72edo: Difference between revisions

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[[toc|flat]]
----
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 [[xenharmonic/24edo|24-tone equal temperament]], a common and standard tuning of [[xenharmonic/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 [[xenharmonic/96edo|96-edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.

72-tone equal temperament approximates [[xenharmonic/11-limit|11-limit just intonation]] exceptionally well, is consistent in the [[xenharmonic/17-limit|17-limit]], and is the ninth [[xenharmonic/The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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.

72 is an excellent tuning for [[xenharmonic/Gamelismic clan|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[xenharmonic/Marvel family#Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.

=Commas and Temperaments= 

72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 
the kleisma, 15625/15552, 
the ampersand, 34171875/33554432 = |-25 7 6>, 
the graviton, 129140163/128000000 = |-13 17 -6>, and 
the ennealimma, 7629394531250/7625597484987 = |1 -27 18> in the 5-limit. 

The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.

72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. 

For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.


It 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.

=Harmonic Scale= 
Mode 8 of the harmonic series -- [[xenharmonic/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 ||
|| ...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 ||   ||

=Intervals= 
|| degrees || cents value || approximate ratios (11-limit) ||||= [[xenharmonic/Ups and Downs Notation|ups and downs ]][[xenharmonic/Ups and Downs Notation|notation]] ||
|| 0 || 0 || 1/1 ||= P1 ||= D ||
|| 1 || 16.667 || 81/80 ||= ^1 (up unison) ||= D^ ||
|| 2 || 33.333 || 45/44 ||= ^^1(double-up unison) ||= D^^ ||
|| 3 || 50 || 33/32 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2 || D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 4 || 66.667 || 25/24 ||= vvm2 ||= Ebvv ||
|| 5 || 83.333 || 21/20 ||= vm2 (downminor 2nd) ||= Ebv ||
|| 6 || 100 || 35/33 ||= m2 ||= Eb ||
|| 7 || 116.667 || 15/14 ||= ^m2 (upminor 2nd) ||= Eb^ ||
|| 8 || 133.333 || 27/25 ||= v~2 ||= Eb^^ ||
|| 9 || 150 || 12/11 ||= ~2 (mid 2nd) ||= Ev<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 10 || 166.667 || 11/10 ||= ^~2 (upmid 2nd) ||= Evv ||
|| 11 || 183.333 || 10/9 ||= vM2 ||= Ev ||
|| 12 || 200 || 9/8 ||= M2 ||= E ||
|| 13 || 216.667 || 25/22 ||= ^M2 ||= E^ ||
|| 14 || 233.333 || 8/7 ||= ^^M2 ||= E^^ ||
|| 15 || 250 || 81/70 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3 ||= E^<span style="font-size: 90%; vertical-align: super;">3</span>, Fv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 16 || 266.667 || 7/6 ||= vvm3 ||= Fvv ||
|| 17 || 283.333 || 33/28 ||= vm3 ||= Fv ||
|| 18 || 300 || 25/21 ||= m3 ||= F ||
|| 19 || 316.667 || 6/5 ||= ^m3 ||= F^ ||
|| 20 || 333.333 || 40/33 ||= v~3 ||= F^^ ||
|| 21 || 350 || 11/9 ||= ~3 ||= F^<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 22 || 366.667 || 99/80 ||= ^~3 ||= F#vv ||
|| 23 || 383.333 || 5/4 ||= vM3 ||= F#v ||
|| 24 || 400 || 44/35 ||= M3 ||= F# ||
|| 25 || 416.667 || 14/11 ||= ^M3 ||= F#^ ||
|| 26 || 433.333 || 9/7 ||= ^^M3 ||= F#^^ ||
|| 27 || 450 || 35/27 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4 ||= F#^<span style="font-size: 90%; vertical-align: super;">3</span>, Gv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 28 || 466.667 || 21/16 ||= vv4 ||= Gvv ||
|| 29 || 483.333 || 33/25 ||= v4 ||= Gv ||
|| 30 || 500 || 4/3 ||= P4 ||= G ||
|| 31 || 516.667 || 27/20 ||= ^4 ||= G^ ||
|| 32 || 533.333 || 15/11 ||= ^^4 ||= G^^ ||
|| 33 || 550 || 11/8 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>4 ||= G^<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 34 || 566.667 || 25/18 ||= vvA4 ||= G#vv ||
|| 35 || 583.333 || 7/5 ||= vA4, vd5 ||= G#v, Abv ||
|| 36 || 600 || 99/70 ||= A4, d5 ||= G#, Ab ||
|| 37 || 616.667 || 10/7 ||= ^A4, ^d5 ||= G#^, Ab^ ||
|| 38 || 633.333 || 36/25 ||= ^^d5 ||= Ab^^ ||
|| 39 || 650 || 16/11 ||= v<span style="font-size: 90%; vertical-align: super;">3</span>5 ||= Av<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 40 || 666.667 || 22/15 ||= vv5 ||= Avv ||
|| 41 || 683.333 || 40/27 ||= v5 ||= Av ||
|| 42 || 700 || 3/2 ||= P5 ||= A ||
|| 43 || 716.667 || 50/33 ||= ^5 ||= A^ ||
|| 44 || 733.333 || 32/21 ||= ^^5 ||= A^^ ||
|| 45 || 750 || 54/35 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6 ||= A^<span style="font-size: 90%; vertical-align: super;">3</span>, Bbv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 46 || 766.667 || 14/9 ||= vvm6 ||= Bbvv ||
|| 47 || 783.333 || 11/7 ||= vm6 ||= Bbv ||
|| 48 || 800 || 35/22 ||= m6 ||= Bb ||
|| 49 || 816.667 || 8/5 ||= ^m6 ||= Bb^ ||
|| 50 || 833.333 || 81/50 ||= v~6 ||= Bb^^ ||
|| 51 || 850 || 18/11 ||= ~6 ||= Bv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 52 || 866.667 || 33/20 ||= ^~6 ||= Bvv ||
|| 53 || 883.333 || 5/3 ||= vM6 ||= Bv ||
|| 54 || 900 || 27/16 ||= M6 ||= B ||
|| 55 || 916.667 || 56/33 ||= ^M6 ||= B^ ||
|| 56 || 933.333 || 12/7 ||= ^^M6 ||= B^^ ||
|| 57 || 950 || 121/70 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7 ||= B^<span style="font-size: 90%; vertical-align: super;">3</span>, Cv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 58 || 966.667 || 7/4 ||= vvm7 ||= Cvv ||
|| 59 || 983.333 || 44/25 ||= vm7 ||= Cv ||
|| 60 || 1000 || 16/9 ||= m7 ||= C ||
|| 61 || 1016.667 || 9/5 ||= ^m7 ||= C^ ||
|| 62 || 1033.333 || 20/11 ||= v~7 ||= C^^ ||
|| 63 || 1050 || 11/6 ||= ~7 ||= C^<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 64 || 1066.667 || 50/27 ||= ^~7 ||= C#vv ||
|| 65 || 1083.333 || 15/8 ||= vM7 ||= C#v ||
|| 66 || 1100 || 66/35 ||= M7 ||= C# ||
|| 67 || 1116.667 || 21/11 ||= ^M7 ||= C#^ ||
|| 68 || 1133.333 || 27/14 ||= ^^M7 ||= C#^^ ||
|| 69 || 1150 || 35/18 ||= ^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8 ||= C#^<span style="font-size: 90%; vertical-align: super;">3</span>, Dv<span style="font-size: 90%; vertical-align: super;">3</span> ||
|| 70 || 1166.667 || 49/25 ||= vv8 ||= Dvv ||
|| 71 || 1183.333 || 99/50 ||= v8 ||= Dv ||
|| 72 || 1200 || 2/1 ||= P8 ||= D ||
Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].

