80edo: Difference between revisions

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Though a strange tuning* in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of (2, 4, 5,) 8, 10, 16 and 20. Initially this doesn't seem very useful unless one is interested in learning those edos, but 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations w.r.t. (integer multiples of) their period, see [[#Consistent circles|the section on consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.

<nowiki>*~~</nowiki>~~ The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).

<nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).

==== Based on subsets ====

Line 204:

| …

|}

<nowiki>*~~</nowiki> based on treating 80edo as a~~ no-31's [[37-limit]] ~~temperament; other approaches are possible.~~ Inconsistent interpretations in italic.

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

== Approximation to JI ==

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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.

{| class="wikitable center-1 center-2 center-3"

|+80-note circles by gen. with related temperaments organized by period

|+ style="font-size: 105%;" | 80-note circles by gen. with related temperaments organized by period

|-

! [[Interval]]

! [[Closing error|Closing Error]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! 1\1

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{| class="wikitable center-all left-5"

|+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]]

! [[Closing error|Closing Error]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! Associated Edostep

! Associated Edostep

|-

| [[28/27]]

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{| class="wikitable center-all left-5"

|+10-note circles by gen. with related [[10th-octave temperaments]]

|+ style="font-size: 105%;" | 10-note circles by gen. with related [[10th-octave temperaments]]

|-

! [[Interval]]

! [[Closing error|Closing Error]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! Associated Edostep

! Associated Edostep

! [[Regular temperament|Temperaments]]

|-

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{| class="wikitable center-all left-5"

|+8-note circles by gen. with related [[8th-octave temperaments]]

|+ style="font-size: 105%;" | 8-note circles by gen. with related [[8th-octave temperaments]]

|-

! [[Interval]]

! [[Closing error|Closing Error]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! Associated Edostep

! Associated Edostep

! [[Regular temperament|Temperaments]]

|-

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{| class="wikitable center-all left-5"

|+5-note circles by gen. with related [[5th-octave temperaments]]

|+ style="font-size: 105%;" | 5-note circles by gen. with related [[5th-octave temperaments]]

|-

! [[Interval]]

! [[Closing error|Closing Error]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! Associated Edostep

! Associated Edostep

! [[Regular temperament|Temperaments]]

|-

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{| class="wikitable center-all left-5"

|+4-note circles with related [[4th-octave temperaments]]

|+ style="font-size: 105%;" | 4-note circles with related [[4th-octave temperaments]]

|-

! [[Interval]]

! [[Closing error|Closing Error]]

! [[Circle#Definitions|Consistency]]

! Associated Edostep

! Associated Edostep

! [[Regular temperament|Temperaments]]

|-

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== Regular temperament properties ==

{~~| class="wikitable center-4 center-5 center-6"~~

~~! rowspan="2" | [[Subgroup]]~~

~~! rowspan="2" | [[Comma list|Comma List]]~~

~~! rowspan="2" | [[Mapping]]~~

~~! rowspan="2" | Optimal 8ve Stretch (¢)~~

~~! colspan="2" | Tuning Error~~

|-

~~! [[TE error|Absolute]] (¢)~~

~~! [[TE simple badness|Relative]] (%)~~

|-

| 2.3

| {{monzo| 127 -80 }}

| {{mapping| 80 127 }}

| −0.961

| 0.960

| 6.40

Line 511:

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| 2048/2025, 390625000/387420489

| {{mapping| 80 127 186}}

| −1.169

| 0.837

| 5.59

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| 1728/1715, 2048/2025, 3136/3125

| {{mapping| 80 127 186 225 }}

| −1.426

| 0.851

| 5.68

Line 525:

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| 176/175, 540/539, 896/891, 1331/1323

| {{mapping| 80 127 186 225 277 }}

| −1.353

| 0.775

| 5.17

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| 169/168, 176/175, 325/324, 364/363, 540/539

| {{mapping| 80 127 186 225 277 296 }}

| −1.105

| 0.901

| 6.01

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| 136/135, 169/168, 176/175, 221/220, 364/363, 540/539

| {{mapping| 80 127 186 225 277 296 327 }}

| −0.949

| 0.917

| 6.12

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| 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399

| {{mapping| 80 127 186 225 277 296 327 340 }}

| −0.903

| 0.867

| 5.78

|}

=== Rank-2 temperaments ===

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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.

