View source for Equal temperament

{{Short description|Musical tuning system with constant ratios between notes}}
{{Wide image|Equal Temper w limits.svg|300px|A comparison of some equal temperaments.{{efn|
name=Sethares|
{{harvp|Sethares|2005}} compares several equal temperaments in a graph with axes reversed from the axes in the first comparison of equal temperaments, and identical axes of the second.<ref>{{harvp|Sethares|2005|at=[https://books.google.com/books?id=KChoKKhjOb0C&pg=PA58&vq=%22tuning+of+one+octave+of+notes%22&source=gbs_search_r&cad=1_1&sig=ACfU3U2dq8ONjfazmWxdojVsQ3TFyz-kTg  fig.&nbsp;4.6, p.&nbsp;58] }}</ref>
}} The graph spans one [[octave]] horizontally (open the image to view the full width), and each shaded rectangle is the width of one step in a scale. The [[just interval]] ratios are separated in rows by their [[prime limit]]s.|300px|right}}
[[File:Chromatische toonladder.png|thumb|300px|12&nbsp;tone equal temperament chromatic scale on {{sc|'''C'''}}, one full octave ascending, notated only with sharps. {{audio|ChromaticScaleUpDown.ogg|Play ascending and descending}}]]

An '''equal temperament''' is a [[musical temperament]] or [[Musical tuning#Tuning systems|tuning system]] that approximates [[Just intonation|just intervals]] by dividing an [[octave]] (or other interval) into steps such that the ratio of the [[frequency|frequencies]] of any adjacent pair of notes is the same. This system yields [[Pitch (music)|pitch]] steps perceived as equal in size, due to the [[logarithm]]ic changes in pitch frequency.<ref>{{cite web |last1=O'Donnell |first1=Michael |title=Perceptual Foundations of Sound |url=http://people.cs.uchicago.edu/~odonnell/Scholar/Work_in_progress/Digital_Sound_Modelling/lectnotes/node4.html |access-date=2017-03-11 |df=dmy-all}}</ref>

In [[classical music]] and Western music in general, the most common tuning system since the 18th&nbsp;century has been [[12 equal temperament]] (also known as ''12&nbsp;tone equal temperament'', ''{{nobr|12 {{sc|TET}}}}'' or ''{{nobr|12 {{sc|ET}}}}'', informally abbreviated as ''12&nbsp;equal''), which divides the octave into 12&nbsp;parts, all of which are equal on a [[logarithmic scale]], with a ratio equal to the [[twelfth root of two|12th root of 2]], (<math display=inline>\sqrt[12]{2}</math> ≈ 1.05946). That resulting smallest interval, {{sfrac|1|12}} the width of an octave, is called a [[semitone]] or half step. In [[Western world|Western countries]] the term ''equal temperament'', without qualification, generally means ''{{nobr|12 {{sc|TET}}}}''.

In modern times, {{nobr|12 {{sc|TET}}}} is usually tuned relative to a [[standard pitch]] of 440&nbsp;Hz, called [[A440 (pitch standard)|A 440]], meaning one note, [[A (musical note)|{{sc|'''A'''}}]], is tuned to 440&nbsp;[[hertz]] and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440&nbsp;Hz; it has varied considerably and generally risen over the past few hundred years.<ref>
{{cite book
 |first1=H.   |last1=Helmholtz |author1-link=Hermann von Helmholtz
 |first2=A.J. |last2=Ellis     |author2-link=Alexander J. Ellis
 |translator-first=A.J. |translator-last=Ellis |translator-link=Alexander J. Ellis
 |section=The History of Musical Pitch in Europe
 |pages=493–511
 |title=On the Sensations of Tone
 |title-link=On the Sensations of Tone
 |place=New York, NY
 |publisher=Dover
 |edition=reprint
}}
</ref>

Other equal temperaments divide the octave differently. For example, some music has been written in [[19 equal temperament|{{nobr|19 {{sc|TET}}}}]] and [[31 equal temperament|{{nobr|31 {{sc|TET}}}}]], while the [[Arab tone system]] uses {{nobr|24 {{sc|TET}}.}}

Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the [[Bohlen–Pierce scale]], which divides the just interval of an octave and a fifth (ratio&nbsp;3:1), called a "tritave" or a "[[pseudo-octave]]" in that system, into 13&nbsp;equal parts.

For tuning systems that divide the octave equally, but are not approximations of just intervals, the term '''equal division of the octave''', or ''{{sc|EDO}}'' can be used.

Unfretted [[string ensemble]]s, which can adjust the tuning of all notes except for [[open string (music)|open strings]], and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to [[just intonation]] for acoustic reasons. Other instruments, such as some [[Wind instruments|wind]], [[Keyboard instruments|keyboard]], and [[fret]]ted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings.<ref>
{{cite journal
 | last1=Varieschi | first1=Gabriele U.
 | last2=Gower     | first2=Christina M.
 | year=2010
 | title=Intonation and compensation of fretted string instruments
 | journal=[[American Journal of Physics]]
 | volume=78 | issue=1 | pages=47–55
 | arxiv=0906.0127 | bibcode=2010AmJPh..78...47V
 | s2cid=20827087  | doi=10.1119/1.3226563
}}
</ref>
Some wind instruments that can easily and spontaneously bend their tone, most notably [[trombone]]s, use tuning similar to string ensembles and vocal groups.

== General properties ==
{{Unreferenced section|date=June 2011}}
{{Wide image|EDO errors.png|700px|A comparison of equal temperaments between {{nobr|10 {{sc|TET}}}} and {{nobr|60 {{sc|TET}}}} on each main interval of small prime limits (red: {{sfrac| 3 | 2 }}, green: {{sfrac| 5 | 4 }}, indigo: {{sfrac| 7 | 4 }}, yellow: {{sfrac| 11 | 8 }}, cyan: {{sfrac| 13 | 8 }}). Each colored graph shows how much error occurs (in cents) on the nearest approximation of the corresponding just interval (the black line on the center). Two black curves surrounding the graph on both sides represent the maximum possible error, while the gray ones inside of them indicate the half of it.|400px|right}}

In an equal temperament, the distance between two adjacent steps of the scale is the same [[Interval (music)|interval]]. Because the perceived identity of an interval depends on its [[ratio]], this scale in even steps is a [[geometric sequence]] of multiplications. (An [[arithmetic sequence]] of intervals would not sound evenly spaced and would not permit [[Transposition (music)|transposition]] to different [[Key (music)|keys]].) Specifically, the smallest [[Interval (music)|interval]] in an equal-tempered scale is the ratio:

:<math>\ r^n = p\ </math>
:<math>\ r = \sqrt[n]{p\ }\ </math>

where the ratio {{mvar|r}} divides the ratio {{mvar|p}} (typically the octave, which is 2:1) into {{mvar|n}} equal parts. (''See [[#Twelve-tone equal temperament|Twelve-tone equal temperament]] below.'')

