Editing Equal temperament (section)

==Twelve-tone equal temperament==
{{main|12 equal temperament}}
12&nbsp;tone equal temperament, which divides the octave into 12&nbsp;intervals of equal size, is the musical system most widely used today, especially in Western music.

=== 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.

=== Mathematics ===
{{anchor|12TET}}
[[File:Monochord ET.png|250px|thumb|One octave of 12&nbsp;{{sc|tet}} on a monochord]]

In 12&nbsp;tone equal temperament, which divides the octave into 12&nbsp;equal parts, the width of a [[semitone]], i.e. the [[Interval ratio|frequency ratio]] of the interval between two adjacent notes, is the [[twelfth root of two]]:

:<math> \sqrt[12]{2\ } = 2^{\tfrac{1}{12}} \approx 1.059463 </math>

This interval is divided into 100&nbsp;cents.

==== Calculating absolute frequencies ====
{{See also|Piano key frequencies}}
To find the frequency, {{math|''P{{sub|n}}''}}, of a note in 12&nbsp;{{sc|TET}}, the following formula may be used:

:<math>\ P_n = P_a\ \cdot\ \Bigl(\ \sqrt[12]{2\ }\ \Bigr)^{ n-a }\ </math>

In this formula {{math|''P{{sub|n}}''}} represents the pitch, or frequency (usually in [[hertz]]), you are trying to find. {{math|''P{{sub|a}}''}} is the frequency of a reference pitch. The indes numbers {{mvar|n}} and {{mvar|a}} are the labels assigned to the desired pitch ({{mvar|n}}) and the reference pitch ({{mvar|a}}). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A{{sub|4}} (the reference pitch) is the 49th&nbsp;key from the left end of a piano (tuned to [[A440 (pitch standard)|440&nbsp;Hz]]), and C{{sub|4}} ([[middle C]]), and F{{music|#}}{{sub|4}} are the 40th and 46th&nbsp;keys, respectively. These numbers can be used to find the frequency of C{{sub|4}} and F{{music|#}}{{sub|4}}:

:<math>P_{40} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(40-49)} \approx 261.626\ \text{Hz}\ </math>
:<math>P_{46} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(46-49)} \approx 369.994\ \text{Hz}\ </math>

==== Converting frequencies to their equal temperament counterparts ====
To convert a frequency (in Hz) to its equal 12&nbsp;{{sc|TET}} counterpart, the following formula can be used:

:<math>\ E_n = E_a\ \cdot\ 2^{\ x}\ \quad </math> where in general <math> \quad\ x\ \equiv\ \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12\log_{2} \left(\frac{\ n\ }{ a }\right) \Biggr) ~.</math>

[[File:12ed2-5Limit.svg|250px|thumb|Comparison of intervals in 12-TET with just intonation]]

{{math|''E{{sub|n}}''}} is the frequency of a pitch in equal temperament, and {{math|''E{{sub|a}}''}} is the frequency of a reference pitch. For example, if we let the reference pitch equal 440&nbsp;Hz, we can see that {{sc|'''E'''}}{{sub|5}} and {{sc|'''C'''}}{{music|#}}{{sub|5}} have the following frequencies, respectively:

: <math>E_{660} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 7 }{\ 12\ }\right)}\ \approx\ 659.255\ \mathsf{Hz}\ \quad </math> where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl(\ 12 \log_{2}\left(\frac{\ 660\ }{ 440 }\right)\ \Biggr) = \frac{ 7 }{\ 12\ } ~.</math>

: <math>E_{550} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 1 }{\ 3\ }\right)}\ \approx\ 554.365\ \mathsf{Hz}\ \quad </math>  where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12 \log_{2}\left(\frac{\ 550\ }{ 440 }\right)\Biggr) = \frac{ 4 }{\ 12\ } = \frac{ 1 }{\ 3\ } ~.</math>

==== Comparison with just intonation ====
The intervals of 12&nbsp;{{sc|TET}} closely approximate some intervals in [[just intonation]].<ref>
{{cite book
 |last=Partch |first=Harry
 |year=1979
 |title=Genesis of a Music |edition=2nd
 |publisher=Da Capo Press
 |isbn=0-306-80106-X
 |page=[https://archive.org/details/genesismusicacco00part/page/n167 134]
 |url=https://archive.org/details/genesismusicacco00part
 |url-access=limited
}}
</ref>
The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.