{~~| class="wikitable center-all left-5"~~

~~|+Rank~~-2 ~~temperaments by generator~~

~~! Periods per 8ve~~

! Generator*

! Cents*

! Associated Ratio*

~~! Temperament~~

|-

| 1

Line 617:

Line 604:

|-

| 2

| 21\80 (19\80)

| 21\80 (19\80)

| 315 (285)

| 315 (285)

| 6/5 (33/28)

| 6/5 (33/28)

| [[Semiparakleismic]]

|-

| 2

| 29\80 (11\80)

| 29\80 (11\80)

| 435 (165)

| 435 (165)

| 9/7 (11/10)

| 9/7 (11/10)

| [[Echidna]] [[Semisupermajor]]

| [[Echidna]] [[Semisupermajor]]

|-

| 2

| 33\80 (7\80)

| 33\80 (7\80)

| 495 (105)

| 495 (105)

| 4/3 (17/16)

| 4/3 (17/16)

| [[Srutal]]

|-

| 4

| 33\80 (7\80)

| 33\80 (7\80)

| 495 (105)

| 495 (105)

| 4/3 (17/16)

| 4/3 (17/16)

| [[Bidia]]

|-

| 5

| 15\80 (1\80)

| 15\80 (1\80)

| 225 (15)

| 225 (15)

| 8/7 (64/63)

| 8/7 (64/63)

| [[Pentorwell]]

|-

| 5

| 37\80 (5\80)

| 37\80 (5\80)

| 555 (75)

| 555 (75)

| 11/8 (25/24)

| 11/8 (25/24)

| [[Trisedodge]] / [[countdown]]

|-

| 8

| 39\80 (1\80)

| 39\80 (1\80)

| 585 (15)

| 585 (15)

| 7/5 (99/98~100/99)

| 7/5 (99/98~100/99)

| [[Octopus]]

|-

| 10

| 21\80 (3\80)

| 21\80 (3\80)

| 315 (45)

| 315 (45)

| 6/5 (40/39)

| 6/5 (40/39)

| [[Deca]]

|-

| 20

| 33\80 (1\80)

| 33\80 (1\80)

| 495 (15)

| 495 (15)

| 4/3 (99/98~100/99)

| 4/3 (99/98~100/99)

| [[Degrees]]

