How to use it
Nothing plays when the page opens. Press “Turn sound on”, a piano key or “Play”. The tuning buttons are A (equal temperament), B (just intonation) and C (Pythagorean tuning). Under “Intervals and chords” choose a major triad, press Play and switch between A, B and C while it sounds. In A the major third beats about ten times a second; in B the beating is gone. The beats panel shows which two harmonics are responsible.
Controls
- Tonic: the note the scale starts on. Just and Pythagorean ratios are measured from it.
- A4: the reference pitch, 415 to 466 Hz, default 440 Hz. Frequencies scale with it; cents do not change.
- Tuning A, B, C: click, tap, or press 1, 2, 3.
- Keyboard: thirteen keys from the tonic to the octave. Use the mouse, touch, or the keys A W S E D F T G Y H U J K.
- Interval or chord: choose the type, and the note of the scale it is built on.
- Esc or Stop all sound silences everything. Sound also stops when you leave the tab. Settings are kept in this browser.
How it works
The size of an interval in cents is 1200 times the base-2 logarithm of its frequency ratio. An octave (ratio 2) is 1200 cents, and one equal-tempered semitone is 100 cents.
Equal temperament divides the octave into twelve identical steps, so note k above the tonic has the ratio 2k/12. Only the octave is pure.
5-limit just intonation uses small whole-number ratios built from the primes 2, 3 and 5: 3/2 for the fifth, 5/4 for the major third, 6/5 for the minor third. A major triad is 4:5:6. This page uses one common version (9/8, 9/5 and 45/32 for the second, minor seventh and tritone); other versions exist.
Pythagorean tuning stacks pure 3/2 fifths and folds the result back into one octave. Here the chain of eleven fifths runs from D♭ to F♯. Its major third, 81/64, is 407.8 cents.
Twelve pure fifths overshoot seven octaves, because (3/2)12 is about 129.75 while 27 is 128. The ratio, 531441/524288, is the Pythagorean comma, about 23.46 cents. The chain therefore needs one fifth that is that much too narrow, the wolf fifth (about 678.5 cents from F♯ to C♯ here). The syntonic comma, 81/80 or about 21.51 cents, is the gap between the Pythagorean major third 81/64 and the pure 5/4. A just scale therefore has fifths 21.5 cents flat, such as D to A (40/27).
Two tones close in frequency make beats: the combined loudness rises and falls |fa − fb| times per second. Musical notes contain harmonics at whole-number multiples of their frequency. If two notes have a ratio close to m/n, the mth harmonic of the lower note nearly coincides with the nth of the upper note:
For a major third m = 5 and n = 4. With A4 = 440 Hz, the equal-tempered C4 and E4 give 5 × 261.63 = 1308.1 Hz against 4 × 329.63 = 1318.5 Hz, so about 10.4 beats per second. In just intonation the two are exactly equal. The tone here has twelve harmonics, the nth with 1/n of the fundamental’s amplitude, so these beats are really in the sound.
Things to try
- Play the major triad on the tonic and switch between A and B. The roughness of A disappears in B.
- In C, play the major third on the tonic. It beats about 16 times a second, rougher than in A. The Pythagorean fifth is pure, so it sounds smooth.
- Select B, the fifth and the root D. This D–A fifth is 21.5 cents flat and beats about 11 times a second. In A the same fifth beats about once a second.
- In the fifths table, find the Pythagorean fifth marked ✗ and press its note to hear the wolf.
- Lower A4 to 415 Hz. All pitches drop together, but the cents stay the same.
Frequently asked questions
What is a cent?
A cent is 1/100 of an equal-tempered semitone, or 1/1200 of an octave. The same number of cents is the same musical distance at any pitch. Whether a few cents are audible depends on the pitch, the loudness and the listener.
Why did equal temperament win on keyboards?
A keyboard has fixed pitches. Tuning it with pure thirds and fifths for one key leaves other keys badly out of tune. Equal temperament spreads the error evenly, so every key is usable. The cost is thirds about 14 cents wider than pure and fifths about 2 cents narrower. Singers and players of instruments that can bend pitch can adjust to pure intervals as they play.
Can one scale have only pure intervals?
No. Pure thirds and fifths cannot be combined into a closed twelve-note scale, because the commas above are left over. Every such tuning is a compromise.
Why can I choose the A4 reference?
440 Hz is the widely used modern reference, but ensembles tune somewhat higher or lower, and groups playing period instruments often use about 415 Hz.