Limmic temperaments

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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

POTE generator: ~5/4 = 399.594

Vals5, 10, 15

Badness: 0.063760

7-limit

blacksmith10.jpg
Lattice of blacksmith

Subgroup: 2.3.5.7

Comma list: 28/27, 49/48

Mapping: [5 8 0 14], 0 0 1 0]]

Mapping generators: ~7/6, ~5

Wedgie⟨⟨0 5 0 8 0 -14]]

POTE generator: ~5/4 = 392.767

Vals5, 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]]

POTE generator: ~5/4 = 394.948

Vals: 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]]

POTE generator: ~5/4 = 391.037

Vals: 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]]

POTE generator: ~5/4 = 398.070

Vals: 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]]

POTE generator: ~5/4 = 396.812

Vals: 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]]

POTE generator: ~5/4 = 374.763

Vals: 5e, 10

Badness: 0.030883

Blackweed

Blackweed is so named because the 20EDO tuning has 4\20 as the period and 420¢ as the generator.

Subgroup: 2.3.14/11

Comma list: 256/243

Mapping: [5 8 0], 0 0 1]]

Mapping generators: ~9/8, ~14/11

POTE generator: ~14/11 = 413.7785

Vals20, 35b