Editing Perfect fifth (section)

==Pitch ratio==
[[File:Just perfect fifth on D.png|thumb|Just perfect fifth on D. The perfect fifth above D (A+, 27/16) is a [[syntonic comma]] (81/80 or 21.5 cents) higher than the [[just major sixth]] above middle C: (A{{music|natural}}, 5/3).<ref name="Fonville">{{cite journal|author-link=John Fonville|author=John Fonville|title=[[Ben Johnston (composer)|Ben Johnston]]'s Extended Just Intonation: A Guide for Interpreters|pages=109 (106–137)|journal=[[Perspectives of New Music]]|volume=29|issue=2|date=Summer 1991|doi=10.2307/833435 |jstor=833435}}</ref>[[File:Just perfect fifth on D.mid]]]]
[[File:Just perfect fifth below A.png|thumb|Just perfect fifth below A. The perfect fifth below A (D-, 10/9) is a syntonic comma lower than the just/Pythagorean major second above middle C: (D{{music|natural}}, 9/8).<ref name="Fonville"/>[[File:Just perfect fifth below A.mid]]]]

The [[Just intonation|justly tuned]] [[interval ratio|pitch ratio]] of a perfect fifth is 3:2 (also known, in early music theory, as a ''[[hemiola]]''),<ref>
{{cite dictionary
 | title = Harvard Dictionary of Music
 | edition = 2nd
 | author = [[Willi Apel]]
 | location=Cambridge, Massachusetts
 | publisher = Harvard University Press
 | year = 1972
 | isbn = 0-674-37501-7 <!--did not originally have isbn-13-->
 | page = 382
 | entry = Hemiola, hemiolia
 | entry-url = https://archive.org/details/harvarddictionar0000apel/page/382
 | entry-url-access = registration
 }}</ref><ref>{{cite dictionary |editor-first=Don Michael |editor-last=Randel|editor-link=Don Michael Randel|date=2003 |entry=Hemiola, hemiola |dictionary=Harvard Dictionary of Music |edition=4th |location=Cambridge, Massachusetts |publisher=Harvard University Press |page=389 |entry-url=https://books.google.com/books?id=02rFSecPhEsC&pg=PA389 |isbn=0-674-01163-5|title=The Harvard Dictionary of Music: Fourth Edition }}</ref> meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two. The just perfect fifth can be heard when a [[violin]] is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune.

Keyboard instruments such as the [[piano]] normally use an [[Equal temperament|equal-tempered]] version of the perfect fifth, enabling the instrument to play in all [[Key (music)|keys]]. In 12-tone equal temperament, the frequencies of the tempered perfect fifth are in the ratio <math>(\sqrt [12]{2})^7</math> or approximately 1.498307. An equally tempered perfect fifth, defined as 700 [[Cent (music)|cents]], is about two cents narrower than a just perfect fifth, which is approximately 701.955 cents.

[[Johannes Kepler|Kepler]] explored [[musical tuning]] in terms of integer ratios, and defined a "lower imperfect fifth" as a 40:27 pitch ratio, and a "greater imperfect fifth" as a 243:160 pitch ratio.<ref>{{cite book|title=Harmonies of the World|author=[[Johannes Kepler]]|editor=[[Stephen Hawking]]|publisher=Running Press|year=2004|isbn=0-7624-2018-9|page=22}}</ref> His lower perfect fifth ratio of 1.48148 (680 cents) is much more "imperfect" than the equal temperament tuning (700 cents) of 1.4983 (relative to the ideal 1.50). [[Hermann von Helmholtz]] uses the ratio 301:200 (708 cents) as an example of an imperfect fifth; he contrasts the ratio of a fifth in equal temperament (700 cents) with a "perfect fifth" (3:2), and discusses the audibility of the [[beat (acoustics)|beats]] that result from such an "imperfect" tuning.<ref>{{cite book | title = On the Sensations of Tone as a Physiological Basis for the Theory of Music |author=[[Hermann von Helmholtz]]| publisher = Longmans, Green | year = 1912 | url = https://archive.org/details/onsensationston01helmgoog | quote = perfect fifth imperfect fifth Helmholtz tempered| pages = [https://archive.org/details/onsensationston01helmgoog/page/n220 199], 313|isbn=9781419178931 }}</ref>