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♮).

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.

Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).

^ 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..."
^ Ginsburg, Jekuthiel (2003). Scripta Mathematica, p. 287. ISBN 978-0-7661-3835-3.
^ Coyne, Richard (2010). The Tuning of Place: Sociable Spaces and Pervasive Digital Media, p. 45. ISBN 978-0-262-01391-8.
^ Kottick, Edward L. (1992). The Harpsichord Owner's Guide, p. 151. ISBN 0-8078-4388-1.

[2] 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..."

[3] Ginsburg, Jekuthiel (2003). Scripta Mathematica, p. 287. ISBN 978-0-7661-3835-3.

[4] Coyne, Richard (2010). The Tuning of Place: Sociable Spaces and Pervasive Digital Media, p. 45. ISBN 978-0-262-01391-8.

[5] Kottick, Edward L. (1992). The Harpsichord Owner's Guide, p. 151. ISBN 0-8078-4388-1.

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