=Linear temperaments= 
||~ Periods per octave ||~ Generator ||~ Names ||
|| 1 || 1\72 || [[xenharmonic/quincy|quincy]] ||
|| 1 || 5\72 || [[marvolo]] ||
|| 1 || 7\72 || [[xenharmonic/miracle|miracle]]/benediction/manna ||
|| 1 || 11\72 ||   ||
|| 1 || 13\72 ||   ||
|| 1 || 17\72 || [[xenharmonic/neominor|neominor]] ||
|| 1 || 19\72 || [[xenharmonic/catakleismic|catakleismic]] ||
|| 1 || 23\72 ||   ||
|| 1 || 25\72 || [[xenharmonic/sqrtphi|sqrtphi]] ||
|| 1 || 29\72 ||   ||
|| 1 || 31\72 || [[xenharmonic/marvo|marvo]]/zarvo ||
|| 1 || 35\72 || [[xenharmonic/cotritone|cotritone]] ||
|| 2 || 1\72 ||   ||
|| 2 || 5\72 || [[xenharmonic/harry|harry]] ||
|| 2 || 7\72 ||   ||
|| 2 || 11\72 || [[xenharmonic/unidec|unidec]]/hendec ||
|| 2 || 13\72 || [[xenharmonic/wizard|wizard]]/lizard/gizzard ||
|| 2 || 17\72 ||   ||
|| 3 || 1\72 ||   ||
|| 3 || 5\72 || [[xenharmonic/tritikleismic|tritikleismic]] ||
|| 3 || 7\72 ||   ||
|| 3 || 11\72 || [[xenharmonic/mirkat|mirkat]] ||
|| 4 || 1\72 || [[xenharmonic/quadritikleismic|quadritikleismic]] ||
|| 4 || 5\72 ||   ||
|| 4 || 7\72 ||   ||
|| 6 || 1\72 ||   ||
|| 6 || 5\72 ||   ||
|| 8 || 1\72 || [[xenharmonic/octoid|octoid]] ||
|| 8 || 2\72 || [[xenharmonic/octowerck|octowerck]] ||
|| 8 || 4\72 ||   ||
|| 9 || 1\72 ||   ||
|| 9 || 3\72 || [[xenharmonic/ennealimmal|ennealimmal]]/ennealimmic ||
|| 12 || 1\72 || [[xenharmonic/compton|compton]] ||
|| 18 || 1\72 || [[xenharmonic/hemiennealimmal|hemiennealimmal]] ||
|| 24 || 1\72 || [[xenharmonic/hours|hours]] ||
|| 36 || 1\72 ||   ||

=Z function= 
72edo is the ninth [[xenharmonic/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 [[xenharmonic/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:xenharmonic/plot72.png]]

=Music= 
[[http://www.archive.org/details/Kotekant|Kotekant]] //[[http://www.archive.org/download/Kotekant/kotekant.mp3|play]]// by [[xenharmonic/Gene Ward Smith|Gene Ward Smith]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3|Twinkle canon – 72 edo]]// by [[http://soonlabel.com/xenharmonic/archives/573|Claudi Meneghin]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3|Lazy Sunday]]// by [[Jake Freivald]] in the [[lazysunday]] scale.
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3|June Gloom #9]]// by Prent Rodgers

