Pythagorean Scale

A series of intervals devised to create a scale dividing the octave into more or less equal steps on the basis of powers of 2/3 or 3/2, i.e. downward or upward intervals of the fifth respectively. Given the series:

and reducing them to intervals lying within the octave, the scale becomes:

In this scale, the ratios between successive steps are either the whole tone 9/8 (204 cents) or the diatonic semitone 256/243 (90 cents). However, the corresponding disadvantage is that no matter how many fifths (3/2 intervals) one takes, either above or below a given note, one never arrives at an octave multiple of that note.

If the power series begun above is extended to the sixth power of (3/2) on the right and the sixth power of (2/3) on the left, then a chromatic scale of 12 tones is produced. However, as with the scale of just intonation, there is a difference between a raised semitone (sharp) and a lowered one (flat). Each is above and below its related note by the interval 2187/2048 (the chromatic semitone) in the Pythagorean scale, thus making C sharp distinct from D flat, for instance.

The interval between the diatonic and chromatic semitone, which is the same as that between the sixth power of (3/2) and the exact octave, is called the Pythagorean comma and has the ratio 531441/524288.

Natural harmonics of a pipe or string fit more closely to a scale of just intonation than to the Pythagorean.

See: Appendix C. Compare: Equal Temperament, Intonation, Tempered Tuning.