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80edo: Difference between revisions

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<nowiki />* {{sg|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.
<nowiki/>* {{sg|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.


== Approximation to JI ==
== Approximation to JI ==
=== 23-odd-limit interval mappings ===
{{Q-odd-limit intervals|80|23}}
=== Consistent circles ===
=== Consistent circles ===
80edo is home to a staggering amount of [[consistent circle]]s, both ones closing after generating all 80 notes and ones closing after generating a subset edo like 2, 4, 5, 8, 10, 16 or 20.
80edo is home to a staggering amount of [[consistent circle]]s, both ones closing after generating all 80 notes and ones closing after generating a subset edo like 2, 4, 5, 8, 10, 16 or 20.
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|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing<br />Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle #Definitions|Consistency]]
! 1\1
! 1\1
! 1\2
! 1\2
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 20-note circles by gen. – '''all are related to [[Degrees]]'''
|+ style="font-size: 105%;" | 20-note circles by gen. – all are related to [[degrees]]
|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing<br />Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle #Definitions|Consistency]]
! Associated<br />Edostep
! Associated<br />Edostep
|-
|-
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|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing<br />Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle #Definitions|Consistency]]
! Associated<br />Edostep
! Associated<br>edostep
! [[Regular temperament|Temperaments]]
! [[Regular temperament|Temperaments]]
|-
|-
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|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing<br />Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle #Definitions|Consistency]]
! Associated<br />Edostep
! Associated<br>edostep
! [[Regular temperament|Temperaments]]
! [[Regular temperament|Temperaments]]
|-
|-
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|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing<br />Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle#Definitions|Consistency]]
! Associated<br />Edostep
! Associated<br>edostep
! [[Regular temperament|Temperaments]]
! [[Regular temperament|Temperaments]]
|-
|-
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|-
|-
! [[Interval]]
! [[Interval]]
! [[Closing error|Closing Error]]
! [[Closing error|Closing<br>error]]
! [[Circle#Definitions|Consistency]]
! [[Circle #Definitions|Consistency]]
! Associated<br />Edostep
! Associated<br>edostep
! [[Regular temperament|Temperaments]]
! [[Regular temperament|Temperaments]]
|-
|-
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|}
|}


=== 23-odd-limit interval mappings ===
=== Zeta peak index ===
{{Q-odd-limit intervals|80|23}}
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[435zpi]]
| 80.0731374302484
| 14.9862992572924
| 7.873146
| 1.247325
| 17.087322
| 80edo
| 1198.90394058339
| 12
| 12
|}


== Regular temperament properties ==
== Regular temperament properties ==
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80et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored. We might mention:
80et [[support]]s a profusion of 19-limit (and lower) rank-2 temperaments which have mostly not been explored. We might mention:


* {{nowrap|31 &amp; 80}} {{multival| 7 6 15 27 -24 -23 -20 … }}
* {{nowrap|31 & 80}} {{multival| 7 6 15 27 -24 -23 -20 … }}
* {{nowrap|72 &amp; 80}} {{multival| 24 30 40 24 32 24 0 … }}
* {{nowrap|72 & 80}} {{multival| 24 30 40 24 32 24 0 … }}
* {{nowrap|34 &amp; 80}} {{multival| 2 -4 -50 22 16 2 -40 … }}
* {{nowrap|34 & 80}} {{multival| 2 -4 -50 22 16 2 -40 … }}
* {{nowrap|46 &amp; 80}} {{multival| 2 -4 30 22 16 2 40 … }}
* {{nowrap|46 & 80}} {{multival| 2 -4 30 22 16 2 40 … }}
* {{nowrap|29 &amp; 80}} {{multival| 3 34 45 33 24 -37 20 … }}
* {{nowrap|29 & 80}} {{multival| 3 34 45 33 24 -37 20 … }}
* {{nowrap|12 &amp; 80}} {{multival| 4 -8 -20 -36 32 4 0 … }}
* {{nowrap|12 & 80}} {{multival| 4 -8 -20 -36 32 4 0 … }}
* {{nowrap|22 &amp; 80}} {{multival| 6 -10 12 -14 -32 6 -40 … }}
* {{nowrap|22 & 80}} {{multival| 6 -10 12 -14 -32 6 -40 … }}
* {{nowrap|58 &amp; 80}} {{multival| 6 -10 12 -14 -32 6 40 … }}
* {{nowrap|58 & 80}} {{multival| 6 -10 12 -14 -32 6 40 … }}
* {{nowrap|41 &amp; 80}} {{multival| 7 26 25 -3 -24 -33 20 … }}
* {{nowrap|41 & 80}} {{multival| 7 26 25 -3 -24 -33 20 … }}


