Editing Pythagorean tuning (section)

{{Short description|Method of tuning a musical instrument}}
[[File:Syntonic tuning continuum.svg|right|250px|thumb|The syntonic tuning continuum, showing Pythagorean tuning at 702 cents.<ref name=Milne2007 />]]
{{multiple image
 | align = 
 | direction = vertical
 | width = 330
 | image1 = Pythagorean diatonic scale on C.png
 | caption1 = A series of fifths generated can give seven notes: a [[Diatonic scale|diatonic]] [[major scale]] on C in Pythagorean tuning {{audio|Pythagorean diatonic scale on C.mid|Play}}.
 | image2 = Diatonic scale on C.png
 | caption2 = Diatonic scale on C {{audio|Diatonic scale on C.mid|Play}} 12-tone equal tempered and{{audio|Just diatonic scale on C.mid|Play}} just intonation.
}}
[[Image:Pythagorean major chord on C.png|thumb|Pythagorean (tonic) major chord on C {{audio|Pythagorean major chord on C.mid|Play}} (compare{{audio|Major triad on C.mid|Play}} equal tempered and {{audio|Just major triad on C.mid|Play}} just).]]
[[File:Music intervals frequency ratio equal tempered pythagorean comparison.svg|thumb|450px|
Comparison of equal-tempered (black) and Pythagorean (green) intervals showing the relationship between frequency ratio and the intervals' values, in cents.]]

'''Pythagorean tuning''' is a system of [[musical tuning]] in which the [[frequency ratio]]s of all [[interval (music)|intervals]] are determined by choosing a sequence of  [[Perfect fifths|fifths]]<ref name="B&S">Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: McGraw-Hill). Vol. I: p.&nbsp;56. {{ISBN|978-0-07-294262-0}}</ref> which are "[[Five-limit tuning#The justest ratios|pure]]" or [[perfect fifth|perfect]], with ratio <math>3:2</math>. This is chosen because it is the next [[harmonic]] of a vibrating string, after the octave (which is the ratio <math>2:1</math>), and hence is the next most [[consonance and dissonance|consonant]] "pure" interval, and the easiest to tune by ear.  As [[Novalis]] put it, "The musical proportions seem to me to be particularly correct natural proportions."<ref>Kenneth Sylvan Guthrie, David R. Fideler (1987). ''The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings which Relate to Pythagoras and Pythagorean Philosophy'', p.&nbsp;24. Red Wheel/Weiser. {{ISBN|9780933999510}}.</ref> Alternatively, it can be described as the tuning of the [[Regular diatonic tuning#Syntonic temperament and timbre|syntonic temperament]]<ref name=Milne2007>{{cite journal
| first1 = Andrew
| last1 = Milne
| author2 = Sethares, W.A.
| author2-link = William Sethares
| author3 = Plamondon, J.
|date=December 2007
| title = Invariant Fingerings Across a Tuning Continuum
| journal = Computer Music Journal
| volume = 31
| issue = 4
| pages = 15–32
| url = http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15
| access-date = 2013-07-11
| doi = 10.1162/comj.2007.31.4.15
| s2cid = 27906745
| doi-access = free
}}</ref> in which the [[generator (music)|generator]] is the ratio [[Perfect fifth|3:2]] (i.e., the untempered [[perfect fifth]]), which is ≈&nbsp;702 [[cent (music)|cents]] wide.

The system dates back to Ancient Mesopotamia;.{{sfn|Dumbrill|1998|p=18}} (See {{slink|Music of Mesopotamia|Music theory}}.) It is named, and has been widely misattributed, to [[Music of ancient Greece|Ancient Greeks]], notably [[Pythagoras]] (sixth century BC) by modern authors of music theory. [[Ptolemy]], and later [[Boethius]], ascribed the division of the [[tetrachord]] by only two intervals, called "semitonium" and "tonus" in Latin (256:243&nbsp;×&nbsp;9:8&nbsp;×&nbsp;9:8), to [[Eratosthenes]]. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords [may be considered] a dissonance."<ref name="B&S"/>

The '''Pythagorean scale''' is any [[scale (music)|scale]] which can be constructed from only pure perfect fifths (3:2) and octaves (2:1).<ref>Sethares, William A. (2005). ''Tuning, Timbre, Spectrum, Scale'', p.&nbsp;163. {{ISBN|1-85233-797-4}}.</ref> In Greek music it was used to [[Tetrachord#Pythagorean tunings|tune tetrachords]], which were composed into scales spanning an octave.<ref>{{cite web |first=Peter A. |last=Frazer |url=http://midicode.com/tunings/Tuning10102004.pdf |title=The Development of Musical Tuning Systems |date=April 2001 |archive-url=https://web.archive.org/web/20060506221411/http://www.midicode.com/tunings/Tuning10102004.pdf |archive-date=2006-05-06 |access-date=2014-02-02}}</ref> A distinction can be made between extended Pythagorean tuning and a 12-tone Pythagorean temperament. Extended Pythagorean tuning corresponds 1-on-1 with western music notation and there is no limit to the number of fifths. In 12-tone Pythagorean temperament however one is limited by 12-tones per octave and one cannot play most music according to the Pythagorean system corresponding to the enharmonic notation. Instead one finds that for instance the diminished sixth becomes a "wolf fifth".