Editing Meantone temperament (section)

{{Short description|Musical tuning system}}

'''Meantone temperaments''' are [[musical temperaments]];<ref name = "Barbour">{{cite book |last=Barbour |first=James Murray |year=1951 |title=Tuning and Temperament: A historical survey, Chapters III, IV and VII |publisher=Dover Publications (2013)| series =Dover Books On Music: History |isbn=9780486434063 }}</ref> that is, a variety of [[Musical tuning#Tuning systems|tuning systems]] constructed, similarly to [[Pythagorean tuning]], as a sequence of equal fifths, both rising and descending, scaled to remain within the same octave. But rather than using [[perfect fifth]]s, consisting of frequency ratios of value <math>3:2</math>, these are ''tempered'' by a suitable factor that narrows them to ratios that are slightly less than <math>3:2</math>, in order to bring the major or minor thirds closer to [[Just intonation|the just intonation ratio]] of <math> 5:4 </math> or <math> 6:5 </math>, respectively. Among temperaments constructed as a sequence of fifths, a [[regular temperament]] is one in which all the fifths are chosen to be of the same size.

'''Twelve-tone [[equal temperament]]'''  ({{nobr|12 {{sc|TET}}}}) is obtained by making all semitones the same size, with each equal to one-twelfth of an octave; i.e. with ratios {{nobr|{{math|{{radic|2 |12}} : 1}}}}. Relative to  [[Pythagorean tuning]], it narrows the [[perfect fifth]]s by about 2&nbsp;[[cents (music)|cents]] or {{nobr|{{sfrac| 1 | 12 }}}} of a [[Pythagorean comma]] to give a frequency ratio of <math> 2^{7/12}:1</math>. This produces major thirds that are wide by about 13&nbsp;[[cents (music)|cents]], or {{nobr|{{sfrac| 1 | 8 }}}} of a semitone. Twelve-tone equal temperament is almost exactly the same as {{nobr|{{sfrac| 1 | 11 }}}} [[syntonic comma]] meantone tuning (1.955 [[cents (music)|cents]] vs. 1.95512).