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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]].
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]].
=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|72|columns=11}}
{{Harmonics in equal|72|columns=11}}
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! Cents
! Cents
! Approximate Ratios *
! Approximate Ratios *
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation|Ups and Downs Notation]]
|-
|-
| 0
| 0
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<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament
<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
=== Interval quality and chord names in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
! [[Kite's color notation|color]]
! [[Color notation|Color]]
! monzo format
! Monzo Format
! examples
! Examples
|-
|-
| dudminor
| dudminor
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|-
|-
| rowspan="2" |downmid
| rowspan="2" |downmid
|luyo
| luyo
|(a b 1 0 -1)
| (a b 1 0 -1)
|15/11
| 15/11
|-
|-
|tho
| tho
|(a b 0 0 0 1)
| (a b 0 0 0 1)
|13/8, 13/9
| 13/8, 13/9
|-
|-
| rowspan="2" | mid
| rowspan="2" | mid
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|-
|-
| rowspan="2" |upmid
| rowspan="2" |upmid
|logu
| logu
|(a b -1 0 1)
| (a b -1 0 1)
|11/10
| 11/10
|-
|-
|thu
| thu
|(a b 0 0 0 -1)
| (a b 0 0 0 -1)
|16/13, 18/13
| 16/13, 18/13
|-
|-
| downmajor
| downmajor
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| 9/7, 12/7
| 9/7, 12/7
|}
|}
All 72-edo 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:
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"
{| class="wikitable center-all"
|-
|-
! [[Kite's color notation|color of the 3rd]]
! [[Kite's color notation|Color of the 3rd]]
! JI chord
! JI Chord
! notes as edosteps
! Notes as Edosteps
! notes of C chord
! Notes of C Chord
! written name
! Written Name
! spoken name
! Spoken Name
|-
|-
| zo
| zo
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|}
|}
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].  
For a more complete list, see [[Ups and Downs Notation #Chord names in other EDOs]].  
=== Remembering the pitch structure ===
The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; ''id est'', every 6 degrees is the 3-limit.
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
* -1 degree corrects 5/4 (80/81)
* -2 degrees corrects 7/4 (63/64)
* +3 degrees corrects 11/8 (33/32)
* +2 degrees corrects 13/8 (1053/1024)
* 0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)


== Notations ==
== Notations ==
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|}
|}


72et is lower in relative error than any previous equal temperaments in the 7-, 11-, 13-, 17-, and 19-limit. The next ETs doing better in these subgroups are 99, 270, 224, 494, and 217, respectively.  
* 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 ===
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| {{Monzo| -5 2 2 -1 }}
| {{Monzo| -5 2 2 -1 }}
| 7.71
| 7.71
| Septimal kleisma, Marvel comma
| Marvel comma
|-
|-
| 7
| 7
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| {{Monzo| -3 4 -2 -2 2 }}
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| 0.18
| Kalisma, Gauss' comma
| Kalisma
|-
|-
| 11
| 11
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| {{Monzo| -3 -1 0 -1 0 2 }}
| {{Monzo| -3 -1 0 -1 0 2 }}
| 10.27
| 10.27
| Buzurgisma, Dhanvantarisma
| Buzurgisma
|-
|-
| 13
| 13
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| {{Monzo| 2 -3 -2 0 0 2 }}
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| 2.56
| Island comma, Parizeksma
| Island comma
|-
|-
| 13
| 13
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| 2
| 2
| 11\72
| 11\72
| [[Unidec]]/hendec
| [[Unidec]] / hendec
|-
|-
| 2
| 2
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== Scales ==
== Scales ==
* [[smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[Smithgw72a]], [[smithgw72b]], [[smithgw72c]], [[smithgw72d]], [[smithgw72e]], [[smithgw72f]], [[smithgw72g]], [[smithgw72h]], [[smithgw72i]], [[smithgw72j]]
* [[blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[Blackjack]], [[miracle_8]], [[miracle_10]], [[miracle_12]], [[miracle_12a]], [[miracle_24hi]], [[miracle_24lo]]
* [[keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[Keenanmarvel]], [[xenakis_chrome]], [[xenakis_diat]], [[xenakis_schrome]]
* [[genus24255et72|Euler(24255) genus in 72 equal]]
* [[Genus24255et72|Euler(24255) genus in 72 equal]]
* [[JuneGloom]]
* [[JuneGloom]]
* [[Aurora scale|Aurora]], [[Blackened skies]], [[Lost spirit]]
* [[Aurora scale|Aurora]], [[Blackened skies]], [[Lost spirit]]
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|  
|  
|}
|}
== Remembering the pitch structure ==
The pitch structure is very easy to remember. In 72tet, 12edo is the Pythagorian ring; ''id est'', every 6 degrees is the 3-limit.
Then, after each subsequent degree in reverse, a new prime limit is unveiled from it:
-1 degree corrects 5/4 (80/81)
-2 degrees corrects 7/4 (63/64)
+3 degrees corrects 11/8 (33/32)
+2 degrees corrects 13/8 (1053/1024)
0 degree corrects 17/16 and 19/16 (4131/4096 and 513/512)


== Music ==
== Music ==
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; [[Xeno Ov Eleas]]
; [[Xeno Ov Eleas]]
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)
* [https://www.youtube.com/watch?v=cx7I0NWem5w ''Χenomorphic Ghost Storm''] (2022)
== See also ==
* [[Joe Maneri]]


== External links ==
== External links ==
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[[Category:Listen]]
[[Category:Listen]]
[[Category:Moria]]
[[Category:Marvel]]
[[Category:Marvel]]
[[Category:Miracle]]
[[Category:Miracle]]
[[Category:Moria]]
[[Category:Prodigy]]
[[Category:Prodigy]]
[[Category:Wizard]]
[[Category:Wizard]]
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