In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.
{{clear}}
:{| class="wikitable"  style="margin:auto;text-align:center;"
|-
! Interval Name
! Exact value in 12&nbsp;{{sc|TET}}
! Decimal value in 12&nbsp;{{sc|TET}}
! Pitch in 
! Just intonation interval
! Cents in just intonation
! 12&nbsp;{{sc|TET}} cents<br/>tuning error
|-
| Unison ([[C (musical note)|{{sc|'''C'''}}]])
| {{big|2}}{{sup|{{frac|0|12}}}} = {{big|1}}
| 1
| 0
| {{sfrac|1|1}} = {{big|1}}
| 0
| 0
|-
| Minor second ([[D♭ (musical note)|{{sc|'''D'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|1|12}}}} = <math>\sqrt[12]{2}</math>
| {{#expr:2^(1/12) round 6}}
| 100
| {{sfrac|16|15}} = {{big|1.06666...}}
| {{#expr:1200*ln(16/15)/ln2 round 2}}
| {{#expr:100-1200*ln(16/15)/ln2 round 2}}
|-
| Major second ([[D (musical note)|{{sc|'''D'''}}]])
| {{big|2}}{{sup|{{frac|2|12}}}} = <math>\sqrt[6]{2}</math>
| {{#expr:2^(1/6) round 6}}
| 200
| {{sfrac|9|8}} = {{big|1.125}}
| {{#expr:1200*ln(9/8)/ln2 round 2}}
| {{#expr:200-1200*ln(9/8)/ln2 round 2}}
|-
| Minor third ([[E♭ (musical note)|{{sc|'''E'''}}{{music|flat}}]])
| {{big|2}}{{sup|{{frac|3|12}}}} = <math>\sqrt[4]{2}</math>
| {{#expr:2^(1/4) round 6}}
| 300
| {{sfrac|6|5}} = {{big|1.2}}
| {{#expr:1200*ln(6/5)/ln2 round 2}}
| {{#expr:300-1200*ln(6/5)/ln2 round 2}}
|-
| Major third ([[E (musical note)|{{sc|'''E'''}}]])
| {{big|2}}{{sup|{{frac|4|12}}}} = <math>\sqrt[3]{2}</math>
| {{#expr:2^(1/3) round 6}}
| 400
| {{sfrac|5|4}} = {{big|1.25}}
| {{#expr:1200*ln(5/4)/ln2 round 2}}
| +{{#expr:400-1200*ln(5/4)/ln2 round 2}}
|-
| Perfect fourth ([[F (musical note)|{{sc|'''F'''}}]])
| {{big|2}}{{sup|{{frac|5|12}}}} = <math>\sqrt[12]{32}</math>
| {{#expr:2^(5/12) round 6}}
| 500
| {{sfrac|4|3}} = {{big|1.33333...}}
| {{#expr:1200*ln(4/3)/ln2 round 2}}
| +{{#expr:500-1200*ln(4/3)/ln2 round 2}}
|-
| Tritone ([[G♭ (musical note)|{{sc|'''G'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|6|12}}}} = <math>\sqrt{2}</math>
| {{#expr:2^(1/2) round 6}}
| 600
| {{sfrac|64|45}}= {{big|1.42222...}}
| {{#expr:1200*ln(64/45)/ln2 round 2}}
| {{#expr:600-1200*ln(64/45)/ln2 round 2}}
|-
| Perfect fifth ([[G (musical note)|{{sc|'''G'''}}]])
| {{big|2}}{{sup|{{frac|7|12}}}} = <math>\sqrt[12]{128}</math>
| {{#expr:2^(7/12) round 6}}
| 700
| {{sfrac|3|2}} = {{big|1.5}}
| {{#expr:1200*ln(3/2)/ln2 round 2}}
| {{#expr:700-1200*ln(3/2)/ln2 round 2}}
|-
| Minor sixth ([[A♭ (musical note)|{{sc|'''A'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|8|12}}}} = <math>\sqrt[3]{4}</math>
| {{#expr:2^(2/3) round 6}}
| 800
| {{sfrac|8|5}} = {{big|1.6}}
| {{#expr:1200*ln(8/5)/ln2 round 2}}
| {{#expr:800-1200*ln(8/5)/ln2 round 2}}
|-
| Major sixth ([[A (musical note)|{{sc|'''A'''}}]])
| {{big|2}}{{sup|{{frac|9|12}}}} = <math>\sqrt[4]{8}</math>
| {{#expr:2^(3/4) round 6}}
| 900
| {{sfrac|5|3}} = {{big|1.66666...}}
| {{#expr:1200*ln(5/3)/ln2 round 2}}
| +{{#expr:900-1200*ln(5/3)/ln2 round 2}}
|-
| Minor seventh ([[B♭ (musical note)|{{sc|'''B'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|10|12}}}} = <math>\sqrt[6]{32}</math>
| {{#expr:2^(5/6) round 6}}
| 1000
| {{sfrac|16|9}} = {{big|1.77777...}}
| {{#expr:1200*ln(16/9)/ln2 round 2}}
| +{{#expr:1000-1200*ln(16/9)/ln2 round 2}}
|-
| Major seventh ([[B (musical note)|{{sc|'''B'''}}]])
| {{big|2}}{{sup|{{frac|11|12}}}} = <math>\sqrt[12]{2048}</math>
| {{#expr:2^(11/12) round 6}}
| 1100
| {{sfrac|15|8}} = {{big|1.875}}
| {{#expr:1200*ln(15/8)/ln2 round 2}}0
| +{{#expr:1100-1200*ln(15/8)/ln2 round 2}}
|-
| Octave ([[C (musical note)|{{sc|'''c'''}}]])
| {{big|2}}{{sup|{{frac|12|12}}}} = {{big|2}}
| {{big|2}}
| 1200
| {{sfrac|2|1}} = {{big|2}}
| 1200.00
| 0
|}

=== Seven-tone equal division of the fifth ===
Violins, violas, and cellos are tuned in perfect fifths ({{sc|'''G D A E'''}} for violins and {{sc|'''C G D A'''}} for violas and cellos), which suggests that their semitone ratio is slightly higher than in conventional 12&nbsp;tone equal temperament. Because a perfect fifth is in 3:2&nbsp;relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of <math display=inline>\sqrt[7]{3/2}</math> to the next (100.28&nbsp;cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with a {{nobr|ratio of ≈ 517:258 or ≈ 2.00388:1}} rather than the usual 2:1, because 12&nbsp;perfect fifths do not equal seven octaves.<ref>
{{cite web
 |last=Cordier |first=Serge
 |title=Le tempérament égal à quintes justes |lang=fr
 |publisher=Association pour la Recherche et le Développement de la Musique
 |website=aredem.online.fr
 |url=http://aredem.online.fr/aredem/page_cordier.html
 |access-date=2010-06-02
}}
</ref> During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2&nbsp;ratio.