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{{Technical data page}}
The '''pelogic family''' tempers out [[135/128]], the pelogic comma, also known as the major chroma or major limma. The [[5-limit]] temperament is (5-limit) '''mavila''', so named after the Chopi village where it was discovered. The generator for all of these is a very flat fifth, lying on the spectrum between 7-equal and 9-equal.
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 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]]).  
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]]).  


One of the most common temperaments talked about in the pelogic family is '''mavila''', the 5-limit temperament eliminating 135/128, from which higher-limit extensions are derived.
== Mavila ==
{{Main| Mavila }}


'Pelogic' (from the Indonesian word ''[[Wikipedia: Pelog|pelog]]'') should probably be pronounced /pɛˈlɒgɪk/ ''pell-LOG-ik''.
[[Subgroup]]: 2.3.5


= Mavila =
[[Comma list]]: 135/128


Subgroup: 2.3.5
{{Mapping|legend=1| 1 0 7 | 0 1 -3 }}
: mapping generators: ~2, ~3


[[Comma list]]: 135/128
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1208.287{{c}}, ~3/2 = 684.501{{c}}
: [[error map]]: {{val| +8.287 -9.167 -6.667 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.111{{c}}
: error map: {{val| 0.000 -22.844 -23.648 }}
 
[[Tuning ranges]]:
* [[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.)
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 23, 30bc }}
 
[[Badness]] (Sintel): 0.928
 
=== Overview to 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 [[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
* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]
* ''[[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]]
* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]
 
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: {{mapping| 1 0 7 5 | 0 1 -3 -1 }}
 
Gencom mapping: {{mapping| 1 0 7 0 5 | 0 1 -3 0 -1 }}
 
Optimal tunings:
* WE: ~2 = 1208.454{{c}}, ~3/2 = 684.577{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 678.978{{c}}
 
{{Optimal ET sequence|legend=0| 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 {{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 }}
 
[[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 }}
 
[[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]] (Sintel): 1.24
 
=== 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 }}
 
Optimal tunings:
* WE: ~2 = 1205.460{{c}}, ~3/2 = 676.873{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.952{{c}}
 
{{Optimal ET sequence|legend=0| 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


[[POTE_tuning|POTE generator]]: ~3/2 = 679.806
Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}


[[Mapping]]: [{{val|1 0 7}}, {{val|0 1 -3}}]
Optimal tunings:  
* WE: ~2 = 1205.396{{c}}, ~3/2 = 676.792{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.988{{c}}


[[Tuning ranges]]:
{{Optimal ET sequence|legend=0| 7, 9, 16 }}
* valid range: [514.286, 600.000] (3\7 to 1\2)


EDOs: {{EDOs| 7, 9, 16, 23, 30bc }}
Badness (Sintel): 0.800


= Septimal mavila =
==== Armodog ====
[[Comma|Commas]]: 126/125, 135/128
Subgroup: 2.3.5.7.11.13.19


[[POTE_tuning|POTE generator]]: ~3/2 = 677.912
Comma list: 27/26, 33/32, 36/35, 39/38, 45/44


Map: [&lt;1 0 7 20|, &lt;0 1 -3 -11|]
Subgroup-val mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}


EDOs: {{EDOs| 7d, 16, 23d }}
Optimal tunings:  
* WE: ~2 = 1204.838{{c}}, ~3/2 = 675.997{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.540{{c}}


Badness: 0.0890
{{Optimal ET sequence|legend=0| 7, 9, 16, 25bf }}


= Pelogic =
Badness (Sintel): 0.830
[[Comma|Commas]]: 21/20, 135/128


[[POTE_tuning|POTE generator]]: ~3/2 = 672.853
== 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.  


Map: [&lt;1 0 7 9|, &lt;0 1 -3 -4|]
This temperament was formerly known as ''septimal mavila'', but decanonicalized in 2025 per community consensus.


[[Wedgie]]: &lt;&lt;1 -3 -4 -7 -9 -1||
[[Subgroup]]: 2.3.5.7


EDOs: {{EDOs| 7d, 9, 16d }}
[[Comma list]]: 126/125, 135/128


Badness: 0.0387
{{Mapping|legend=1| 1 0 7 20 | 0 1 -3 -11 }}


== 11-limit ==
[[Optimal tuning]]s:
Commas: 21/20, 33/32, 45/44
* [[WE]]: ~2 = 1208.187{{c}}, ~3/2 = 682.538{{c}}
: [[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 }}


POTE generator: ~3/2 = 672.644
[[Tuning ranges]]:
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = [675.000, 678.261] (9\16 to 13\23)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [671.229, 701.955]


Map: [&lt;1 0 7 9 5|, &lt;0 1 -3 -4 -1|]
{{Optimal ET sequence|legend=1| 7d, 16, 23d }}


EDOs: {{EDOs| 7d, 9, 16d }}
[[Badness]] (Sintel): 2.25


Badness: 0.0228
=== 11-limit ===
Subgroup: 2.3.5.7.11


= Armodue =
Comma list: 33/32, 45/44, 126/125
This temperament is also known as '''hexadecimal'''.


