Mavila family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
m Recategorize
 
(32 intermediate revisions by 7 users not shown)
Line 1: Line 1:
The '''pelogic family''' tempers out [[135/128]], the pelogic comma, also known as the major chroma or major limma. The [[5-limit]] temperament is [[mavila]], so named after the Chopi village where it was discovered, and is the base from which higher limit temperaments are derived. The generator for all of these is a very flat fifth, lying on the spectrum between [[7edo|7-equal]] and [[9edo|9-equal]].
{{Technical data page}}
The '''mavila family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[135/128]], the mavila comma, also known as the major chroma or major limma. The [[5-limit]] temperament is [[mavila]], so named after the Chopi village where it was discovered, and is the base from which higher limit temperaments are derived. The generator for all of these is a very flat fifth, lying on the spectrum between [[7edo]] and [[9edo]].


One of the most salient and characteristic features of pelogic temperament is that when you stack 4 of the tempered fifths you get to a minor third instead of the usual major third that you would get if the fifths were pure. This also means that the arrangement of small and large steps in a 7-note mavila scale is the inverse of a diatonic scale of 2 small steps and 5 large steps; mavila has 2 large steps and 5 small steps (see [[2L 5s]]).  
One of the most salient and characteristic features of mavila temperaments is that when you stack 4 of the tempered fifths you get to a minor third instead of the usual major third that you would get if the fifths were pure. This also means that the arrangement of small and large steps in a 7-note mavila scale is the inverse of a diatonic scale of 2 small steps and 5 large steps; mavila has 2 large steps and 5 small steps (see [[2L 5s]]).  


Another salient feature of pelogic temperament is the fact that 9-note [[mos scale]]s may be produced, thus giving us three different mos scales to choose from that are not decidedly chromatic in nature (5-, 7-, and 9-note scales). This is reflected in the design of the 9 + 7 layout of the Goldsmith keyboard for 16 tone equal temperament (see [[7L 2s]]).
Another salient feature of mavila temperaments is the fact that 9-note [[mos scale]]s may be produced, thus giving us three different mos scales to choose from that are not decidedly chromatic in nature (5-, 7-, and 9-note scales). This is reflected in the design of the 9 + 7 layout of the Goldsmith keyboard for 16-tone equal temperament (see [[7L 2s]]).  
 
'Pelogic' (from the Indonesian word ''[[Wikipedia: Pelog|pelog]]'') should probably be pronounced /pɛˈlɒgɪk/ ''pell-LOG-ik''.


== Mavila ==
== Mavila ==
Line 15: Line 14:


{{Mapping|legend=1| 1 0 7 | 0 1 -3 }}
{{Mapping|legend=1| 1 0 7 | 0 1 -3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 677.145
* [[WE]]: ~2 = 1208.287{{c}}, ~3/2 = 684.501{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 679.806
: [[error map]]: {{val| +8.287 -9.167 -6.667 }}
* [[DKW theory|DKW]]: ~2 = 1\1, ~3/2 = 675.46
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.111{{c}}
: error map: {{val| 0.000 -22.844 -23.648 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~3/2 = [600.000, 685.714] (1\2 to 4\7)
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000, 685.714] (1\2 to 4\7)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [671.229, 701.955] (1/3-comma to Pyth.)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [671.229, 701.955] (1/3-comma to Pyth.)


