# Horwell temperaments

(Redirected from Fifthplus)

Horwell temperaments temper out the horwell comma, [-16 1 5 1 = 65625/65536.

Discussed elsewhere are bisupermajor, countercata, eris, escaped, hemithirds, keen, mabila, maquiloid, narayana, orwell, paramity, pontiac, tertiaseptal, and worschmidt.

## Mutt

Main article: Mutt temperament

Subgroup: 2.3.5

Comma list: [-44 -3 21

Mapping: [3 5 7], 0 -7 -1]]

POTE generator: ~5/4 = 385.980

### 7-limit

Subgroup: 2.3.5.7

Comma list: 65625/65536, 250047/250000

Mapping: [3 5 7 8], 0 -7 -1 12]]

Wedgie⟨⟨21 3 -36 -44 -116 -92]]

POTE generator: ~5/4 = 385.964

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4356, 16384/16335

Mapping: [3 5 7 8 10], 0 -7 -1 12 11]]

POTE generator: ~5/4 = 386.020

Optimal GPV sequence: 3, 84, 87, 171, 258, 429e

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 625/624, 2200/2197

Mapping: [3 5 7 8 10 11], 0 -7 -1 12 11 3]]

POTE generator: ~5/4 = 386.022

Optimal GPV sequence: 3, 84, 87, 171, 258, 429ef

## Fifthplus

Fifthplus (22&171) tempers out the sesesix comma, [-74 13 23 in the 5-limit. The name "fifthplus" means using a sharp fifth interval (such as superpyth fifth) as a generator.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 420175/419904

Mapping: [1 11 -3 20], 0 -23 13 -42]]

Wedgie⟨⟨23 -13 42 -74 2 134]]

POTE generator: ~5488/3645 = 708.774

## Emkay

Emkay (87&224) tempers out the same 5-limit comma as the emka temperament (37&50), but with the horwell (65625/65536) rather than the hemimean (3136/3125) tempered out.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 244140625/243045684

Mapping: [1 14 6 -28], 0 -27 -8 67]]

Wedgie⟨⟨27 8 -67 -50 -182 -178]]

POTE generator: ~3125/2268 = 551.7745

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 65625/65536

Mapping: [1 14 6 -28 3], 0 -27 -8 67 1]]

POTE generator: ~11/8 = 551.7746

Optimal GPV sequence: 87, 137, 224, 311, 535, 1381ce, 1916ce

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197

Mapping: [1 14 6 -28 3 6], 0 -27 -8 67 1 -5]]

POTE generator: ~11/8 = 551.7749

Optimal GPV sequence: 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff

## Kastro

Subgroup: 2.3.5.7

Comma list: 65625/65536, 117649/116640

Mapping: [1 5 1 6], 0 -31 12 -29]]

POTE generator: ~3375/3136 = 132.1845

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 12005/11979

Mapping: [1 5 1 6 5], 0 -31 12 -29 -14]]

POTE generator: ~121/112 = 132.1864

Optimal GPV sequence: 109, 118, 345de, 463de, 581dde

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 364/363, 385/384, 3388/3375

Mapping: [1 5 1 6 5 7], 0 -31 12 -29 -14 -30]]

POTE generator: ~13/12 = 132.1789

Optimal GPV sequence: 109, 118f, 227f

## Oquatonic

The oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the dimcomp (390625/388962), as well as the hemfiness (4096000/4084101, saquinru-atriyo). In this temperament, major third of 5/4 is mapped into 9\28.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 390625/388962

Mapping: [28 0 65 123], 0 1 0 -1]]

Mapping generators: ~128/125, ~3

Wedgie⟨⟨28 0 -28 -65 -123 -65]]

POTE generator: ~3/2 = 702.1137

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 65625/65536

Mapping: [28 0 65 123 230], 0 1 0 -1 -3]]

POTE generator: ~3/2 = 702.0186

Optimal GPV sequence: 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197

Mapping: [28 0 65 123 230 148], 0 1 0 -1 -3 -1]]

POTE generator: ~3/2 = 702.0288

Optimal GPV sequence: 84, 140, 224, 364, 588