Editing Semitone (section)

===Pythagorean tuning===
<!--[[Minor semitone]] and [[major semitone]] link directly here.-->
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| image1 = Pythagorean limma on C.png
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| caption1 = Pythagorean limma on C[[File:Pythagorean minor semitone on C.mid|90px]]
| image2 = Pythagorean apotome on C.png
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| caption2 = Pythagorean apotome on C[[File:Pythagorean apotome on C.mid|90px]]
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{{multiple image
   | width1    = 200
   | image1    = Pythagorean limma.png
   | caption1  = Pythagorean limma as five descending just perfect fifths from C (the inverse is B+)
   | width2    = 200
   | image2    = Pythagorean apotome.png
   | caption2  = Pythagorean apotome as seven just perfect fifths
}}

Like meantone temperament, [[Pythagorean tuning]] is a broken [[circle of fifths]]. This creates two distinct semitones, but because Pythagorean tuning is also a form of 3-limit [[just intonation]], these semitones are rational. Also, unlike most meantone temperaments, the chromatic semitone is larger than the diatonic.

The '''Pythagorean diatonic semitone''' has a ratio of 256/243 ({{Audio|Pythagorean minor semitone on C.mid|play}}), and is often called the '''Pythagorean limma'''. It is also sometimes called the ''Pythagorean minor semitone''. It is about 90.2 cents.
:<math>\frac{256}{243} = \frac{2^8}{3^5} \approx 90.2 \text{ cents}</math>

It can be thought of as the difference between three [[octaves]] and five [[perfect fifth|just fifths]], and functions as a [[#Minor second|diatonic semitone]] in a [[Pythagorean tuning]].

The '''Pythagorean chromatic semitone''' has a ratio of 2187/2048 ({{Audio|Pythagorean apotome on C.mid|play}}). It is about 113.7 [[Cent (music)|cents]]. It may also be called the '''Pythagorean apotome'''<ref name="Rashed">Rashed, Roshdi (ed.) (1996). ''Encyclopedia of the History of Arabic Science, Volume 2'', pp. 588, 608. Routledge. {{ISBN|0-415-12411-5}}.</ref><ref>[[Hermann von Helmholtz]] (1885). ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', p. 454.</ref><ref>Benson, Dave (2006). ''Music: A Mathematical Offering'', p. 369. {{ISBN|0-521-85387-7}}.</ref> or the ''Pythagorean major semitone''. (''See [[Pythagorean interval]]''.)
:<math>\frac{2187}{2048} = \frac{3^7}{2^{11}} \approx 113.7\text{ cents}</math>

It can be thought of as the difference between four perfect [[octave]]s and seven [[perfect fifth|just fifths]], and functions as a [[chromatic semitone]] in a [[Pythagorean tuning]].

The Pythagorean limma and Pythagorean apotome are [[enharmonic]] equivalents (chromatic semitones) and only a [[Pythagorean comma]] apart, in contrast to diatonic and chromatic semitones in [[meantone temperament]] and 5-limit [[just intonation]].