=Scales= 
[[xenharmonic/smithgw72a|smithgw72a]], [[xenharmonic/smithgw72b|smithgw72b]], [[xenharmonic/smithgw72c|smithgw72c]], [[xenharmonic/smithgw72d|smithgw72d]], [[xenharmonic/smithgw72e|smithgw72e]], [[xenharmonic/smithgw72f|smithgw72f]], [[xenharmonic/smithgw72g|smithgw72g]], [[xenharmonic/smithgw72h|smithgw72h]], [[xenharmonic/smithgw72i|smithgw72i]], [[xenharmonic/smithgw72j|smithgw72j]]
[[xenharmonic/blackjack|blackjack]], [[xenharmonic/miracle_8|miracle_8]], [[xenharmonic/miracle_10|miracle_10]], [[xenharmonic/miracle_12|miracle_12]], [[xenharmonic/miracle_12a|miracle_12a]], [[xenharmonic/miracle_24hi|miracle_24hi]], [[xenharmonic/miracle_24lo|miracle_24lo]]
[[xenharmonic/keenanmarvel|keenanmarvel]], [[xenharmonic/xenakis_chrome|xenakis_chrome]], [[xenharmonic/xenakis_diat|xenakis_diat]], [[xenharmonic/xenakis_schrome|xenakis_schrome]]
[[xenharmonic/genus24255et72|Euler(24255) genus in 72 equal]]
[[JuneGloom]]

=External links= 
* [[http://en.wikipedia.org/wiki/72_tone_equal_temperament|Wikipedia article on 72edo]]
* [[http://orthodoxwiki.org/Byzantine_Chant|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/|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
* [[@http://www.myspace.com/dawier|Danny Wier, composer and musician who specializes in 72-edo]]

Original HTML content:

<html><head><title>72edo</title></head><body><!-- ws:start:WikiTextTocRule:16:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:16 --><!-- ws:start:WikiTextTocRule:17: --><a href="#Commas and Temperaments">Commas and Temperaments</a><!-- ws:end:WikiTextTocRule:17 --><!-- ws:start:WikiTextTocRule:18: --> | <a href="#Harmonic Scale">Harmonic Scale</a><!-- ws:end:WikiTextTocRule:18 --><!-- ws:start:WikiTextTocRule:19: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:19 --><!-- ws:start:WikiTextTocRule:20: --> | <a href="#Linear temperaments">Linear temperaments</a><!-- ws:end:WikiTextTocRule:20 --><!-- ws:start:WikiTextTocRule:21: --> | <a href="#Z function">Z function</a><!-- ws:end:WikiTextTocRule:21 --><!-- ws:start:WikiTextTocRule:22: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:22 --><!-- ws:start:WikiTextTocRule:23: --> | <a href="#Scales">Scales</a><!-- ws:end:WikiTextTocRule:23 --><!-- ws:start:WikiTextTocRule:24: --> | <a href="#External links">External links</a><!-- ws:end:WikiTextTocRule:24 --><!-- ws:start:WikiTextTocRule:25: -->
<!-- ws:end:WikiTextTocRule:25 --><hr />
72-tone equal temperament (or 72-edo) divides the octave into 72 steps or <em>moria</em>. 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 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/24edo">24-tone equal temperament</a>, a common and standard tuning of <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Arabic%2C%20Turkish%2C%20Persian">Arabic</a> music, and has itself been used to tune Turkish music.<br />
<br />
Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with <a class="wiki_link" href="http://xenharmonic.wikispaces.com/96edo">96-edo</a>), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.<br />
<br />
72-tone equal temperament approximates <a class="wiki_link" href="http://xenharmonic.wikispaces.com/11-limit">11-limit just intonation</a> exceptionally well, is consistent in the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/17-limit">17-limit</a>, and is the ninth <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">Zeta integral tuning</a>. 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.<br />
<br />
72 is an excellent tuning for <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Gamelismic%20clan">miracle temperament</a>, especially the 11-limit version, and the related rank three temperament <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Marvel%20family#Prodigy">prodigy</a>, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Commas and Temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Commas and Temperaments</h1>
 <br />
72et tempers out the Pythagorean comma, 531441/524288, in the 3-limit and <br />
the kleisma, 15625/15552, <br />
the ampersand, 34171875/33554432 = |-25 7 6&gt;, <br />
the graviton, 129140163/128000000 = |-13 17 -6&gt;, and <br />
the ennealimma, 7629394531250/7625597484987 = |1 -27 18&gt; in the 5-limit. <br />
<br />
The 7-limit commas include 225/224, 1029/1024, 16875/16807, 19683/19600, 420175/419904, 2401/2400, 4375/4374 and 250047/250000.<br />
<br />
72et shines in the 11-limit, with commas 243/242, 385/384, 441/440, 540/539, 1375/1372, 6250/6237, 4000/3993, 3025/3024 and 9801/9800. <br />
<br />
For the 13-limit, it tempers out 169/168, 325/324, 351/350, 364/363, 625/624, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079 and 6656/6655.<br />
<br />
<br />
It 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:2 -->Harmonic Scale</h1>
 Mode 8 of the harmonic series -- <a class="wiki_link" href="http://xenharmonic.wikispaces.com/overtone%20scales">overtones 8 through 16</a>, 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).<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>13<br />
</td>
        <td><br />
</td>
        <td>14<br />
</td>
        <td><br />
</td>
        <td>15<br />
</td>
        <td><br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td><br />
</td>
        <td>9/8<br />
</td>
        <td><br />
</td>
        <td>5/4<br />
</td>
        <td><br />
</td>
        <td>11/8<br />
</td>
        <td><br />
</td>
        <td>3/2<br />
</td>
        <td><br />
</td>
        <td>13/8<br />
</td>
        <td><br />
</td>
        <td>7/4<br />
</td>
        <td><br />
</td>
        <td>15/8<br />
</td>
        <td><br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>203.9<br />
</td>
        <td><br />
</td>
        <td>386.3<br />
</td>
        <td><br />
</td>
        <td>551.3<br />
</td>
        <td><br />
</td>
        <td>702.0<br />
</td>
        <td><br />
</td>
        <td>840.5<br />
</td>
        <td><br />
</td>
        <td>968.8<br />
</td>
        <td><br />
</td>
        <td>1088.3<br />
</td>
        <td><br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 72edo:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>23<br />
</td>
        <td><br />
</td>
        <td>33<br />
</td>
        <td><br />
</td>
        <td>42<br />
</td>
        <td><br />
</td>
        <td>50<br />
</td>
        <td><br />
</td>
        <td>58<br />
</td>
        <td><br />
</td>
        <td>65<br />
</td>
        <td><br />
</td>
        <td>72<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td><br />
</td>
        <td>200.0<br />
</td>
        <td><br />
</td>
        <td>383.3<br />
</td>
        <td><br />
</td>
        <td>550.0<br />
</td>
        <td><br />
</td>
        <td>700.0<br />
</td>
        <td><br />
</td>
        <td>833.3<br />
</td>
        <td><br />
</td>
        <td>966.7<br />
</td>
        <td><br />
</td>
        <td>1083.3<br />
</td>
        <td><br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Steps as Freq. Ratio:<br />
</td>
        <td><br />
</td>
        <td>9:8<br />
</td>
        <td><br />
</td>
        <td>10:9<br />
</td>
        <td><br />
</td>
        <td>11:10<br />
</td>
        <td><br />
</td>
        <td>12:11<br />
</td>
        <td><br />
</td>
        <td>13:12<br />
</td>
        <td><br />
</td>
        <td>14:13<br />
</td>
        <td><br />
</td>
        <td>15:14<br />
</td>
        <td><br />