{{Optimal ET sequence|legend=1| 7, 9, 16, 23, 30bc }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 23, 30bc }}


[[Badness]]: 0.039556
[[Badness]] (Sintel): 0.928


=== Overview to extensions ===
=== Overview to 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 septimal mavila, [[21/20]] for pelogic, [[36/35]] for armodue, [[875/864]] for hornbostel, [[49/48]] for superpelog, and [[50/49]] for bipelog.
==== 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 [[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
* ''[[Wallaby]]'' (+28/27) → [[Trienstonic clan #Wallaby|Trienstonic clan]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[Superpelog]]'' (+49/48) → [[Slendro clan #Superpelog|Slendro clan]]
* ''[[Wallaby]]'' (+28/27) → [[Very low accuracy temperaments #Wallaby|Very low accuracy temperaments]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Clyndro]]'' (+360/343) → [[Gamelismic clan #Clyndro|Gamelismic clan]]
* ''[[Clyndro]]'' (+360/343) → [[Gamelismic clan #Clyndro|Gamelismic clan]]
* ''[[Jamesbond]]'' (+25/24) → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]
 
Considered below are mavling, pelogic, armodue, hornbostel, bipelog, and mohavila.


== Septimal mavila ==
==== Subgroup extensions ====
[[Subgroup]]: 2.3.5.7
Mavila naturally extends to the 2.3.5.11 subgroup, with the generator standing in for ~16/11 and ~22/15, as is given right below.
 
=== 2.3.5.11 subgroup ===
Subgroup: 2.3.5.11
 
Comma list: 33/32, 45/44


[[Comma list]]: 126/125, 135/128
Subgroup-val mapping: {{mapping| 1 0 7 5 | 0 1 -3 -1 }}


{{Mapping|legend=1| 1 0 7 20 | 0 1 -3 -11 }}
Gencom mapping: {{mapping| 1 0 7 0 5 | 0 1 -3 0 -1 }}


{{Multival|legend=1| 1 -3 -11 -7 -20 -17 }}
Optimal tunings:
* WE: ~2 = 1208.454{{c}}, ~3/2 = 684.577{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 678.978{{c}}


: mapping generators: ~2, ~3
{{Optimal ET sequence|legend=0| 7, 16, 23e, 30bce }}


[[Optimal tuning]]s
Badness (Sintel): 0.424
* [[CTE]]: ~2 = 1\1, ~3/2 = 675.7492
* [[POTE]]: ~2 = 1\1, ~3/2 = 677.913


[[Tuning ranges]]:
== Armodue ==
* 7-odd-limit [[diamond monotone]]: ~3/2 = [675.000, 678.261] (9\16 to 13\23)
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.  
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [671.229, 701.955]


{{Optimal ET sequence|legend=1| 7d, 16, 23d }}
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>.


[[Badness]]: 0.089013
[[Subgroup]]: 2.3.5.7


=== 11-limit ===
[[Comma list]]: 36/35, 135/128
Subgroup: 2.3.5.7.11


Comma list: 33/32, 45/44, 126/125
{{Mapping|legend=1| 1 0 7 -5 | 0 1 -3 5 }}


Mapping: {{mapping| 1 0 7 20 5 | 0 1 -3 -11 -1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.996{{c}}, ~3/2 = 676.803{{c}}
: [[error map]]: {{val| +4.996 -20.157 +3.261 +15.187 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 674.220{{c}}
: error map: {{val| 0.000 -27.735 -8.974 +2.275 }}


: mapping generators: ~2, ~3
[[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 tunings:
{{Optimal ET sequence|legend=1| 7, 9, 16 }}
* CTE: ~2 = 1\1, ~3/2 = 675.6200
* POTE: ~2 = 1\1, ~3/2 = 677.924


Optimal ET sequence: {{Optimal ET sequence| 7d, 16, 23de }}
[[Badness]] (Sintel): 1.24


Badness: 0.042049
=== 11-limit ===
Subgroup: 2.3.5.7.11


== Pelogic ==
Comma list: 33/32, 36/35, 45/44
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 21/20, 135/128
Mapping: {{mapping| 1 0 7 -5 5 | 0 1 -3 5 -1 }}


{{Mapping|legend=1| 1 0 7 9 | 0 1 -3 -4}}
Optimal tunings:
* WE: ~2 = 1205.460{{c}}, ~3/2 = 676.873{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.952{{c}}


