Pythagorean comma

Pythagorean comma (531441:524288) on C
{ \magnifyStaff #3/2 \omit Score.TimeSignature \relative c' <c! \tweak Accidental.stencil #ly:text-interface::print \tweak Accidental.text \markup { \concat { \lower #1 "+++" \sharp}} bis>1 }
Pythagorean comma on C using Ben Johnston's notation. The note depicted as lower on the staff (B♯+++) is slightly higher in pitch (than C♮).
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♭).

In musical tuning, the Pythagorean comma (or ditonic comma[a]), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯.[1] It is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73[2]). The comma that musical temperaments often "temper" is the Pythagorean comma.[3]

The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma[4] (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning); the difference between 12 just perfect fifths and seven octaves; or the difference between three Pythagorean ditones and one octave. (This is why the Pythagorean comma is also called a ditonic comma.)

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 (e.g. from C♯ to D♭), equal to about −23.46 cents.


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  1. ^ Apel, Willi (1969). Harvard Dictionary of Music, p. 188. ISBN 978-0-674-37501-7. "...the difference between the two semitones of the Pythagorean scale..."
  2. ^ Ginsburg, Jekuthiel (2003). Scripta Mathematica, p. 287. ISBN 978-0-7661-3835-3.
  3. ^ Coyne, Richard (2010). The Tuning of Place: Sociable Spaces and Pervasive Digital Media, p. 45. ISBN 978-0-262-01391-8.
  4. ^ Kottick, Edward L. (1992). The Harpsichord Owner's Guide, p. 151. ISBN 0-8078-4388-1.