</td>
        <td>16:15<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td><br />
</td>
        <td>203.9<br />
</td>
        <td><br />
</td>
        <td>182.4<br />
</td>
        <td><br />
</td>
        <td>165.0<br />
</td>
        <td><br />
</td>
        <td>150.6<br />
</td>
        <td><br />
</td>
        <td>138.6<br />
</td>
        <td><br />
</td>
        <td>128.3<br />
</td>
        <td><br />
</td>
        <td>119.4<br />
</td>
        <td><br />
</td>
        <td>111.7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 72edo:<br />
</td>
        <td><br />
</td>
        <td>12<br />
</td>
        <td><br />
</td>
        <td>11<br />
</td>
        <td><br />
</td>
        <td>10<br />
</td>
        <td><br />
</td>
        <td>9<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>8<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
        <td>7<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td><br />
</td>
        <td>200.0<br />
</td>
        <td><br />
</td>
        <td>183.3<br />
</td>
        <td><br />
</td>
        <td>166.7<br />
</td>
        <td><br />
</td>
        <td>150.0<br />
</td>
        <td><br />
</td>
        <td>133.3<br />
</td>
        <td><br />
</td>
        <td>133.3<br />
</td>
        <td><br />
</td>
        <td>116.7<br />
</td>
        <td><br />
</td>
        <td>116.7<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:4 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <td>degrees<br />
</td>
        <td>cents value<br />
</td>
        <td>approximate ratios (11-limit)<br />
</td>
        <td colspan="2" style="text-align: center;"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups and downs </a><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a><br />
</td>
    </tr>
    <tr>
        <td>0<br />
</td>
        <td>0<br />
</td>
        <td>1/1<br />
</td>
        <td style="text-align: center;">P1<br />
</td>
        <td style="text-align: center;">D<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>16.667<br />
</td>
        <td>81/80<br />
</td>
        <td style="text-align: center;">^1 (up unison)<br />
</td>
        <td style="text-align: center;">D^<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>33.333<br />
</td>
        <td>45/44<br />
</td>
        <td style="text-align: center;">^^1(double-up unison)<br />
</td>
        <td style="text-align: center;">D^^<br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>50<br />
</td>
        <td>33/32<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2<br />
</td>
        <td>D^<span style="font-size: 90%; vertical-align: super;">3</span>, Ebv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>66.667<br />
</td>
        <td>25/24<br />
</td>
        <td style="text-align: center;">vvm2<br />
</td>
        <td style="text-align: center;">Ebvv<br />
</td>
    </tr>
    <tr>
        <td>5<br />
</td>
        <td>83.333<br />
</td>
        <td>21/20<br />
</td>
        <td style="text-align: center;">vm2 (downminor 2nd)<br />
</td>
        <td style="text-align: center;">Ebv<br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>100<br />
</td>
        <td>35/33<br />
</td>
        <td style="text-align: center;">m2<br />
</td>
        <td style="text-align: center;">Eb<br />
</td>
    </tr>
    <tr>
        <td>7<br />
</td>
        <td>116.667<br />
</td>
        <td>15/14<br />
</td>
        <td style="text-align: center;">^m2 (upminor 2nd)<br />
</td>
        <td style="text-align: center;">Eb^<br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>133.333<br />
</td>
        <td>27/25<br />
</td>
        <td style="text-align: center;">v~2<br />
</td>
        <td style="text-align: center;">Eb^^<br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>150<br />
</td>
        <td>12/11<br />
</td>
        <td style="text-align: center;">~2 (mid 2nd)<br />
</td>
        <td style="text-align: center;">Ev<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>10<br />
</td>
        <td>166.667<br />
</td>
        <td>11/10<br />
</td>
        <td style="text-align: center;">^~2 (upmid 2nd)<br />
</td>
        <td style="text-align: center;">Evv<br />
</td>
    </tr>
    <tr>
        <td>11<br />
</td>
        <td>183.333<br />
</td>
        <td>10/9<br />
</td>
        <td style="text-align: center;">vM2<br />
</td>
        <td style="text-align: center;">Ev<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>200<br />
</td>
        <td>9/8<br />
</td>
        <td style="text-align: center;">M2<br />
</td>
        <td style="text-align: center;">E<br />
</td>
    </tr>
    <tr>
        <td>13<br />
</td>
        <td>216.667<br />
</td>
        <td>25/22<br />
</td>
        <td style="text-align: center;">^M2<br />
</td>
        <td style="text-align: center;">E^<br />
</td>
    </tr>
    <tr>
        <td>14<br />
</td>
        <td>233.333<br />
</td>
        <td>8/7<br />
</td>
        <td style="text-align: center;">^^M2<br />
</td>
        <td style="text-align: center;">E^^<br />
</td>
    </tr>
    <tr>
        <td>15<br />
</td>
        <td>250<br />
</td>
        <td>81/70<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3<br />
</td>
        <td style="text-align: center;">E^<span style="font-size: 90%; vertical-align: super;">3</span>, Fv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>16<br />
</td>
        <td>266.667<br />
</td>
        <td>7/6<br />
</td>
        <td style="text-align: center;">vvm3<br />
</td>
        <td style="text-align: center;">Fvv<br />
</td>
    </tr>
    <tr>
        <td>17<br />
</td>
        <td>283.333<br />
</td>
        <td>33/28<br />
</td>
        <td style="text-align: center;">vm3<br />
</td>
        <td style="text-align: center;">Fv<br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>300<br />
</td>
        <td>25/21<br />
</td>
        <td style="text-align: center;">m3<br />
</td>
        <td style="text-align: center;">F<br />
</td>
    </tr>
    <tr>
        <td>19<br />
</td>
        <td>316.667<br />
</td>
        <td>6/5<br />
</td>
