Pelogic family

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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

Subgroup: 2.3.5

Comma list: 135/128

Mapping[1 0 7], 0 1 -3]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 677.145
  • POTE: ~2 = 1\1, ~3/2 = 679.806

Tuning ranges:

Optimal ET sequence7, 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

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 = 1\1, ~3/2 = 675.7492
  • POTE: ~2 = 1\1, ~3/2 = 677.913

Tuning ranges:

Optimal ET sequence7d, 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]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 667.5573
  • POTE: ~2 = 1\1, ~3/2 = 672.853

Tuning ranges:

Optimal ET sequence7d, 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]]

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 675.0988
  • POTE: ~2 = 1\1, ~3/2 = 673.997

Tuning ranges:

Optimal ET sequence7, 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

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 = 1\1, ~3/2 = 675.2877
  • POTE: ~2 = 1\1, ~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 = 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 tunings:

  • CTE: ~2 = 1\1, ~3/2 = 680.3705
  • POTE: ~2 = 1\1, ~3/2 = 678.947

Optimal ET sequence7, 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 tunings:

  • CTE: ~7/5 = 1\2, ~3/2 = 677.114
  • POTE: ~7/5 = 1\2, ~3/2 = 681.195

Optimal ET sequence14c, 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 tunings:

  • CTE: ~2 = 1\1, ~25/21 = 336.1216
  • POTE: ~2 = 1\1, ~25/21 = 337.658

Optimal ET sequence7d, 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

Gene Ward Smith
Mike Battaglia
John Moriarty