Mavila family

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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:

  • CTE: ~2 = 1200.000, ~3/2 = 677.145
error map: ⟨0.000 -24.810 -17.749]
  • POTE: ~2 = 1200.000, ~3/2 = 679.806
error map: ⟨0.000 -22.149 -25.732]

Tuning ranges:

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

Badness: 0.039556

Overview to extensions

The second comma of the 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.

Temperaments discussed elsewhere include

Considered below are septimal mavila, pelogic, armodue, hornbostel, bipelog, and mohavila.

2.3.5.11 subgroup

Subgroup: 2.3.5.11

Comma list: 33/32, 45/44

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

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

gencom: [2 3; 33/32, 45/44]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 676.039
  • POTE: ~2 = 1200.000, ~3/2 = 679.788

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

RMS error: 4.705 cents

Septimal mavila

Subgroup: 2.3.5.7

Comma list: 126/125, 135/128

Mapping: [⟨1 0 7 20], ⟨0 1 -3 -11]]

Wedgie: ⟨⟨ 1 -3 -11 -7 -20 -17 ]]

mapping generators: ~2, ~3

Optimal tunings

  • CTE: ~2 = 1200.000, ~3/2 = 675.749
error map: ⟨0.000 -26.206 -13.561 -2.067]
  • POTE: ~2 = 1200.000, ~3/2 = 677.913
error map: ⟨0.000 -24.042 -20.052 -25.866]

Tuning ranges:

Optimal ET sequence: 7d, 16, 23d

Badness: 0.089013

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]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 675.620
  • POTE: ~2 = 1200.000, ~3/2 = 677.924

Optimal ET sequence: 7d, 16, 23de

Badness: 0.042049

Pelogic

'Pelogic' (from the Indonesian word pelog) should probably be pronounced /pɛˈlɒgɪk/ pell-LOG-ik.

Subgroup: 2.3.5.7

Comma list: 21/20, 135/128

Mapping: [⟨1 0 7 9], ⟨0 1 -3 -4]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -3 -4 -7 -9 -1 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 667.557
error map: ⟨0.000 -34.398 +11.014 -39.055]
  • POTE: ~2 = 1200.000, ~3/2 = 672.853
error map: ⟨0.000 -29.102 -4.873 -60.239]

Tuning ranges:

Optimal ET sequence: 7d, 9, 16d

Badness: 0.038661

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]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 667.180
  • POTE: ~2 = 1200.000, ~3/2 = 672.644

Optimal ET sequence: 7d, 9, 16d

Badness: 0.022753

Armodue

This temperament is also known as hexadecimal.

Subgroup: 2.3.5.7

Comma list: 36/35, 135/128

Mapping: [⟨1 0 7 -5], ⟨0 1 -3 5]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -3 5 -7 5 20 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 675.099
error map: ⟨0.000 -26.856 -11.610 +6.668]
  • POTE: ~2 = 1200.000, ~3/2 = 673.997
error map: ⟨0.000 -27.958 -8.304 +1.158]

Tuning ranges:

Optimal ET sequence: 7, 9, 16

Badness: 0.049038

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]]

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: 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: [⟨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: 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: [⟨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: 7, 9, 16, 25bf

Badness: 0.0160

Hornbostel

Subgroup: 2.3.5.7

Comma list: 135/128, 729/700

Mapping: [⟨1 0 7 -16], ⟨0 1 -3 12]]

mapping generators: ~2, ~3

Wedgie: ⟨⟨ 1 -3 12 -7 16 36 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 680.371
error map: ⟨0.000 -21.584 -27.425 -4.380]
  • POTE: ~2 = 1200.000, ~3/2 = 678.947
error map: ⟨0.000 -23.008 -23.156 -21.456]

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

Badness: 0.121294

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]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1200.000, ~3/2 = 680.241
  • POTE: ~2 = 1200.000, ~3/2 = 678.909

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

Badness: 0.055036

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

Wedgie: ⟨⟨ 2 -6 -6 -14 -15 3 ]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~3/2 = 677.114
error map: ⟨0.000 -24.841 -17.656 -0.168]
  • POTE: ~7/5 = 600.000, ~3/2 = 681.195
error map: ⟨0.000 -20.760 -29.900 -12.412]

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

Badness: 0.074703

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]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~3/2 = 676.393
  • POTE: ~7/5 = 600.000, ~3/2 = 681.280

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

Badness: 0.035694

Mohavila

Named by Mike Battaglia in 2012[1], 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

Wedgie: ⟨⟨ 2 -6 -15 -14 -29 -18 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~25/21 = 336.122
error map: ⟨0.000 -29.712 -3.043 -10.649]
  • POTE: ~2 = 1200.000, ~25/21 = 337.658
error map: ⟨0.000 -26.638 -12.264 -33.701]

Optimal ET sequence: 7d, 25b, 32bd

Badness: 0.222377

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]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • CTE: ~2 = 1200.000, ~25/21 = 336.016
  • POTE: ~2 = 1200.000, ~25/21 = 337.633

Optimal ET sequence: 7d, 25b, 32bde

Badness: 0.092074

Listening examples

Gene Ward Smith
Mike Battaglia
John Moriarty

Notes

  1. ↑ Yahoo! Tuning Group | Mohavila temperament
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