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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 72-EDO
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-10-12 11:43:18 UTC</tt>.<br>
| en = 72edo
: The original revision id was <tt>264022697</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja =
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]
{{Wikipedia|72 equal temperament}}
----
{{ED intro}}
72-tone equal temperament (or 72-edo) divides the octave into 72 steps or //moria//. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic, Turkish, Persian|Arabic]] music, and has itself been used to tune Turkish music.


Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200{{c}}, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|Arabic music]], and has itself been used to tune Turkish music.


72-tone equal temperament approximates 11-limit [[JustIntonation|just intonation]] exceptionally well, and is the ninth [[The Riemann Zeta Function+and Tuning#Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
Composers that used 72edo include [[Ivan Wyschnegradsky]], [[Julián Carrillo]] (who is better associated with [[96edo]]), [[Georg Friedrich Haas]], [[Ezra Sims]], [[Rick Tagawa]], [[James Tenney]], and the jazz musician [[Joe Maneri]].


72 is an excellent tuning for [[Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
== Theory ==
72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and is the ninth [[zeta integral edo]]. It is the second edo (after [[58edo|58]]) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, and the first edo to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]].  


=Harmonic Scale=
The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
Mode 8 of the harmonic series -- [[overtone scales|overtones 8 through 16]], octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).


|| Overtones in "Mode 8": || 8 ||  || 9 ||  || 10 ||  || 11 ||  || 12 ||  || 13 ||  || 14 ||  || 15 ||  || 16 ||
72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s. It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
|| ...as JI Ratio from 1/1: || 1/1 ||  || 9/8 ||  || 5/4 ||  || 11/8 ||  || 3/2 ||  || 13/8 ||  || 7/4 ||  || 15/8 ||  || 2/1 ||
|| ...in cents: || 0 ||  || 203.9 ||  || 386.3 ||  || 551.3 ||  || 702.0 ||  || 840.5 ||  || 968.8 ||  || 1088.3 ||  || 1200.0 ||
|| Nearest degree of 72edo: || 0 ||  || 12 ||  || 23 ||  || 33 ||  || 42 ||  || 50 ||  || 58 ||  || 65 ||  || 72 ||
|| ...in cents: || 0 ||  || 200.0 ||  || 383.3 ||  || 550.0 ||  || 700.0 ||  || 833.3 ||  || 966.7 ||  || 1083.3 ||  || 1200.0 ||
|| Steps as Freq. Ratio: ||  || 9:8 ||  || 10:9 ||  || 11:10 ||  || 12:11 ||  || 13:12 ||  || 14:13 ||  || 15:14 ||  || 16:15 ||  ||
|| ...in cents: ||  || 203.9 ||  || 182.4 ||  || 165.0 ||  || 150.6 ||  || 138.6 ||  || 128.3 ||  || 119.4 ||  || 111.7 ||  ||
|| Nearest degree of 72edo: ||  || 12 ||  || 11 ||  || 10 ||  || 9 ||  || 8 ||  || 8 ||  || 7 ||  || 7 ||  ||
|| ...in cents: ||  || 200.0 ||  || 183.3 ||  || 166.7 ||  || 150.0 ||  || 133.3 ||  || 133.3 ||  || 116.7 ||  || 116.7 ||  ||


=Z function=
The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.
72edo is the ninth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[The Riemann Zeta Function and Tuning#The%20Z%20function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.


[[image:plot72.png]]
72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].


=Music=  
=== Prime harmonics ===
[[http://www.archive.org/details/Kotekant|Kotekant]] [[http://www.archive.org/download/Kotekant/kotekant.mp3|play]] by [[Gene Ward Smith]]
{{Harmonics in equal|72|columns=9}}
{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}


=External links=  
=== Octave stretch ===
* [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|Wikipedia article on 72edo]]
72edo's approximations of harmonics 3, 5, 7, 11, 13 and 17 can all be improved by slightly [[stretched and compressed tuning|stretching the octave]], using tunings such as [[114edt]] or [[186ed6]]. 114edt is quite hard and might be best for the 13- or 17-limit specifically. 186ed6 is milder and less disruptive, suitable for 11-limit and/or full 19-limit harmonies.  
* [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|OrthodoxWiki Article on Byzantine chant, which uses 72edo]]
* [[http://en.wikipedia.org/wiki/Joe_Maneri|Wikipedia article on Joe Maneri (1927-2009)]]
* [[http://www.ekmelic-music.org/en/index.htmmusik/|Ekmelic Music Society/Gesellschaft für Ekmelische Musik]], a group of composers and researchers dedicated to 72edo music
* [[http://sonic-arts.org/tagawa/72edo.htm|Rick Tagawa's 72edo site]], including theory and composers' list
* [[http://dannywier.ucoz.com|Danny Wier, composer and musician who specializes in 72-edo]]</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;72edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Harmonic Scale"&gt;Harmonic Scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Z function"&gt;Z function&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Music"&gt;Music&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt; | &lt;a href="#External links"&gt;External links&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;&lt;hr /&gt;
72-tone equal temperament (or 72-edo) divides the octave into 72 steps or &lt;em&gt;moria&lt;/em&gt;. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of &lt;a class="wiki_link" href="/24edo"&gt;24-tone equal temperament&lt;/a&gt;, a common and standard tuning of &lt;a class="wiki_link" href="/Arabic%2C%20Turkish%2C%20Persian"&gt;Arabic&lt;/a&gt; music, and has itself been used to tune Turkish music.&lt;br /&gt;
&lt;br /&gt;
Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with &lt;a class="wiki_link" href="/96edo"&gt;96-edo&lt;/a&gt;), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.&lt;br /&gt;
&lt;br /&gt;
72-tone equal temperament approximates 11-limit &lt;a class="wiki_link" href="/JustIntonation"&gt;just intonation&lt;/a&gt; exceptionally well, and is the ninth [[The Riemann Zeta Function+and Tuning#Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.&lt;br /&gt;
&lt;br /&gt;
72 is an excellent tuning for &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle temperament&lt;/a&gt;, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Harmonic Scale&lt;/h1&gt;
Mode 8 of the harmonic series -- &lt;a class="wiki_link" href="/overtone%20scales"&gt;overtones 8 through 16&lt;/a&gt;, octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).&lt;br /&gt;
&lt;br /&gt;


=== Subsets and supersets ===
Since 72 factors into primes as {{nowrap| 2<sup>3</sup> × 3<sup>2</sup> }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13.