        <td style="text-align: center;">^m3<br />
</td>
        <td style="text-align: center;">F^<br />
</td>
    </tr>
    <tr>
        <td>20<br />
</td>
        <td>333.333<br />
</td>
        <td>40/33<br />
</td>
        <td style="text-align: center;">v~3<br />
</td>
        <td style="text-align: center;">F^^<br />
</td>
    </tr>
    <tr>
        <td>21<br />
</td>
        <td>350<br />
</td>
        <td>11/9<br />
</td>
        <td style="text-align: center;">~3<br />
</td>
        <td style="text-align: center;">F^<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>22<br />
</td>
        <td>366.667<br />
</td>
        <td>99/80<br />
</td>
        <td style="text-align: center;">^~3<br />
</td>
        <td style="text-align: center;">F#vv<br />
</td>
    </tr>
    <tr>
        <td>23<br />
</td>
        <td>383.333<br />
</td>
        <td>5/4<br />
</td>
        <td style="text-align: center;">vM3<br />
</td>
        <td style="text-align: center;">F#v<br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>400<br />
</td>
        <td>44/35<br />
</td>
        <td style="text-align: center;">M3<br />
</td>
        <td style="text-align: center;">F#<br />
</td>
    </tr>
    <tr>
        <td>25<br />
</td>
        <td>416.667<br />
</td>
        <td>14/11<br />
</td>
        <td style="text-align: center;">^M3<br />
</td>
        <td style="text-align: center;">F#^<br />
</td>
    </tr>
    <tr>
        <td>26<br />
</td>
        <td>433.333<br />
</td>
        <td>9/7<br />
</td>
        <td style="text-align: center;">^^M3<br />
</td>
        <td style="text-align: center;">F#^^<br />
</td>
    </tr>
    <tr>
        <td>27<br />
</td>
        <td>450<br />
</td>
        <td>35/27<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4<br />
</td>
        <td style="text-align: center;">F#^<span style="font-size: 90%; vertical-align: super;">3</span>, Gv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>28<br />
</td>
        <td>466.667<br />
</td>
        <td>21/16<br />
</td>
        <td style="text-align: center;">vv4<br />
</td>
        <td style="text-align: center;">Gvv<br />
</td>
    </tr>
    <tr>
        <td>29<br />
</td>
        <td>483.333<br />
</td>
        <td>33/25<br />
</td>
        <td style="text-align: center;">v4<br />
</td>
        <td style="text-align: center;">Gv<br />
</td>
    </tr>
    <tr>
        <td>30<br />
</td>
        <td>500<br />
</td>
        <td>4/3<br />
</td>
        <td style="text-align: center;">P4<br />
</td>
        <td style="text-align: center;">G<br />
</td>
    </tr>
    <tr>
        <td>31<br />
</td>
        <td>516.667<br />
</td>
        <td>27/20<br />
</td>
        <td style="text-align: center;">^4<br />
</td>
        <td style="text-align: center;">G^<br />
</td>
    </tr>
    <tr>
        <td>32<br />
</td>
        <td>533.333<br />
</td>
        <td>15/11<br />
</td>
        <td style="text-align: center;">^^4<br />
</td>
        <td style="text-align: center;">G^^<br />
</td>
    </tr>
    <tr>
        <td>33<br />
</td>
        <td>550<br />
</td>
        <td>11/8<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>4<br />
</td>
        <td style="text-align: center;">G^<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>34<br />
</td>
        <td>566.667<br />
</td>
        <td>25/18<br />
</td>
        <td style="text-align: center;">vvA4<br />
</td>
        <td style="text-align: center;">G#vv<br />
</td>
    </tr>
    <tr>
        <td>35<br />
</td>
        <td>583.333<br />
</td>
        <td>7/5<br />
</td>
        <td style="text-align: center;">vA4, vd5<br />
</td>
        <td style="text-align: center;">G#v, Abv<br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>600<br />
</td>
        <td>99/70<br />
</td>
        <td style="text-align: center;">A4, d5<br />
</td>
        <td style="text-align: center;">G#, Ab<br />
</td>
    </tr>
    <tr>
        <td>37<br />
</td>
        <td>616.667<br />
</td>
        <td>10/7<br />
</td>
        <td style="text-align: center;">^A4, ^d5<br />
</td>
        <td style="text-align: center;">G#^, Ab^<br />
</td>
    </tr>
    <tr>
        <td>38<br />
</td>
        <td>633.333<br />
</td>
        <td>36/25<br />
</td>
        <td style="text-align: center;">^^d5<br />
</td>
        <td style="text-align: center;">Ab^^<br />
</td>
    </tr>
    <tr>
        <td>39<br />
</td>
        <td>650<br />
</td>
        <td>16/11<br />
</td>
        <td style="text-align: center;">v<span style="font-size: 90%; vertical-align: super;">3</span>5<br />
</td>
        <td style="text-align: center;">Av<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>40<br />
</td>
        <td>666.667<br />
</td>
        <td>22/15<br />
</td>
        <td style="text-align: center;">vv5<br />
</td>
        <td style="text-align: center;">Avv<br />
</td>
    </tr>
    <tr>
        <td>41<br />
</td>
        <td>683.333<br />
</td>
        <td>40/27<br />
</td>
        <td style="text-align: center;">v5<br />
</td>
        <td style="text-align: center;">Av<br />
</td>
    </tr>
    <tr>
        <td>42<br />
</td>
        <td>700<br />
</td>
        <td>3/2<br />
</td>
        <td style="text-align: center;">P5<br />
</td>
        <td style="text-align: center;">A<br />
</td>
    </tr>
    <tr>
        <td>43<br />
</td>
        <td>716.667<br />
</td>
        <td>50/33<br />
</td>
        <td style="text-align: center;">^5<br />
</td>
        <td style="text-align: center;">A^<br />
</td>
    </tr>
    <tr>
        <td>44<br />
</td>
        <td>733.333<br />
</td>
        <td>32/21<br />
</td>
        <td style="text-align: center;">^^5<br />
</td>
        <td style="text-align: center;">A^^<br />
</td>
    </tr>
    <tr>
        <td>45<br />
</td>
        <td>750<br />
</td>
        <td>54/35<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6<br />
</td>
        <td style="text-align: center;">A^<span style="font-size: 90%; vertical-align: super;">3</span>, Bbv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>46<br />
</td>
        <td>766.667<br />
</td>
        <td>14/9<br />