Scales are often measured in [[cent (music)|cents]], which divide the octave into 1200&nbsp;equal intervals (each called a cent). This [[logarithm]]ic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in [[ethnomusicology]]. The basic step in cents for any equal temperament can be found by taking the width of {{mvar|p}} above in cents (usually the octave, which is 1200&nbsp;cents wide), called below {{mvar|w}}, and dividing it into {{mvar|n}} parts:
:<math>\ c = \frac{\ w\ }{ n }\ </math>

In musical analysis, material belonging to an equal temperament is often given an [[Musical notation#Integer notation|integer notation]], meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking the [[logarithm]] of a multiplication reduces it to addition. Furthermore, by applying the [[modular arithmetic]] where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to [[pitch class]]es, which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., {{mvar|c}} is 0 regardless of octave register. The [[MIDI]] encoding standard uses integer note designations.

=== General formulas for the equal-tempered interval ===
{{Missing information|section|the general formulas for the equal-tempered interval|date=February 2019}}

==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]]), that is to be calculated. {{math|''P{{sub|a}}''}} is the frequency of a reference pitch. The index 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.

==Other equal temperaments==<!--[[10 equal temperament]], etc. redirect directly here-->
{{See also|Sonido 13}}

=== Five-, seven-, and nine-tone temperaments in ethnomusicology ===<!--[[7 equal temperament]] and [[5 equal temperament]] redirect directly here-->
[[File:7-tet scale on C.png|thumb|300px|Approximation of {{nobr|7 {{sc|tet}}}}]]

Five- and seven-tone equal temperament (''{{nobr|5 {{sc|TET}}}}'' {{audio|5-tet scale on C.mid|Play}} and ''7 {{sc|TET}}''{{audio|7-tet scale on C.mid|Play}} ), with 240&nbsp;cent {{Audio|1 step in 5-et on C.mid|Play}} and 171&nbsp;cent {{Audio|1 step in 7-et on C.mid|Play}} steps, respectively, are fairly common.

{{nobr|5 {{sc|TET}}}} and {{nobr|7 {{sc|TET}}}} mark the endpoints of the [[syntonic temperament]]'s valid tuning range, as shown in [[#Figure 1|Figure&nbsp;1]].
* In {{nobr|5 {{sc|TET}},}} the tempered perfect fifth is 720&nbsp;cents wide (at the top of the tuning continuum), and marks the endpoint on the tuning continuum at which the width of the minor second shrinks to a width of 0&nbsp;cents.
* In {{nobr|7 {{sc|TET}},}} the tempered perfect fifth is 686&nbsp;cents wide (at the bottom of the tuning continuum), and marks the endpoint on the tuning continuum, at which the minor second expands to be as wide as the major second (at 171&nbsp;cents each).

====5&nbsp;tone and 9&nbsp;tone equal temperament====
According to [[Jaap Kunst|Kunst]] (1949), Indonesian [[gamelan]]s are tuned to {{nobr|5 {{sc|TET}},}} but according to [[Mantle Hood|Hood]] (1966) and [[Colin McPhee|McPhee]] (1966) their tuning varies widely, and according to [[Michael Tenzer|Tenzer]] (2000) they contain [[pseudo-octave|stretched octaves]]. It is now accepted that of the two primary tuning systems in gamelan music, [[slendro]] and [[pelog]], only slendro somewhat resembles five-tone equal temperament, while pelog is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to {{nobr|9 {{sc|TET}}}} (133-cent steps {{Audio|Semitone Maximus on C.mid|Play}}).<ref>{{harvp|Surjodiningrat|Sudarjana|Susanto|1972}}</ref>

====7-tone equal temperament====
A [[Thai music|Thai]] xylophone measured by Morton in 1974 "varied only plus or minus 5&nbsp;cents" from {{nobr|7 {{sc|TET}}}}.<ref>{{harvp|Morton|1980}}</ref> According to Morton,
: "Thai instruments of fixed pitch are tuned to an equidistant system of seven pitches per octave ... As in Western traditional music, however, all pitches of the tuning system are not used in one mode (often referred to as 'scale'); in the Thai system five of the seven are used in principal pitches in any mode, thus establishing a pattern of nonequidistant intervals for the mode."<ref>{{cite book |last=Morton |first=David |year=1980 |title=The Music of Thailand |series=Musics of Many Cultures |page=70 |editor-last=May |editor-first=Elizabeth |isbn=0-520-04778-8 }}</ref> {{audio|Thai pentatonic scale mode 1.mid|Play}}

A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175&nbsp;cent seven-tone equal temperament, which stretches the octave slightly, as with instrumental gamelan music.<ref>{{harvp|Boiles|1969}}</ref>

[[Music of China|Chinese music]] has traditionally used {{nobr|7 {{sc|TET}}}}.{{efn|
'Hepta-equal temperament' in our folk music has always been a controversial issue.<!-- mostly from Google translate, please verify --><ref>{{cite web |url=http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm |script-title=zh:有关"七平均律"新文献著作的发现 |language=zh |trans-title=Findings of new literatures concerning the hepta – equal temperament |archive-url=https://web.archive.org/web/20071027064731/http://www.wanfangdata.com.cn/qikan/periodical.Articles/ZHONGUOYY/ZHON2004/0404/040425.htm |archive-date=2007-10-27}}</ref>
}}{{efn|From the flute for two thousand years of the production process, and the Japanese shakuhachi remaining in the production of Sui and Tang Dynasties and the actual temperament, identification of people using the so-called 'Seven Laws' at least two thousand years of history; and decided that this law system associated with the flute law.<!-- from Google translate, please verify --><ref>{{cite web |url=http://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005 |script-title=zh:七平均律"琐谈--兼及旧式均孔曲笛制作与转调 |language=zh |trans-title=abstract of ''About "Seven- equal- tuning System"''  |access-date=2007-06-25 |archive-url=https://web.archive.org/web/20070930155436/http://scholar.ilib.cn/Abstract.aspx?A=xhyyxyxb200102005 |archive-date=2007-09-30 |url-status=dead }}</ref>
}}