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{| class="wikitable center-all right-2 left-3"
        &lt;td&gt;Overtones in &amp;quot;Mode 8&amp;quot;:&lt;br /&gt;
|-
&lt;/td&gt;
! #
        &lt;td&gt;8&lt;br /&gt;
! Cents
&lt;/td&gt;
! Approximate ratios<ref group="note">{{sg|limit=19-limit}} For lower limits see [[Table of 72edo intervals]].</ref>
        &lt;td&gt;&lt;br /&gt;
! colspan="3" | [[Ups and downs notation]]
&lt;/td&gt;
! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]
        &lt;td&gt;9&lt;br /&gt;
! (K, S, U)
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 0
&lt;/td&gt;
| 0.0
        &lt;td&gt;10&lt;br /&gt;
| 1/1
&lt;/td&gt;
| P1
        &lt;td&gt;&lt;br /&gt;
| perfect unison
&lt;/td&gt;
| D
        &lt;td&gt;11&lt;br /&gt;
| P1
&lt;/td&gt;
| perfect unison
        &lt;td&gt;&lt;br /&gt;
| D
&lt;/td&gt;
| D
        &lt;td&gt;12&lt;br /&gt;
|-
&lt;/td&gt;
| 1
        &lt;td&gt;&lt;br /&gt;
| 16.7
&lt;/td&gt;
| 81/80, 91/90, 99/98, 100/99, 105/104
        &lt;td&gt;13&lt;br /&gt;
| ^1
&lt;/td&gt;
| up unison
        &lt;td&gt;&lt;br /&gt;
| ^D
&lt;/td&gt;
| K1, L1
        &lt;td&gt;14&lt;br /&gt;
| comma-wide unison, large unison
&lt;/td&gt;
| KD, LD
        &lt;td&gt;&lt;br /&gt;
| KD
&lt;/td&gt;
|-
        &lt;td&gt;15&lt;br /&gt;
| 2
&lt;/td&gt;
| 33.3
        &lt;td&gt;&lt;br /&gt;
| 45/44, 49/48, 50/49, 55/54, 64/63
&lt;/td&gt;
| ^^
        &lt;td&gt;16&lt;br /&gt;
| dup unison
&lt;/td&gt;
| ^^D
    &lt;/tr&gt;
| S1, O1
    &lt;tr&gt;
| super unison, on unison
        &lt;td&gt;...as JI Ratio from 1/1:&lt;br /&gt;
| SD, OD
&lt;/td&gt;
| SD
        &lt;td&gt;1/1&lt;br /&gt;
|-
&lt;/td&gt;
| 3
        &lt;td&gt;&lt;br /&gt;
| 50.0
&lt;/td&gt;
| 33/32, 36/35, 40/39
        &lt;td&gt;9/8&lt;br /&gt;
| ^<sup>3</sup>1, v<sup>3</sup>m2
&lt;/td&gt;
| trup unison, trudminor 2nd
        &lt;td&gt;&lt;br /&gt;
| ^<sup>3</sup>D, v<sup>3</sup>Eb
&lt;/td&gt;
| U1, H1, hm2
        &lt;td&gt;5/4&lt;br /&gt;
| uber unison, hyper unison, hypominor 2nd
&lt;/td&gt;
| UD, HD, uEb
        &lt;td&gt;&lt;br /&gt;
| UD, uEb
&lt;/td&gt;
|-
        &lt;td&gt;11/8&lt;br /&gt;
| 4
&lt;/td&gt;
| 66.7
        &lt;td&gt;&lt;br /&gt;
| 25/24, 26/25, 27/26, 28/27
&lt;/td&gt;
| vvm2
        &lt;td&gt;3/2&lt;br /&gt;
| dudminor 2nd
&lt;/td&gt;
| vvEb
        &lt;td&gt;&lt;br /&gt;
| kkA1, sm2
&lt;/td&gt;
| classic aug unison, subminor 2nd
        &lt;td&gt;13/8&lt;br /&gt;
| kkD#, sEb
&lt;/td&gt;
| sD#, (kkD#), sEb
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 5
        &lt;td&gt;7/4&lt;br /&gt;
| 83.3
&lt;/td&gt;
| 20/19, 21/20, 22/21
        &lt;td&gt;&lt;br /&gt;
| vm2
&lt;/td&gt;
| downminor 2nd
        &lt;td&gt;15/8&lt;br /&gt;
| vEb
&lt;/td&gt;
| kA1, lm2
        &lt;td&gt;&lt;br /&gt;
| comma-narrow aug unison, little minor 2nd
&lt;/td&gt;
| kD#, lEb
        &lt;td&gt;2/1&lt;br /&gt;
| kD#, kEb
&lt;/td&gt;
|-
    &lt;/tr&gt;
| 6
    &lt;tr&gt;
| 100.0
        &lt;td&gt;...in cents:&lt;br /&gt;
| 17/16, 18/17, 19/18
&lt;/td&gt;
| m2
        &lt;td&gt;0&lt;br /&gt;
| minor 2nd
&lt;/td&gt;
| Eb
        &lt;td&gt;&lt;br /&gt;
| m2
&lt;/td&gt;
| minor 2nd
        &lt;td&gt;203.9&lt;br /&gt;
| Eb
&lt;/td&gt;
| Eb
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 7
        &lt;td&gt;386.3&lt;br /&gt;
| 116.7
&lt;/td&gt;
| 15/14, 16/15
        &lt;td&gt;&lt;br /&gt;
| ^m2
&lt;/td&gt;
| upminor 2nd
        &lt;td&gt;551.3&lt;br /&gt;
| ^Eb
&lt;/td&gt;
| Km2
        &lt;td&gt;&lt;br /&gt;
| classic minor 2nd
&lt;/td&gt;
| KEb
        &lt;td&gt;702.0&lt;br /&gt;
| KEb
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 8
&lt;/td&gt;
| 133.3
        &lt;td&gt;840.5&lt;br /&gt;
| 13/12, 14/13, 27/25
&lt;/td&gt;
| ^^m2, v~2
        &lt;td&gt;&lt;br /&gt;
| dupminor 2nd, downmid 2nd
&lt;/td&gt;
| ^^Eb
        &lt;td&gt;968.8&lt;br /&gt;
| Om2
&lt;/td&gt;
| on minor 2nd
        &lt;td&gt;&lt;br /&gt;
| OEb
&lt;/td&gt;
| SEb
        &lt;td&gt;1088.3&lt;br /&gt;
|-
&lt;/td&gt;
| 9
        &lt;td&gt;&lt;br /&gt;
| 150.0
&lt;/td&gt;
| 12/11
        &lt;td&gt;1200.0&lt;br /&gt;
| ~2
&lt;/td&gt;
| mid 2nd
    &lt;/tr&gt;
| v<sup>3</sup>E
    &lt;tr&gt;
| N2
        &lt;td&gt;Nearest degree of 72edo:&lt;br /&gt;
| neutral 2nd
&lt;/td&gt;
| UEb/uE
        &lt;td&gt;0&lt;br /&gt;
| UEb/uE
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 10
&lt;/td&gt;
| 166.7
        &lt;td&gt;12&lt;br /&gt;
| 11/10
&lt;/td&gt;
| ^~2, vvM2
        &lt;td&gt;&lt;br /&gt;
| upmid 2nd, dudmajor 2nd
&lt;/td&gt;
| vvE
        &lt;td&gt;23&lt;br /&gt;
| oM2
&lt;/td&gt;
| off major 2nd
        &lt;td&gt;&lt;br /&gt;
| oE
&lt;/td&gt;
| sE
        &lt;td&gt;33&lt;br /&gt;
|-
&lt;/td&gt;
| 11
        &lt;td&gt;&lt;br /&gt;
| 183.3
&lt;/td&gt;
| 10/9
        &lt;td&gt;42&lt;br /&gt;
| vM2
&lt;/td&gt;
| downmajor 2nd
        &lt;td&gt;&lt;br /&gt;
| vE
&lt;/td&gt;
| kM2
        &lt;td&gt;50&lt;br /&gt;
| classic/comma-narrow major 2nd
&lt;/td&gt;
| kE
        &lt;td&gt;&lt;br /&gt;
| kE
&lt;/td&gt;
|-
        &lt;td&gt;58&lt;br /&gt;
| 12
&lt;/td&gt;
| 200.0
        &lt;td&gt;&lt;br /&gt;
| 9/8
&lt;/td&gt;
| M2
        &lt;td&gt;65&lt;br /&gt;
| major 2nd
&lt;/td&gt;
| E
        &lt;td&gt;&lt;br /&gt;
| M2
&lt;/td&gt;
| major 2nd
        &lt;td&gt;72&lt;br /&gt;
| E
&lt;/td&gt;
| E
    &lt;/tr&gt;
|-
    &lt;tr&gt;
| 13
        &lt;td&gt;...in cents:&lt;br /&gt;
| 216.7
&lt;/td&gt;
| 17/15, 25/22
