Editing Pythagorean comma (section)

{{Short description|Small interval between musical notes}}
{{Image frame|width=220
| content = Pythagorean comma (531441:524288) on C
| caption = {{center|<score>{ \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
}</score>}}Pythagorean comma on C using [[Ben Johnston notation|Ben Johnston's notation]]. The note depicted as lower on the staff (B[[semitone#Just intonation|{{music|#}}]][[syntonic comma|+++]]) is slightly higher in pitch (than C{{music|natural}}).[[File:Pythagorean comma on C.mid]]
}}

[[File:Pythagorean comma (difference A1-m2).PNG|thumb|right|450px|Pythagorean comma ('''PC''') defined in [[Pythagorean tuning]] as difference between semitones (A1&nbsp;–&nbsp;m2), or interval between [[enharmonic|enharmonically equivalent]] notes (from D{{Music|b}} to C{{Music|#}}). The [[diminished second]] has the same width but an opposite direction (from to C{{Music|#}} to D{{Music|b}}).]]

In [[musical tuning]], the '''Pythagorean comma''' (or '''ditonic comma'''{{efn|1=not to be confused with the diatonic comma, better known as ''[[syntonic comma]]'', equal to the frequency ratio 81:80, or around 21.51 cents. See: [[Ben Johnston (composer)|Johnston, Ben]] (2006). ''"Maximum Clarity" and Other Writings on Music'', edited by [[Bob Gilmore]]. Urbana: University of Illinois Press. {{ISBN|0-252-03098-2}}.}}), named after the ancient mathematician and philosopher [[Pythagoras]], is the small [[Interval (music)|interval]] (or [[Comma (music)|comma]]) existing in [[Pythagorean tuning]] between two [[enharmonic|enharmonically equivalent]] notes such as C and B{{Music|#}}, or D{{Music|b}} and C{{Music|#}}.<ref>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..."</ref> It is equal to the [[Interval ratio|frequency ratio]] {{frac|(1.5)<sup>12</sup>|2<sup>7</sup>}} = {{frac|531441|524288}} [[&asymp;]] 1.01364, or about 23.46 [[Cent (music)|cents]], roughly a quarter of a [[semitone]] (in between 75:74 and 74:73<ref>[[Jekuthiel Ginsburg|Ginsburg, Jekuthiel]] (2003). ''[[Scripta Mathematica]]'', p. 287. {{ISBN|978-0-7661-3835-3}}.</ref>). The comma that [[musical temperament]]s often "temper" is the Pythagorean comma.<ref>[[Richard Coyne|Coyne, Richard]] (2010). ''The Tuning of Place: Sociable Spaces and Pervasive Digital Media'', p. 45. {{ISBN|978-0-262-01391-8}}.</ref>

The Pythagorean comma can be also defined as the difference between a [[Pythagorean apotome]] and a [[Pythagorean limma]]<ref>Kottick, Edward L. (1992). ''The Harpsichord Owner's Guide'', p. 151. {{ISBN|0-8078-4388-1}}.</ref> (i.e., between a chromatic and a diatonic [[semitone]], as determined in Pythagorean tuning); the difference between 12 [[Just intonation|just]] [[perfect fifth]]s and seven [[octave]]s; or the difference between three Pythagorean [[ditone]]s 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{{Music|#}} to D{{Music|b}}), equal to about &minus;23.46 cents.