Editing Ditone (section)

==Pythagorean tuning==
The '''Pythagorean ditone''' is the major third in [[Pythagorean tuning]], which has an interval ratio of 81:64,<ref>[[James Murray Barbour]], ''Tuning and Temperament: A Historical Survey'' (East Lansing: Michigan State College Press, 1951): v. Paperback reprint (Mineola, NY: Dover Books, 2004). {{ISBN|978-0-486-43406-3}}.</ref> which is 407.82 [[Cent (music)|cents]]. The Pythagorean ditone is evenly divisible by two [[major tone]]s (9/8 or 203.91 cents) and is wider than a just major third (5/4, 386.31 cents) by a [[syntonic comma]] (81/80, 21.51 cents). Because it is a comma wider than a "perfect" major third of 5:4, it is called a "comma-redundant" interval.<ref>Abraham Rees, "Inconcinnous", in ''The Cyclopædia, or Universal Dictionary of Arts, Sciences, and Literature. In Thirty-Nine Volumes'', vol. 13 (London: Longman, Hurst, Rees, Orme, & Brown, 1819) [not paginated].</ref> {{Audio|Pythagorean major chord on C.mid|Play}}

"The major third that appears commonly in the [Pythagorean] system (C–E, D–F{{music|#}}, etc.) is more properly known as the Pythagorean ditone and consists of two major and two minor semitones (2M+2m). This is the interval that is extremely sharp, at 408c (the ''pure'' major third is only 386c)."<ref>[[Jeffery T. Kite-Powell|Jeffrey T. Kite-Powel]]<nowiki/>l, ''A Performer's Guide to Renaissance Music'', second edition, revised and expanded; Publications of the Early Music Institute (Bloomington and Indianapolis: Indiana University Press, 2007), p.281. {{ISBN|978-0-253-34866-1}}.</ref>

It may also be thought of as four justly tuned [[Perfect fifth|fifths]] minus two [[octave]]s. 

The [[prime factor]]ization of the 81:64 ditone is 3^4/2^6 (or 3/1 * 3/1 * 3/1 * 3/1 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2).