        &lt;td&gt;0&lt;br /&gt;
| ^M2
&lt;/td&gt;
| upmajor 2nd
        &lt;td&gt;&lt;br /&gt;
| ^E
&lt;/td&gt;
| LM2
        &lt;td&gt;200.0&lt;br /&gt;
| large major 2nd
&lt;/td&gt;
| LE
        &lt;td&gt;&lt;br /&gt;
| KE
&lt;/td&gt;
|-
        &lt;td&gt;383.3&lt;br /&gt;
| 14
&lt;/td&gt;
| 233.3
        &lt;td&gt;&lt;br /&gt;
| 8/7
&lt;/td&gt;
| ^^M2
        &lt;td&gt;550.0&lt;br /&gt;
| dupmajor 2nd
&lt;/td&gt;
| ^^E
        &lt;td&gt;&lt;br /&gt;
| SM2
&lt;/td&gt;
| supermajor 2nd
        &lt;td&gt;700.0&lt;br /&gt;
| SE
&lt;/td&gt;
| SE
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 15
        &lt;td&gt;833.3&lt;br /&gt;
| 250.0
&lt;/td&gt;
| 15/13, 22/19
        &lt;td&gt;&lt;br /&gt;
| ^<sup>3</sup>M2, <br>v<sup>3</sup>m3
&lt;/td&gt;
| trupmajor 2nd,<br>trudminor 3rd
        &lt;td&gt;966.7&lt;br /&gt;
| ^<sup>3</sup>E, <br>v<sup>3</sup>F
&lt;/td&gt;
| HM2, hm3
        &lt;td&gt;&lt;br /&gt;
| hypermajor 2nd, hypominor 3rd
&lt;/td&gt;
| HE, hF
        &lt;td&gt;1083.3&lt;br /&gt;
| UE, uF
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 16
&lt;/td&gt;
| 266.7
        &lt;td&gt;1200.0&lt;br /&gt;
| 7/6
&lt;/td&gt;
| vvm3
    &lt;/tr&gt;
| dudminor 3rd
    &lt;tr&gt;
| vvF
        &lt;td&gt;Steps as Freq. Ratio:&lt;br /&gt;
| sm3
&lt;/td&gt;
| subminor 3rd
        &lt;td&gt;&lt;br /&gt;
| sF
&lt;/td&gt;
| sF
        &lt;td&gt;9:8&lt;br /&gt;
|-
&lt;/td&gt;
| 17
        &lt;td&gt;&lt;br /&gt;
| 283.3
&lt;/td&gt;
| 13/11, 20/17
        &lt;td&gt;10:9&lt;br /&gt;
| vm3
&lt;/td&gt;
| downminor 3rd
        &lt;td&gt;&lt;br /&gt;
| vF
&lt;/td&gt;
| lm3
        &lt;td&gt;11:10&lt;br /&gt;
| little minor 3rd
&lt;/td&gt;
| lF
        &lt;td&gt;&lt;br /&gt;
| kF
&lt;/td&gt;
|-
        &lt;td&gt;12:11&lt;br /&gt;
| 18
&lt;/td&gt;
| 300.0
        &lt;td&gt;&lt;br /&gt;
| 19/16, 25/21, 32/27
&lt;/td&gt;
| m3
        &lt;td&gt;13:12&lt;br /&gt;
| minor 3rd
&lt;/td&gt;
| F
        &lt;td&gt;&lt;br /&gt;
| m3
&lt;/td&gt;
| minor 3rd
        &lt;td&gt;14:13&lt;br /&gt;
| F
&lt;/td&gt;
| F
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 19
        &lt;td&gt;15:14&lt;br /&gt;
| 316.7
&lt;/td&gt;
| 6/5
        &lt;td&gt;&lt;br /&gt;
| ^m3
&lt;/td&gt;
| upminor 3rd
        &lt;td&gt;16:15&lt;br /&gt;
| ^F
&lt;/td&gt;
| Km3
        &lt;td&gt;&lt;br /&gt;
| classic minor 3rd
&lt;/td&gt;
| KF
    &lt;/tr&gt;
| KF
    &lt;tr&gt;
|-
        &lt;td&gt;...in cents:&lt;br /&gt;
| 20
&lt;/td&gt;
| 333.3
        &lt;td&gt;&lt;br /&gt;
| 17/14, 39/32, 40/33
&lt;/td&gt;
| ^^m3, v~3
        &lt;td&gt;203.9&lt;br /&gt;
| dupminor 3rd, downmid 3rd
&lt;/td&gt;
| ^^F
        &lt;td&gt;&lt;br /&gt;
| Om3
&lt;/td&gt;
| on minor third
        &lt;td&gt;182.4&lt;br /&gt;
| OF
&lt;/td&gt;
| SF
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 21
        &lt;td&gt;165.0&lt;br /&gt;
| 350.0
&lt;/td&gt;
| 11/9, 27/22
        &lt;td&gt;&lt;br /&gt;
| ~3
&lt;/td&gt;
| mid 3rd
        &lt;td&gt;150.6&lt;br /&gt;
| ^<sup>3</sup>F
&lt;/td&gt;
| N3
        &lt;td&gt;&lt;br /&gt;
| neutral 3rd
&lt;/td&gt;
| UF/uF#
        &lt;td&gt;138.6&lt;br /&gt;
| UF/uF#
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 22
&lt;/td&gt;
| 366.7
        &lt;td&gt;128.3&lt;br /&gt;
| 16/13, 21/17, 26/21
&lt;/td&gt;
| ^~3, vvM3
        &lt;td&gt;&lt;br /&gt;
| upmid 3rd, dudmajor 3rd
&lt;/td&gt;
| vvF#
        &lt;td&gt;119.4&lt;br /&gt;
| oM3
&lt;/td&gt;
| off major 3rd
        &lt;td&gt;&lt;br /&gt;
| oF#
&lt;/td&gt;
| sF#
        &lt;td&gt;111.7&lt;br /&gt;
|-
&lt;/td&gt;
| 23
        &lt;td&gt;&lt;br /&gt;
| 383.3
&lt;/td&gt;
| 5/4
    &lt;/tr&gt;
| vM3
    &lt;tr&gt;
| downmajor 3rd
        &lt;td&gt;Nearest degree of 72edo:&lt;br /&gt;
| vF#
&lt;/td&gt;
| kM3
        &lt;td&gt;&lt;br /&gt;
| classic major 3rd
&lt;/td&gt;
| kF#
        &lt;td&gt;12&lt;br /&gt;
| kF#
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 24
&lt;/td&gt;
| 400.0
        &lt;td&gt;11&lt;br /&gt;
| 24/19
&lt;/td&gt;
| M3
        &lt;td&gt;&lt;br /&gt;
| major 3rd
&lt;/td&gt;
| F#
        &lt;td&gt;10&lt;br /&gt;
| M3
&lt;/td&gt;
| major 3rd
        &lt;td&gt;&lt;br /&gt;
| F#
&lt;/td&gt;
| F#
        &lt;td&gt;9&lt;br /&gt;
|-
&lt;/td&gt;
| 25
        &lt;td&gt;&lt;br /&gt;
| 416.7
&lt;/td&gt;
| 14/11
        &lt;td&gt;8&lt;br /&gt;
| ^M3
&lt;/td&gt;
| upmajor 3rd
        &lt;td&gt;&lt;br /&gt;
| ^F#
&lt;/td&gt;
| LM3
        &lt;td&gt;8&lt;br /&gt;
| large major 3rd
&lt;/td&gt;
| LF#
        &lt;td&gt;&lt;br /&gt;
| KF#
&lt;/td&gt;
|-
        &lt;td&gt;7&lt;br /&gt;
| 26
&lt;/td&gt;
| 433.3
        &lt;td&gt;&lt;br /&gt;
| 9/7
&lt;/td&gt;
| ^^M3
        &lt;td&gt;7&lt;br /&gt;
| dupmajor 3rd
&lt;/td&gt;
| ^^F#
        &lt;td&gt;&lt;br /&gt;
| SM3
&lt;/td&gt;
| supermajor 3rd
    &lt;/tr&gt;
| SF#
    &lt;tr&gt;
| SF#
        &lt;td&gt;...in cents:&lt;br /&gt;
|-
&lt;/td&gt;
| 27
        &lt;td&gt;&lt;br /&gt;
| 450.0
&lt;/td&gt;
| 13/10, 22/17
        &lt;td&gt;200.0&lt;br /&gt;
| ^<sup>3</sup>M3, v<sup>3</sup>4
&lt;/td&gt;
| trupmajor 3rd, trud 4th
        &lt;td&gt;&lt;br /&gt;
| ^<sup>3</sup>F#, v<sup>3</sup>G
&lt;/td&gt;
| HM3, h4
        &lt;td&gt;183.3&lt;br /&gt;
| hypermajor 3rd, hypo 4th
&lt;/td&gt;
| HF#, hG
        &lt;td&gt;&lt;br /&gt;
| UF#, uG
&lt;/td&gt;
|-
        &lt;td&gt;166.7&lt;br /&gt;
| 28
&lt;/td&gt;
| 466.7
        &lt;td&gt;&lt;br /&gt;
| 17/13, 21/16
&lt;/td&gt;
| vv4
        &lt;td&gt;150.0&lt;br /&gt;
