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{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}
{{Optimal ET sequence|legend=1| 10, 26, 36, 154f, 190ffg, 226ffg }}
== Mothra ==
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 &amp; 31}}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]](, [[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 81/80, 1029/1024
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
: mapping generators: ~2, ~8/7
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.400
: [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }}
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.193
: error map: {{val| 0.000 -5.375 +0.005 -1.019 }}
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
[[Badness]] (Smith): 0.037146
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 385/384
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
{{Multival|legend=1| 3 12 -1 -8 12 -10 -23 -36 -60 -19 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.203
* POTE: ~2 = 1200.000, ~8/7 = 232.031
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
Badness (Smith): 0.025642
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 105/104, 144/143
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.993
* POTE: ~2 = 1200.000, ~8/7 = 231.811
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
Badness (Smith): 0.023954
; Music
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning
=== Cynder ===
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 1029/1024
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.566
* POTE: ~2 = 1200.000, ~8/7 = 231.317
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
Badness (Smith): 0.055706
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 78/77, 81/80, 640/637
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.546
* POTE: ~2 = 1200.000, ~8/7 = 231.293
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
Badness (Smith): 0.034124
=== Mosura ===
The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]],) [[176/175|S8/S10]]}.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 540/539
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
Wedgie: {{Multival| 3 12 -1 23 12 -10 26 -36 12 68 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.557
* POTE: ~2 = 1200.000, ~8/7 = 232.419
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
Badness (Smith): 0.031334
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 144/143, 176/175, 196/195
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.635
* POTE: ~2 = 1200.000, ~8/7 = 232.640
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
Badness (Smith): 0.036857


== Rodan ==
== Rodan ==
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Badness (Smith): 0.028444
Badness (Smith): 0.028444
== Mothra ==
Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap|26 &amp; 31}}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].
Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]](, [[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 81/80, 1029/1024
{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}
: mapping generators: ~2, ~8/7
{{Multival|legend=1| 3 12 -1 12 -10 -36 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.400
: [[error map]]: {{val| 0.000 -4.756 +2.482 -1.226 }}
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.193
: error map: {{val| 0.000 -5.375 +0.005 -1.019 }}
[[Algebraic generator]]: Rabrindanath, largest real root of ''x''<sup>8</sup> - 3''x''<sup>2</sup> + 1, or 232.0774 cents.
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 1 0 1/4 0 }}, {{monzo| 0 0 1 0 }}, {{monzo| 3 0 -1/12 0 }}]
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}
[[Badness]] (Smith): 0.037146
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 385/384
Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}
{{Multival|legend=1| 3 12 -1 -8 12 -10 -23 -36 -60 -19 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.203
* POTE: ~2 = 1200.000, ~8/7 = 232.031
{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}
Badness (Smith): 0.025642
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 105/104, 144/143
Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.993
* POTE: ~2 = 1200.000, ~8/7 = 231.811
{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}
Badness (Smith): 0.023954
; Music
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning
=== Cynder ===
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 1029/1024
Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.566
* POTE: ~2 = 1200.000, ~8/7 = 231.317
{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}
Badness (Smith): 0.055706
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 78/77, 81/80, 640/637
Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 231.546
* POTE: ~2 = 1200.000, ~8/7 = 231.293
{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}
Badness (Smith): 0.034124
=== Mosura ===
The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]],) [[176/175|S8/S10]]}.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 540/539
Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}
Wedgie: {{Multival| 3 12 -1 23 12 -10 26 -36 12 68 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.557
* POTE: ~2 = 1200.000, ~8/7 = 232.419
{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}
Badness (Smith): 0.031334
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 144/143, 176/175, 196/195
Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.635
* POTE: ~2 = 1200.000, ~8/7 = 232.640
{{Optimal ET sequence|legend=0| 31, 67, 98 }}
Badness (Smith): 0.036857


== Gorgo ==
== Gorgo ==
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