Editing Equal temperament (section)

=== History ===
The two figures frequently credited with the achievement of exact calculation of equal temperament are [[Zhu Zaiyu]] (also romanized as Chu-Tsaiyu. Chinese: {{lang|zh|朱載堉}}) in 1584 and [[Simon Stevin]] in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu,<ref name=Kuttner163/> it is known that Zhu "presented a highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later."

The developments occurred independently.<ref>
{{cite journal
 |first=Fritz A. |last=Kuttner
 |date=May 1975
 |title=Prince Chu Tsai-Yü's life and work: A re-evaluation of his contribution to equal temperament theory
 |journal=[[Ethnomusicology (journal)|Ethnomusicology]]
 |volume=19 |issue=2 |pages=163–206
|doi=10.2307/850355
 |jstor=850355
 }}
</ref>{{rp|style=ama|page=200}}

Kenneth Robinson credits the invention of equal temperament to Zhu<ref name=Robinson>
{{cite book
 |first=Kenneth |last=Robinson
 |year=1980
 |title=A critical study of Chu Tsai-yü's contribution to the theory of equal temperament in Chinese music
 |series=Sinologica Coloniensia |volume=9
 |place=Wiesbaden, DE
 |publisher=Franz Steiner Verlag
 |page={{mvar|vii}}
}}
</ref>{{efn|
"[[Zhu Zaiyu|Chu-Tsaiyu]] {{grey|[was]}} the first formulator of the mathematics of 'equal temperament' anywhere in the {{nobr|world." — {{harvp|Robinson|1980|p={{mvar|vii}} }}<ref name=Robinson/>}}
}}
and provides textual quotations as evidence.<ref name=Robinson221>
{{cite book
 |last1=Robinson |first1=Kenneth G.
 |first2=Joseph  |last2=Needham
 |year=1962–2004
 |title=Physics and Physical Technology
 |series=Science and Civilisation in China |volume=4
 |section=Part&nbsp;1: Physics
 |editor-last=Needham |editor-first=Joseph
 |place=Cambridge, UK
 |publisher=University Press
 |page=221
}}
</ref> In 1584 Zhu wrote:
: I have founded a new system. I establish one foot as the number from which the others are to be extracted, and using proportions I extract them. Altogether one has to find the exact figures for the pitch-pipers in twelve operations.<ref name=Zhu-1584/><ref name=Robinson221/>

Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications".<ref name=Kuttner163>{{harvp|Kuttner|1975|p=163}}</ref> Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor.<ref name=Kuttner220>{{harvp|Kuttner|1975|p=200}}</ref>

==== China ====
[[File:乐律全书全-1154.jpg|250px|thumb|right|[[Zhu Zaiyu]]'s equal temperament pitch pipes]]
Chinese theorists had previously come up with approximations for {{nobr|12 {{sc|TET}}}}, but Zhu was the first person to mathematically solve 12&nbsp;tone equal temperament,<ref name=Cho>{{cite journal |first=Gene J. |last=Cho |date=February 2010 |title=The significance of the discovery of the musical equal temperament in the cultural history |journal=Journal of Xinghai Conservatory of Music |issn=1000-4270 |url=http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm |archive-url=https://web.archive.org/web/20120315013436/http://en.cnki.com.cn/Article_en/CJFDTOTAL-XHYY201002002.htm |archive-date=2012-03-15 |df=dmy-all}}</ref> which he described in two books, published in 1580<ref name=Zhu-1580>
{{cite book
 |last=Zhu |first=Zaiyu |author-link=Zhu Zaiyu
 |year=1580
 |trans-title=Fusion of Music and Calendar
 |script-title=zh:律暦融通
 |title=Lǜ lì róng tōng |lang=zh
}}
</ref> and 1584.<ref name=Zhu-1584>
{{cite book
 |last=Zhu |first=Zaiyu |author-link=Zhu Zaiyu
 |year=1584
 |trans-title=Complete Compendium of Music and Pitch
 |script-title=zh:樂律全書
 |title=Yuè lǜ quán shū |lang=zh
}}
</ref><ref>
{{cite web
 |title=Quantifying ritual: Political cosmology, courtly music, and precision mathematics in seventeenth-century China
 |series=Roger Hart Departments of History and Asian Studies, University of Texas, Austin
 |website=uts.cc.utexas.edu
 |url=http://uts.cc.utexas.edu/~rhart/papers/quantifying.html
 |access-date=2012-03-20 |url-status=dead
 |archive-url=https://web.archive.org/web/20120305174554/http://uts.cc.utexas.edu/~rhart/papers/quantifying.html
 |archive-date=2012-03-05
}}
</ref> Needham also gives an extended account.<ref>{{harvp|Robinson|Needham|1962–2004|p=220&nbsp;ff}}</ref>

