Pelogic 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, 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 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).
'Pelogic' (from the Indonesian word pelog) should probably be pronounced /pɛˈlɒgɪk/ pell-LOG-ik.
Mavila
- Main article: Mavila
Subgroup: 2.3.5
Comma list: 135/128
Mapping: [⟨1 0 7], ⟨0 1 -3]]
- mapping generators: ~2, ~3
- 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: 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
- Wallaby (+28/27) → Trienstonic clan
- Medusa (+15/14) → Very low accuracy temperaments
- Superpelog (+49/48) → Slendro clan
- Clyndro (+360/343) → Gamelismic clan
- Jamesbond (+25/24) → 7th-octave temperaments
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
- 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]
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 = 1\1, ~3/2 = 675.6200
- POTE: ~2 = 1\1, ~3/2 = 677.924
Optimal ET sequence: 7d, 16, 23de
Badness: 0.042049
Pelogic
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]]
- 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: 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 = 1\1, ~3/2 = 667.1801
- POTE: ~2 = 1\1, ~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]]
- 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: 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 = 1\1, ~3/2 = 674.6841
- POTE: ~2 = 1\1, ~3/2 = 673.807
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 = 1\1, ~3/2 = 675.2877
- POTE: ~2 = 1\1, ~3/2 = 673.763
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 = 1\1, ~3/2 = 675.1703
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 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 = 1\1, ~3/2 = 680.2409
- POTE: ~2 = 1\1, ~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 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 = 1\2, ~3/2 = 676.3926
- POTE: ~7/5 = 1\2, ~3/2 = 681.280
Optimal ET sequence: 14c, 30bce, 44bccdee
Badness: 0.035694
Mohavila
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 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 = 1\1, ~25/21 = 336.0156
- POTE: ~2 = 1\1, ~25/21 = 337.633
Optimal ET sequence: 7d, 25b, 32bde
Badness: 0.092074
Listening examples
- Mysterious Mush (spectrally mapped)
- Mysterious Mush (unmapped)
- Hopper by Singer-Medora-White-Smith; in f^4-10f+10=0 equal-beating mavila