# Limmic temperaments

This **limmic temperaments** page collects various temperaments tempering out the Pythagorean limma, 256/243. As a consequence, 3/2 is always represented by 3\5, 720 cents assuming pure octaves. While quite sharp, this is close enough to a just fifth to serve as a fifth, and some people are fond of it.

## Blacksmith

### 5-limit (blackwood)

Subgroup: 2.3.5

Comma list: 256/243

Mapping: [⟨5 8 0], ⟨0 0 1]]

- mapping generators: ~9/8, ~5

Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 399.594

Optimal ET sequence: 5, 10, 15

Badness: 0.063760

### 7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [⟨5 8 0 14], ⟨0 0 1 0]]

Wedgie: ⟨⟨0 5 0 8 0 -14]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 392.767

Optimal ET sequence: 5, 10, 15, 40b, 55b

Badness: 0.025640

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 55/54

Mapping: [⟨5 8 0 14 29], ⟨0 0 1 0 -1]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 394.948

Optimal ET sequence: 5, 10, 15, 40be, 55be, 70bde, 85bcde

Badness: 0.024641

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 55/54

Mapping: [⟨5 8 0 14 29 7], ⟨0 0 1 0 -1 1]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 391.037

Optimal ET sequence: 5, 10, 15, 25e, 40bef

Badness: 0.020498

### Farrier

Subgroup: 2.3.5.7.11

Comma list: 28/27, 49/48, 77/75

Mapping: [⟨5 8 0 14 -6], ⟨0 0 1 0 2]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 398.070

Optimal ET sequence: 5e, 10e, 15

Badness: 0.029200

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 49/48, 66/65

Mapping: [⟨5 8 0 14 -6 7], ⟨0 0 1 0 2 1]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 396.812

Optimal ET sequence: 5e, 10e, 15

Badness: 0.022325

### Ferrum

Subgroup: 2.3.5.7.11

Comma list: 28/27, 35/33, 49/48

Mapping: [⟨5 8 0 14 6], ⟨0 0 1 0 1]]

Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 374.763

Badness: 0.030883

## Blackweed

Blackweed is a variant of blackwood as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420¢ as the generator.

Subgroup: 2.3.11/7

Comma list: [8 -5⟩ = 256/243

Sval mapping: [⟨5 8 0], ⟨0 0 1]]

- sval mapping generators: ~9/8, ~11/7

Optimal tuning (subgroup POTE): ~11/7 = 786.2215