Zhu obtained his result by dividing the length of string and pipe successively by {{nobr|<math display=inline>\sqrt[12]{2}</math> ≈ 1.059463}}, and for pipe length by {{nobr|<math>\sqrt[24]{2}</math> ≈ 1.029302}},<ref>{{cite book |title=The Shorter Science & Civilisation in China |edition=abridgemed |editor-first=Colin |editor-last=Ronan |page=385}} — reduced version of the original {{harvp|Robinson|Needham|1962–2004}}.</ref> such that after 12&nbsp;divisions (an octave), the length was halved.

Zhu created several instruments tuned to his system, including bamboo pipes.<ref>
{{cite book
 |first=Lau |last=Hanson
 |script-title=zh:劳汉生 《珠算与实用数学》 389页
 |trans-title=Abacus and Practical Mathematics
 |page=389
}}
</ref>

==== Europe ====
Some of the first Europeans to advocate equal temperament were lutenists [[Vincenzo Galilei]], [[Giacomo Gorzanis]], and [[Francesco Spinacino]], all of whom wrote music in it.<ref>
{{cite book
 |last=Galilei |first=V.  |author-link=Vincenso Galilei
 |year=1584
 |title=Il Fronimo ... Dialogo sopra l'arte del bene intavolare |lang=it
 |trans-title=The Fronimo ... Dialogue on the art of a good beginning
 |publisher=[[Girolamo Scotto]]
 |place=Venice, IT
 |pages=80–89
}}
</ref><ref>
{{cite web
 |title=Resound – corruption of music
 |website=Philresound.co.uk
 |url=http://www.philresound.co.uk/page4.htm
 |access-date=2012-03-20 |url-status=dead
 |archive-url=https://web.archive.org/web/20120324234829/http://www.philresound.co.uk/page4.htm
 |archive-date=2012-03-24
}}
</ref><ref>
{{cite book
 |first=Giacomo |last=Gorzanis
 |year=1982 |orig-year={{circa|1525~1575}}
 |title=Intabolatura di liuto |lang=it
 |trans-title=Lute tabulation |edition=reprint
 |place=Geneva, CH |publisher=Minkoff 
}}
</ref><ref>
{{cite web
 |title=Spinacino 1507a: Thematic Index
 |publisher=Appalachian State University
 |url=http://www.sunstar.com.ph/cebu/business/bickering-blocking-cebu-s-progress
 |url-status=dead |access-date=2012-06-14 |df=dmy-all
 |archive-url=https://web.archive.org/web/20110725182053/http://www.library.appstate.edu/music/lute/16index/tspi07a.html
 |archive-date=2011-07-25
}}
</ref>

[[Simon Stevin]] was the first to develop 12&nbsp;{{sc|TET}} based on the [[twelfth root of two]], which he described in ''van de Spiegheling der singconst'' ({{circa|1605}}), published posthumously in 1884.<ref>
{{cite book
 |first=Simon |last=Stevin  |author-link=Simon Stevin
 |orig-date={{circa|1605}}
 |date=2009-06-30 |df=dmy-all
 |title=Van de Spiegheling der singconst
 |editor-first=Rudolf |editor-last=Rasch
 |publisher=The Diapason Press
 |url=http://diapason.xentonic.org/ttl/ttl21.html
 |via=diapason.xentonic.org  |access-date=2012-03-20 |url-status=dead
 |archive-url=https://web.archive.org/web/20110717015203/http://diapason.xentonic.org/ttl/ttl21.html
 |archive-date=2011-07-17
}}
</ref>

Plucked instrument players (lutenists and guitarists) generally favored equal temperament,<ref>
{{cite book
 |first=Mark |last=Lindley
 |title=Lutes, Viols, Temperaments
 |isbn=978-0-521-28883-5
}}
</ref> while others were more divided.<ref>
{{cite book
 |first=Andreas |last=Werckmeister |author-link=Andreas Werckmeister 
 |year=1707
 |title=Musicalische paradoxal-Discourse |lang=de
 |trans-title=Paradoxical Musical Discussion
}}
</ref> In the end, 12-tone equal temperament won out. This allowed [[enharmonic modulation]], new styles of symmetrical tonality and [[polytonality]], [[atonality|atonal music]] such as that written with the [[Twelve-tone technique|12-tone technique]] or [[serialism]], and [[jazz]] (at least its piano component) to develop and flourish.