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.
Remarkably, trisedodge admits an extension to the full 29-limit which, except for prime 13, is surprisingly obvious/simple a way to extend the 11-limit representation.
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]]
Wedgie: ⟨⟨ 15 10 -15 -19 -66 -63 ]]
- 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
13-limit
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
17-limit
We extend to prime 17 by using the sharp tendency of prime 5 to justify tempering out (16/15)/(17/16) = S16. Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings.
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255
Mapping: [⟨5 1 7 21 15 37 32], ⟨0 3 2 -3 1 -8 -5]]
Optimal tunings:
- CTE: ~55/48 = 1\5, ~11/8 = 554.722 (~25/24 = 74.722)
- CWE: ~55/48 = 1\5, ~11/8 = 554.614 (~25/24 = 74.614)
Optimal ET sequence: 15f, 50dfg, 65d, 80, 145d
Badness: ?
Badness (Dirichlet): 1.609
19-limit
We extend to prime 19 by tempering out 361/360 = S19 or equivalently 400/399 = S20, whose naturalness becomes much clearer when we consider it in the 23-limit as the result of tempering out (23/19)/(22/20)2 = S20/S22, relying on the surprisingly obvious mapping of 23/16 as one period above 5/4 so that ~23/20 = 1\5.
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189,
Mapping: [⟨5 1 7 21 15 37 32 12], ⟨0 3 2 -3 1 -8 -5 4]]
Optimal tunings:
- CTE: ~55/48 = 1\5, ~11/8 = 554.698 (~25/24 = 74.698)
- CWE: ~55/48 = 1\5, ~11/8 = 554.667 (~25/24 = 74.667)
Optimal ET sequence: 15f, 65d, 80
Badness: ?
Badness (Dirichlet): 1.542
23-limit
As mentioned, prime 23 is found as prime 5 plus a period (up to octave-equivalence). This id done by tempering out (55/48)/(23/20) = 276/275 which is 3025/3024 flat of 253/252. Curiously, the CTE and CWE tunings are almost exactly the same here (different by about a hundredth of a cent).
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189, 253/252
Mapping: [⟨5 1 7 21 15 37 32 12 18], ⟨0 3 2 -3 1 -8 -5 4 2]]
Optimal tunings:
- CTE: ~23/20 = 1\5, ~11/8 = 554.689 (~24/23 = 74.689)
- CWE: ~23/20 = 1\5, ~11/8 = 554.688 (~24/23 = 74.688)
Optimal ET sequence: 15f, 65d, 80
Badness: ?
Badness (Dirichlet): 1.463
29-limit
There's a surprisingly obvious mapping of 29/16 as two periods above 11/8 so that ~29/22 = 2\5 and meaning equating ~32/29 with ~11/10, the generator. This defines trisedodge as being an unambiguously full 29-limit temperament, with an interesting feature of having two possible mappings of prime 7 and 13; prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as it is approximated in 5edo, while prime 13 can alternately be found as 8 generators up instead of down, corresponding to #Trisey, though using both of those mappings simultaneously only really makes sense in 80edo, which is a reasonable edo tuning for it and happens to correspond to the 80-note MOS of trisedodge required for finding every prime relative to the same root, though note that 11/10 is practically just there so that intervals of 29 require error cancellation of the oversharp 29th harmonic to help justify harmonically
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189, 253/252, 232/231
Mapping: [⟨5 1 7 21 15 37 32 12 18 22], ⟨0 3 2 -3 1 -8 -5 4 2 1]]
Optimal tunings:
- CTE: ~23/20 = 1\5, ~11/8 = 554.673 (~24/23 = 74.673)
- CWE: ~23/20 = 1\5, ~11/8 = 554.684 (~24/23 = 74.684)
Optimal ET sequence: 15f, 65d, 80
Badness: ?
Badness (Dirichlet): 1.380
Trisey
Note that trisey can be extended to the full 29-limit by following canonical trisedodge extension path; 80edo is a good tuning for merging trisedodge and 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
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