Mavila family: Difference between revisions

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I still find all these mapping generators redundant. + units
Switch to Sintel's badness, WE & CWE tunings, per community consensus
Line 18: Line 18:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 677.145{{c}}
* [[WE]]: ~2 = 1208.287{{c}}, ~3/2 = 684.501{{c}}
: [[error map]]: {{val| 0.000 -24.810 -17.749 }}
: [[error map]]: {{val| +8.287 -9.167 -6.667 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.806{{c}}
: error map: {{val| 0.000 -22.844 -23.648 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 677.145{{c}}
: error map: {{val| 0.000 -24.810 -17.749 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.806{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.806{{c}}
: error map: {{val| 0.000 -22.149 -25.732 }}
: error map: {{val| 0.000 -22.149 -25.732 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 29: Line 33:
{{Optimal ET sequence|legend=1| 7, 9, 16, 23, 30bc }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 23, 30bc }}


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


=== Overview to extensions ===
=== Overview to extensions ===
Line 48: Line 52:


=== 2.3.5.11 subgroup ===
=== 2.3.5.11 subgroup ===
[[Subgroup]]: 2.3.5.11
Subgroup: 2.3.5.11


[[Comma list]]: 33/32, 45/44
Comma list: 33/32, 45/44


{{Mapping|legend=2| 1 0 7 5 | 0 1 -3 -1 }}
Subgoup-val mapping: {{mapping| 1 0 7 5 | 0 1 -3 -1 }}


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


[[Optimal tuning]]s:  
Optimal tunings:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 676.039{{c}}
* WE: ~2 = 1208.454{{c}}, ~3/2 = 684.577{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 679.788{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 678.978{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 676.039{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 679.788{{c}} -->


{{Optimal ET sequence|legend=1| 7, 16, 23e, 30bce }}
{{Optimal ET sequence|legend=0| 7, 16, 23e, 30bce }}


[[Tp tuning #T2 tuning|RMS error]]: 4.705 cents
Badness (Sintel): 0.424


== Armodue ==
== Armodue ==
Line 76: Line 82:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 675.099{{c}}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 675.099{{c}}
: [[error map]]: {{val| 0.000 -26.856 -11.610 +6.668 }}
: [[error map]]: {{val| 0.000 -26.856 -11.610 +6.668 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 673.997{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 673.997{{c}}
: error map: {{val| 0.000 -27.958 -8.304 +1.158 }}
: error map: {{val| 0.000 -27.958 -8.304 +1.158 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 87: Line 97:
{{Optimal ET sequence|legend=1| 7, 9, 16 }}
{{Optimal ET sequence|legend=1| 7, 9, 16 }}


[[Badness]] (Smith): 0.049038
[[Badness]] (Sintel): 1.24


=== 11-limit ===
=== 11-limit ===
Line 97: Line 107:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 674.684{{c}}
* WE: ~2 = 1205.460{{c}}, ~3/2 = 676.873{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 673.807{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.952{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 674.684{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 673.807{{c}} -->


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


Badness (Smith): 0.027211
Badness (Sintel): 0.900


=== 13-limit ===
=== 13-limit ===
Line 112: Line 124:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.288{{c}}
* WE: ~2 = 1205.396{{c}}, ~3/2 = 676.792{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 673.763{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.988{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.288{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 673.763{{c}} -->


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


Badness (Smith): 0.019351
Badness (Sintel): 0.800


==== Armodog ====
==== Armodog ====
Line 124: Line 138:
Comma list: 27/26, 33/32, 36/35, 39/38, 45/44
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 }}
Subgroup-val mapping: {{mapping| 1 0 7 -5 5 -1 -2 | 0 1 -3 5 -1 3 4 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.170{{c}}
* WE: ~2 = 1204.838{{c}}, ~3/2 = 675.997{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.540{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 673.540{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.170{{c}} -->


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


Badness (Smith): 0.0160
Badness (Sintel): 0.830


== Mavling ==
== Mavling ==
Line 145: Line 160:
{{Mapping|legend=1| 1 0 7 20 | 0 1 -3 -11 }}
{{Mapping|legend=1| 1 0 7 20 | 0 1 -3 -11 }}


[[Optimal tuning]]s
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 675.749{{c}}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 675.749{{c}}
: [[error map]]: {{val| 0.000 -26.206 -13.561 -2.067 }}
: [[error map]]: {{val| 0.000 -26.206 -13.561 -2.067 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 677.913{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 677.913{{c}}
: error map: {{val| 0.000 -24.042 -20.052 -25.866 }}
: error map: {{val| 0.000 -24.042 -20.052 -25.866 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 157: Line 176:
{{Optimal ET sequence|legend=1| 7d, 16, 23d }}
{{Optimal ET sequence|legend=1| 7d, 16, 23d }}


[[Badness]] (Smith): 0.089013
[[Badness]] (Sintel): 2.25


=== 11-limit ===
=== 11-limit ===
Line 167: Line 186:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.620{{c}}
* WE: ~2 = 1208.243{{c}}, ~3/2 = 682.582{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 677.924{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 677.434{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 675.620{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 677.924{{c}} -->


