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