Editing Meantone temperament (section)

==History of meantone temperament and its practical implementation==
Mention of tuning systems that could possibly refer to meantone were published as early as 1496 ([[Franchinus Gaffurius|Gaffurius]]).<ref>{{cite book |first=Franchinus |last=Gaffurius |author-link=Franchinus Gaffurius |year=1496 |title=Practicae musica |publisher = Gulielmum signer Rothomagensem|location = Milan |language=it|url=https://imslp.org/wiki/Practica_musicae_(Gaffurius,_Franchinus)}}</ref> [[Pietro Aron]]<ref>{{cite book |first=Pietro |last=Aron |author-link=Pietro Aron |year=1523 |location =Venice |title=Thoscanello de la musica |publisher= Marchio Sessa|language=it|url=https://imslp.org/wiki/Thoscanello_de_la_musica_(Aron%2C_Pietro)}}</ref>  (Venice, 1523) was unmistakably discussing quarter-comma meantone.  Lodovico Fogliani <ref name="Fogliani">{{cite book | 
first= Lodovico |last = Fogliani |title= Musica theorica |publisher = Civico Museo Bibliografico Musicale | series= Bibliotheca Musica Bononiensis|volume= II/13, 88 pp. |location =Bologna | edition = Line-cut of the Venice, 1529}}</ref> mentioned the quarter-comma system, but offered no discussion of it. The first mathematically precise meantone tuning descriptions are to be found in late 16th century treatises by [[Gioseffo Zarlino|Zarlino]]<ref>{{cite book |first=Gioseffo |last=Zarlino |author-link=Gioseffo Zarlino |year=1558 |title=Le istitutioni harmoniche |language=it | location = Venice | url=https://imslp.org/wiki/Le_Istitutioni_Harmoniche_(Zarlino%2C_Gioseffo)}}</ref> and [[Francisco de Salinas|de&nbsp;Salinas]].<ref>{{cite book |first=Francisco |last=de&nbsp;Salinas |author-link=Francisco de Salinas |year=1577 |title=De musica libri septem |location = Salamanca | publisher = Mathias Gastius | url= https://imslp.org/wiki/De_musica_libri_septem_(Salinas%2C_Francisco_de)}}</ref> Both these authors described the {{nobr|{{sfrac| 1 | 4 }} comma,}} {{nobr|{{sfrac| 1 | 3 }} comma,}} and {{nobr|{{sfrac| 2 | 7 }} comma}} meantone systems.
[[Marin Mersenne]] described various tuning systems in his seminal work on [[music theory]], ''[[Harmonie universelle]]'',<ref>{{cite book | first = Marin | last = Mersenne| title= Harmonie universelle | author-link=Marin Mersenne|date = 1639 | location = Paris |url= http://gallica.bnf.fr/ark:/12148/bpt6k5471093v |publisher = First edition online from [[Bibliothèque nationale de France#Gallica|Gallica]] | trans-title = Translation to English by Roger E. Chapman (The Hague, 1957)}}</ref> including the 31 tone equitempered one, but rejected it on practical grounds.

Meantone temperaments were sometimes referred to under other names or descriptions.  For example, in 1691 [[Christiaan Huygens|Huygens]]<ref name=Huygens-1691>{{cite book |first=Christiaan |last=Huygens |author-link=Christiaan Huygens |year=1691 |title=Lettre à Henri Basnage de Beauval touchant le cycle harmonique, citée dans: "Histoire des Ouvrages des Sçavans" |language=fr |trans-title=Letter concerning the harmonic cycle|location = Rotterdam |url=https://www.huygens-fokker.org/docs/lettre.html}}</ref> advocated the use of the 31 tone equitempered system {{nobr|([[31 equal temperament|31 {{sc|TET}}]])}} as an excellent  approximation for the {{sfrac| 1 | 4 }} comma meantone system, mentioning prior writings of [[Gioseffo Zarlino|Zarlino]] and [[Francisco de Salinas|Salinas]], and dissenting from the negative opinion of [[Mersenne]] (1639). He made a detailed comparison of the frequency ratios in the {{nobr|31 {{sc|TET}}}} system and the [[quarter-comma meantone]] temperament, which he referred to variously as "temperament ordinaire", or "the one that everyone uses". (See references cited in the article [[Temperament Ordinaire]].)

Of course, the quarter-comma meantone system (or any other meantone system) could not have been implemented with complete accuracy until much later, since devices that could accurately measure all pitch frequencies didn't exist until the mid-19th century.  But tuners could apply the same methods that "by ear" tuners have always used: go up by fifths, and down by octaves, or down by fifths, and up by octaves, tempering the fifths so they are slightly smaller than the [[Just tuning|just]] {{sfrac| 3 | 2 }} ratio. How tuners could identify a "quarter comma" reliably by ear is a bit more subtle. Since this amounts to about 0.3% of the frequency which, near [[middle C|middle {{sc|'''C'''}} (~264&nbsp;Hz)]], is about one [[Hertz (unit)|hertz]], they could do it by using perfect fifths as a reference and adjusting the tempered note to produce beats at this rate. However, the frequency of the beats would have to be slightly adjusted, proportionately to the frequency of the note. Alternatively the diatonic scale major thirds can be adjusted to [[Just tuning|just]] major thirds, of ratio {{sfrac| 5 | 4 }}, by eliminating the beats.

For 12&nbsp;tone [[equal temperament|equally-tempered tuning]], the fifths have to be tempered by considerably less than a {{nobr|{{sfrac|1|4}} comma}} (very close to a {{nobr|{{sfrac|1|11}} [[syntonic comma]]}}, or a {{nobr|{{sfrac|1|12}} [[Pythagorean comma]]}}), since they must form a perfect cycle, with no gap at the end, whereas {{nobr|{{sfrac|1|4}} comma}} meantone tuning, as mentioned above, has a residual gap that is twice as large as the Pythagorean one, in the opposite direction.

Although meantone is best known as a tuning system associated with earlier music of the Renaissance and Baroque, there is evidence of its continuous usage as a keyboard temperament well into the 19th&nbsp;century.
: "The mode of tuning which prevailed before the introduction of equal temperament, is called the Meantone System. It has hardly yet died out in England, for it may still be heard on a few organs in country churches. According to Don B. Yñiguez, organist of Seville Cathedral, the meantone system is generally maintained on Spanish organs, even at the present day." — [[George Grove|G. Grove]] (1890)<ref>{{cite encyclopedia |first=G. |last=Grove |author-link=George Grove |year=1890 |title={{grey|[no title cited]}} |encyclopedia=[[A Dictionary of Music and Musicians]] |publisher=Macmillan |place=London, UK |volume=IV |edition=1st |page=72}}</ref>

It has had a considerable revival for early music performance in the late 20th&nbsp;century and in newly composed works specifically demanding meantone by some composers, such as [[John Adams (composer)|Adams]], [[György Ligeti|Ligeti]], and [[Douglas Leedy|Leedy]].