: mapping generators: ~2, ~3
{{Optimal ET sequence|legend=0| 7, 9, 16 }}


{{Multival|legend=1| 1 -3 -4 -7 -9 -1 }}
Badness (Sintel): 0.900


[[Optimal tuning]]s:
=== 13-limit ===
* [[CTE]]: ~2 = 1\1, ~3/2 = 667.5573
Subgroup: 2.3.5.7.11.13
* [[POTE]]: ~2 = 1\1, ~3/2 = 672.853


[[Tuning ranges]]:  
Comma list: 27/26, 33/32, 36/35, 45/44
* 7-odd-limit [[diamond monotone]]: ~3/2 = 666.667 (5\9)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [617.488, 701.955]


{{Optimal ET sequence|legend=1| 7d, 9, 16d }}
Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}


[[Badness]]: 0.038661
Optimal tunings:  
* WE: ~2 = 1205.396{{c}}, ~3/2 = 676.792{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.988{{c}}


=== 11-limit ===
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
Subgroup: 2.3.5.7.11


Comma list: 21/20, 33/32, 45/44
Badness (Sintel): 0.800


Mapping: {{mapping| 1 0 7 9 5 | 0 1 -3 -4 -1 }}
==== Armodog ====
Subgroup: 2.3.5.7.11.13.19


: mapping generators: ~2, ~3
Comma list: 27/26, 33/32, 36/35, 39/38, 45/44


Optimal tunings:
Subgroup-val mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}
* CTE: ~2 = 1\1, ~3/2 = 667.1801
* POTE: ~2 = 1\1, ~3/2 = 672.644


Optimal ET sequence: {{Optimal ET sequence| 7d, 9, 16d }}
Optimal tunings:  
* WE: ~2 = 1204.838{{c}}, ~3/2 = 675.997{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.540{{c}}


Badness: 0.022753
{{Optimal ET sequence|legend=0| 7, 9, 16, 25bf }}


== Armodue ==
Badness (Sintel): 0.830
This temperament is also known as '''hexadecimal'''.  


[[Subgroup]]: 2.3.5.7
== Mavling ==
Mavling tempers out 126/125 and may be described as the {{nowrap| 7d & 16 }} temperament. The 7/4 is mapped to the augmented sixth of the antidiatonic scale.  


[[Comma list]]: 36/35, 135/128
This temperament was formerly known as ''septimal mavila'', but decanonicalized in 2025 per community consensus.


{{Mapping|legend=1| 1 0 7 -5 | 0 1 -3 5 }}
[[Subgroup]]: 2.3.5.7


: mapping generators: ~2, ~3
[[Comma list]]: 126/125, 135/128


{{Multival|legend=1| 1 -3 5 -7 5 20 }}
{{Mapping|legend=1| 1 0 7 20 | 0 1 -3 -11 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 675.0988
* [[WE]]: ~2 = 1208.187{{c}}, ~3/2 = 682.538{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 673.997
: [[error map]]: {{val| +8.187 -11.230 -1.178 -3.057 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 677.350{{c}}
: error map: {{val| 0.000 -24.605 -18.363 -19.672 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 7-odd-limit [[diamond monotone]]: ~3/2 = [666.667, 675.000] (5\9 to 9\16)
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = [675.000, 678.261] (9\16 to 13\23)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [666.718, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [671.229, 701.955]


{{Optimal ET sequence|legend=1| 7, 9, 16 }}
{{Optimal ET sequence|legend=1| 7d, 16, 23d }}


[[Badness]]: 0.049038
[[Badness]] (Sintel): 2.25


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 36/35, 45/44
Comma list: 33/32, 45/44, 126/125


Mapping: {{mapping| 1 0 7 -5 5 | 0 1 -3 5 -1 }}
Mapping: {{mapping| 1 0 7 20 5 | 0 1 -3 -11 -1 }}
 
: mapping generators: ~2, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 674.6841
* WE: ~2 = 1208.243{{c}}, ~3/2 = 682.582{{c}}
* POTE: ~2 = 1\1, ~3/2 = 673.807
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 677.434{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 9, 16 }}
{{Optimal ET sequence|legend=0| 7d, 16, 23de }}


Badness: 0.027211
Badness (Sintel): 1.39


=== 13-limit ===
== Pelogic ==
Subgroup: 2.3.5.7.11.13
''Pelogic'' (from the Indonesian word ''[[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"/>.
 