| dud 4th
&lt;/td&gt;
| vvG
        &lt;td&gt;&lt;br /&gt;
| s4
&lt;/td&gt;
| sub 4th
        &lt;td&gt;133.3&lt;br /&gt;
| sG
&lt;/td&gt;
| sG
        &lt;td&gt;&lt;br /&gt;
|-
&lt;/td&gt;
| 29
        &lt;td&gt;133.3&lt;br /&gt;
| 483.3
&lt;/td&gt;
| 33/25
        &lt;td&gt;&lt;br /&gt;
| v4
&lt;/td&gt;
| down 4th
        &lt;td&gt;116.7&lt;br /&gt;
| vG
&lt;/td&gt;
| l4
        &lt;td&gt;&lt;br /&gt;
| little 4th
&lt;/td&gt;
| lG
        &lt;td&gt;116.7&lt;br /&gt;
| kG
&lt;/td&gt;
|-
        &lt;td&gt;&lt;br /&gt;
| 30
&lt;/td&gt;
| 500.0
    &lt;/tr&gt;
| 4/3
&lt;/table&gt;
| P4
| perfect 4th
| G
| P4
| perfect 4th
| G
| G
|-
| 31
| 516.7
| 27/20
| ^4
| up 4th
| ^G
| K4
| comma-wide 4th
| KG
| KG
|-
| 32
| 533.3
| 15/11, 19/14, ''26/19''
| ^^4, v~4
| dup 4th, downmid 4th
| ^^G
| O4
| on 4th
| OG
| SG
|-
| 33
| 550.0
| 11/8
| ~4
| mid 4th
| ^<sup>3</sup>G
| U4/N4
| uber 4th / neutral 4th
| UG
| UG
|-
| 34
| 566.7
| 18/13, 25/18
| ^~4, vvA4
| upmid 4th, dudaug 4th
| vvG#
| kkA4, sd5
| classic aug 4th, sub dim 5th
| kkG#, sAb
| SG#, (kkG#), sAb
|-
| 35
| 583.3
| 7/5
| vA4, vd5
| downaug 4th, <br>downdim 5th
| vG#, vAb
| kA4, ld5
| comma-narrow aug 4th, little dim 5th
| kG#, lAb
| kG#, kAb
|-
| 36
| 600.0
| 17/12, 24/17
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| A4, d5
| aug 4th, dim 5th
| G#, Ab
| G#, Ab
|-
| 37
| 616.7
| 10/7
| ^A4, ^d5
| upaug 4th, updim 5th
| ^G#, ^Ab
| LA4, Kd5
| large aug 4th, comma-wide dim 5th
| LG#, KAb
| KG#, KAb
|-
| 38
| 633.3
| 13/9, 36/25
| v~5, ^^d5
| downmid 5th, <br>dupdim 5th
| ^^Ab
| SA4, KKd5
| super aug 4th, classic dim 5th
| SG#, KKAb
| SG#, SAb, (KKAb)
|-
| 39
| 650.0
| 16/11
| ~5
| mid 5th
| v<sup>3</sup>A
| u5/N5
| unter 5th / neutral 5th
| uA
| uA
|-
| 40
| 666.7
| ''19/13'', 22/15, 28/19
| vv5, ^~5
| dud 5th, upmid 5th
| vvA
| o5
| off 5th
| oA
| sA
|-
| 41
| 683.3
| 40/27
| v5
| down 5th
| vA
| k5
| comma-narrow 5th
| kA
| kA
|-
| 42
| 700.0
| 3/2
| P5
| perfect 5th
| A
| P5
| perfect 5th
| A
| A
|-
| 43
| 716.7
| 50/33
| ^5
| up 5th
| ^A
| L5
| large fifth
| LA
| KA
|-
| 44
| 733.3
| 26/17, 32/21
| ^^5
| dup 5th
| ^^A
| S5
| super fifth
| SA
| SA
|-
| 45
| 750.0
| 17/11, 20/13
| ^<sup>3</sup>5, v<sup>3</sup>m6
| trup 5th, trudminor 6th
| ^<sup>3</sup>A, v<sup>3</sup>Bb
| H5, hm6
| hyper fifth, hypominor 6th
| HA, hBb
| UA, uBb
|-
| 46
| 766.7
| 14/9
| vvm6
| dudminor 6th
| vvBb
| sm6
| superminor 6th
| sBb
| sBb
|-
| 47
| 783.3
| 11/7
| vm6
| downminor 6th
| vBb
| lm6
| little minor 6th
| lBb
| kBb
|-
| 48
| 800.0
| 19/12
| m6
| minor 6th
| Bb
| m6
| minor 6th
| Bb
| Bb
|-
| 49
| 816.7
| 8/5
| ^m6
| upminor 6th
| ^Bb
| Km6
| classic minor 6th
| kBb
| kBb
|-
| 50
| 833.3
| 13/8, 21/13, 34/21
| ^^m6, v~6
| dupminor 6th, downmid 6th
| ^^Bb
| Om6
| on minor 6th
| oBb
| sBb
|-
| 51
| 850.0
| 18/11, 44/27
| ~6
| mid 6th
| v<sup>3</sup>B
| N6
| neutral 6th
| UBb, uB
| UBb, uB
|-
| 52
| 866.7
| 28/17, 33/20, 64/39
| ^~6, vvM6
| upmid 6th, dudmajor 6th
| vvB
| oM6
| off major 6th
| oB
| sB
|-
| 53
| 883.3
| 5/3
| vM6
| downmajor 6th
| vB
| kM6
| classic major 6th
| kB
| kB
|-
| 54
| 900.0
| 27/16, 32/19, 42/25
| M6
| major 6th
| B
| M6
| major 6th
| B
| B
|-
| 55
| 916.7
| 17/10, 22/13
| ^M6
| upmajor 6th
| ^B
| LM6
| large major 6th
| LB
| KB
|-
| 56
| 933.3
| 12/7
| ^^M6
| dupmajor 6th
| ^^B
| SM6
| supermajor 6th
| SB
| SB
|-
| 57
| 950.0
| 19/11, 26/15
| ^<sup>3</sup>M6, <br>v<sup>3</sup>m7
| trupmajor 6th,<br>trudminor 7th
| ^<sup>3</sup>B, <br>v<sup>3</sup>C
| HM6, hm7
| hypermajor 6th, hypominor 7th
| HB, hC
| UB, uC
|-
| 58
| 966.7
| 7/4
| vvm7
| dudminor 7th
| vvC
| sm7
| subminor 7th
| sC
| sC
|-
| 59
| 983.3
| 30/17, 44/25
| vm7
| downminor 7th
| vC
| lm7
| little minor 7th
| lC
| kC
|-
| 60
| 1000.0
| 16/9
| m7
| minor 7th
| C
| m7
| minor 7th
| C
| C
|-
| 61
| 1016.7
| 9/5
| ^m7
| upminor 7th
| ^C
| Km7
| classic/comma-wide minor 7th
| KC
| KC
|-
| 62
| 1033.3
| 20/11
| ^^m7, v~7
| dupminor 7th, downmid 7th
| ^^C
| Om7
| on minor 7th
| OC
| SC
|-
| 63
| 1050.0
| 11/6
| ~7
| mid 7th
| ^<sup>3</sup>C
| N7, hd8
| neutral 7th, hypo dim 8ve
| UC/uC#, hDb
| UC/uC#, uDb
|-
| 64
| 1066.7
| 13/7, 24/13, 50/27
| ^~7, vvM7
| upmid 7th, dudmajor 7th
| vvC#
| oM7, sd8
| off major 7th, sub dim 8ve
| oC#, sDb
| sC#, sDb
|-
| 65
| 1083.3
| 15/8, 28/15
| vM7
| downmajor 7th
| vC#
| kM7, ld8
| classic major 7th, little dim 8ve
| kC#, lDb
| kC#, kDb
|-
| 66
| 1100.0
| 17/9, 32/17, 36/19
| M7
| major 7th
| C#
| M7, d8
| major 7th, dim 8ve
| C#, Db
| C#, Db
|-
| 67
| 1116.7
| 19/10, 21/11, 40/21
| ^M7
| upmajor 7th
| ^C#
| LM7, Kd8
| large major 7th, comma-wide dim 8ve
| LC#, KDb
| KC#, KDb
|-
| 68
| 1133.3
| 25/13, 27/14, 48/25, 52/27
| ^^M7
| dupmajor 7th
| ^^C#
| SM7, KKd8
| supermajor 7th, classic dim 8ve
| SC#, KKDb
| SC#, SDb, (KKDb)
|-
| 69
| 1150.0
| 35/18, 39/20, 64/33
| ^<sup>3</sup>M7, v<sup>3</sup>8
| trupmajor 7th, trud octave
| ^<sup>3</sup>C#, v<sup>3</sup>D
| HM7, u8, h8
| hypermajor 7th, unter 8ve, hypo 8ve
| HC#, uD, hD
| UC#, uDb, uD
|-
| 70
| 1166.7
| 49/25, 55/28, 63/32, 88/45, 96/49
| vv8
| dud octave
| vvD
| s8, o8
| sub 8ve, off 8ve
| sD, oD
| sD
|-
| 71
| 1183.3
| 99/50, 160/81, 180/91, 196/99, 208/105
| v8
| down octave
| vD
| k8, l8
| comma-narrow 8ve, little 8ve
| kD, lD
| kD
|-
| 72
| 1200.0
| 2/1
| P8
| perfect octave
| D
| P8
| perfect octave
| D
| D
|}
<references group="note" />


