Mavila family
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. The generator for all of these is a very flat fifth, lying on the spectrum between 7-equal and 9-equal.
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).
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).
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.
'Pelogic' (from the Indonesian word pelog) should probably be pronounced /pɛˈlɒgɪk/ pell-LOG-ik.
Mavila
Subgroup: 2.3.5
Comma list: 135/128
Mapping: [⟨1 0 7], ⟨0 1 -3]]
POTE generator: ~3/2 = 679.806
- 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]
- 5-odd-limit diamond monotone and tradeoff: ~3/2 = [671.229, 685.714]
Badness: 0.039556
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 wallaby and jamesbond.
Septimal mavila
Comma list: 126/125, 135/128
Mapping: [⟨1 0 7 20], ⟨0 1 -3 -11]]
Wedgie: ⟨⟨ 1 -3 -11 -7 -20 -17 ]]
POTE generator: ~3/2 = 677.913
- 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]
- 7-odd-limit diamond monotone and tradeoff: ~3/2 = [675.000, 678.261]
Badness: 0.089013
11-limit
Comma list: 33/32, 45/44, 126/125
Mapping: [⟨1 0 7 20 5], ⟨0 1 -3 -11 -1]]
POTE generator: ~3/2 = 677.924
Vals: Template:Val list
Badness: 0.042049
Pelogic
Comma list: 21/20, 135/128
Mapping: [⟨1 0 7 9], ⟨0 1 -3 -4]]
Wedgie: ⟨⟨ 1 -3 -4 -7 -9 -1 ]]
POTE generator: ~3/2 = 672.853
- 7-odd-limit diamond monotone: ~3/2 = 666.667 (5\9)
- 7-odd-limit diamond tradeoff: ~3/2 = [617.488, 701.955]
- 7-odd-limit diamond monotone and tradeoff: ~3/2 = 666.667
Badness: 0.038661
11-limit
Comma list: 21/20, 33/32, 45/44
Mapping: [⟨1 0 7 9 5], ⟨0 1 -3 -4 -1]]
POTE generator: ~3/2 = 672.644
Vals: Template:Val list
Badness: 0.022753
Armodue
This temperament is also known as hexadecimal.
Comma list: 36/35, 135/128
Mapping: [⟨1 0 7 -5], ⟨0 1 -3 5]]
Wedgie: ⟨⟨ 1 -3 5 -7 5 20 ]]
POTE generator: ~3/2 = 673.997
Badness: 0.049038
11-limit
Comma list: 33/32, 36/35, 45/44
Mapping: [⟨1 0 7 -5 5], ⟨0 1 -3 5 -1]]
POTE generator: ~3/2 = 673.807
Vals: Template:Val list
Badness: 0.027211
13-limit
Comma list: 27/26, 33/32, 36/35, 45/44
Mapping: [⟨1 0 7 -5 5 -1], ⟨0 1 -3 5 -1 3]]
POTE generator: ~3/2 = 673.763
Vals: Template:Val list
Badness: 0.019351
Hornbostel
Comma list: 135/128, 875/864
Mapping: [⟨1 0 7 -16], ⟨0 1 -3 12]]
Wedgie: ⟨⟨ 1 -3 12 -7 16 36 ]]
POTE generator: ~3/2 = 678.947
Badness: 0.121394
Superpelog
Comma list: 49/48, 135/128
Mapping: [⟨1 0 7 2], ⟨0 2 -6 1]]
Wedgie: ⟨⟨ 2 -6 1 -14 -4 19 ]]
POTE generator: ~8/7 = 259.952
Badness: 0.058216
11-limit
Comma list: 33/32, 45/44, 49/48
Mapping: [⟨1 0 7 2 5], ⟨0 2 -6 1 -2]]
POTE generator: ~8/7 = 259.959
Vals: Template:Val list
Badness: 0.028535
Mindaugas Rex Lithuaniae by Chris Vaisvil (in 5\23 tuning)
Bipelog
Comma list: 50/49, 135/128
Mapping: [⟨2 0 14 15], ⟨0 1 -3 -3]]
Wedgie: ⟨⟨ 2 -6 -6 -14 -15 3 ]]
POTE generator: ~3/2 = 681.195
Badness: 0.074703
11-limit
Comma list: 33/32, 45/44, 50/49
Mapping: [⟨2 0 14 15 10], ⟨0 1 -3 -3 -1]]
POTE generator: ~3/2 = 681.280
Vals: Template:Val list
Badness: 0.035694
Mohavila
Comma list: 135/128, 1323/1250
Mapping: [⟨1 1 4 7], ⟨0 2 -6 -15]]
Wedgie: ⟨⟨ 2 -6 -15 -14 -29 -18 ]]
POTE generator: ~25/21 = 337.658
Badness: 0.222377
11-limit
Comma list: 33/32, 45/44, 1323/1250
Mapping: [⟨1 1 4 7 4], ⟨0 2 -6 -15 -2]]
POTE generator: ~25/21 = 337.633
Vals: Template:Val list
Badness: 0.092074
Mavila listening examples
- Mysterious Mush (spectrally mapped)
- Mysterious Mush (unmapped)
- Hopper by Singer-Medora-White-Smith; in f^4-10f+10=0 equal-beating mavila