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== Notations ==
== Notations ==
 
=== Sagittal ===
From the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], a diagram of how to notate 72-EDO in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 72-EDO in the Revo flavor of Sagittal:


[[File:72edo Sagittal.png|800px]]
[[File:72edo Sagittal.png|800px]]


== Just approximation ==
== JI approximation ==
=== Z function ===
72edo is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|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.


=== Temperament measures ===
[[File:plot72.png|alt=plot72.png|plot72.png]]
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 72et.
 
{| class="wikitable center-all"
== Regular temperament properties ==
! colspan="2" |
{| class="wikitable center-4 center-5 center-6"
! 3-limit
! rowspan="2" | Subgroup
! 5-limit
! rowspan="2" | [[Comma list]]
! 7-limit
! rowspan="2" | [[Mapping]]
! 11-limit
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 13-limit
! colspan="2" | Tuning error
! 17-limit
|-
! 19-limit
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3.5
| +0.617
| 15625/15552, 531441/524288
| [{{val| 72 114 167 }}]
| +0.839
| +0.839
| 0.594
| 3.56
|-
| 2.3.5.7
| 225/224, 1029/1024, 4375/4374
| [{{val| 72 114 167 202 }}]
| +0.822
| +0.822
| 0.515
| 3.09
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 4375/4356
| [{{val| 72 114 167 202 249 }}]
| +0.734
| +0.734
| 0.493
| 2.96
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 243/242, 325/324, 385/384
| [{{val| 72 114 167 202 266 249 }}]
| +0.936
| +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
| [{{val| 72 114 167 202 249 266 294 }}]
| +0.975
| +0.975
| +0.780
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 0.594
| 0.515
| 0.493
| 0.638
| 0.599
| 0.599
| 0.762
|-
! [[TE simple badness|relative]] (%)
| 3.70
| 3.56
| 3.09
| 2.96
| 3.82
| 3.59
| 3.59
| 4.57
|}
|}
* 72et has a lower relative error than any previous ETs in the 7-, 11-, 13-, 17-, and 19-limit. The next ET that does better in these subgroups is 99, 270, 224, 494, and 217, respectively.
=== Z function ===
72edo is the ninth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|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]]
72et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next ETs better in these subgroups are 99, 270, 224, 494, and 217, respectively.  


== Commas ==
=== Commas ===


Commas tempered out by 72edo include…
Commas tempered out by 72edo include…
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<references/>
<references/>


== Temperaments ==
=== Rank-2 temperaments ===
 
* [[List of edo-distinct 72et rank two temperaments]]
* [[List of edo-distinct 72et rank two temperaments]]


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== Music ==
== Music ==
[http://www.archive.org/details/Kotekant Kotekant] ''[http://www.archive.org/download/Kotekant/kotekant.mp3 play]'' by [[Gene_Ward_Smith|Gene Ward Smith]]
* [http://www.archive.org/details/Kotekant ''Kotekant''] [http://www.archive.org/download/Kotekant/kotekant.mp3 play] by [[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/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]]
''[http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Lazy%20Sunday.mp3 Lazy Sunday]'' by [[Jake_Freivald|Jake Freivald]] in the [[lazysunday|lazysunday]] scale.
 
''[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/drum12a-c-t9.mp3 June Gloom #9]'' by Prent Rodgers


== External links ==
== Links ==
* [[Wikipedia:72_equal_temperament|72 equal temperament - Wikipedia]]
* [[Wikipedia: 72 equal temperament]]
* [[Wikipedia: Joe Maneri]] – Wikipedia article on Joe Maneri (1927-2009)
* [http://orthodoxwiki.org/Byzantine_Chant OrthodoxWiki Article on Byzantine chant, which uses 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://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://72note.com/site/original.html Rick Tagawa's 72edo site], including theory and composers' list
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