly so every key is usable, at the cost that every interval except the octave is slightly off.
For a single interval see the musical interval calculator. For note frequencies use the note frequency converter, for scale notes the musical scale generator, and for key relationships the circle of fifths.
Reading the cents
Differences of a few cents are small, but the ear notices an out-of-tune third at around 10 to 15 cents, and beats in sustained chords are the result. The just column shows the largest differences in the thirds and sixths, and the Pythagorean column in the thirds as well, which is why Renaissance music moved away from pure fifths toward tunings that improved the thirds.
Assumptions and limits
- The tonic is taken in the octave from C4 to B4, tuned in equal temperament from the A4 you enter.
- The just intonation set is the common 5-limit set with 45/32 for the tritone and 9/5 for the minor seventh. Other choices exist.
- The Pythagorean set uses fifths extended up from the tonic and folded into one octave, with the tritone as 729/512.
- Reference pitch must be between 20 and 4000 Hz. Results are in hertz and cents.
Frequently asked questions
What is just intonation?
It is a tuning where intervals are simple whole-number ratios of frequencies, such as 3/2 for a fifth and 5/4 for a major third. It sounds very pure in one key and drifts out of tune in others.
What is the difference between Pythagorean and equal temperament?
Pythagorean tuning keeps fifths at exactly 3/2 and has wide major thirds. Equal temperament divides the octave into 12 equal parts, so no interval but the octave is pure, but all keys sound alike.
What is a cent?
A cent is one hundredth of an equal-tempered semitone, so an octave is 1200 cents. A just fifth is 701.955 cents, about two cents wider than the 700 of equal temperament.
Why does 432 Hz sound different?
Only because every note is slightly lower in pitch than at 440 Hz, by about 31.8 cents. The ratios between notes stay the same, so the relationships between notes are unchanged.