Editing Equal temperament (section)

==== 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>