The 7/4 is mapped to the major sixth of the antidiatonic scale.  


Comma list: 27/26, 33/32, 36/35, 45/44
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}
[[Comma list]]: 21/20, 135/128


: mapping generators: ~2, ~3
{{Mapping|legend=1| 1 0 7 9 | 0 1 -3 -4 }}


Optimal tunings:  
[[Optimal tuning]]s:  
* CTE: ~2 = 1\1, ~3/2 = 675.2877
* [[WE]]: ~2 = 1210.184{{c}}, ~3/2 = 678.563{{c}}
* POTE: ~2 = 1\1, ~3/2 = 673.763
: [[error map]]: {{val| +10.184 -13.208 +18.732 -32.160 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 671.548{{c}}
: error map: {{val| 0.000 -30.407 -0.957 -55.017 }}


Optimal ET sequence: {{Optimal ET sequence| 7, 9, 16 }}
[[Tuning ranges]]:  
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = 666.667 (5\9)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [617.488, 701.955]


Badness: 0.019351
{{Optimal ET sequence|legend=1| 7d, 9, 16d }}


==== Armodog ====
[[Badness]] (Sintel): 0.978
Subgroup: 2.3.5.7.11.13.19


Comma list: 27/26, 33/32, 36/35, 39/38, 45/44
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}
Comma list: 21/20, 33/32, 45/44


: mapping generators: ~2, ~3
Mapping: {{mapping| 1 0 7 9 5 | 0 1 -3 -4 -1 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 675.1703
* WE: ~2 = 1209.379{{c}}, ~3/2 = 677.901{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 671.507{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 9, 16, 25bf }}
{{Optimal ET sequence|legend=0| 7d, 9, 16d }}


Badness: 0.0160
Badness (Sintel): 0.752


== Hornbostel ==
== Hornbostel ==
Hornbostel tempers out 729/700 and may be described as the {{nowrap| 7 & 23d }} temperament. The 7/4 is mapped to the diminished seventh of the antidiatonic scale.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 202: Line 221:


{{Mapping|legend=1| 1 0 7 -16 | 0 1 -3 12 }}
{{Mapping|legend=1| 1 0 7 -16 | 0 1 -3 12 }}
: mapping generators: ~2, ~3
{{Multival|legend=1| 1 -3 12 -7 16 36 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~3/2 = 680.3705
* [[WE]]: ~2 = 1207.970{{c}}, ~3/2 = 683.457{{c}}
* [[POTE]]: ~2 = 1\1, ~3/2 = 678.947
: [[error map]]: {{val| +7.970 -10.529 -4.805 +0.775 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.270{{c}}
: error map: {{val| 0.000 -22.685 -24.124 -17.583 }}


{{Optimal ET sequence|legend=1| 7, 16d, 23d, 53bbccd }}
{{Optimal ET sequence|legend=1| 7, 16d, 23d, 53bbccd }}


[[Badness]]: 0.121294
[[Badness]] (Sintel): 3.07


=== 11-limit ===
=== 11-limit ===
Line 221: Line 238:


Mapping: {{mapping| 1 0 7 -16 5 | 0 1 -3 12 -1 }}
Mapping: {{mapping| 1 0 7 -16 5 | 0 1 -3 12 -1 }}
: mapping generators: ~2, ~3


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~3/2 = 680.2409
* WE: ~2 = 1208.145{{c}}, ~3/2 = 683.517{{c}}
* POTE: ~2 = 1\1, ~3/2 = 678.909
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 679.150{{c}}