=== Various equal temperaments ===
{{more citations needed section|date=March 2020}}
[[File:16-tet scale on C.png|400px|thumb|[[Easley Blackwood, Jr.|Easley Blackwood]]'s notation system for 16&nbsp;equal temperament: Intervals are notated similarly to those they approximate and there are fewer [[enharmonic]] equivalents.<ref>{{cite book |first=Myles Leigh |last=Skinner |year=2007 |title=Toward a Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky |page=55 |isbn=9780542998478}}</ref> {{audio|16-tet scale on C.mid|Play}}]]

[[File:Equal temperaments comparison diagram.svg|thumb|Comparison of equal temperaments from 9 to 25<ref>{{harvp|Sethares|2005|p=58}}</ref>{{efn|name=Sethares}}]]

; [[19 equal temperament|19&nbsp;EDO]]: Many instruments have been built using [[19 equal temperament|19&nbsp;EDO]] tuning. Equivalent to {{nobr|{{sfrac| 1 | 3 }} comma}} meantone, it has a slightly flatter perfect fifth (at 695&nbsp;cents), but its minor third and major sixth are less than one-fifth of a cent away from just, with the lowest EDO that produces a better minor third and major sixth than 19&nbsp;EDO being 232&nbsp;EDO. Its [[perfect fourth]] (at 505&nbsp;cents), is seven cents sharper than just intonation's and five cents sharper than 12&nbsp;EDO's.

; [[22 equal temperament|22&nbsp;EDO]]: [[22 equal temperament|22&nbsp;EDO]] is one of the most accurate EDOs to represent "superpythagorean" temperament (where 7:4 and 16:9 are the same interval). The perfect fifth is tuned sharp, resulting in four fifths and three fourths reaching supermajor thirds (9/7) and subminor thirds (7/6). One step closer to each other are the classical major and minor thirds (5/4 and 6/5).

; [[23 equal temperament|23&nbsp;EDO]]: [[23 equal temperament|23&nbsp;EDO]] is the largest EDO that fails to approximate the 3rd, 5th, 7th, and 11th&nbsp;harmonics (3:2, 5:4, 7:4, 11:8) within 20&nbsp;cents, but it does approximate some ratios between them (such as the 6:5 minor third) very well, making it attractive to microtonalists seeking unusual harmonic territory.

; [[Quarter tone|24&nbsp;EDO]]: [[Quarter tone|24&nbsp;EDO]], the [[quarter tone scale|quarter-tone scale]], is particularly popular, as it represents a convenient access point for composers conditioned on standard Western 12&nbsp;EDO pitch and notation practices who are also interested in microtonality. Because 24&nbsp;EDO contains all the pitches of 12&nbsp;EDO, musicians employ the additional colors without losing any tactics available in 12&nbsp;tone harmony. That 24 is a multiple of 12 also makes 24&nbsp;EDO easy to achieve instrumentally by employing two traditional 12&nbsp;EDO instruments tuned a quarter-tone apart, such as two pianos, which also allows each performer (or one performer playing a different piano with each hand) to read familiar 12&nbsp;tone notation. Various composers, including [[Charles Ives]], experimented with music for quarter-tone pianos. 24&nbsp;EDO also approximates the 11th and 13th harmonics very well, unlike 12&nbsp;EDO.

; 26&nbsp;EDO: 26 is the denominator of a convergent to log<sub>2</sub>(7), tuning the [[Harmonic seventh|7th harmonic]] (7:4) with less than half a cent of error. Although it is a meantone temperament, it is a very flat one, with four of its perfect fifths producing a major third 17&nbsp;cents flat (equated with the 11:9 neutral third). 26&nbsp;EDO has two minor thirds and two minor sixths and could be an alternate temperament for [[Close and open harmony|barbershop harmony]].

; 27&nbsp;EDO: 27 is the lowest number of equal divisions of the octave that uniquely represents all intervals involving the first eight harmonics. It tempers out the [[septimal comma]] but not the [[syntonic comma]].

; [[58 equal temperament|29&nbsp;EDO]]: [[58 equal temperament|29]] is the lowest number of equal divisions of the octave whose perfect fifth is closer to just than in 12&nbsp;EDO, in which the fifth is 1.5&nbsp;cents sharp instead of 2&nbsp;cents flat. Its classic major third is roughly as inaccurate as 12&nbsp;EDO, but is tuned 14&nbsp;cents flat rather than 14&nbsp;cents sharp. It also tunes the 7th, 11th, and 13th&nbsp;harmonics flat by roughly the same amount, allowing 29&nbsp;EDO to match intervals such as 7:5, 11:7, and 13:11 very accurately. Cutting all 29 intervals in half produces [[58 equal temperament|58 EDO]], which allows for lower errors for some just tones.

; [[31 equal temperament|31&nbsp;EDO]]: [[31 equal temperament|31&nbsp;EDO]] was advocated by [[Christiaan Huygens]] and [[Adriaan Fokker]] and represents a rectification of [[quarter-comma meantone]] into an equal temperament. 31&nbsp;EDO does not have as accurate a perfect fifth as 12&nbsp;EDO (like 19&nbsp;EDO), but its major thirds and minor sixths are less than 1&nbsp;cent away from just. It also provides good matches for harmonics up to 11, of which the seventh harmonic is particularly accurate.

; [[34 equal temperament|34&nbsp;EDO]]: [[34 equal temperament|34&nbsp;EDO]] gives slightly lower total combined errors of approximation to 3:2, 5:4, 6:5, and their inversions than 31&nbsp;EDO does, despite having a slightly less accurate fit for 5:4. 34&nbsp;EDO does not accurately approximate the seventh harmonic or ratios involving 7, and is not meantone since its fifth is sharp instead of flat. It enables the 600&nbsp;cent tritone, since 34 is an even number.

; [[41 equal temperament|41&nbsp;EDO]]: [[41 equal temperament|41]] is the next EDO with a better perfect fifth than 29&nbsp;EDO and 12&nbsp;EDO. Its classical major third is also more accurate, at only six cents flat. It is not a meantone temperament, so it distinguishes 10:9 and 9:8, along with the classic and Pythagorean major thirds, unlike 31&nbsp;EDO. It is more accurate in the 13&nbsp;limit than 31&nbsp;EDO.