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


Badness (Smith): 0.042049
Badness (Sintel): 1.39


== Pelogic ==
== Pelogic ==
Line 186: Line 207:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 667.557{{c}}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 667.557{{c}}
: [[error map]]: {{val| 0.000 -34.398 +11.014 -39.055 }}
: [[error map]]: {{val| 0.000 -34.398 +11.014 -39.055 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 672.853{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 672.853{{c}}
: error map: {{val| 0.000 -29.102 -4.873 -60.239 }}
: error map: {{val| 0.000 -29.102 -4.873 -60.239 }} -->


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 197: Line 222:
{{Optimal ET sequence|legend=1| 7d, 9, 16d }}
{{Optimal ET sequence|legend=1| 7d, 9, 16d }}


[[Badness]] (Smith): 0.038661
[[Badness]] (Sintel): 0.978


=== 11-limit ===
=== 11-limit ===
Line 206: Line 231:
Mapping: {{mapping| 1 0 7 9 5 | 0 1 -3 -4 -1 }}
Mapping: {{mapping| 1 0 7 9 5 | 0 1 -3 -4 -1 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~3/2 = 667.180{{c}}
* WE: ~2 = 1209.379{{c}}, ~3/2 = 677.901{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 672.644{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 671.507{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 667.180{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 672.644{{c}} -->


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


Badness (Smith): 0.022753
Badness (Sintel): 0.752


== Hornbostel ==
== Hornbostel ==
Line 224: Line 251:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 680.371{{c}}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 680.371{{c}}
: [[error map]]: {{val| 0.000 -21.584 -27.425 -4.380 }}
: [[error map]]: {{val| 0.000 -21.584 -27.425 -4.380 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 678.947{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 678.947{{c}}
: error map: {{val| 0.000 -23.008 -23.156 -21.456 }}
: error map: {{val| 0.000 -23.008 -23.156 -21.456 }} -->


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


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


=== 11-limit ===
=== 11-limit ===
Line 241: Line 272:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~3/2 = 680.241
* WE: ~2 = 1208.145{{c}}, ~3/2 = 683.517{{c}}
* POTE: ~2 = 1200.000, ~3/2 = 678.909
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 679.150{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 680.241{{c}}
* POTE: ~2 = 1200.000{{c}}, ~3/2 = 678.909{{c}} -->


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


Badness (Smith): 0.055036
Badness (Sintel): 1.82


== Bipelog ==
== Bipelog ==
Line 258: Line 291:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 677.114{{c}}
* [[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 }}
<!-- * [[CTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 677.114{{c}}
: [[error map]]: {{val| 0.000 -24.841 -17.656 -0.168 }}
: [[error map]]: {{val| 0.000 -24.841 -17.656 -0.168 }}
* [[POTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 681.195{{c}}
* [[POTE]]: ~7/5 = 600.000{{c}}, ~3/2 = 681.195{{c}}
: error map: {{val| 0.000 -20.760 -29.900 -12.412 }}
: error map: {{val| 0.000 -20.760 -29.900 -12.412 }} -->


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


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


=== 11-limit ===
=== 11-limit ===
Line 274: Line 311:
Mapping: {{mapping| 2 0 14 15 10 | 0 1 -3 -3 -1 }}
Mapping: {{mapping| 2 0 14 15 10 | 0 1 -3 -3 -1 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~7/5 = 600.000{{c}}, ~3/2 = 676.393{{c}}
* WE: ~7/5 = 603.958{{c}}, ~3/2 = 685.773{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~3/2 = 681.280{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 680.267{{c}}
<!-- * CTE: ~7/5 = 600.000{{c}}, ~3/2 = 676.393{{c}}
* POTE: ~7/5 = 600.000{{c}}, ~3/2 = 681.280{{c}} -->


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


Badness (Smith): 0.035694
Badness (Sintel): 1.18


== Mohavila ==
== Mohavila ==
Line 294: Line 333:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~25/21 = 336.122{{c}}
* [[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 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~25/21 = 336.122{{c}}
: [[error map]]: {{val| 0.000 -29.712 -3.043 -10.649 }}
: [[error map]]: {{val| 0.000 -29.712 -3.043 -10.649 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/21 = 337.658{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~25/21 = 337.658{{c}}
: error map: {{val| 0.000 -26.638 -12.264 -33.701 }}
: error map: {{val| 0.000 -26.638 -12.264 -33.701 }} -->


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


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


=== 11-limit ===
=== 11-limit ===
Line 311: Line 354:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000{{c}}, ~25/21 = 336.016{{c}}
* WE: ~2 = 1208.211{{c}}, ~25/21 = 339.943{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/21 = 337.633{{c}}
* CWE: ~2 = 1200.000{{c}}, ~25/21 = 337.286{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~25/21 = 336.016{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/21 = 337.633{{c}} -->


{{Optimal ET sequence|legend=0| 7d, 25b, 32bde }}
{{Optimal ET sequence|legend=0| 7d, 25b, 32bde }}


Badness (Smith): 0.092074
Badness (Sintel): 3.04


== Listening examples ==
== Listening examples ==

Revision as of 13:20, 6 September 2025

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 = 679.806 ¢
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

Subgoup-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

Listening examples

Gene Ward Smith
Mike Battaglia
John Moriarty

Notes

  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
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