Optimal ET sequence: {{Optimal ET sequence| 7, 16d, 23de, 53bbccdee }}
{{Optimal ET sequence|legend=0| 7, 16d, 23de, 53bbccdee }}


Badness: 0.055036
Badness (Sintel): 1.82


== Bipelog ==
== Bipelog ==
Line 240: Line 255:


: mapping generators: ~7/5, ~3
: mapping generators: ~7/5, ~3
{{Multival|legend=1| 2 -6 -6 -14 -15 3 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 1\2, ~3/2 = 677.114
* [[WE]]: ~7/5 = 603.757{{c}}, ~3/2 = 685.461{{c}}
* [[POTE]]: ~7/5 = 1\2, ~3/2 = 681.195
: [[error map]]: {{val| +7.514 -8.980 -12.641 +8.604 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 680.206{{c}}
: error map: {{val| 0.000 -21.749 -26.932 -9.444 }}


{{Optimal ET sequence|legend=1| 14c, 30bc, 44bccd }}
{{Optimal ET sequence|legend=1| 14c, 30bc, 44bccd }}


[[Badness]]: 0.074703
[[Badness]] (Sintel): 1.89


=== 11-limit ===
=== 11-limit ===
Line 258: Line 273:
Mapping: {{mapping| 2 0 14 15 10 | 0 1 -3 -3 -1 }}
Mapping: {{mapping| 2 0 14 15 10 | 0 1 -3 -3 -1 }}


: mapping generators: ~7/5, ~3
Optimal tunings:  
 
* WE: ~7/5 = 603.958{{c}}, ~3/2 = 685.773{{c}}
Optimal tunings:
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 680.267{{c}}
* CTE: ~7/5 = 1\2, ~3/2 = 676.3926
* POTE: ~7/5 = 1\2, ~3/2 = 681.280


Optimal ET sequence: {{Optimal ET sequence| 14c, 30bce, 44bccdee }}
{{Optimal ET sequence|legend=0| 14c, 30bce, 44bccdee }}


Badness: 0.035694
Badness (Sintel): 1.18


== Mohavila ==
== Mohavila ==
Named by Mike Battaglia in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104134.html Yahoo! Tuning Group | ''Mohavila temperament'']</ref>, mohavila splits the mavila fifth in two. Unlike [[mohaha]], this generator is not used as an ~11/9. In fact, the prime 11 is the same as in mavila, so the ~11/9 is the major third, tempered together with ~5/4. The fifth is only split to derive septimal intervals.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 276: Line 291:


: mapping generators: ~2, ~25/21
: mapping generators: ~2, ~25/21
{{Multival|legend=1| 2 -6 -15 -14 -29 -18 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~25/21 = 336.1216
* [[WE]]: ~2 = 1208.410{{c}}, ~25/21 = 340.025{{c}}
* [[POTE]]: ~2 = 1\1, ~25/21 = 337.658
: [[error map]]: {{val| +8.410 -13.496 +7.177 -10.327 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~25/21 = 337.260{{c}}
: error map: {{val| 0.000 -27.435 -9.872 -27.722 }}


{{Optimal ET sequence|legend=1| 7d, 25b, 32bd }}
{{Optimal ET sequence|legend=1| 7d, 25b, 32bd }}


[[Badness]]: 0.222377
[[Badness]] (Sintel): 5.63


=== 11-limit ===
=== 11-limit ===
Line 293: Line 308:


Mapping: {{mapping| 1 1 4 7 4 | 0 2 -6 -15 -2 }}
Mapping: {{mapping| 1 1 4 7 4 | 0 2 -6 -15 -2 }}
: mapping generators: ~2, ~25/21


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~25/21 = 336.0156
* WE: ~2 = 1208.211{{c}}, ~25/21 = 339.943{{c}}
* POTE: ~2 = 1\1, ~25/21 = 337.633
* CWE: ~2 = 1200.000{{c}}, ~25/21 = 337.286{{c}}
 