[[Comma|Commas]]: 36/35, 135/128
Mapping: {{mapping| 1 0 7 20 5 | 0 1 -3 -11 -1 }}


[[POTE_tuning|POTE generator]]: ~3/2 = 673.997
Optimal tunings:  
* WE: ~2 = 1208.243{{c}}, ~3/2 = 682.582{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 677.434{{c}}


Map: [&lt;1 0 7 -5|, &lt;0 1 -3 5|]
{{Optimal ET sequence|legend=0| 7d, 16, 23de }}


[[Wedgie]]: &lt;&lt;1 -3 5 -7 5 20||
Badness (Sintel): 1.39


EDOs: {{EDOs| 7, 9, 16, 41b, 57bb }}
== Pelogic ==
''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"/>.


Badness: 0.0490
The 7/4 is mapped to the major sixth of the antidiatonic scale.  


== 11-limit ==
[[Subgroup]]: 2.3.5.7
Commas: 33/32, 36/35, 45/44


POTE generator: ~3/2 = 673.807
[[Comma list]]: 21/20, 135/128


Map: [&lt;1 0 7 -5 5|, &lt;0 1 -3 5 -1|]
{{Mapping|legend=1| 1 0 7 9 | 0 1 -3 -4 }}


EDOs: {{EDOs| 7, 9, 16, 25b, 41be, 57bbee }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1210.184{{c}}, ~3/2 = 678.563{{c}}
: [[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 }}


Badness: 0.0272
[[Tuning ranges]]:  
* [[7-odd-limit]] [[diamond monotone]]: ~3/2 = 666.667 (5\9)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [617.488, 701.955]


== 13-limit ==
{{Optimal ET sequence|legend=1| 7d, 9, 16d }}
Commas: 27/26, 33/32, 36/35, 45/44


POTE generator: ~3/2 = 673.763
[[Badness]] (Sintel): 0.978


Map: [&lt;1 0 7 -5 5 -1|, &lt;0 1 -3 5 -1 3|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 7, 9, 16, 41bef, 57bbeef }}
Comma list: 21/20, 33/32, 45/44


Badness: 0.0194
Mapping: {{mapping| 1 0 7 9 5 | 0 1 -3 -4 -1 }}


= Hornbostel =
Optimal tunings:
[[Comma|Commas]]: 135/128, 875/864
* WE: ~2 = 1209.379{{c}}, ~3/2 = 677.901{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 671.507{{c}}


[[POTE_tuning|POTE generator]]: ~3/2 = 678.947
{{Optimal ET sequence|legend=0| 7d, 9, 16d }}


Map: [&lt;1 0 7 -16|, &lt;0 1 -3 12|]
Badness (Sintel): 0.752


[[wedgie|Wedgie]]: &lt;&lt;1 -3 12 -7 16 36||
== 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.


EDOs: {{EDOs| 7, 16d, 23d }}
[[Subgroup]]: 2.3.5.7


Badness: 0.1213
[[Comma list]]: 135/128, 729/700


= Superpelog =
{{Mapping|legend=1| 1 0 7 -16 | 0 1 -3 12 }}
[[Comma|Commas]]: 49/48, 135/128


[[POTE_tuning|POTE generator]]: ~8/7 = 259.952
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1207.970{{c}}, ~3/2 = 683.457{{c}}
: [[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 }}


Map: [&lt;1 0 7 2|, &lt;0 2 -6 1|]
{{Optimal ET sequence|legend=1| 7, 16d, 23d, 53bbccd }}


[[wedgie|Wedgie]]: &lt;&lt;2 -6 1 -14 -4 19||
[[Badness]] (Sintel): 3.07


EDOs: {{EDOs| 9, 14c, 23d, 37bcd, 60bbccdd }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0582
Comma list: 33/32, 45/44, 729/700


== 11-limit ==
Mapping: {{mapping| 1 0 7 -16 5 | 0 1 -3 12 -1 }}
Commas: 33/32, 45/44, 49/48