|}

~~<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct~~

== Detemperaments ==

@@ Line 11: / Line 11: @@
 Though a strange tuning* in lower prime limits, 80edo offers a very unique composite structure that can aid with familiarization/conceptualization by way of its subset edos of (2, 4, 5,) 8, 10, 16 and 20. Initially this doesn't seem very useful unless one is interested in learning those edos, but 80edo supports a plethora of multiperiod temperaments with accurate JI interval interpretations based on these edos; to see a fairly comprehensive list of these temperaments and of their most accurate JI interpretations w.r.t. (integer multiples of) their period, see [[#Consistent circles|the section on consistent circles]]. These represent a large number of practically completely unexplored and novel high-limit temperaments with varying musical potential.
-<nowiki>*</nowiki> The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).
+<nowiki />* The strangeness of its tuning can largely be explained by the addition of vals [[80edo]] = [[53edo]] + [[27edo]], where [[27edo]] exaggerates the idiosyncratic mapping of the 2.3.5.7.13 subgroup, as while 53edo tempers {[[625/624|S25]], [[676/675|S26]], [[729/728|S27]]} (supporting [[catakleismic]]) and {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]} (supporting [[buzzard]]), 27edo tempers {[[1728/1715|S6/S7]], [[64/63|S8]], [[325/324|S25*S26]], [[351/350|S26*S27]]} [[Square superparticular|implying]] {[[4375/4374|S25/S27]], [[169/168|S13]]} but maps S25~S27 positively and S26 negatively, which 80et thus inherits though with less damage. This is not insignificant, because this plays a special role (as we'll see in the next section on subsets).
 ==== Based on subsets ====
@@ Line 204: / Line 204: @@
 | …
 |}
-<nowiki>*</nowiki> based on treating 80edo as a no-31's [[37-limit]] temperament; other approaches are possible. Inconsistent interpretations in italic.
+<nowiki />* {{sg|no-31's [[37-limit]]}} Inconsistent interpretations in ''italic''.
 == Approximation to JI ==
@@ Line 210: / Line 210: @@
 edo 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.
 {| class="wikitable center-1 center-2 center-3"
-|+80-note circles by gen. with related temperaments organized by period
+|+ style="font-size: 105%;" | 80-note circles by gen. with related temperaments organized by period
 |-
 ! [[Interval]]
-! [[Closing error|Closing<br>Error]]
+! [[Closing error|Closing<br />Error]]
 ! [[Circle#Definitions|Consistency]]
 ! 1\1
@@ Line 274: / Line 274: @@
 {| class="wikitable center-all left-5"
-|+20-note circles by gen. — '''all are related to [[Degrees]]'''
+|+ style="font-size: 105%;" | 20-note circles by gen. &ndash; '''all are related to [[Degrees]]'''
+|-
 ! [[Interval]]
-! [[Closing error|Closing<br>Error]]
+! [[Closing error|Closing<br />Error]]
 ! [[Circle#Definitions|Consistency]]
-! Associated<br>Edostep
+! Associated<br />Edostep
 |-
 | [[28/27]]
@@ Line 337: / Line 338: @@
 {| class="wikitable center-all left-5"
-|+10-note circles by gen. with related [[10th-octave temperaments]]
+|+ style="font-size: 105%;" | 10-note circles by gen. with related [[10th-octave temperaments]]
 |-
 ! [[Interval]]
-! [[Closing error|Closing<br>Error]]
+! [[Closing error|Closing<br />Error]]
 ! [[Circle#Definitions|Consistency]]
-! Associated<br>Edostep
+! Associated<br />Edostep
 ! [[Regular temperament|Temperaments]]
 |-
@@ Line 371: / Line 372: @@
 {| class="wikitable center-all left-5"
-|+8-note circles by gen. with related [[8th-octave temperaments]]
+|+ style="font-size: 105%;" | 8-note circles by gen. with related [[8th-octave temperaments]]
 |-
 ! [[Interval]]
-! [[Closing error|Closing<br>Error]]
+! [[Closing error|Closing<br />Error]]
 ! [[Circle#Definitions|Consistency]]
-! Associated<br>Edostep
+! Associated<br />Edostep
 ! [[Regular temperament|Temperaments]]
 |-
@@ Line 423: / Line 424: @@
 {| class="wikitable center-all left-5"
-|+5-note circles by gen. with related [[5th-octave temperaments]]
+|+ style="font-size: 105%;" | 5-note circles by gen. with related [[5th-octave temperaments]]
 |-
 ! [[Interval]]
-! [[Closing error|Closing<br>Error]]
+! [[Closing error|Closing<br />Error]]
 ! [[Circle#Definitions|Consistency]]
-! Associated<br>Edostep
+! Associated<br />Edostep
 ! [[Regular temperament|Temperaments]]
 |-
@@ Line 463: / Line 464: @@
 {| class="wikitable center-all left-5"
-|+4-note circles with related [[4th-octave temperaments]]
+|+ style="font-size: 105%;" | 4-note circles with related [[4th-octave temperaments]]
 |-
 ! [[Interval]]
 ! [[Closing error|Closing Error]]
 ! [[Circle#Definitions|Consistency]]
-! Associated<br>Edostep
+! Associated<br />Edostep
 ! [[Regular temperament|Temperaments]]
 |-
@@ Line 491: / Line 492: @@
 == Regular temperament properties ==
-{| class="wikitable center-4 center-5 center-6"
+{{comma basis begin}}
-! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning Error
-|-
-! [[TE error|Absolute]] (¢)
-! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo| 127 -80 }}
 | {{mapping| 80 127 }}
-| −0.961
+| &minus;0.961
 | 0.960
 | 6.40
@@ Line 511: / Line 504: @@
 | 2048/2025, 390625000/387420489
 | {{mapping| 80 127 186}}
-| −1.169
+| &minus;1.169
 | 0.837
 | 5.59
@@ Line 518: / Line 511: @@
 | 1728/1715, 2048/2025, 3136/3125
 | {{mapping| 80 127 186 225 }}
-| −1.426
+| &minus;1.426
 | 0.851
 | 5.68
@@ Line 525: / Line 518: @@
 | 176/175, 540/539, 896/891, 1331/1323
 | {{mapping| 80 127 186 225 277 }}
-| −1.353
+| &minus;1.353
 | 0.775
 | 5.17
@@ Line 532: / Line 525: @@
 | 169/168, 176/175, 325/324, 364/363, 540/539
 | {{mapping| 80 127 186 225 277 296 }}
-| −1.105
+| &minus;1.105
 | 0.901
 | 6.01
@@ Line 539: / Line 532: @@
 | 136/135, 169/168, 176/175, 221/220, 364/363, 540/539
 | {{mapping| 80 127 186 225 277 296 327 }}
-| −0.949
+| &minus;0.949
 | 0.917
 | 6.12
@@ Line 546: / Line 539: @@
 | 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399
 | {{mapping| 80 127 186 225 277 296 327 340 }}
-| −0.903
+| &minus;0.903
 | 0.867
 | 5.78
-|}
+{{comma basis end}}
 === Rank-2 temperaments ===
@@ Line 566: / Line 559: @@
 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.
-{| class="wikitable center-all left-5"
+{{rank-2 begin}}
-|+Rank-2 temperaments by generator
-! Periods<br>per 8ve
-! Generator*
-! Cents*
-! Associated<br>Ratio*
-! Temperament
 |-
 | 1
@@ Line 617: / Line 604: @@
 |-
 | 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]]
 |-
 | 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
-| 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
 |-
 | 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]]
-|}
+{{rank-2 end}}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+{{orf}}
 == Detemperaments ==