Optimal ET sequence: {{Optimal ET sequence| 7d, 25b, 32bde }}
 
Badness: 0.092074


== Listening examples ==
{{Optimal ET sequence|legend=0| 7d, 25b, 32bde }}
; [[Gene Ward Smith]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/mushc.ogg Mysterious Mush (spectrally mapped)]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/mush.ogg Mysterious Mush (unmapped)]
* ''[http://micro.soonlabel.com/gene_ward_smith/transformers/hopper.mp3 Hopper]'' by Singer-Medora-White-Smith; in f^4-10f+10=0 equal-beating mavila


; [[Mike Battaglia]]
Badness (Sintel): 3.04
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-9-edo The Mavila Experiments - 9-EDO Version]
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-16-edo The Mavila Experiments - 16-EDO Version]
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments The Mavila Experiments - 23-EDO Version]
* [https://soundcloud.com/mikebattagliaexperiments/sets/the-mavila-experiments-25-edo The Mavila Experiments - 25-EDO Version]


; [[John Moriarty]]
== References ==
* ''[http://clones.soonlabel.com/public/micro/j_l_moriat/Mavila.mp3 Mavila]''


[[Category:Mavila family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pelogic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Pelogic| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Listen]]

Latest revision as of 12:09, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The mavila family of temperaments tempers out 135/128, the mavila comma, also known as the major chroma or major limma. The 5-limit temperament is mavila, so named after the Chopi village where it was discovered, and is the base from which higher limit temperaments are derived. The generator for all of these is a very flat fifth, lying on the spectrum between 7edo and 9edo.

One of the most salient and characteristic features of mavila temperaments is that when you stack 4 of the tempered fifths you get to a minor third instead of the usual major third that you would get if the fifths were pure. This also means that the arrangement of small and large steps in a 7-note mavila scale is the inverse of a diatonic scale of 2 small steps and 5 large steps; mavila has 2 large steps and 5 small steps (see 2L 5s).

Another salient feature of mavila temperaments is the fact that 9-note mos scales may be produced, thus giving us three different mos scales to choose from that are not decidedly chromatic in nature (5-, 7-, and 9-note scales). This is reflected in the design of the 9 + 7 layout of the Goldsmith keyboard for 16-tone equal temperament (see 7L 2s).

Mavila

Main article: Mavila

Subgroup: 2.3.5

Comma list: 135/128

Mapping: [⟨1 0 7], ⟨0 1 -3]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1208.287 ¢, ~3/2 = 684.501 ¢
error map: ⟨+8.287 -9.167 -6.667]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 679.111 ¢
error map: ⟨0.000 -22.844 -23.648]

Tuning ranges:

Optimal ET sequence: 7, 9, 16, 23, 30bc

Badness (Sintel): 0.928

Overview to extensions

7-limit extensions

The second comma of the 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

Considered below are mavling, pelogic, armodue, hornbostel, bipelog, and mohavila.

Subgroup extensions

Mavila naturally extends to the 2.3.5.11 subgroup, with the generator standing in for ~16/11 and ~22/15, as is given right below.

2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 33/32, 45/44

Subgroup-val mapping: [⟨1 0 7 5], ⟨0 1 -3 -1]]

Gencom mapping: [⟨1 0 7 0 5], ⟨0 1 -3 0 -1]]

Optimal tunings:

  • WE: ~2 = 1208.454 ¢, ~3/2 = 684.577 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 678.978 ¢

Optimal ET sequence: 7, 16, 23e, 30bce

Badness (Sintel): 0.424

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 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[1]. The alternative name hexadecimal was attested as early as 2004[2][3].