; 46&nbsp;EDO: 46&nbsp;EDO provides major thirds and perfect fifths that are both slightly sharp of just, and many{{who|date=September 2024}} say that this gives major triads a characteristic bright sound. The prime harmonics up to 17 are all within 6&nbsp;cents of accuracy, with 10:9 and 9:5 a fifth of a cent away from pure. As it is not a meantone system, it distinguishes 10:9 and 9:8.

; [[53 equal temperament|53&nbsp;EDO]]: [[53 equal temperament|53&nbsp;EDO]] has only had occasional use, but is better at approximating the traditional [[just intonation|just]] consonances than 12, 19 or 31&nbsp;EDO. Its extremely accurate [[perfect fifth]]s make it equivalent to an extended [[Pythagorean tuning]], as 53 is the denominator of a convergent to log<sub>2</sub>(3). With its accurate cycle of fifths and multi-purpose comma step, 53&nbsp;EDO has been used in [[Turkish music]] theory. It is not a meantone temperament, which put good thirds within easy reach by stacking fifths; instead, like all [[schismatic temperament|schismatic temperaments]], the very consonant thirds are represented by a Pythagorean diminished fourth (C-F{{music|b}}), reached by stacking eight perfect fourths. It also tempers out the [[kleisma]], allowing its fifth to be reached by a stack of six minor thirds (6:5).

; [[58 equal temperament|58&nbsp;EDO]]: [[58 equal temperament]] is a duplication of 29&nbsp;EDO, which it contains as an embedded temperament. Like 29&nbsp;EDO it can match intervals such as 7:4, 7:5, 11:7, and 13:11 very accurately, as well as better approximating just thirds and sixths.

; [[72 equal temperament|72&nbsp;EDO]]: [[72 equal temperament|72&nbsp;EDO]] approximates many [[just intonation]] intervals well, providing near-just equivalents to the 3rd, 5th, 7th, and 11th harmonics. 72&nbsp;EDO has been taught, written and performed in practice by [[Joe Maneri]] and his students (whose atonal inclinations typically avoid any reference to [[just intonation]] whatsoever). As it is a multiple of 12, 72&nbsp;EDO can be considered an extension of 12&nbsp;EDO, containing six copies of 12&nbsp;EDO starting on different pitches, three copies of 24&nbsp;EDO, and two copies of 36&nbsp;EDO.

; [[96 equal temperament|96&nbsp;EDO]]: [[96 equal temperament|96&nbsp;EDO]] approximates all intervals within 6.25&nbsp;cents, which is barely distinguishable. As an eightfold multiple of 12, it can be used fully like the common 12&nbsp;EDO. It has been advocated by several composers, especially [[Julián Carrillo]].<ref>{{cite web |last1=Monzo |first1=Joe |title=Equal-temperament |url=http://tonalsoft.com/enc/e/equal-temperament.aspx#edo-table |website=Tonalsoft Encyclopedia of Microtonal Music Theory |publisher=Joe Monzo |access-date=26 February 2019 |date=2005}}</ref>

Other equal divisions of the octave that have found occasional use include [[13 equal temperament|13&nbsp;EDO]], [[15 equal temperament|15&nbsp;EDO]], [[17 equal temperament|17&nbsp;EDO]], and 55&nbsp;EDO.

2, 5, 12, 41, 53, 306, 665 and 15601 are [[denominator]]s of first [[convergent (continued fraction)|convergents]] of log{{sub|2}}(3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601 [[just intonation|just]] twelfths/fifths than in any equal temperament with fewer tones.<ref>{{cite web |title=665&nbsp;edo |website=xenoharmonic (microtonal wiki) |url=http://xenharmonic.wikispaces.com/665edo |access-date=2014-06-18 |archive-date=2015-11-18 |archive-url=https://web.archive.org/web/20151118233400/http://xenharmonic.wikispaces.com/665edo |url-status=dead }}</ref><ref>{{cite web |title=convergents log2(3), 10 |publisher=[[WolframAlpha]] |url=http://www.wolframalpha.com/input/?i=convergents%28log2%283%29%2C+10%29 |access-date=2014-06-18}}</ref>

1, 2, 3, 5, 7, 12, 29, 41, 53, 200,&nbsp;... {{OEIS|A060528}} is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote.{{efn|
OEIS sequences that contain divisions of the octave that provide improving approximations of just intervals:
: {{OEIS|A060528}} — 3:2
: {{OEIS|A054540}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3
: {{OEIS|A060525}} — 3:2 and 4:3, 5:4 and 8:5
: {{OEIS|A060526}} — 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7
: {{OEIS|A060527}} — 3:2 and 4:3, 5:4 and 8:5, 7:4 and 8:7, 16:11 and 11:8
: {{OEIS|A060233}} — 4:3 and 3:2, 5:4 and 8:5, 6:5 and 5:3, 7:4 and 8:7, 16:11 and 11:8, 16:13 and 13:8
: {{OEIS|A061920}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45
: {{OEIS|A061921}} — 3:2 and 4:3, 5:4 and 8:5, 6:5 and 5:3, 9:8 and 16:9, 10:9 and 9:5, 16:15 and 15:8, 45:32 and 64:45, 27:20 and 40:27, 32:27 and 27:16, 81:64 and 128:81, 256:243 and 243:128
: {{OEIS|A061918}} — 5:4 and 8:5
: {{OEIS|A061919}} — 6:5 and 5:3
: {{OEIS|A060529}} — 6:5 and 5:3, 7:5 and 10:7, 7:6 and 12:7
: {{OEIS|A061416}} — 11:8 and 16:11
}}

=== Equal temperaments of non-octave intervals ===
The equal-tempered version of the [[Bohlen–Pierce scale]] consists of the ratio 3:1 (1902&nbsp;cents) conventionally a [[perfect fifth]] plus an [[octave]] (that is, a perfect twelfth), called in this theory a [[tritave]] ({{Audio|Tritave on C.mid|play}}), and split into 13&nbsp;equal parts. This provides a very close match to [[just intonation|justly tuned]] ratios consisting only of odd numbers. Each step is 146.3 cents ({{Audio|BP scale et.mid|play}}), or <math display=inline>\sqrt[13]{3}</math>.