&lt;br /&gt;
=== Interval quality and chord names in color notation ===
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Z function"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Z function&lt;/h1&gt;
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
72edo is the ninth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&gt;zeta integral edo&lt;/a&gt;, as well as being a peak and gap edo, and the maximum value of the &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function"&gt;Z function&lt;/a&gt; in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.&lt;br /&gt;
 
&lt;br /&gt;
{| class="wikitable center-all"
&lt;!-- ws:start:WikiTextLocalImageRule:373:&amp;lt;img src=&amp;quot;/file/view/plot72.png/219772696/plot72.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:373 --&gt;&lt;br /&gt;
|-
&lt;br /&gt;
! Quality
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Music"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Music&lt;/h1&gt;
! [[Color notation|Color]]
&lt;a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow"&gt;Kotekant&lt;/a&gt; &lt;a class="wiki_link_ext" href="http://www.archive.org/download/Kotekant/kotekant.mp3" rel="nofollow"&gt;play&lt;/a&gt; by &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt;&lt;br /&gt;
! Monzo format
&lt;br /&gt;
! Examples
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="External links"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;External links&lt;/h1&gt;
|-
&lt;ul&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow"&gt;Wikipedia article on 72edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow"&gt;OrthodoxWiki Article on Byzantine chant, which uses 72edo&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Joe_Maneri" rel="nofollow"&gt;Wikipedia article on Joe Maneri (1927-2009)&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://www.ekmelic-music.org/en/index.htmmusik/" rel="nofollow"&gt;Ekmelic Music Society/Gesellschaft für Ekmelische Musik&lt;/a&gt;, a group of composers and researchers dedicated to 72edo music&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://sonic-arts.org/tagawa/72edo.htm" rel="nofollow"&gt;Rick Tagawa's 72edo site&lt;/a&gt;, including theory and composers' list&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link_ext" href="http://dannywier.ucoz.com" rel="nofollow"&gt;Danny Wier, composer and musician who specializes in 72-edo&lt;/a&gt;&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
| dudminor
| zo
| (a b 0 1)
| 7/6, 7/4
|-
| minor
| fourthward wa
| (a b), b < -1
| 32/27, 16/9
|-
| upminor
| gu
| (a b -1)
| 6/5, 9/5
|-
| rowspan="2" | dupminor, <br>downmid
| luyo
| (a b 1 0 -1)
| 15/11
|-
| tho
| (a b 0 0 0 1)
| 13/8, 13/9
|-
| rowspan="2" | mid
| ilo
| (a b 0 0 1)
| 11/9, 11/6
|-
| lu
| (a b 0 0 -1)
| 12/11, 18/11
|-
| rowspan="2" | upmid, <br>dudmajor
| logu
| (a b -1 0 1)
| 11/10
|-
| thu
| (a b 0 0 0 -1)
| 16/13, 18/13
|-
| downmajor
| yo
| (a b 1)
| 5/4, 5/3
|-
| major
| fifthward wa
| (a b), b > 1
| 9/8, 27/16
|-
| dupmajor
| ru
| (a b 0 -1)
| 9/7, 12/7
|-
| rowspan="2" | trupmajor, <br>trudminor
| thogu
| (a b -1 0 0 1)
| 13/10
|-
| thuyo
| (a b 1 0 0 -1)
| 15/13
|}
All 72edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
 