</td>
        <td style="text-align: center;">vvm6<br />
</td>
        <td style="text-align: center;">Bbvv<br />
</td>
    </tr>
    <tr>
        <td>47<br />
</td>
        <td>783.333<br />
</td>
        <td>11/7<br />
</td>
        <td style="text-align: center;">vm6<br />
</td>
        <td style="text-align: center;">Bbv<br />
</td>
    </tr>
    <tr>
        <td>48<br />
</td>
        <td>800<br />
</td>
        <td>35/22<br />
</td>
        <td style="text-align: center;">m6<br />
</td>
        <td style="text-align: center;">Bb<br />
</td>
    </tr>
    <tr>
        <td>49<br />
</td>
        <td>816.667<br />
</td>
        <td>8/5<br />
</td>
        <td style="text-align: center;">^m6<br />
</td>
        <td style="text-align: center;">Bb^<br />
</td>
    </tr>
    <tr>
        <td>50<br />
</td>
        <td>833.333<br />
</td>
        <td>81/50<br />
</td>
        <td style="text-align: center;">v~6<br />
</td>
        <td style="text-align: center;">Bb^^<br />
</td>
    </tr>
    <tr>
        <td>51<br />
</td>
        <td>850<br />
</td>
        <td>18/11<br />
</td>
        <td style="text-align: center;">~6<br />
</td>
        <td style="text-align: center;">Bv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>52<br />
</td>
        <td>866.667<br />
</td>
        <td>33/20<br />
</td>
        <td style="text-align: center;">^~6<br />
</td>
        <td style="text-align: center;">Bvv<br />
</td>
    </tr>
    <tr>
        <td>53<br />
</td>
        <td>883.333<br />
</td>
        <td>5/3<br />
</td>
        <td style="text-align: center;">vM6<br />
</td>
        <td style="text-align: center;">Bv<br />
</td>
    </tr>
    <tr>
        <td>54<br />
</td>
        <td>900<br />
</td>
        <td>27/16<br />
</td>
        <td style="text-align: center;">M6<br />
</td>
        <td style="text-align: center;">B<br />
</td>
    </tr>
    <tr>
        <td>55<br />
</td>
        <td>916.667<br />
</td>
        <td>56/33<br />
</td>
        <td style="text-align: center;">^M6<br />
</td>
        <td style="text-align: center;">B^<br />
</td>
    </tr>
    <tr>
        <td>56<br />
</td>
        <td>933.333<br />
</td>
        <td>12/7<br />
</td>
        <td style="text-align: center;">^^M6<br />
</td>
        <td style="text-align: center;">B^^<br />
</td>
    </tr>
    <tr>
        <td>57<br />
</td>
        <td>950<br />
</td>
        <td>121/70<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7<br />
</td>
        <td style="text-align: center;">B^<span style="font-size: 90%; vertical-align: super;">3</span>, Cv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>58<br />
</td>
        <td>966.667<br />
</td>
        <td>7/4<br />
</td>
        <td style="text-align: center;">vvm7<br />
</td>
        <td style="text-align: center;">Cvv<br />
</td>
    </tr>
    <tr>
        <td>59<br />
</td>
        <td>983.333<br />
</td>
        <td>44/25<br />
</td>
        <td style="text-align: center;">vm7<br />
</td>
        <td style="text-align: center;">Cv<br />
</td>
    </tr>
    <tr>
        <td>60<br />
</td>
        <td>1000<br />
</td>
        <td>16/9<br />
</td>
        <td style="text-align: center;">m7<br />
</td>
        <td style="text-align: center;">C<br />
</td>
    </tr>
    <tr>
        <td>61<br />
</td>
        <td>1016.667<br />
</td>
        <td>9/5<br />
</td>
        <td style="text-align: center;">^m7<br />
</td>
        <td style="text-align: center;">C^<br />
</td>
    </tr>
    <tr>
        <td>62<br />
</td>
        <td>1033.333<br />
</td>
        <td>20/11<br />
</td>
        <td style="text-align: center;">v~7<br />
</td>
        <td style="text-align: center;">C^^<br />
</td>
    </tr>
    <tr>
        <td>63<br />
</td>
        <td>1050<br />
</td>
        <td>11/6<br />
</td>
        <td style="text-align: center;">~7<br />
</td>
        <td style="text-align: center;">C^<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>64<br />
</td>
        <td>1066.667<br />
</td>
        <td>50/27<br />
</td>
        <td style="text-align: center;">^~7<br />
</td>
        <td style="text-align: center;">C#vv<br />
</td>
    </tr>
    <tr>
        <td>65<br />
</td>
        <td>1083.333<br />
</td>
        <td>15/8<br />
</td>
        <td style="text-align: center;">vM7<br />
</td>
        <td style="text-align: center;">C#v<br />
</td>
    </tr>
    <tr>
        <td>66<br />
</td>
        <td>1100<br />
</td>
        <td>66/35<br />
</td>
        <td style="text-align: center;">M7<br />
</td>
        <td style="text-align: center;">C#<br />
</td>
    </tr>
    <tr>
        <td>67<br />
</td>
        <td>1116.667<br />
</td>
        <td>21/11<br />
</td>
        <td style="text-align: center;">^M7<br />
</td>
        <td style="text-align: center;">C#^<br />
</td>
    </tr>
    <tr>
        <td>68<br />
</td>
        <td>1133.333<br />
</td>
        <td>27/14<br />
</td>
        <td style="text-align: center;">^^M7<br />
</td>
        <td style="text-align: center;">C#^^<br />
</td>
    </tr>
    <tr>
        <td>69<br />
</td>
        <td>1150<br />
</td>
        <td>35/18<br />
</td>
        <td style="text-align: center;">^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8<br />
</td>
        <td style="text-align: center;">C#^<span style="font-size: 90%; vertical-align: super;">3</span>, Dv<span style="font-size: 90%; vertical-align: super;">3</span><br />
</td>
    </tr>
    <tr>
        <td>70<br />
</td>
        <td>1166.667<br />
</td>
        <td>49/25<br />
</td>
        <td style="text-align: center;">vv8<br />
</td>
        <td style="text-align: center;">Dvv<br />
</td>
    </tr>
    <tr>
        <td>71<br />
</td>
        <td>1183.333<br />
</td>
        <td>99/50<br />
</td>
        <td style="text-align: center;">v8<br />
</td>
        <td style="text-align: center;">Dv<br />
</td>
    </tr>
    <tr>
        <td>72<br />
</td>
        <td>1200<br />
</td>
        <td>2/1<br />
</td>
        <td style="text-align: center;">P8<br />
</td>
        <td style="text-align: center;">D<br />
</td>
    </tr>
</table>