POTE generator: ~8/7 = 259.959
Optimal tunings:  
* WE: ~2 = 1208.145{{c}}, ~3/2 = 683.517{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 679.150{{c}}


Map: [&lt;1 0 7 2 5|, &lt;0 2 -6 1 -2|]
{{Optimal ET sequence|legend=0| 7, 16d, 23de, 53bbccdee }}


EDOs: {{EDOs| 9, 14c, 23de, 37bcde }}
Badness (Sintel): 1.82


Badness: 0.0285
== Bipelog ==
[[Subgroup]]: 2.3.5.7


''[http://micro.soonlabel.com/MOS/20120418-9mos-mindaugas.mp3 Mindaugas Rex Lithuaniae]'' by [http://chrisvaisvil.com/?p=2267 Chris Vaisvil] (in 5\23 tuning)
[[Comma list]]: 50/49, 135/128


= Bipelog =
{{Mapping|legend=1| 2 0 14 15 | 0 1 -3 -3 }}
[[Comma|Commas]]: 50/49, 135/128


[[POTE_tuning|POTE generator]]: ~3/2 = 681.195
: mapping generators: ~7/5, ~3


Map: [&lt;2 0 14 15|, &lt;0 1 -3 -3|]
[[Optimal tuning]]s:  
* [[WE]]: ~7/5 = 603.757{{c}}, ~3/2 = 685.461{{c}}
: [[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 }}


[[wedgie|Wedgie]]: &lt;&lt;2 -6 -6 -14 -15 3||
{{Optimal ET sequence|legend=1| 14c, 30bc, 44bccd }}


EDOs: {{EDOs| 14c, 16, 30bc, 44bccd }}
[[Badness]] (Sintel): 1.89


Badness: 0.0747
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 33/32, 45/44, 50/49
Commas: 33/32, 45/44, 50/49


POTE generator: ~3/2 = 681.280
Mapping: {{mapping| 2 0 14 15 10 | 0 1 -3 -3 -1 }}


Map: [&lt;2 0 14 15 10|, &lt;0 1 -3 -3 -1|]
Optimal tunings:  
* WE: ~7/5 = 603.958{{c}}, ~3/2 = 685.773{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 680.267{{c}}


EDOs: {{EDOs| 14c, 16, 30bce, 44bccdee }}
{{Optimal ET sequence|legend=0| 14c, 30bce, 44bccdee }}


Badness: 0.0357
Badness (Sintel): 1.18


= Mohavila =
== Mohavila ==
Commas: 135/128, 1323/1250
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.


POTE generator: ~25/21 = 337.658
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 1 4 7|, &lt;0 2 -6 -15|]
[[Comma list]]: 135/128, 1323/1250


Wedgie: &lt;&lt;2 -6 -15 -14 -29 -18||
{{Mapping|legend=1| 1 1 4 7 | 0 2 -6 -15 }}


EDOs: {{EDOs| 32bd, 36 }}
: mapping generators: ~2, ~25/21


Badness: 0.2224
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1208.410{{c}}, ~25/21 = 340.025{{c}}
: [[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 }}


== 11-limit ==
{{Optimal ET sequence|legend=1| 7d, 25b, 32bd }}
Commas: 33/32, 45/44, 1323/1250


POTE generator: ~25/21 = 337.633
[[Badness]] (Sintel): 5.63


Map: [&lt;1 1 4 7 4|, &lt;0 2 -6 -15 -2|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: {{EDOs| 32bde }}
Comma list: 33/32, 45/44, 1323/1250


Badness: 0.0921
Mapping: {{mapping| 1 1 4 7 4 | 0 2 -6 -15 -2 }}


= Mavila listening examples =
Optimal tunings:
'''[[Gene Ward Smith]]'''
* WE: ~2 = 1208.211{{c}}, ~25/21 = 339.943{{c}}
*[http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/mushc.ogg Mysterious Mush (spectrally mapped)]
* CWE: ~2 = 1200.000{{c}}, ~25/21 = 337.286{{c}}
*[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]]'''
{{Optimal ET sequence|legend=0| 7d, 25b, 32bde }}
*[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]]'''
Badness (Sintel): 3.04
*''[http://clones.soonlabel.com/public/micro/j_l_moriat/Mavila.mp3 Mavila]''


[[Category:Theory]]
== References ==
[[Category:Temperament family]]
[[Category:Pelogic]]
[[Category:Pelogic family| ]] <!-- main article -->


[[Category:Rank 2]]
[[Category:Mavila family| ]] <!-- main article -->
[[Category:Listen]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Mavila_family"