Just intonation calculator
Pick a root note and octave and the calculator lists all twelve chromatic intervals in 5-limit just intonation or Pythagorean tuning: the frequency ratio, the frequency on your root, the size in cents and how many cents it lies above or below equal temperament. Type any ratio, such as 7/4, to see its size too.
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How to use
- Choose the root note and its octave; middle C is C4.
- Leave the reference at A4 = 440 Hz or change it, for example to 442 Hz.
- Choose 5-limit just intonation or Pythagorean tuning.
- Read the table: for each interval the note name, ratio, frequency, size in cents and, in brackets, the difference from equal temperament.
- To check any other ratio, such as 7/4 or 11/8, type it into the ratio box.
How it works
The root frequency is equal-tempered: f = A4 × 2^((n − 69) ÷ 12), with n the MIDI note number. Each interval’s frequency is root × ratio, its size is 1200 × log₂(ratio) cents, and the difference is that size minus 100 cents per semitone.
The 5-limit ratios used are 1/1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8 and 2/1. The Pythagorean ones are 1/1, 256/243, 9/8, 32/27, 81/64, 4/3, 729/512, 3/2, 128/81, 27/16, 16/9, 243/128 and 2/1. Other sets exist; the minor seventh, for example, is also given as 16/9 in just intonation.
Example
Root C4 with A4 = 440 Hz:
- C4 = 440 × 2^(−9/12) = 261.63 Hz;
- just major third 5/4: 261.63 × 1.25 = 327.03 Hz, 1200 × log₂ 1.25 = 386.31 cents, which is 13.69 cents below the equal-tempered 400;
- just fifth 3/2: 392.44 Hz, 701.96 cents, 1.96 cents above the equal-tempered 700.
Tips
- Singers, string players and brass players drift toward just intervals in sustained chords; the table shows by how much.
- To tune a synthesizer with per-note tuning, use the difference in brackets as each note’s offset in cents from its equal-tempered pitch.
- Just intervals depend on the root, so retune the table when the music changes key.
Frequently asked questions
What is just intonation?
A tuning in which intervals are small whole-number frequency ratios: a fifth is 3/2, a major third 5/4, a minor third 6/5. Such intervals sound beat-free and very smooth. The catch is that they do not add up evenly around the octave, so a keyboard tuned this way sounds pure in one key and out of tune in distant ones.
How far is equal temperament from just intonation?
Equal temperament splits the octave into 12 equal semitones of 100 cents. Its fifth, 700 cents, is only 1.96 cents narrower than a pure 3/2, but its major third, 400 cents, is 13.69 cents wider than a pure 5/4 at 386.31 cents. That is why sustained thirds on a piano have a slight shimmer.
What is the difference between just and Pythagorean tuning?
Pythagorean tuning builds everything from pure fifths, 3/2, folded back into the octave. Its fifths and fourths are pure, but its major third is 81/64, 407.82 cents, even wider than equal temperament. 5-limit just intonation uses the 5/4 third instead, at the cost of one fifth in the scale that is not pure.
How are cents calculated?
Cents = 1200 × log₂(ratio). An octave, 2/1, is 1200 cents, and an equal-tempered semitone is 100. For the ratio 7/4: 1200 × log₂ 1.75 = 968.83 cents, 31.17 cents below the equal-tempered minor seventh.