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]]

POTE generator: ~3/2 = 679.806

Tuning ranges:

  • 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]

Optimal GPV sequence7, 9, 16, 23, 30bc

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, medusa, clyndro and jamesbond.

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]]

POTE generator: ~3/2 = 677.913

Tuning ranges:

  • 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]

Optimal GPV 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]]

POTE generator: ~3/2 = 677.924

Optimal GPV 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]]

Wedgie⟨⟨1 -3 -4 -7 -9 -1]]

POTE generator: ~3/2 = 672.853

Tuning ranges:

  • 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

Optimal GPV 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]]

POTE generator: ~3/2 = 672.644

Optimal GPV 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]]

Wedgie⟨⟨1 -3 5 -7 5 20]]

POTE generator: ~3/2 = 673.997

Tuning ranges:

  • 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]
  • 7-odd-limit diamond monotone and tradeoff: ~3/2 = [666.718, 675.000]

Optimal GPV sequence7, 9, 16, 41b, 57bb

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]]

POTE generator: ~3/2 = 673.807

Optimal GPV sequence: 7, 9, 16, 25b, 41be, 57bbee

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]]

POTE generator: ~3/2 = 673.763

Optimal GPV sequence: 7, 9, 16, 41bef, 57bbeef

Badness: 0.019351

Hornbostel

Subgroup: 2.3.5.7

Comma list: 135/128, 729/700

Mapping: [1 0 7 -16], 0 1 -3 12]]

Wedgie⟨⟨1 -3 12 -7 16 36]]

POTE generator: ~3/2 = 678.947

Optimal GPV sequence7, 16d, 23d, 53bbccd, 76bbccdd

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]]

POTE generator: ~3/2 = 678.909

Optimal GPV sequence: 7, 16d, 23de

Badness: 0.055036

Superpelog

Subgroup: 2.3.5.7

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

Optimal GPV sequence9, 14c, 23d, 37bcd, 60bbccdd

Badness: 0.058216

11-limit

Subgroup: 2.3.5.7.11

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

Optimal GPV sequence: 9, 14c, 23de, 37bcde

Badness: 0.028535

Music
Mindaugas Rex Lithuaniae by Chris Vaisvil (in 5\23 tuning)

Bipelog

Subgroup: 2.3.5.7

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

Optimal GPV sequence14c, 16, 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]]

POTE generator: ~3/2 = 681.280

Optimal GPV sequence: 14c, 16, 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]]

Wedgie⟨⟨2 -6 -15 -14 -29 -18]]

POTE generator: ~25/21 = 337.658

Optimal GPV sequence7d, 18b, 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]]

POTE generator: ~25/21 = 337.633

Optimal GPV sequence: 7d, 18b, 25b, 32bde

Badness: 0.092074

Listening examples

Gene Ward Smith
Mike Battaglia
John Moriarty