# 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

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]

Optimal GPV sequence: 7, 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

- 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 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]]

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

- 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 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]]

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

- 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 sequence: 7, 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 sequence: 7, 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 sequence: 9, 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 sequence: 14c, 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 sequence: 7d, 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

- Mysterious Mush (spectrally mapped)
- Mysterious Mush (unmapped)
*Hopper*by Singer-Medora-White-Smith; in f^4-10f+10=0 equal-beating mavila