[[Wendy Carlos]] created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120&nbsp;cents. These were called ''[[alpha scale|alpha]]'', ''[[beta scale|beta]]'', and ''[[gamma scale|gamma]]''. They can be considered equal divisions of the perfect fifth. Each of them provides a very good approximation of several just intervals.<ref>{{cite web |last1=Carlos |first1=Wendy |title=Three Asymmetric Divisions of the Octave |url=http://www.wendycarlos.com/resources/pitch.html |website=wendycarlos.com |publisher=Serendip LLC |access-date=2016-09-01}}</ref> Their step sizes:
* ''alpha'': <math display=inline>\sqrt[9]{\frac{3}{2}}</math> (78.0&nbsp;cents) {{audio|Alpha scale step on C.mid|Play}}
* ''beta'': <math display=inline>\sqrt[11]{\frac{3}{2}}</math> (63.8&nbsp;cents) {{audio|Beta scale step on C.mid|Play}}
* ''gamma'': <math display=inline>\sqrt[20]{\frac{3}{2}}</math> (35.1&nbsp;cents) {{audio|Gamma scale step on C.mid|Play}}
Alpha and beta may be heard on the title track of Carlos's 1986 album ''[[Beauty in the Beast]]''.

=== Equal temperament with a non-integral number of notes per octave ===
While traditional equal temperaments—such as 12‑TET, 19‑TET, or 31‑TET—divide the octave into an integral number of equal parts, it is also possible to explore systems that divide the octave into a non-integral (often irrational) number. In such temperaments, the interval between successive pitches is defined by the ratio 2^(1/N), where N is not an integer. This results in irrational step sizes, meaning their multiples never exactly equal an octave.

Such tunings are of interest because, by deliberately sacrificing the octave (i.e., the second harmonic), they can yield a system that offers an improved overall approximation of other intervals in the harmonic series.

For example, in a tuning system based on 18.911‑EDO, the step size is 1200⁄18.911 ≈ 63.45 cents. Approximating the just perfect fifth (with a ratio of 3:2, or about 701.96 cents) requires about 11 steps:

* ''11 steps × 63.45 cents ≈ 698.95 cents,''
yielding an error of roughly 3 cents.

Similarly, for the just major third (with a ratio of 5:4, or about 386.31 cents), 6 steps are used:

* ''6 steps × 63.45 cents ≈ 380.70 cents,''
resulting in an error of approximately 5.61 cents.

Thus, although a perfect octave is absent, the consonance of many other intervals in these systems can be significantly higher than in integer-based equal temperaments.

=== Proportions between semitone and whole tone ===
{{More citations needed section|date=August 2017}}
In this section, ''semitone'' and ''whole tone'' may not have their usual 12&nbsp;EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let the number of steps in a semitone be {{mvar|s}}, and the number of steps in a tone be {{mvar|t}}.

There is exactly one family of equal temperaments that fixes the semitone to any [[proper fraction]] of a whole tone, while keeping the notes in the right order (meaning that, for example, {{sc|'''C'''}}, {{sc|'''D'''}}, {{sc|'''E'''}}, {{sc|'''F'''}}, and {{sc|'''F'''}}{{music|#}} are in ascending order if they preserve their usual relationships to {{sc|'''C'''}}). That is, fixing {{mvar|q}} to a proper fraction in the relationship {{nobr|{{math|''q t'' {{=}} ''s''}} }} also defines a unique family of one equal temperament and its multiples that fulfil this relationship.

For example, where {{mvar|k}} is an integer, {{nobr|12{{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|2}}}},}} {{nobr|19 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac|1|3}}}},}} and {{nobr|31 {{mvar|k}} {{sc|EDO}} }} sets {{nobr|{{math|''q'' {{=}} {{sfrac| 2 | 5 }} }}.}} The smallest multiples in these families (e.g. 12, 19 and 31 above) has the additional property of having no notes outside the [[circle of fifths]]. (This is not true in general; in 24&nbsp;{{sc|EDO}}, the half-sharps and half-flats are not in the circle of fifths generated starting from {{sc|'''C'''}}.) The extreme cases are {{nobr|5 {{mvar|k}} {{sc|EDO}},}} where {{nobr|{{math|''q'' {{=}} 0}} }} and the semitone becomes a unison, and {{nobr|7 {{mvar|k}} {{sc|EDO}} }}, where {{nobr|{{math|''q'' {{=}} 1}} }} and the semitone and tone are the same interval.

Once one knows how many steps a semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into {{nobr|{{math| 7 ''t'' − 2 ''s''}} steps}} and the perfect fifth into {{nobr|{{math| 4 ''t'' − ''s'' }} steps.}} If there are notes outside the circle of fifths, one must then multiply these results by {{mvar|n}}, the number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24&nbsp;{{sc|EDO}}, six in 72&nbsp;{{sc|EDO}}). (One must take the small semitone for this purpose: 19&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 1 | 3 }} tone and the other being {{sfrac| 2 | 3 }}. Similarly, 31&nbsp;{{sc|EDO}} has two semitones, one being {{sfrac| 2 | 5 }} tone and the other being {{sfrac| 3 | 5 }}).

The smallest of these families is {{nobr|12 {{mvar|k}} {{sc|EDO}},}} and in particular, 12&nbsp;{{sc|EDO}} is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12&nbsp;{{sc|EDO}} has become the most commonly used equal temperament. (Another reason is that 12&nbsp;EDO is the smallest equal temperament to closely approximate 5&nbsp;limit harmony, the next-smallest being 19&nbsp;EDO.)

Each choice of fraction {{mvar|q}} for the relationship results in exactly one equal temperament family, but the converse is not true: 47&nbsp;{{sc|EDO}} has two different semitones, where one is {{sfrac| 1 | 7 }} tone and the other is {{sfrac| 8 | 9 }}, which are not complements of each other like in 19&nbsp;{{sc|EDO}} ({{sfrac| 1 | 3 }} and {{sfrac| 2 | 3 }}). Taking each semitone results in a different choice of perfect fifth.

== Related tuning systems ==
Equal temperament systems can be thought of in terms of the spacing of three intervals found in [[just intonation]], ''most'' of whose chords are harmonically perfectly in tune—a good property not quite achieved between almost all pitches in almost all equal temperaments. Most just chords sound amazingly consonant, and most equal-tempered chords sound at least slightly dissonant. In C&nbsp;major those three intervals are:<ref name=Milne-Sethares-Plamdon-2007/>
* the [[whole tone|greater tone]] {{nobr|{{math| ''T'' {{=}} {{sfrac| 9 | 8 }}  {{=}} }}}} the interval from C:D, F:G, and A:B;
* the [[whole tone|lesser tone]] {{nobr|{{math| ''t'' {{=}} {{sfrac| 10 | 9 }}  {{=}} }}}} the interval from D:E and G:A;
* the [[semitone|diatonic semitone]] {{nobr|{{math| ''s'' {{=}} {{sfrac| 16 | 15 }}  {{=}} }}}} the interval from E:F and B:C.