Subgroup: 2.3.5.7

Comma list: 36/35, 135/128

Mapping: [⟨1 0 7 -5], ⟨0 1 -3 5]]

Optimal tunings:

  • WE: ~2 = 1204.996 ¢, ~3/2 = 676.803 ¢
error map: ⟨+4.996 -20.157 +3.261 +15.187]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 674.220 ¢
error map: ⟨0.000 -27.735 -8.974 +2.275]

Tuning ranges:

Optimal ET sequence: 7, 9, 16

Badness (Sintel): 1.24

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 36/35, 45/44

Mapping: [⟨1 0 7 -5 5], ⟨0 1 -3 5 -1]]

Optimal tunings:

  • WE: ~2 = 1205.460 ¢, ~3/2 = 676.873 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 673.952 ¢

Optimal ET sequence: 7, 9, 16

Badness (Sintel): 0.900

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 33/32, 36/35, 45/44

Mapping: [⟨1 0 7 -5 5 -1], ⟨0 1 -3 5 -1 3]]

Optimal tunings:

  • WE: ~2 = 1205.396 ¢, ~3/2 = 676.792 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 673.988 ¢

Optimal ET sequence: 7, 9, 16

Badness (Sintel): 0.800

Armodog

Subgroup: 2.3.5.7.11.13.19

Comma list: 27/26, 33/32, 36/35, 39/38, 45/44

Subgroup-val mapping: [⟨1 0 7 -5 5 -1 -2], ⟨0 1 -3 5 -1 3 4]]

Optimal tunings:

  • WE: ~2 = 1204.838 ¢, ~3/2 = 675.997 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 673.540 ¢

Optimal ET sequence: 7, 9, 16, 25bf

Badness (Sintel): 0.830

Mavling

Mavling tempers out 126/125 and may be described as the 7d & 16 temperament. The 7/4 is mapped to the augmented sixth of the antidiatonic scale.

This temperament was formerly known as septimal mavila, but decanonicalized in 2025 per community consensus.

Subgroup: 2.3.5.7

Comma list: 126/125, 135/128

Mapping: [⟨1 0 7 20], ⟨0 1 -3 -11]]

Optimal tunings:

  • WE: ~2 = 1208.187 ¢, ~3/2 = 682.538 ¢
error map: ⟨+8.187 -11.230 -1.178 -3.057]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 677.350 ¢
error map: ⟨0.000 -24.605 -18.363 -19.672]

Tuning ranges:

Optimal ET sequence: 7d, 16, 23d

Badness (Sintel): 2.25

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 126/125

Mapping: [⟨1 0 7 20 5], ⟨0 1 -3 -11 -1]]

Optimal tunings:

  • WE: ~2 = 1208.243 ¢, ~3/2 = 682.582 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 677.434 ¢

Optimal ET sequence: 7d, 16, 23de

Badness (Sintel): 1.39

Pelogic

Pelogic (from the Indonesian word pelog) should probably be pronounced /pɛˈlɒgɪk/ pell-LOG-ik. This name dates back to as early as 2004[2][3] and has been approved of by Mike Battaglia in 2011, reasoning that Pelog is supposed to be flatter than 16- or 23edo, and this temperament, tempering out 21/20 and described as the 7d & 9 temperament, tends towards such a tuning[1].

The 7/4 is mapped to the major sixth of the antidiatonic scale.

Subgroup: 2.3.5.7

Comma list: 21/20, 135/128

Mapping: [⟨1 0 7 9], ⟨0 1 -3 -4]]

Optimal tunings:

  • WE: ~2 = 1210.184 ¢, ~3/2 = 678.563 ¢
error map: ⟨+10.184 -13.208 +18.732 -32.160]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 671.548 ¢
error map: ⟨0.000 -30.407 -0.957 -55.017]

Tuning ranges:

Optimal ET sequence: 7d, 9, 16d

Badness (Sintel): 0.978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 33/32, 45/44

Mapping: [⟨1 0 7 9 5], ⟨0 1 -3 -4 -1]]

Optimal tunings:

  • WE: ~2 = 1209.379 ¢, ~3/2 = 677.901 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 671.507 ¢

Optimal ET sequence: 7d, 9, 16d

Badness (Sintel): 0.752

Hornbostel

Hornbostel tempers out 729/700 and may be described as the 7 & 23d temperament. The 7/4 is mapped to the diminished seventh of the antidiatonic scale.