In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.  
In each case, the numbers joined by an ampersand represent 19-limit [[patent val]]s (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.  
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
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|-
|-
| 2
| 2
| 21\80<br />(19\80)
| 21\80<br>(19\80)
| 315<br />(285)
| 315<br>(285)
| 6/5<br />(33/28)
| 6/5<br>(33/28)
| [[Semiparakleismic]]
| [[Semiparakleismic]]
|-
|-
| 2
| 2
| 29\80<br />(11\80)
| 29\80<br>(11\80)
| 435<br />(165)
| 435<br>(165)
| 9/7<br />(11/10)
| 9/7<br>(11/10)
| [[Echidna]]<br />[[Semisupermajor]]
| [[Echidna]]<br>[[Semisupermajor]]
|-
|-
| 2
| 2
| 33\80<br />(7\80)
| 33\80<br>(7\80)
| 495<br />(105)
| 495<br>(105)
| 4/3<br />(17/16)
| 4/3<br>(17/16)
| [[Srutal]]
| [[Srutal]]
|-
|-
| 4
| 4
| 33\80<br />(7\80)
| 33\80<br>(7\80)
| 495<br />(105)
| 495<br>(105)
| 4/3<br />(17/16)
| 4/3<br>(17/16)
| [[Bidia]]
| [[Bidia]]
|-
|-
| 5
| 5
| 15\80<br />(1\80)
| 15\80<br>(1\80)
| 225<br />(15)
| 225<br>(15)
| 8/7<br />(64/63)
| 8/7<br>(64/63)
| [[Pentorwell]]
| [[Pentorwell]]
|-
|-
| 5
| 5
| 37\80<br />(5\80)
| 37\80<br>(5\80)
| 555<br />(75)
| 555<br>(75)
| 11/8<br />(25/24)
| 11/8<br>(25/24)
| [[Trisedodge]] / [[countdown]]
| [[Trisedodge]] / [[countdown]]
|-
|-
| 8
| 8
| 39\80<br />(1\80)
| 39\80<br>(1\80)
| 585<br />(15)
| 585<br>(15)
| 7/5<br />(99/98~100/99)
| 7/5<br>(99/98~100/99)
| [[Octopus]]
| [[Octopus]]
|-
|-
| 10
| 10
| 21\80<br />(3\80)
| 21\80<br>(3\80)
| 315<br />(45)
| 315<br>(45)
| 6/5<br />(40/39)
| 6/5<br>(40/39)
| [[Deca]]
| [[Deca]]
|-
|-
| 20
| 20
| 33\80<br />(1\80)
| 33\80<br>(1\80)
| 495<br />(15)
| 495<br>(15)
| 4/3<br />(99/98~100/99)
| 4/3<br>(99/98~100/99)
| [[Degrees]]
| [[Degrees]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[435zpi]]
| 80.0731374302484
| 14.9862992572924
| 7.873146
| 1.247325
| 17.087322
| 80edo
| 1198.90394058339
| 12
| 12
|}


== Detemperaments ==
== Detemperaments ==
Retrieved from "https://en.xen.wiki/w/80edo"