Analyzing an equal temperament in terms of how it modifies or adapts these three intervals provides a quick way to evaluate how consonant various chords can possibly be in that temperament, based on how distorted these intervals are.<ref name=Milne-Sethares-Plamdon-2007/>{{efn|
For 12&nbsp;pitch systems, either for a whole 12&nbsp;note scale, for or 12&nbsp;note subsequences embedded inside some larger scale,<ref name=Milne-Sethares-Plamdon-2007 /> use this analysis as a way to program software to microtune an electronic keyboard dynamically, or 'on the fly', while a musician is playing. The object is to fine tune the notes momentarily in use, and any likely subsequent notes involving consonant chords, to always produce pitches that are harmonically in-tune, inspired by how orchestras and choruses constantly re-tune their overall pitch on long-duration chords for greater consonance than possible with strict 12&nbsp;TET.<ref name=Milne-Sethares-Plamdon-2007/>
}}

=== Regular diatonic tunings ===
{{anchor|Figure 1}}[[File:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg|right|250px|thumb|Figure 1: The [[regular diatonic tuning]]s continuum, which includes many notable "equal temperament" tunings<ref name=Milne-Sethares-Plamdon-2007>
{{cite journal
 |last1=Milne |first1=A.
 |last2=Sethares |first2=W.A. |author2-link=William Sethares
 |last3=Plamondon |first3=J.
 |title=Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum
 |journal=[[Computer Music Journal]]
 |date=Winter 2007
 |volume=31 |issue=4 |pages=15–32
 |issn=0148-9267 |doi=10.1162/comj.2007.31.4.15 |doi-access=free
 |df=dmy-all
}} Online: {{ISSN|1531-5169}}</ref>]]
The diatonic tuning in ''12&nbsp;tone equal temperament'' {{nobr|(12 {{sc|TET}})}} can be generalized to any regular diatonic tuning dividing the octave as a sequence of steps {{nobr| {{mvar|T t s T t T s}} }} (or some [[circular shift]] or "rotation" of it). To be called a ''regular'' diatonic tuning, each of the two semitones ({{mvar| s }}) must be smaller than either of the tones ([[major tone|greater tone]], {{mvar| T }}, and [[minor tone|lesser tone]], {{mvar| t }}). The comma {{mvar|κ}} is implicit as the size ratio between the greater and lesser tones: Expressed as frequencies {{nobr|{{math|''κ'' {{=}} {{sfrac| ''T'' | ''t'' }}}} ,}} or as [[cent (music)|cents]] {{nobr|{{math| ''κ'' {{=}} ''T''  − ''t'' }} }}.

The notes in a regular diatonic tuning are connected in a "spiral of fifths" that does ''not'' close (unlike the [[circle of fifths]] in {{nobr|12 {{sc|TET}}).}} Starting on the subdominant {{sc|'''F'''}} (in the [[C major|key of C]]) there are three [[perfect fifth]]s in a row—{{sc|'''F'''}}–{{sc|'''C'''}}, {{sc|'''C'''}}–{{sc|'''G'''}}, and {{sc|'''G'''}}–{{sc|'''D'''}}—each a composite of some [[permutation]] of the smaller intervals {{nobr| {{mvar|T T t s}} .}} The three in-tune fifths are interrupted by the [[List of pitch intervals|grave fifth]] {{sc|'''D'''}}–{{sc|'''A'''}} {{=}} {{nobr| {{mvar|T t t s}} }}([[List of pitch intervals|''grave'']] means "flat by a [[comma (music)|comma]]"), followed by another perfect fifth, {{sc|'''E'''}}–{{sc|'''B'''}}, and another grave fifth, {{sc|'''B'''}}–{{sc|'''F'''}}{{music|#}}, and then restarting in the sharps with {{sc|'''F'''}}{{music|#}}–{{sc|'''C'''}}{{music|#}}; the same pattern repeats through the sharp notes, then the double-sharps, and so on, indefinitely. But each octave of all-natural or all-sharp or all-double-sharp notes flattens by two commas with every transition from naturals to sharps, or single sharps to double sharps, etc. The pattern is also reverse-symmetric in the flats: Descending by [[perfect fourth|fourths]] the pattern reciprocally sharpens notes by two commas with every transition from natural notes to flattened notes, or flats to double flats, etc. If left unmodified, the two grave fifths in each block of all-natural notes, or all-sharps, or all-flat notes, are [[wolf interval|"wolf" intervals]]: Each of the grave fifths out of tune by a [[diatonic comma]].

Since the comma, {{mvar|κ}}, expands the [[minor tone|lesser tone]] {{nobr|&thinsp;{{mvar|t {{=}} s c}} ,}} into the [[major tone|greater tone]], {{nobr|&thinsp;{{mvar|T {{=}} s c κ}} ,}} a [[just intonation|just]] octave {{nobr| {{mvar|T t s T t T s}} }} can be broken up into a sequence {{nobr|&thinsp;{{mvar|s c κ &thinsp; s c &thinsp; s &thinsp; s c κ &thinsp; s c &thinsp; s c κ &thinsp; s}} ,}} (or a [[circular shift]] of it) of 7&nbsp;diatonic semitones {{mvar|s}}, 5&nbsp;chromatic semitones {{mvar|c}}, and 3&nbsp;[[syntonic comma|commas]] {{nobr|&thinsp;{{mvar|κ}} .}} Various equal temperaments alter the interval sizes, usually breaking apart the three commas and then redistributing their parts into the seven diatonic semitones {{mvar|s}}, or into the five chromatic semitones {{mvar|c}}, or into both {{mvar|s}} and {{mvar|c}}, with some fixed proportion for each type of semitone.

The sequence of intervals {{mvar|s}}, {{mvar|c}}, and {{mvar|κ}} can be repeatedly appended to itself into a greater [[circle of fifths|spiral of 12&nbsp;fifths]], and made to connect at its far ends by slight adjustments to the size of one or several of the intervals, or left unmodified with occasional less-than-perfect fifths, flat by a comma.

