page_title: Pythagorean comma
section_title: Circle of fifths and enharmonic change
hierarchical_section_title: Pythagorean comma / Circle of fifths and enharmonic change
caption_attribution_description: English: Pythagorean comma.
context_page_description: In musical tuning, the Pythagorean comma, named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯. It is equal to the frequency ratio ​⁄₂⁷ = ​⁵³¹⁴⁴¹⁄₅₂₄₂₈₈ ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone. The comma, which musical temperaments often refer to tempering, is the Pythagorean comma.
  The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma, or the difference between twelve just perfect fifths and seven octaves, or the difference between three Pythagorean ditones and one octave.
  The diminished second, in Pythagorean tuning, is defined as the difference between limma and apotome. It coincides, therefore, with the opposite of a Pythagorean comma, and can be viewed as a descending Pythagorean comma, equal to about −23.46 cents.
context_section_description: The Pythagorean comma can also be thought of as the discrepancy between twelve justly tuned perfect fifths (ratio 3:2) (play ) and seven octaves (ratio 2:1):
  In the following table of musical scales in the circle of fifths, the Pythagorean comma is visible as the small interval between e.g. F♯ and G♭.
  The 6♭ and the 6♯ scales* are not identical - even though they are on the piano keyboard - but the ♭ scales are one Pythagorean comma lower. Disregarding this difference leads to enharmonic change.
  * The 7♭ and 5♯, respectively 5♭ and 7♯ scales differ in the same way by one Pythagorean comma. Scales with seven accidentals are seldom used, because the enharmonic scales with five accidentals are treated as equivalent.
  This interval has serious implications for the various tuning schemes of the chromatic scale, because in Western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used 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 (out by about 0.1%). Take the just fifth to the twelfth power, then subtract seven octaves, and you get the Pythagorean comma (about 1.4% difference).
caption: Pythagorean comma as twelve (eleven shown?) justly tuned perfect fifths in Ben Johnston notation.
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