Trisedodge family: Difference between revisions
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→Septimal trisedodge: add 13-limit & CTE tunings |
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{{Mapping|legend=1| 5 1 7 21 | 0 3 2 -3 }} | {{Mapping|legend=1| 5 1 7 21 | 0 3 2 -3 }} | ||
[[Optimal tuning]] ([[POTE]] | [[Optimal tuning]]s (all have ~144/125 = 1\5): | ||
* [[POTE]]: ~25/24 = 74.9480 | |||
* [[CTE]]: ~25/24 = 74.515 | |||
{{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | ||
Badness (Dirichlet): 3.485 | |||
=== 11-limit === | === 11-limit === | ||
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Mapping: {{mapping| 5 1 7 21 15 | 0 3 2 -3 1 }} | Mapping: {{mapping| 5 1 7 21 15 | 0 3 2 -3 1 }} | ||
Optimal | Optimal tunings: (all have 1\5 = ~55/48) | ||
* POTE: ~25/24 = 74.9401 | |||
* CTE: ~25/24 = 74.466 | |||
{{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | ||
Badness: 0. | Badness (Dirichlet): 1.438 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 176/175, 352/351, 1040/1029, 3025/3024 | |||
Mapping: {{mapping| 5 1 7 21 15 37 | 0 3 2 -3 1 -8 }} | |||
Optimal tuning ([[CTE]]): ~25/24 = 74.6 | |||
{{Optimal ET sequence|legend=1| 15f, 50df, 65d, 80, 145d }} | |||
Badness (Dirichlet): 1.843 | |||
== Coblack == | == Coblack == | ||
Revision as of 20:29, 15 June 2024
The trisedodge family tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15⟩.
Temperaments discussed elsewhere include quindecic and decistearn.
Trisedodge
Subgroup: 2.3.5
Comma list: 30958682112/30517578125
Mapping: [⟨5 1 7], ⟨0 3 2]]
- mapping generators: ~144/125, ~864/625
Optimal tuning (POTE): ~144/125 = 1\5, ~25/24 = 74.0077
Optimal ET sequence: 15, 50, 65, 340c, 405c, 470c, 535c, 600c
Badness: 0.252724
Countdown
Subgroup: 2.3.5.11
Comma list: 4000/3993, 6912/6875
Sval mapping: [⟨5 1 7 15], ⟨0 3 2 1]]
Optimal tuning (POTE): ~55/48 = 1\5, ~25/24 = 74.1247
Optimal ET sequence: 15, 35, 50, 65, 210e, 275e, 340ce
RMS error: 0.3198 cents
Septimal trisedodge
Subgroup: 2.3.5.7
Comma list: 4000/3969, 110592/109375
Mapping: [⟨5 1 7 21], ⟨0 3 2 -3]]
Optimal tunings (all have ~144/125 = 1\5):
Optimal ET sequence: 15, 50d, 65d, 80
Badness (Dirichlet): 3.485
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 1331/1323, 2560/2541
Mapping: [⟨5 1 7 21 15], ⟨0 3 2 -3 1]]
Optimal tunings: (all have 1\5 = ~55/48)
- POTE: ~25/24 = 74.9401
- CTE: ~25/24 = 74.466
Optimal ET sequence: 15, 50d, 65d, 80
Badness (Dirichlet): 1.438
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 352/351, 1040/1029, 3025/3024
Mapping: [⟨5 1 7 21 15 37], ⟨0 3 2 -3 1 -8]]
Optimal tuning (CTE): ~25/24 = 74.6
Optimal ET sequence: 15f, 50df, 65d, 80, 145d
Badness (Dirichlet): 1.843
Coblack
In addition to 126/125, the coblack temperament tempers out the cloudy comma, 16807/16384, which is the amount by which five septimal supermajor seconds (8/7) fall short of an octave.
Subgroup: 2.3.5.7
Comma list: 126/125, 16807/16384
Mapping: [⟨5 1 7 14], ⟨0 3 2 0]]
Optimal tuning (POTE): ~8/7 = 1\5, ~21/20 = 73.044
Optimal ET sequence: 15, 35, 50, 65, 115d
Badness: 0.107282
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/242, 385/384
Mapping: [⟨5 1 7 14 15], ⟨0 3 2 0 1]]
Optimal tuning (POTE): ~8/7 = 1\5, ~21/20 = 73.264
Optimal ET sequence: 15, 35, 50, 65, 115d
Badness: 0.045070