Pythagorean tuning

Pythagorean tuning is the tuning system in which only 3-limit just intonation intervals are used - that is, intervals generated by stacking perfect fifths of 3/2 and octaves of 2/1 up and down. Pythagorean tuning is a rank-2 system that does not include any tempering, and is thus useful as a basis for notation. When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the chain of fifths. Simple Pythagorean intervals include 3/2, 9/8, 32/27, 81/64, and their octave complements.

Pythagorean tuning and temperaments

Note that Pythagorean tuning often refers to the tuning, not the interpretation, and this is its distinction from the 3-limit - that is, some people consider regular temperaments that are well-tuned in Pythagorean tuning to, themselves, count as Pythagorean.

Schismic and garibaldi

The most notable example of this is schismic temperament , which equates the moderately complex Pythagorean interval 8192/6561, the diatonic diminished fourth, to 5/4, which when tuned to just Pythagorean tuning has only 2 cents of error, and its extension garibaldi, which further equates the double-diminished octave to 7/4, with only 4 cents of error when the former is tuned just.