Pythagorean comma

As described in the introduction, the Pythagorean comma may be derived in multiple ways: A just perfect fifth has a frequency ratio of 3:2.

Equal temperament, today the most common tuning system in the West, reconciled this by flattening each fifth by a twelfth of a Pythagorean comma (approximately 2 cents), thus producing perfect octaves.

Another way to express this is that the just fifth has a frequency ratio (compared to the tonic) of 3:2 or 1.5 to 1, whereas the seventh semitone (based on 12 equal logarithmic divisions of an octave) is the seventh power of the twelfth root of two or 1.4983... to 1, which is not quite the same (a difference of about 0.1%).

Take the just fifth to the 12th power, then subtract seven octaves, and you get the Pythagorean comma (about a 1.4% difference).

In George Russell's Lydian Chromatic Concept of Tonal Organization (1953), the half step between the Lydian Tonic and ♭2 in his Altered Major and Minor Auxiliary Diminished Blues scales is theoretically based on the Pythagorean comma.

Pythagorean comma ( PC ) defined in Pythagorean tuning as difference between semitones (A1 – m2), or interval between enharmonically equivalent notes (from D ♭ to C ♯ ). The diminished second has the same width but an opposite direction (from to C ♯ to D ♭ ).
Octaves (1 × 1200 = 1200) versus ditones (3 × 407.82 = 1223.46), depicted as with Cuisenaire rods (red (2) is used for 1200, magenta (4) is used for 407.82).
The Pythagorean comma shown as the gap (on the right side) which causes a 12-pointed star to fail to close, which star represents the Pythagorean scale; each line representing a just perfect fifth. That gap has a central angle of 7.038 degrees, which is 23.46% of 30 degrees.