Subgroup: 2.3.5.7

Comma list: 135/128, 729/700

Mapping: [⟨1 0 7 -16], ⟨0 1 -3 12]]

Optimal tunings:

  • WE: ~2 = 1207.970 ¢, ~3/2 = 683.457 ¢
error map: ⟨+7.970 -10.529 -4.805 +0.775]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 679.270 ¢
error map: ⟨0.000 -22.685 -24.124 -17.583]

Optimal ET sequence: 7, 16d, 23d, 53bbccd

Badness (Sintel): 3.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 729/700

Mapping: [⟨1 0 7 -16 5], ⟨0 1 -3 12 -1]]

Optimal tunings:

  • WE: ~2 = 1208.145 ¢, ~3/2 = 683.517 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 679.150 ¢

Optimal ET sequence: 7, 16d, 23de, 53bbccdee

Badness (Sintel): 1.82

Bipelog

Subgroup: 2.3.5.7

Comma list: 50/49, 135/128

Mapping: [⟨2 0 14 15], ⟨0 1 -3 -3]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • WE: ~7/5 = 603.757 ¢, ~3/2 = 685.461 ¢
error map: ⟨+7.514 -8.980 -12.641 +8.604]
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 680.206 ¢
error map: ⟨0.000 -21.749 -26.932 -9.444]

Optimal ET sequence: 14c, 30bc, 44bccd

Badness (Sintel): 1.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 50/49

Mapping: [⟨2 0 14 15 10], ⟨0 1 -3 -3 -1]]

Optimal tunings:

  • WE: ~7/5 = 603.958 ¢, ~3/2 = 685.773 ¢
  • CWE: ~7/5 = 600.000 ¢, ~3/2 = 680.267 ¢

Optimal ET sequence: 14c, 30bce, 44bccdee

Badness (Sintel): 1.18

Mohavila

Named by Mike Battaglia in 2012[4], mohavila splits the mavila fifth in two. Unlike mohaha, this generator is not used as an ~11/9. In fact, the prime 11 is the same as in mavila, so the ~11/9 is the major third, tempered together with ~5/4. The fifth is only split to derive septimal intervals.

Subgroup: 2.3.5.7

Comma list: 135/128, 1323/1250

Mapping: [⟨1 1 4 7], ⟨0 2 -6 -15]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • WE: ~2 = 1208.410 ¢, ~25/21 = 340.025 ¢
error map: ⟨+8.410 -13.496 +7.177 -10.327]
  • CWE: ~2 = 1200.000 ¢, ~25/21 = 337.260 ¢
error map: ⟨0.000 -27.435 -9.872 -27.722]

Optimal ET sequence: 7d, 25b, 32bd

Badness (Sintel): 5.63

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 1323/1250

Mapping: [⟨1 1 4 7 4], ⟨0 2 -6 -15 -2]]

Optimal tunings:

  • WE: ~2 = 1208.211 ¢, ~25/21 = 339.943 ¢
  • CWE: ~2 = 1200.000 ¢, ~25/21 = 337.286 ¢

Optimal ET sequence: 7d, 25b, 32bde

Badness (Sintel): 3.04

References

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | Higher-limit mavila extension spring cleaning
  2. ↑ 2.0 2.1 Yahoo! Tuning Group | 114 7-limit temperaments
  3. ↑ 3.0 3.1 Yahoo! Tuning Group | Pelogic and "hexidecimal"
  4. ↑ Yahoo! Tuning Group | Mohavila temperament
Retrieved from "https://en.xen.wiki/index.php?title=Mavila_family&oldid=235675"