Limmic temperaments: Difference between revisions

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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 [[temperament]]s [[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 ==
== Blacksmith ==
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[[Comma list]]: 256/243
[[Comma list]]: 256/243


[[Mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=1| 5 8 0 | 0 0 1 }}


Mapping generators: ~9/8, ~5
: mapping generators: ~9/8, ~5


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~5/4 = 399.594
[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~5/4 = 399.594
Line 24: Line 24:
[[Comma list]]: 28/27, 49/48
[[Comma list]]: 28/27, 49/48


[[Mapping]]: [{{val| 5 8 0 14 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 5 8 0 14 | 0 0 1 0 }}
 
Mapping generators: ~7/6, ~5


{{Multival|legend=1| 0 5 0 8 0 -14 }}
{{Multival|legend=1| 0 5 0 8 0 -14 }}


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\5, ~5/4 = 392.767
[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 392.767


{{Optimal ET sequence|legend=1| 5, 10, 15, 40b, 55b }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 40b, 55b }}
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Comma list: 28/27, 49/48, 55/54
Comma list: 28/27, 49/48, 55/54


Mapping: [{{val| 5 8 0 14 29 }}, {{val| 0 0 1 0 -1 }}]
Mapping: {{mapping| 5 8 0 14 29 | 0 0 1 0 -1 }}


Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 394.948
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 394.948


{{Optimal ET sequence|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 40be, 55be, 70bde, 85bcde }}
Line 54: Line 52:
Comma list: 28/27, 40/39, 49/48, 55/54
Comma list: 28/27, 40/39, 49/48, 55/54


Mapping: [{{val| 5 8 0 14 29 7 }}, {{val| 0 0 1 0 -1 1 }}]
Mapping: {{mapping| 5 8 0 14 29 7 | 0 0 1 0 -1 1 }}


Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 391.037
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 391.037


{{Optimal ET sequence|legend=1| 5, 10, 15, 25e, 40bef }}
{{Optimal ET sequence|legend=1| 5, 10, 15, 25e, 40bef }}
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Comma list: 28/27, 49/48, 77/75
Comma list: 28/27, 49/48, 77/75


Mapping: [{{val| 5 8 0 14 -6 }}, {{val| 0 0 1 0 2 }}]
Mapping: {{mapping| 5 8 0 14 -6 | 0 0 1 0 2 }}


Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 398.070
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 398.070


{{Optimal ET sequence|legend=1| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=1| 5e, 10e, 15 }}
Line 80: Line 78:
Comma list: 28/27, 40/39, 49/48, 66/65
Comma list: 28/27, 40/39, 49/48, 66/65


Mapping: [{{val| 5 8 0 14 -6 7 }}, {{val| 0 0 1 0 2 1 }}]
Mapping: {{mapping| 5 8 0 14 -6 7 | 0 0 1 0 2 1 }}


Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 396.812
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 396.812


{{Optimal ET sequence|legend=1| 5e, 10e, 15 }}
{{Optimal ET sequence|legend=1| 5e, 10e, 15 }}
Line 93: Line 91:
Comma list: 28/27, 35/33, 49/48
Comma list: 28/27, 35/33, 49/48


Mapping: [{{val| 5 8 0 14 6 }}, {{val| 0 0 1 0 1 }}]
Mapping: {{mapping| 5 8 0 14 6 | 0 0 1 0 1 }}


Optimal tuning (POTE): ~9/8 = 1\5, ~5/4 = 374.763
Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 374.763


{{Optimal ET sequence|legend=1| 5e, 10 }}
{{Optimal ET sequence|legend=1| 5e, 10 }}
Line 108: Line 106:
[[Comma list]]: {{monzo| 8 -5 }} = 256/243
[[Comma list]]: {{monzo| 8 -5 }} = 256/243


[[Sval]] [[mapping]]: [{{val| 5 8 0 }}, {{val| 0 0 1 }}]
{{Mapping|legend=2| 5 8 0 | 0 0 1 }}


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


[[Optimal tuning]] ([[subgroup POTE]]): ~11/7 = 786.2215
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~11/7 = 786.2215


{{Optimal ET sequence|legend=1| 15, 20, 35b }}
{{Optimal ET sequence|legend=1| 15, 20, 35b }}

Revision as of 06:55, 22 September 2023

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

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]]

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

Optimal ET sequence: 5e, 10

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

Subgroup-val mapping: [⟨5 8 0], ⟨0 0 1]]

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

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

Optimal ET sequence: 15, 20, 35b

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