Mavila family: Difference between revisions

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 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 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]]).  

{{Main| Mavila }}

[[Subgroup]]: 2.3.5

[[Comma list]]: 135/128

{{Mapping|legend=1| 1 0 7 | 0 1 -3 }}

: mapping generators: ~2, ~3

[[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}}

* [[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.

* ''[[Medusa]]'' (+15/14) → [[Very low accuracy temperaments #Medusa|Very low accuracy temperaments]]

* ''[[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.

Subgroup: 2.3.5.11

Comma list: 33/32, 45/44

Gencom mapping: {{mapping| 1 0 7 0 5 | 0 1 -3 0 -1 }}

Optimal tunings:  

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 678.978{{c}}

{{Optimal ET sequence|legend=0| 7, 16, 23e, 30bce }}

 

Badness (Sintel): 0.424

 

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 }}

 

* [[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 }}

 

* [[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 }}

 

* 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 }}

 

 

=== 13-limit ===

 

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

 

Mapping: {{mapping| 1 0 7 -5 5 -1 | 0 1 -3 5 -1 3 }}

 

* WE: ~2 = 1205.396{{c}}, ~3/2 = 676.792{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.988{{c}}

 

{{Optimal ET sequence|legend=0| 7, 9, 16 }}

 

Badness (Sintel): 0.800

 

==== Armodog ====

 

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

 

Subgroup-val mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}

 

Optimal tunings:  

* WE: ~2 = 1204.838{{c}}, ~3/2 = 675.997{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.540{{c}}

 

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

 

 

== 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.

 

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|legend=1| 1 0 7 20 | 0 1 -3 -11 }}

 

* [[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 }}

 

[[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]

 

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

 

* WE: ~2 = 1208.243{{c}}, ~3/2 = 682.582{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 677.434{{c}}

 

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

 

 

== 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"/>.

 

 

[[Subgroup]]: 2.3.5.7

 

[[Comma list]]: 21/20, 135/128

 

{{Mapping|legend=1| 1 0 7 9 | 0 1 -3 -4 }}

 

[[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}}

 

[[Tuning ranges]]:  

* [[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 }}

 

[[Badness]] (Sintel): 0.978

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

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

 

* WE: ~2 = 1209.379{{c}}, ~3/2 = 677.901{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 671.507{{c}}

 

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

 

 

== 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.

 

 

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

 

{{Mapping|legend=1| 1 0 7 -16 | 0 1 -3 12 }}

 

[[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 }}

 

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

 

[[Badness]] (Sintel): 3.07

 

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 0 7 -16 5 | 0 1 -3 12 -1 }}

 

* WE: ~2 = 1208.145{{c}}, ~3/2 = 683.517{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 679.150{{c}}

 

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

 

 

== Bipelog ==

 

[[Comma list]]: 50/49, 135/128

 

{{Mapping|legend=1| 2 0 14 15 | 0 1 -3 -3 }}

 

: mapping generators: ~7/5, ~3

 

* [[WE]]: ~7/5 = 603.757{{c}}, ~3/2 = 685.461{{c}}

* [[CWE]]: ~7/5 = 600.000{{c}}, ~3/2 = 680.206{{c}}

 

 

 

 

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

 

 

 

 

 

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.