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Mavila family: Difference between revisions

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=== Overview to extensions ===
=== Overview to extensions ===
==== 7-limit extensions ====
==== 7-limit extensions ====
The second comma of the [[normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. That means [[126/125]] for marling, [[21/20]] for pelogic, [[36/35]] for armodue, [[875/864]] for hornbostel, [[49/48]] for superpelog, [[50/49]] for bipelog, and [[1323/1250]] for mohavila.
The second comma of the [[normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. That means [[36/35]] for armodue, [[126/125]] for mavling, [[21/20]] for pelogic, [[875/864]] for hornbostel, [[49/48]] for superpelog, [[50/49]] for bipelog, and [[1323/1250]] for mohavila.


Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
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[[Tp tuning #T2 tuning|RMS error]]: 4.705 cents
[[Tp tuning #T2 tuning|RMS error]]: 4.705 cents
== Armodue ==
Armodue, also known as '''hexadecimal''', is the main 7-limit extension of mavila, and also the main temperament of [[Armodue theory]]. It tempers out 36/35, and can be described as the {{nowrap| 7 & 9 }} temperament. 7/4 is mapped to the minor seventh of the antidiatonic scale, where we will find 9/5 in the 5-limit. [[16edo]] shows us an obvious tuning.
The name ''armodue'' has been established in 2011 thanks to [[Mike Battaglia]]<ref name="mike's cleanup">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101194.html Yahoo! Tuning Group | ''Higher-limit mavila extension spring cleaning'']</ref>. The alternative name ''hexadecimal'' was attested as early as 2004<ref name="big temp list">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_8809.html Yahoo! Tuning Group | ''114 7-limit temperaments'']</ref><ref name="pelogic & hex">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9670.html Yahoo! Tuning Group | ''Pelogic and "hexidecimal"'']</ref>.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 36/35, 135/128
{{Mapping|legend=1| 1 0 7 -5 | 0 1 -3 5 }}
: mapping generators: ~2, ~3
{{Multival|legend=1| 1 -3 5 -7 5 20 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 675.099
: [[error map]]: {{val| 0.000 -26.856 -11.610 +6.668 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 673.997
: error map: {{val| 0.000 -27.958 -8.304 +1.158 }}
[[Tuning ranges]]:
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = [666.667, 675.000] (5\9 to 9\16)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [666.718, 701.955]
{{Optimal ET sequence|legend=1| 7, 9, 16 }}
[[Badness]]: 0.049038
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 33/32, 36/35, 45/44
Mapping: {{mapping| 1 0 7 -5 5 | 0 1 -3 5 -1 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 674.684
* POTE: ~2 = 1200.000, ~3/2 = 673.807
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
Badness: 0.027211
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 33/32, 36/35, 45/44
Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 675.288
* POTE: ~2 = 1200.000, ~3/2 = 673.763
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
Badness: 0.019351
==== Armodog ====
Subgroup: 2.3.5.7.11.13.19
Comma list: 27/26, 33/32, 36/35, 39/38, 45/44
Mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 675.170
* CWE: ~2 = 1200.000, ~3/2 = 673.540
{{Optimal ET sequence|legend=0| 7, 9, 16, 25bf }}
Badness: 0.0160


== Mavling ==
== Mavling ==
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== Pelogic ==
== Pelogic ==
''Pelogic'' (from the Indonesian word ''[[Wikipedia: Pelog|pelog]]'') should probably be pronounced /pɛˈlɒgɪk/ ''pell-LOG-ik''. This name dates back to as early as 2004<ref name="big temp list">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_8809.html Yahoo! Tuning Group | ''114 7-limit temperaments'']</ref><ref name="pelogic & hex">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9670.html Yahoo! Tuning Group | ''Pelogic and "hexidecimal"'']</ref> and has been approved of by [[Mike Battaglia]] in 2011, reasoning that Pelog is supposed to be flatter than [[16edo|16-]] or [[23edo]], and this temperament, tempering out 21/20 and described as the {{nowrap| 7d & 9 }} temperament, tends towards such a tuning<ref name="mike's cleanup">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101194.html Yahoo! Tuning Group | ''Higher-limit mavila extension spring cleaning'']</ref>.  
''Pelogic'' (from the Indonesian word ''[[Wikipedia: Pelog|pelog]]'') should probably be pronounced /pɛˈlɒgɪk/ ''pell-LOG-ik''. This name dates back to as early as 2004<ref name="big temp list"/><ref name="pelogic & hex"/> and has been approved of by [[Mike Battaglia]] in 2011, reasoning that Pelog is supposed to be flatter than [[16edo|16-]] or [[23edo]], and this temperament, tempering out 21/20 and described as the {{nowrap| 7d & 9 }} temperament, tends towards such a tuning<ref name="mike's cleanup"/>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness: 0.022753
Badness: 0.022753
== Armodue ==
Armodue, also known as '''hexadecimal''', is the main 7-limit extension of mavila, and also the main temperament of [[Armodue theory]]. It tempers out 36/35, and can be described as the {{nowrap| 7 & 9 }} temperament. 7/4 is mapped to the minor seventh of the antidiatonic scale, where we will find 9/5 in the 5-limit. [[16edo]] shows us an obvious tuning.
The name ''armodue'' has been established in 2011 thanks to Mike Battaglia<ref name="mike's cleanup"/>. The alternative name ''hexadecimal'' was attested as early as 2004<ref name="big temp list"/><ref name="pelogic & hex"/>.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 36/35, 135/128
{{Mapping|legend=1| 1 0 7 -5 | 0 1 -3 5 }}
: mapping generators: ~2, ~3
{{Multival|legend=1| 1 -3 5 -7 5 20 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 675.099
: [[error map]]: {{val| 0.000 -26.856 -11.610 +6.668 }}
* [[POTE]]: ~2 = 1200.000, ~3/2 = 673.997
: error map: {{val| 0.000 -27.958 -8.304 +1.158 }}
[[Tuning ranges]]:
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = [666.667, 675.000] (5\9 to 9\16)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [666.718, 701.955]
{{Optimal ET sequence|legend=1| 7, 9, 16 }}
[[Badness]]: 0.049038
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 33/32, 36/35, 45/44
Mapping: {{mapping| 1 0 7 -5 5 | 0 1 -3 5 -1 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 674.684
* POTE: ~2 = 1200.000, ~3/2 = 673.807
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
Badness: 0.027211
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 33/32, 36/35, 45/44
Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 675.288
* POTE: ~2 = 1200.000, ~3/2 = 673.763
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
Badness: 0.019351
==== Armodog ====
Subgroup: 2.3.5.7.11.13.19
Comma list: 27/26, 33/32, 36/35, 39/38, 45/44
Mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}
: mapping generators: ~2, ~3
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 675.170
* CWE: ~2 = 1200.000, ~3/2 = 673.540
{{Optimal ET sequence|legend=0| 7, 9, 16, 25bf }}
Badness: 0.0160


== Hornbostel ==
== Hornbostel ==
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