Trisedodge family
The trisedodge family tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15⟩.
Named by Petr Pařízek in 2011, trisedodge (originally spelt trisedoge) means that three semidiminished octaves add up to 7/1, and that an octave is made of 5 periods[1].
Temperaments discussed elsewhere include quindecic and decistearn. Considered below are trisedodge and coblack.
Trisedodge
The generator of trisedodge is ~864/625 at around 554 cents, which in all 11-limit extensions is used to represent 11/8, and three of them and a period is equal to 3/1. This generator, when reduced to the minimal size, represents 25/24. However, another possible generator is ~6/5, reached by a period plus 25/24, that is, (144/125)(25/24) = 6/5.
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, since 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
Comma list: 30958682112/30517578125
Mapping: [⟨5 1 7], ⟨0 3 2]]
- mapping generators: ~144/125, ~864/625
- CTE: ~144/125 = 1\5, ~864/625 = 553.8249 (~25/24 = 73.8249)
- POTE: ~144/125 = 1\5, ~864/625 = 554.0077 (~25/24 = 74.0077)
Optimal ET sequence: 15, 35, 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]]
- CTE: ~55/48 = 1\5, ~11/8 = 553.7951 (~25/24 = 73.7951)
- POTE: ~55/48 = 1\5, ~11/8 = 554.1247 (~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]]
- CTE: ~144/125 = 1\5, ~175/128 = 554.5146 (~25/24 = 74.5146)
- POTE: ~144/125 = 1\5, ~175/128 = 554.9480 (~25/24 = 74.9480)
Optimal ET sequence: 15, 50d, 65d, 80
Badness: 0.137695
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:
- CTE: ~55/48 = 1\5, ~11/8 = 554.4664 (~25/24 = 74.4664)
- POTE: ~55/48 = 1\5, ~11/8 = 554.9401 (~25/24 = 74.9401)
Optimal ET sequence: 15, 50d, 65d, 80
Badness: 0.043508
Trisey
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 364/363, 640/637
Mapping: [⟨5 1 7 21 15 0], ⟨0 3 2 -3 1 8]]
Optimal tunings:
- CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
- CWE: ~55/48 = 1\5, ~11/8 = 555.1626 (~25/24 = 75.1626)
Optimal ET sequence: 15, 80, 175bcde, 255bcdde
Badness: 0.0380
Dodgy
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 1040/1029, 1331/1323
Mapping: [⟨5 1 7 21 15 37], ⟨0 3 2 -3 1 -8]]
Optimal tunings:
- CTE: ~55/48 = 1\5, ~11/8 = 554.6802 (~25/24 = 74.6802)
- CWE: ~55/48 = 1\5, ~11/8 = 554.6627 (~25/24 = 74.6627)
Optimal ET sequence: 15f, 50df, 65d, 80, 145d
Badness: 0.0446
Coblack
- See also: Cloudy clan #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. Coblack was also named by Petr Pařízek, who considered it a counterpart of blacksmith[1].
Subgroup: 2.3.5.7
Comma list: 126/125, 16807/16384
Mapping: [⟨5 1 7 14], ⟨0 3 2 0]]
Wedgie: ⟨⟨15 10 0 -19 -42 -28]]
- CTE: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)
- POTE: ~8/7 = 1\5, ~48/35 = 553.044 (~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 tunings:
- CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)
- POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)
Optimal ET sequence: 15, 35, 50, 65, 115d
Badness: 0.045070