# 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