View source for Ditone

{{Short description|Interval in music}}
{{see also|Major third}}
[[Image:Ditone on C.png|thumb|Pythagorean ditone on C {{audio|Pythagorean major third on C.mid|Play}}]]
[[Image:Ditone.png|thumb|Pythagorean ditone as four just perfect fifths]]

In [[music]], a '''ditone''' ({{Langx|la|ditonus}}, from {{Langx|grc|δίτονος}}, "of two tones") is the [[interval (music)|interval]] of a [[major third]]. The size of a ditone varies according to the sizes of the two tones of which it is compounded. The largest is the Pythagorean ditone, with a ratio of 81:64, also called a comma-redundant major third; the smallest is the interval with a ratio of 100:81, also called a comma-deficient major third.<ref>Abraham Rees, "Ditone, Ditonum", in ''The Cyclopædia, or Universal Dictionary of Arts, Sciences, and Literature. In Thirty-Nine Volumes'', vol. 12 (London: Longman, Hurst, Rees, Orme, & Brown, 1819) [not paginated].</ref>

==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).

==Just intonation==
In [[Didymus the Musician|Didymus]]'s diatonic and [[Ptolemy]]'s [[Ptolemy's intense diatonic scale|syntonic]] tunings, the ditone is a just major third with a ratio of 5:4, made up of two unequal tones—a [[Major second#Major and minor tones|major and a minor tone]] of 9:8 and 10:9, respectively. The difference between the two systems is that Didymus places the minor tone below the major, whereas Ptolemy does the opposite.<ref>James Murray Barbour, ''Tuning and Temperament: A Historical Survey'' (East Lansing: Michigan State College Press, 1951): 21. Paperback reprint (Mineola, NY: Dover Books, 2004) {{ISBN|978-0-486-43406-3}}.</ref>

==Meantone temperament==
In [[meantone temperament]]s, the major tone and minor tone are replaced by a "mean tone" which is somewhere in between the two. Two of these tones make a ditone or major third. This major third is exactly the just (5:4) major third in quarter-comma meantone. This is the source of the name: the note exactly halfway between the bounding tones of the major third is called the "[[mean]] tone".<ref>Mimi Waitzman, "Meantone Temperament in Theory and Practice", ''In Theory Only'' 5, no. 4 (May 1981): 3–15. Citation on 4.</ref>

==Equal temperament==
Modern writers occasionally use the word "ditone" to describe the interval of a major third in [[equal temperament]].<ref>Anonymous, "Ditonus", ''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)|John Tyrrell]] (London: Macmillan Publishers, 2001).</ref> For example, "In modern acoustics, the equal-tempered semitone has 100 cents, the tone 200 cents, the ditone or major third 400 cents, the perfect fourth 500 cents, and so on. …”<ref>Manuel Pedro Ferreira, "Proportions in Ancient and Medieval Music", in ''Mathematics and Music: A Diderot Mathematical Forum'', edited by Gerard Assayag, Hans Georg Feichtinger, and José Francesco Rodrigues, 1–17 (Berlin, Heidelberg, and New York: Springer, 2002): 5. {{ISBN|3540437274}}.</ref>

==See also==
*[[Tritone]]
*[[Whole tone|Tone]]
*[[Pythagorean interval]]

==References==
{{reflist}}

{{Intervals|state=expanded}}

[[Category:Major intervals]]
[[Category:3-limit tuning and intervals]]
[[Category:Just tuning and intervals]]
[[Category:Thirds (music)]]