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="Linear temperaments"></a><!-- ws:end:WikiTextHeadingRule:6 -->Linear temperaments</h1>
 

<table class="wiki_table">
    <tr>
        <th>Periods per octave<br />
</th>
        <th>Generator<br />
</th>
        <th>Names<br />
</th>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quincy">quincy</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>5\72<br />
</td>
        <td><a class="wiki_link" href="/marvolo">marvolo</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>7\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle">miracle</a>/benediction/manna<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>11\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>13\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>17\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/neominor">neominor</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>19\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/catakleismic">catakleismic</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>23\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>25\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/sqrtphi">sqrtphi</a><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>29\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>31\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/marvo">marvo</a>/zarvo<br />
</td>
    </tr>
    <tr>
        <td>1<br />
</td>
        <td>35\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/cotritone">cotritone</a><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>1\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>5\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/harry">harry</a><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>7\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>11\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/unidec">unidec</a>/hendec<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>13\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/wizard">wizard</a>/lizard/gizzard<br />
</td>
    </tr>
    <tr>
        <td>2<br />
</td>
        <td>17\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>1\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>5\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/tritikleismic">tritikleismic</a><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>7\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>3<br />
</td>
        <td>11\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/mirkat">mirkat</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/quadritikleismic">quadritikleismic</a><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>5\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>4<br />
</td>
        <td>7\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>1\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>6<br />
</td>
        <td>5\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octoid">octoid</a><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>2\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/octowerck">octowerck</a><br />
</td>
    </tr>
    <tr>
        <td>8<br />
</td>
        <td>4\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>1\72<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>9<br />
</td>
        <td>3\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/ennealimmal">ennealimmal</a>/ennealimmic<br />
</td>
    </tr>
    <tr>
        <td>12<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/compton">compton</a><br />
</td>
    </tr>
    <tr>
        <td>18<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hemiennealimmal">hemiennealimmal</a><br />
</td>
    </tr>
    <tr>
        <td>24<br />
</td>
        <td>1\72<br />
</td>
        <td><a class="wiki_link" href="http://xenharmonic.wikispaces.com/hours">hours</a><br />
</td>
    </tr>
    <tr>
        <td>36<br />
</td>
        <td>1\72<br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Z function"></a><!-- ws:end:WikiTextHeadingRule:8 -->Z function</h1>
 72edo is the ninth <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a>, as well as being a peak and gap edo, and the maximum value of the <a class="wiki_link" href="http://xenharmonic.wikispaces.com/The%20Riemann%20Zeta%20Function%20and%20Tuning#The%20Z%20function">Z function</a> 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.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:1571:&lt;img src=&quot;http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="http://xenharmonic.wikispaces.com/file/view/plot72.png/219772696/plot72.png" alt="plot72.png" title="plot72.png" /><!-- ws:end:WikiTextLocalImageRule:1571 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h1&gt; --><h1 id="toc5"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:10 -->Music</h1>
 <a class="wiki_link_ext" href="http://www.archive.org/details/Kotekant" rel="nofollow">Kotekant</a> <em><a class="wiki_link_ext" href="http://www.archive.org/download/Kotekant/kotekant.mp3" rel="nofollow">play</a></em> by <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Gene%20Ward%20Smith">Gene Ward Smith</a><br />
<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-72-edo.mp3" rel="nofollow">Twinkle canon – 72 edo</a></em> by <a class="wiki_link_ext" href="http://soonlabel.com/xenharmonic/archives/573" rel="nofollow">Claudi Meneghin</a><br />
<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3" rel="nofollow">Lazy Sunday</a></em> by <a class="wiki_link" href="/Jake%20Freivald">Jake Freivald</a> in the <a class="wiki_link" href="/lazysunday">lazysunday</a> scale.<br />
<em><a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3" rel="nofollow">June Gloom #9</a></em> by Prent Rodgers<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Scales</h1>
 <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72a">smithgw72a</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72b">smithgw72b</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72c">smithgw72c</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72d">smithgw72d</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72e">smithgw72e</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72f">smithgw72f</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72g">smithgw72g</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72h">smithgw72h</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72i">smithgw72i</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/smithgw72j">smithgw72j</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/blackjack">blackjack</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_8">miracle_8</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_10">miracle_10</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_12">miracle_12</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_12a">miracle_12a</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_24hi">miracle_24hi</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/miracle_24lo">miracle_24lo</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/keenanmarvel">keenanmarvel</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_chrome">xenakis_chrome</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_diat">xenakis_diat</a>, <a class="wiki_link" href="http://xenharmonic.wikispaces.com/xenakis_schrome">xenakis_schrome</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/genus24255et72">Euler(24255) genus in 72 equal</a><br />
<a class="wiki_link" href="/JuneGloom">JuneGloom</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:&lt;h1&gt; --><h1 id="toc7"><a name="External links"></a><!-- ws:end:WikiTextHeadingRule:14 -->External links</h1>
 <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/72_tone_equal_temperament" rel="nofollow">Wikipedia article on 72edo</a></li><li><a class="wiki_link_ext" href="http://orthodoxwiki.org/Byzantine_Chant" rel="nofollow">OrthodoxWiki Article on Byzantine chant, which uses 72edo</a></li><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Joe_Maneri" rel="nofollow">Wikipedia article on Joe Maneri (1927-2009)</a></li><li><a class="wiki_link_ext" href="http://www.ekmelic-music.org/en/" rel="nofollow">Ekmelic Music Society/Gesellschaft für Ekmelische Musik</a>, a group of composers and researchers dedicated to 72edo music</li><li><a class="wiki_link_ext" href="http://72note.com/site/original.html" rel="nofollow">Rick Tagawa's 72edo site</a>, including theory and composers' list</li><li><a class="wiki_link_ext" href="http://www.myspace.com/dawier" rel="nofollow" target="_blank">Danny Wier, composer and musician who specializes in 72-edo</a></li></ul></body></html>

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