{| class="wikitable center-all"
|-
! [[Color notation|Color of the 3rd]]
! JI chord
! Notes as edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo
| 6:7:9
| 0-16-42
| C vvEb G
| Cvvm
| C dudminor
|-
| gu
| 10:12:15
| 0-19-42
| C ^Eb G
| C^m
| C upminor
|-
| ilo
| 18:22:27
| 0-21-42
| C v<span style="font-size: 90%; vertical-align: super;">3</span>E G
| C~
| C mid
|-
| yo
| 4:5:6
| 0-23-42
| C vE G
| Cv
| C downmajor or C down
|-
| ru
| 14:18:27
| 0-26-42
| C ^^E G
| C^^
| C dupmajor or C dup
|}
For a more complete list, see [[Ups and downs notation #Chord names in other EDOs]].
 
=== Relationship between primes and rings ===
In 72tet, there are 6 [[ring number|rings]]. 12edo is the plain ring; thus every 6 degrees is the 3-limit.
 
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* −1 degree (the down ring) corrects 81/64 to 5/4 via 80/81
* −2 degrees (the dud ring) corrects 16/9 to 7/4 via 63/64
* +3 degrees  (the trup ring) corrects 4/3 to 11/8 via 33/32
* +2 degrees (the dup ring) corrects 128/81 to 13/8 via 1053/1024
* 0 degrees (the plain ring) corrects 256/243 to 17/16 via 4131/4096
* 0 degrees (the plain ring) corrects 32/27 to 19/16 via 513/512
Thus the product of a ratio's monzo with {{map| 0 0 -1 -2 3 2 0 0 }}, modulo 6, specifies which ring the ratio lies on.
 
== Notations ==
=== Ups and downs notation ===
72edo can be notated with ups and downs, spoken as up, dup, trup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, trud, dupflat etc.
{{Sharpness-sharp6a}}
 
Half-sharps and half-flats can be used to avoid triple arrows:
{{Sharpness-sharp6b}}
 
[[Alternative symbols for ups and downs notation#Sharp-6| Alternative ups and downs]] have sharps and flats with arrows borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp6}}
 
If double arrows are not desirable, arrows can be attached to quarter-tone accidentals:
{{Sharpness-sharp6-qt}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[65edo#Sagittal notation|65-EDO]] and [[79edo#Sagittal notation|79]], and is a superset of the notations for EDOs [[36edo#Sagittal notation|36]], [[24edo#Sagittal notation|24]], [[18edo#Sagittal notation|18]], [[12edo#Sagittal notation|12]], [[8edo#Sagittal notation|8]], and [[6edo#Sagittal notation|6]].
 
