Trisedodge family: Difference between revisions
→13-limit: correction (logged wrong extension) |
→Septimal trisedodge: we haven't agreed to replace Smith's badness with another. |
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{{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | ||
Badness | Badness: 0.137695 | ||
=== 11-limit === | === 11-limit === | ||
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{{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | {{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }} | ||
Badness | Badness: 0.043508 | ||
=== 13-limit === | === 13-limit === | ||
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{{Optimal ET sequence|legend=1| 15, 80, 175bcde, 255bcdde }} | {{Optimal ET sequence|legend=1| 15, 80, 175bcde, 255bcdde }} | ||
Badness | Badness: 0.0380 | ||
== Coblack == | == Coblack == | ||
Revision as of 13:15, 16 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
The generator of this temperament, when reduced to be as small as possible (by using period-reduction and period-complement-reduction) is what is shown in the optimal tunings, however note that the generator is also interpretable as ~6/5 (reached by a period plus 25/24, that is, 144/125 * 25/24 = 6/5). Other possible generators are described below in the limits in which they appear.
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: 0.137695
11-limit
In the 11-limit the generator can be taken to be ~11/10 (reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10). Therefore, because a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 at 2 gens.
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 = ~63/55~55/48)
- POTE: ~25/24 = 74.9401
- CTE: ~25/24 = 74.466
Optimal ET sequence: 15, 50d, 65d, 80
Badness: 0.043508
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 640/637, 833/825
Mapping: [⟨5 1 7 21 15 0], ⟨0 3 2 -3 1 8]]
Optimal tunings:
- POTE: ~25/24 = 75.261
- CTE: ~25/24 = 74.741
Optimal ET sequence: 15, 80, 175bcde, 255bcdde
Badness: 0.0380
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