=== Morphing diatonic tunings into EDO ===
Various equal temperaments can be understood and analyzed as having made adjustments to the sizes of and subdividing the three intervals—{{mvar| T }}, {{mvar| t }}, and {{mvar| s }}, or at finer resolution, their constituents {{mvar| s }}, {{mvar| c }}, and {{mvar| κ }}. An equal temperament can be created by making the sizes of the [[major tone|major]] and [[minor tone]]s ({{mvar|T}}, {{mvar|t}}) the same (say, by setting {{nobr|{{math|''κ'' {{=}} 0}}}}, with the others expanded to still fill out the octave), and both semitones ([[diatonic semitone|{{mvar|s}}]] and {{mvar|c}}) the same, then 12 equal semitones, two per tone, result. In {{nobr|12 {{sc|TET}}}}, the semitone, {{mvar|s}}, is exactly half the size of the same-size whole tones {{mvar|T}} = {{mvar|t}}.

Some of the intermediate sizes of tones and semitones can also be generated in equal temperament systems, by modifying the sizes of the comma and semitones. One obtains {{nobr|[[7 equal temperament|7 {{sc|TET}}]]}} in the limit as the size of {{mvar|c}} and {{mvar|κ}} tend to zero, with the octave kept fixed, and {{nobr|5 {{sc|TET}}}} in the limit as {{mvar|s}} and {{mvar|κ}} tend to zero; {{nobr|12 {{sc|TET}}}} is of course, the case {{nobr|&thinsp;{{mvar|s {{=}} c}}&thinsp;}} and {{nobr|&thinsp;{{math|''κ'' {{=}} 0}} .}} For instance:

;{{nobr|[[5 equal temperament|5 {{sc|tet}}]]}} and {{nobr|[[7 equal temperament|7 {{sc|tet}}]]}}: There are two extreme cases that bracket this framework: When {{mvar|s}} and {{mvar|κ}} reduce to zero with the octave size kept fixed, the result is {{nobr|{{mvar|t t t t t}} ,}} a 5&nbsp;tone equal temperament. As the {{mvar|s}} gets larger (and absorbs the space formerly used for the comma {{mvar|κ}}), eventually the steps are all the same size, {{nobr|{{mvar|t t t t t t t}} ,}} and the result is seven-tone equal temperament. These two extremes are not included as "regular" diatonic tunings.

;{{nobr|[[19 equal temperament|19 {{sc|tet}}]]}}: If the diatonic semitone is set double the size of the chromatic semitone, i.e. {{nobr|&thinsp;{{mvar|s {{=}} 2 c}} }} (in cents) and {{nobr|&thinsp;{{math|''κ'' {{=}} 0}} ,}} the result is {{nobr|[[19 equal temperament|19 {{sc|tet}}]],}} with one step for the chromatic semitone {{mvar|c}}, two steps for the diatonic semitone {{mvar|s}}, three steps for the tones {{mvar|T}} =  {{mvar|t}}, and the total number of steps {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 9 + 6 + 4}} {{=}}&thinsp;}} 19&nbsp;steps. The imbedded 12&nbsp;tone sub-system closely approximates the historically important {{nobr|{{sfrac| 1 | 3 }} comma}} [[meantone temperament|meantone system]].

;{{nobr|[[31 equal temperament|31 {{sc|tet}}]]}}: If the chromatic semitone is two-thirds the size of the diatonic semitone, i.e. {{nobr|{{math|''c'' {{=}} {{sfrac| 2 | 3 }} ''s''}} ,}} with {{nobr|{{math|''κ'' {{=}} 0}} ,}} the result is [[31 equal temperament|31 {{sc|tet}}]], with two steps for the chromatic semitone, three steps for the diatonic semitone, and five steps for the tone, where {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 15 + 10 + 6}} {{=}}&thinsp;}} [[31 equal temperament|31&nbsp;steps]]. The imbedded 12&nbsp;tone sub-system closely approximates the historically important [[quarter comma meantone|{{nobr|{{sfrac| 1 | 4 }} comma}} meantone]].

;{{nobr|[[43 equal temperament|43 {{sc|tet}}]]}}: If the chromatic semitone is three-fourths the size of the diatonic semitone, i.e. {{nobr|{{math|''c'' {{=}} {{sfrac| 3 | 4 }} ''s''}} ,}} with {{nobr|{{math|''κ'' {{=}} 0}} ,}} the result is [[43 equal temperament|43 {{sc|tet}}]], with three steps for the chromatic semitone, four steps for the diatonic semitone, and seven steps for the tone, where {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 21 + 14 + 8}} {{=}}&thinsp;}} 43. The imbedded 12&nbsp;tone sub-system closely approximates {{nobr|{{sfrac| 1 | 5 }} comma}} meantone.

;{{nobr|[[53 equal temperament|53 {{sc|tet}}]]}}: If the chromatic semitone is made the same size as three commas, {{nobr|&thinsp;{{math|''c'' {{=}} 3 ''κ''}}&thinsp;}} (in cents, in frequency {{nobr|&thinsp;{{math|''c'' {{=}} ''κ''³}}&thinsp;}}) the diatonic the same as five commas, {{nobr|&thinsp;{{math|''s'' {{=}} 5 ''κ''}} ,}} that makes the lesser tone eight commas {{nobr|{{math|''t'' {{=}} ''s'' + ''c'' {{=}} 8 ''κ''}} ,}} and the greater tone nine, {{nobr|&thinsp;{{math|''T'' {{=}} ''s'' + ''c'' + ''κ'' {{=}} 9 ''κ''}} .}} Hence {{math|{{nobr|&thinsp;3 ''T'' + 2 ''t'' + 2 ''s''}} {{=}} {{nobr| 27 ''κ'' + 16 ''κ'' + 10 ''κ''}} {{=}} 53 ''κ''}} for [[53 equal temperament|53&nbsp;steps]] of one comma each. The comma size / step size is {{nobr|&thinsp;{{math|''κ'' {{=}} {{sfrac| 1 200 | 53 }} }}&thinsp;¢}} exactly, or {{nobr|&thinsp;{{math|''κ'' {{=}} 22.642}} ¢}} {{nobr|&thinsp;{{math|≈ 21.506}} ¢ ,}} the [[syntonic comma]]. It is an exceedingly close approximation to 5-limit [[just intonation]] and Pythagorean tuning, and is the basis for [[Turkish makam|Turkish music theory]].

