Porcupine rank three family: Difference between revisions

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These are the rank-3 temperaments tempering out the porcupine comma or maximal diesis, [[250/243]]. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.  
The '''porcupine rank-3 family''' of [[temperament]]s [[tempering out|tempers out]] the porcupine comma or maximal diesis, [[250/243]]. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.  


== Rank-3 porcupine ==
== Rank-3 porcupine ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: [[250/243]]
[[Comma list]]: [[250/243]]


[[Mapping]]: [{{val| 1 2 3 0 }}, {{val| 0 -3 -5 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 1 2 3 0 | 0 -3 -5 0 | 0 0 0 1 }}


Mapping generators: ~2, ~10/9, ~7
: mapping generators: ~2, ~10/9, ~7


[[POTE generator]]s: ~10/9 = 163.9504, ~7/4 = 970.0552
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 163.9504, ~7/4 = 970.0552


{{Optimal ET sequence|legend=1| 7d, 8d, 14c, 15, 22, 37, 51, 73c }}
{{Optimal ET sequence|legend=1| 7d, 8d, 14c, 15, 22, 37, 51, 73c }}
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== Sonic ==
== Sonic ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 55/54, 100/99
[[Comma list]]: 55/54, 100/99


[[Mapping]]: [{{val| 1 2 3 0 4 }}, {{val| 0 -3 -5 0 -4 }}, {{val| 0 0 0 1 0 }}]
{{Mapping|legend=1| 1 2 3 0 4 | 0 -3 -5 0 -4 | 0 0 0 1 0 }}


[[POTE generator]]s: ~11/10 = 164.0777, ~7/4 = 967.8413
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/10 = 164.0777, ~7/4 = 967.8413


{{Optimal ET sequence|legend=1| 7d, 8d, 14c, 15, 22, 37, 51, 88b }}
{{Optimal ET sequence|legend=1| 7d, 8d, 14c, 15, 22, 37, 51, 88b }}
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== Tikal ==
== Tikal ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 56/55, 250/243
[[Comma list]]: 56/55, 250/243


[[Mapping]]: [{{val| 1 2 3 0 0 }}, {{val| 0 -3 -5 0 5 }}, {{val| 0 0 0 1 1 }}]
{{Mapping|legend=1| 1 2 3 0 0 | 0 -3 -5 0 5 | 0 0 0 1 1 }}


[[POTE generator]]s: ~10/9 = 161.5193, ~7/4 = 971.0457
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 161.5193, ~7/4 = 971.0457


{{Optimal ET sequence|legend=1| 7d, 8d, 15, 30dee, 37ee, 52bee }}
{{Optimal ET sequence|legend=1| 7d, 8d, 15, 30dee, 37ee, 52bee }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Porcupine rank-3 family| ]] <!-- main article -->
[[Category:Porcupine rank-3 family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 12:17, 11 September 2023

The porcupine rank-3 family of temperaments tempers out the porcupine comma or maximal diesis, 250/243. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Rank-3 porcupine

Subgroup: 2.3.5.7

Comma list: 250/243

Mapping: [⟨1 2 3 0], ⟨0 -3 -5 0], ⟨0 0 0 1]]

mapping generators: ~2, ~10/9, ~7

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 163.9504, ~7/4 = 970.0552

Optimal ET sequence: 7d, 8d, 14c, 15, 22, 37, 51, 73c

Badness: 0.535 × 10-3

Sonic

Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99

Mapping: [⟨1 2 3 0 4], ⟨0 -3 -5 0 -4], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 164.0777, ~7/4 = 967.8413

Optimal ET sequence: 7d, 8d, 14c, 15, 22, 37, 51, 88b

Badness: 0.523 × 10-3

Tikal

Subgroup: 2.3.5.7.11

Comma list: 56/55, 250/243

Mapping: [⟨1 2 3 0 0], ⟨0 -3 -5 0 5], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 161.5193, ~7/4 = 971.0457

Optimal ET sequence: 7d, 8d, 15, 30dee, 37ee, 52bee

Badness: 1.973 × 10-3

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