==== Evo flavor ====
<imagemap>
File:72-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo_Sagittal.svg]]
</imagemap>
 
==== Revo flavor ====
<imagemap>
File:72-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Revo_Sagittal.svg]]
</imagemap>
 
==== Evo-SZ flavor ====
<imagemap>
File:72-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 711 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:72-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72edo in the Revo flavor of Sagittal:
 
[[File:72edo Sagittal.png|800px]]
 
=== Ivan Wyschnegradsky's notation ===
{{Sharpness-sharp6-iw|72}}
 
== Approximation to JI ==
[[File:72ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected intervals approximated in 72edo]]
 
=== Interval mappings ===
{{Q-odd-limit intervals|72}}
 
=== Zeta properties ===
72edo is the ninth [[zeta integral edo]], as well as being a peak and gap edo, and the maximum value of the [[the Riemann zeta function and tuning#The Z function|Z function]] in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
 
[[File:plot72.png|alt=plot72.png|plot72.png]]
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 15625/15552, 531441/524288
| {{Mapping| 72 114 167 }}
| +0.839
| 0.594
| 3.56
|-
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| {{Mapping| 72 114 167 202 }}
| +0.822
| 0.515
| 3.09
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 4000/3993
| {{Mapping| 72 114 167 202 249 }}
| +0.734
| 0.493
| 2.96
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 243/242, 325/324, 385/384
| {{Mapping| 72 114 167 202 249 266 }}
| +0.936
| 0.638
| 3.82
|-
| 2.3.5.7.11.13.17
| 169/168, 221/220, 225/224, 243/242, 273/272, 325/324
| {{Mapping| 72 114 167 202 249 266 294 }}
| +0.975
| 0.599
| 3.59
|-
| 2.3.5.7.11.13.17.19
| 153/152, 169/168, 210/209, 221/220, 225/224, 243/242, 273/272
| {{Mapping| 72 114 167 202 249 266 294 306 }}
| +0.780
| 0.762
| 4.57
|}
* 72et has lower relative errors than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next equal temperaments doing better in these subgroups are [[99edo|99]], [[270edo|270]], [[224edo|224]], [[494edo|494]], and [[217edo|217]], respectively.
 
=== Commas ===
Commas tempered out by 72edo include…
 
{| class="commatable wikitable center-1 center-2 right-4"
|-
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Pythagorean comma
|-
| 5
| [[15625/15552]]
| {{Monzo| -6 -5 6 }}
| 8.11
| Kleisma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| [[Ampersand comma]]
|-
| 5
| [[129140163/128000000|(18 digits)]]
| {{Monzo| -13 17 -6 }}
| 15.35
| [[Graviton]]
|-
| 5
| <abbr title="7629394531250/7625597484987">(26 digits)</abbr>
| {{Monzo| 1 -27 18 }}
| 0.86
| [[Ennealimma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Marvel comma
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Gamelisma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Breedsma
|-
| 7
| [[4375/4374]]
| {{Monzo| -1 -7 4 1 }}
| 0.40
| Ragisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Mirkwai comma
|-
| 7
| [[19683/19600]]
| {{Monzo| -4 9 -2 -2 }}
| 7.32
| Cataharry comma
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{Monzo | -6 -8 2 5 }}
| 1.12
| [[Wizma]]
|-
| 7
| <abbr title="250047/250000">(12 digits)</abbr>
| {{Monzo| -4 6 -6 3 }}
| 0.33
| [[Landscape comma]]
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2}}
| 7.14
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Swetisma
|-
| 11
| [[1375/1372]]
| {{Monzo| -2 0 3 -3 1 }}
| 3.78
| Moctdel comma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Lehmerisma
|-
| 11
| [[4000/3993]]
| {{Monzo| 5 -1 3 0 -3 }}
| 3.03
| Wizardharry comma
|-
| 11
| [[6250/6237]]
| {{Monzo| 1 -4 5 -1 -1 }}
| 3.60
| Liganellus comma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Kalisma
|-
| 11
| <abbr title="1771561/1769472">(14 digits)</abbr>
| {{Monzo| 16 -3 0 0 6 }}
| 2.04
| [[Nexus comma]]
|-
| 13
| [[169/168]]
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| Buzurgisma
|-
| 13
| [[325/324]]
| {{Monzo| -2 -4 2 0 0 1 }}
| 5.34
| Marveltwin comma
|-
| 13
| [[351/350]]
| {{Monzo| -1 3 -2 -1 0 1 }}
| 4.94
| Ratwolfsma
|-
| 13
| [[364/363]]
| {{Monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Minor minthma
|-
| 13
| [[625/624]]
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.77
| Tunbarsma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Island comma
|-
| 13
| [[729/728]]
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.38
| Squbema
|-
| 13
| [[1001/1000]]
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.73
| Sinbadma
|-
| 13
| [[1575/1573]]
| {{Monzo| 2 2 1 -2 -1 }}
| 2.20
| Nicola
|-
| 13
| [[1716/1715]]
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.01
| Lummic comma
|-
| 13
| [[2080/2079]]
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83
| Ibnsinma
|-
| 13
| [[6656/6655]]
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| Jacobin comma
|}
<references group="note" />
 
=== Rank-2 temperaments ===
* [[List of edo-distinct 72et rank two temperaments]]
 
72edo provides the [[optimal patent val]] for [[miracle]] and [[wizard]] in the 7-limit, miracle, [[catakleismic]], [[bikleismic]], [[compton]], [[ennealimnic]], [[ennealiminal]], [[enneaportent]], [[marvolo]] and [[catalytic]] in the 11-limit, and catakleismic, bikleismic, compton, [[comptone]], [[enneaportent]], [[ennealim]], catalytic, marvolo, [[manna]], [[hendec]], [[lizard]], [[neominor]], [[hours]], and [[semimiracle]] in the 13-limit.
 