== See also ==
{{div col begin|colwidth=14em}}
* [[Just intonation]]
* [[Musical acoustics]]<br/>(the physics of music)
* [[Music and mathematics]]
* [[Microtuner]]
* [[Microtonal music]]
* [[Piano tuning]]
* [[List of meantone intervals]]
* [[Diatonic and chromatic]]
* [[Electronic tuner]]
* [[Musical tuning]]
{{div col end}}

== Footnotes ==
{{notelist}}

== References ==
{{Reflist|25em}}

== Sources ==
{{refbegin|colwidth=25em|small=yes}}
* {{cite journal |last=Boiles |first=J. |date=1969 |title=Terpehua though-song |journal=Ethnomusicology |volume=13 |pages=42–47}}
* {{cite book
 |last=Cho |first=Gene Jinsiong
 |year=2003
 |title=The Discovery of Musical Equal Temperament in China and Europe in the Sixteenth Century
 |place=Lewiston, NY
 |publisher=[[Edwin Mellen Press]]
}}
* {{cite book
 |last=Duffin |first=Ross W.
 |year=2007
 |title=How Equal Temperament Ruined Harmony (and why you should care)
 |place=New York, NY
 |publisher=W.W.Norton & Company
 |isbn=978-0-39306227-4
}}
* {{cite book
 |last=Jorgensen |first=Owen
 |year=1991
 |title=Tuning
 |publisher=Michigan State University Press
 |isbn=0-87013-290-3
}}
* {{cite book
 |last=Sethares |first=William A. |author-link=William Sethares 
 |year=2005
 |title=Tuning, Timbre, Spectrum, Scale |edition=2nd
 |publisher=Springer-Verlag
 |location=London, UK
 |isbn=1-85233-797-4
}}
* {{cite book
 |last1=Surjodiningrat |first1=W.
 |last2=Sudarjana |first2=P.J.
 |last3=Susanto   |first3=A.
 |year=1972
 |title=Tone measurements of outstanding Javanese gamelans in Jogjakarta and Surakarta
 |publisher=Gadjah Mada University Press
 |place=Jogjakarta, IN
}} As cited by {{cite web
 |title=The gamelan pelog scale of Central Java as an example of a non-harmonic musical scale
 |website=telia.com
 |series=Neuroscience of Music
 |url=http://web.telia.com/~u57011259/pelog_main.htm
 |access-date=19 May 2006
 |url-status=dead <!-- presumed -->
 |archive-url=https://web.archive.org/web/20050127000731/http://web.telia.com/~u57011259/pelog_main.htm
 |archive-date=2005-01-27
 |df=dmy-all
 }}
* {{cite report |last=Stewart |first=P.J. |year=2006 |orig-date=January 1999 |title=From galaxy to galaxy: Music of the spheres |url=https://www.academia.edu/8096295 |via=academia.edu |id=8096295 }} {{cite web
 |title=Alt. link 1
 |url=https://www.researchgate.net/publication/269108386
 |via=researchgate.net
 |id=269108386
 }} {{cite web
 |title=Alt. link 2
 |url=https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxncmFucGhpNjE4fGd4OjRhN2VjNTNhOGY1ZmRkNDA
 |via=Google docs
 |url-access=registration
 }}
* {{cite conference
 |last=Khramov
 |first=Mykhaylo
 |date=26–29 July 2008
 |title=Approximation of 5-limit just intonation
 |department=Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave
 |conference=The International Conference SIGMAP-2008
 |place=[[Porto]]
 |pages=181–184
 |isbn=978-989-8111-60-9
 |url=https://www.scitepress.org/papers/2008/19320/19320.pdf
 }}
{{refend}}

== Further reading ==
* {{cite book
 |last=Helmholtz |first=H. |author-link=Hermann von Helmholtz
 |publication-date=2005 
 |orig-year=1877 (4th German ed.), 1885 (2nd English ed.)
 |title=On the Sensations of Tone as a Physiological Basis for the Theory of Music |edition=reprint
 |translator=Ellis, A.J.
 |translator-link=Alexander J. Ellis
 |publication-place=Whitefish, MT 
 |publisher=Kellinger Publishing
 |isbn=978-1-41917893-1  |oclc=71425252 
 |url=https://archive.org/stream/onsensationston01helmgoog#page/n0/mode/2up
 |via=[[Internet Archive]] (archive.org)
}}<br/> — A foundational work on acoustics and the perception of sound. Especially the material in ''Appendix&nbsp;XX: Additions by the translator'', pages&nbsp;430–556, (pdf pages 451–577) (see also wiki article ''[[On Sensations of Tone]]'')

== External links ==
* [https://www.kylegann.com/histune.html An Introduction to Historical Tunings] by [[Kyle Gann]]
* [https://en.xen.wiki/w/EDO_vs_ET Xenharmonic wiki on EDOs vs. Equal Temperaments]
* [http://www.huygens-fokker.org/index_en.html Huygens-Fokker Foundation Centre for Microtonal Music]
* [https://web.archive.org/web/20041011224640/http://www.fortunecity.com/tinpan/lennon/362/english/acoustics.htm A.Orlandini: Music Acoustics]
* [http://digicoll.library.wisc.edu/cgi-bin/HistSciTech/HistSciTech-idx?type=turn&entity=HistSciTech001000270617&isize=M&q1=temperament "Temperament" from ''A supplement to Mr. Chambers's cyclopædia'' (1753)]
* Barbieri, Patrizio. [https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm Enharmonic instruments and music, 1470–1900]. (2008) Latina, Il Levante Libreria Editrice
* [http://www.interdependentscience.com/music/calliopist.html Fractal Microtonal Music], ''Jim Kukula''.
* [https://www.academia.edu/5210832/18th_Century_Quotes_on_J.S._Bachs_Temperament All existing 18th century quotes on J.S. Bach and temperament]
* Dominic Eckersley: "[https://www.academia.edu/3368760/Rosetta_Revisited_Bachs_Very_Ordinary_Temperament Rosetta Revisited: Bach's Very Ordinary Temperament]"
* [http://home.deds.nl/~broekaert/Well%20Tempering.html Well Temperaments, based on the Werckmeister Definition]
* [https://en.xen.wiki/images/b/b3/Zetamusic5.pdf F<small>AVORED</small> C<small>ARDINALITIES</small> O<small>F</small> S<small>CALES</small>] by P<small>ETER</small> B<small>UCH</small>

{{Atonality}}
{{Microtonal music}}
{{Musical tuning}}

[[Category:Equal temperaments| ]]
[[Category:Chinese discoveries]]