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 ==

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[[Comma list]]: 256/243

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

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

: mapping generators: ~9/8, ~5

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

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[[Comma list]]: 28/27, 49/48

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

~~Mapping generators: ~7/6, ~5~~

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

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

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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 }}

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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

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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

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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

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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

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[[Comma list]]: {{monzo| 8 -5 }} = 256/243

~~[[Sval]] [[mapping]]: [~~{{~~val~~| 5 8 0 ~~}}, {{val~~| 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

@@ Line 1: / Line 1: @@
-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 ==
@@ Line 7: / Line 7: @@
 [[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
@@ Line 24: / Line 24: @@
 [[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 }}
-[[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 }}
@@ Line 41: / Line 39: @@
 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 }}
@@ Line 54: / Line 52: @@
 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 }}
@@ Line 67: / Line 65: @@
 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 }}
@@ Line 80: / Line 78: @@
 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 }}
@@ Line 93: / Line 91: @@
 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 }}
@@ Line 108: / Line 106: @@
 [[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 }}