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\72
| 16.7
| 105/104
| [[Quincy]]
|-
| 1
| 5\72
| 83.3
| 21/20
| [[Marvolo]]
|-
| 1
| 7\72
| 116.7
| 15/14
| [[Miracle]] / benediction / manna
|-
| 1
| 17\72
| 283.3
| 13/11
| [[Neominor]]
|-
| 1
| 19\72
| 316.7
| 6/5
| [[Catakleismic]]
|-
| 1
| 25\72
| 416.7
| 14/11
| [[Sqrtphi]]
|-
| 1
| 29\72
| 483.3
| 45/34
| [[Hemiseven]]
|-
| 1
| 31\72
| 516.7
| 27/20
| [[Marvo]] / [[zarvo]]
|-
| 1
| 35\72
| 583.3
| 7/5
| [[Cotritone]]
|-
| 2
| 5\72
| 83.3
| 21/20
| [[Harry]]
|-
| 2
| 7\72
| 116.7
| 15/14
| [[Semimiracle]]
|-
| 2
| 11\72
| 183.3
| 10/9
| [[Unidec]] / hendec
|-
| 2
| 21\72<br>(19\72)
| 316.7<br>(283.3)
| 6/5<br>(13/11)
| [[Bikleismic]]
|-
| 2
| 23\72<br>(13\72)
| 383.3<br>(216.7)
| 5/4<br>(17/15)
| [[Wizard]] / lizard / gizzard
|-
| 3
| 11\72
| 183.3
| 10/9
| [[Mirkat]]
|-
| 3
| 19\72<br>(5\72)
| 316.7<br>(83.3)
| 6/5<br>(21/20)
| [[Tritikleismic]]
|-
| 4
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Quadritikleismic]]
|-
| 8
| 34\72<br>(2\72)
| 566.7<br>(33.3)
| 168/121<br>(55/54)
| [[Octowerck]] / octowerckis
|-
| 8
| 35\72<br>(1\72)
| 583.3<br>(16.7)
| 7/5<br>(100/99)
| [[Octoid]] / octopus
|-
| 9
| 19\72<br>(3\72)
| 316.7<br>(50.0)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimnic
|-
| 9
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Enneaportent]]
|-
| 12
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(100/99)
| [[Compton]] / comptone
|-
| 18
| 19\72<br>(1\72)
| 316.7<br>(16.7)
| 6/5<br>(105/104)
| [[Hemiennealimmal]]
|-
| 24
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(105/104)
| [[Hours]]
|-
| 36
| 23\72<br>(1\72)
| 383.3<br>(16.7)
| 5/4<br>(81/80)
| [[Gamelstearn]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 
== Scales ==
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[Keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[JuneGloom]]
* [[Harry Partch's 43-tone scale]]: 1 2 2 2 2 1 1 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 1 1 2 2 2 2 1
* [[Magnetosphere scale|Magnetosphere]], [[Blackened skies]], [[Lost spirit]]
* [[5- to 10-tone scales in 72edo]]
 
=== Harmonic scale ===
Mode 8 of the harmonic series&mdash;[[overtone scale|harmonics 8 through 16]], octave repeating&mdash;is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
 
{| class="wikitable"
|-
! Harmonics in "Mode 8":
| 8
|
| 9
|
| 10
|
| 11
|
| 12
|
| 13
|
| 14
|
| 15
|
| 16
|-
! …as JI Ratio from 1/1:
| 1/1
|
| 9/8
|
| 5/4
|
| 11/8
|
| 3/2
|
| 13/8
|
| 7/4
|
| 15/8
|
| 2/1
|-
! …in cents:
| 0
|
| 203.9
|
| 386.3
|
| 551.3
|
| 702.0
|
| 840.5
|
| 968.8
|
| 1088.3
|
| 1200.0
|-
! Nearest degree of 72edo:
| 0
|
| 12
|
| 23
|
| 33
|
| 42
|
| 50
|
| 58
|
| 65
|
| 72
|-
! …in cents:
| 0
|
| 200.0
|
| 383.3
|
| 550.0
|
| 700.0
|
| 833.3
|
| 966.7
|
| 1083.3
|
| 1200.0
|-
! Steps as Freq. Ratio:
|
| 9:8
|
| 10:9
|
| 11:10
|
| 12:11
|
| 13:12
|
| 14:13
|
| 15:14
|
| 16:15
|
|-
! …in cents:
|
| 203.9
|
| 182.4
|
| 165.0
|
| 150.6
|
| 138.6
|
| 128.3
|
| 119.4
|
| 111.7
|
|-
! Nearest degree of 72edo:
|
| 12
|
| 11
|
| 10
|
| 9
|
| 8
|
| 8
|
| 7
|
| 7
|
|-
! …in cents:
|
| 200.0
|
| 183.3
|
| 166.7
|
| 150.0
|
| 133.3
|
| 133.3
|
| 116.7
|
| 116.7
|
|}
 
== Instruments ==
If one can get six 12edo instruments tuned a twelfth-tone apart, it is possible to use these instruments in combination to play the full gamut of 72edo (see Music).
 
One can also use a skip fretting system:
* [[Skip fretting system 72 2 27]]
 
Alternatively, an appropriately mapped keyboard of sufficient size is usable for playing 72edo:
* [[Lumatone mapping for 72edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/VwVp3RVao_k ''microtonal improvisation in 72edo''] (2025)
 
; [[Ambient Esoterica]]
* [https://www.youtube.com/watch?v=seWcDAoQjxY ''Goetic Synchronities''] (2023)
* [https://www.youtube.com/watch?v=CrcdM1e2b6Q ''Rainy Day Generative Pillow''] (2024)
 
; [[Jake Freivald]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 ''Lazy Sunday'']{{dead link}} in the [[lazysunday]] scale
 
{{Wikipedia|In vain (Haas)}}
; [[Georg Friedrich Haas]]
* [https://www.youtube.com/watch?v=ix4yA-c-Pi8 ''Blumenstück''] (2000)
* [https://youtu.be/cmX-h7_us7A ''in vain''] (2000) ([https://www.universaledition.com/georg-friedrich-haas-278/works/in-vain-7566 score])
 
; [[Claudi Meneghin]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3 ''Twinkle canon – 72 edo'']{{dead link}}
* [https://www.youtube.com/watch?v=zR0NDgh4944 ''The Miracle Canon'', 3-in-1 on a Ground]
* [https://www.youtube.com/watch?v=w6Bckog1eOM ''Sicilienne in Miracle'']
* [https://www.youtube.com/watch?v=QKeZLtFHfNU ''Arietta with 5 Variations'', for Organ] (2024)
 
; [[Prent Rodgers]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 ''June Gloom #9'']{{dead link}}
 
; [[Gene Ward Smith]]
* [https://www.archive.org/details/Kotekant ''Kotekant''] [https://www.archive.org/download/Kotekant/kotekant.mp3 play] (2010)
 
;[[Ivan Wyschnegradsky]]
* [https://www.youtube.com/watch?v=RCcJHCkYQ6U Arc-en-ciel, for 6 pianos in twelfth tones, Op. 37] (1956)
 
; [[James Tenney]]
* [https://www.youtube.com/watch?v=jGsxqU1PhZs&list=OLAK5uy_mKyMEMZW7noeLncJnu-JT65go8w7403DA ''Changes for Six Harps'']
 
; [[Xeno Ov Eleas]]
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)
 
== External links ==
* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 72edo]
* [http://www.ekmelic-music.org/en/ Ekmelic Music Society/Gesellschaft für Ekmelische Musik], a group of composers and researchers dedicated to 72edo music
* [http://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
* [https://www.myspace.com/dawier Danny Wier, composer and musician who specializes in 72-edo]
* [http://tonalsoft.com/enc/number/72edo.aspx 72-ed2 / 72-edo / 72-ET / 72-tone equal-temperament] on [[Tonalsoft Encyclopedia]]
 
[[Category:Listen]]
[[Category:Compton]]
[[Category:Marvel]]
[[Category:Miracle]]
[[Category:Prodigy]]
[[Category:Wizard]]
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