page_title: Pythagorean comma
section_title: Size
hierarchical_section_title: Pythagorean comma / Size
caption_attribution_description: English: e^(i*log2(3/2)*2*pi*n) The Pythagorean comma, 23.46 cents, 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.
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 size of a Pythagorean comma, measured in cents, is
  or more exactly, in terms of frequency ratios:
caption: 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.
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