Limmic temperaments: Difference between revisions
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Blacksmith is still the only extension of blackwood since blackweed is of a different subgroup |
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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 [[cent]]s 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. | 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 [[cent]]s 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 | Subgroup: 2.3.5 | ||
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[[Badness]]: 0.063760 | [[Badness]]: 0.063760 | ||
== | === 7-limit === | ||
[[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]] | [[File:blacksmith10.jpg|alt=blacksmith10.jpg|thumb|Lattice of blacksmith]] | ||
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Subgroup: 2.3.14/11 | Subgroup: 2.3.14/11 | ||
Comma: 256/243 | [[Comma list]]: 256/243 | ||
Mapping: [{{val|5 8 0}}, {{val|0 0 1}}] | [[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}] | ||
Mapping generators: ~9/8, ~14/11 | Mapping generators: ~9/8, ~14/11 | ||
POTE generator: ~14/11 = 413.7785 | [[POTE generator]]: ~14/11 = 413.7785 | ||
{{Val list|legend=1| 20, 35b }} | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
Revision as of 08:33, 12 October 2021
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
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]]
Mapping generators: ~7/6, ~5
Wedgie: ⟨⟨ 0 5 0 8 0 -14 ]]
POTE generator: ~5/4 = 392.767
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: Template:Val list
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: Template:Val list
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: Template:Val list
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: Template:Val list
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: Template:Val list
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