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Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. Its discovery is generally credited to Pythagoras. more...

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It is the oldest way of tuning the 12-note chromatic scale and, as such, it is the basis for many other methods of tuning. Notably, 53 equal temperament is an almost exact approximation of a very extended Pythagorean tuning.

Method

Pythagorean tuning is based on a stack of perfect fifths, each tuned in the ratio 3:2, the next simplest ratio after 2:1, which is the ratio of an octave. The two notes A and D, for example, are tuned so that their frequencies are in the ratio 3:2 — if D is tuned to 200 Hz, then the A is tuned to 300 Hz. The E a fifth above that A is also tuned in the ratio 3:2 — with the A at 300 Hz, this puts the E at 450 Hz, 9:4 above the original D. When describing tunings, it is usual to speak of all notes as being within an octave of each other, and as this E is over an octave above the original D, it is usual to halve its frequency to move it down an octave. Therefore, the E is tuned to 225 Hz, a 9:8 above the D. The B a 3:2 above that E is tuned to the ratio 27:16 and so on, until the starting note, D, is arrived at again.

In applying this tuning to the chromatic scale, however, a problem arises: no number of 3:2s will fit exactly into an octave. Because of this, the D arrived at after twelve fifths have been tuned up is about a quarter of a semitone sharper than the D used to begin the process. The table below (starting at E flat rather than D) illustrates this, showing the note name, the ratio above D, and the value in cents above the D for each note in the chromatic scale. The cent values of the same notes in equal temperament are also given for comparison (marked in the table below as "et-Cents").

In order to keep the ratios in this table relatively simple, fifths are tuned down from D as well as up. The first note in the circle of fifths given here is E flat (equivalent to D#), from which five perfect fifths are tuned before arriving at D, the nominal unison note.

In equal temperament, and most other modern tunings of the chromatic scale, pairs of enharmonic notes such as E flat and D sharp are thought of as being the same note — however, as the above table indicates, in Pythagorean tuning, they theoretically have different ratios, and are at a different frequency. This discrepancy, of about 23.5 cents, or one quarter of a semitone, is known as a Pythagorean comma.

To get around this problem, Pythagorean tuning uses the above 12 notes from E flat to G sharp shown above, and then places above the G sharp another E flat, starting the sequence again. This leaves the interval G#—Eb sounding badly out of tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out of tune interval such as this one is known as a wolf interval. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.