Trisedodge family

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The trisedodge family of temperaments tempers out the trisedodge comma (monzo: [19 10 -15⟩, ratio: 30 958 682 112 / 30 517 578 125).

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.

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, 6/5 = (144/125)⋅(25/24).

Subgroup: 2.3.5

Comma list: 30958682112/30517578125

Mapping: [⟨5 1 7], ⟨0 3 2]]

mapping generators: ~144/125, ~864/625

Optimal tunings:

• WE: ~144/125 = 239.9482 ¢, ~864/625 = 553.8881 ¢ (~25/24 = 73.9917 ¢)

error map: ⟨-0.259 -0.342 +1.100]

• CWE: ~144/125 = 240.0000 ¢, ~864/625 = 553.9342 ¢ (~25/24 = 73.9342 ¢)

error map: ⟨0.000 -0.152 +1.555]

Optimal ET sequence: 15, 35, 50, 65, 340c, 405c, …, 600c

Badness (Sintel): 5.93

Overview to extensions

The second comma of the comma list defines which 7-limit family member we are looking at. Among these are septimal trisedodge (65d & 80), which adds 4000/3993, and coblack (50 & 65), which adds 126/125. Remarkably, septimal trisedodge admits an extension to the full 29-limit, which, except for prime 13, is obvious and simple a way to extend the 11-limit representation.

Temperaments discussed elsewhere include quindecic and decistearn. Considered below are trisedodge and coblack.

Septimal trisedodge and coblack have the common 2.3.5.11 subgroup restriction, called countdown, considered immediately below. In this temperament, 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 in 2 generator steps.

Countdown

Subgroup: 2.3.5.11

Comma list: 4000/3993, 6912/6875

Subgroup-val mapping: [⟨5 1 7 15], ⟨0 3 2 1]]

Optimal tunings:

• WE: ~55/48 = 240.0000 ¢, ~11/8 = 553.9041 ¢ (~25/24 = 74.0952 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 554.0109 ¢ (~25/24 = 74.0109 ¢)

Optimal ET sequence: 15, 35, 50, 65, 210e, 275e, 340ce

Badness (Sintel): 0.794

Septimal trisedodge

We can extend trisedodge to the 17-limit by using the sharp tendency of prime 5 to justify tempering out 256/255 (S16). Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings. We can then extend it to the 19-limit by tempering out 361/360 (S19) or equivalently 400/399 (S20), whose naturalness becomes much clearer when we consider it in the 23-limit, where we equate 23/19 with a stack of two 11/10's, tempering out 2300/2299 (S20/S22), relying on the obvious mapping of 23/16 as one period above 5/4 so that ~23/20 is tuned to 1\5. The mapping of 23 also implies tempering out 276/275 (the difference between 55/48 and 23/20), which is 3025/3024 flat of 253/252. Finally, there is an obvious mapping for 29/16 as two periods above 11/8 so that ~29/22 is tuned to 2\5 and that ~32/29 is equated with ~11/10, the generator.

This defines trisedodge as being an unambiguously full 29-limit temperament, with an interesting feature of having possible alternative mappings for primes 7 and 13. Prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as in coblack, while prime 13 can alternatively 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 generator chain 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

Comma list: 4000/3969, 110592/109375

Mapping: [⟨5 1 7 21], ⟨0 3 2 -3]]

Optimal tunings:

• WE: ~144/125 = 239.7187 ¢, ~175/128 = 554.2976 ¢ (~25/24 = 74.8601 ¢)

error map: ⟨-1.406 +0.656 +0.312 +2.374]

• CWE: ~144/125 = 240.0000 ¢, ~175/128 = 554.8511 ¢ (~25/24 = 74.8511 ¢)

error map: ⟨0.000 +2.598 +3.388 +6.621]

Optimal ET sequence: 15, 50d, 65d, 80

Badness (Sintel): 3.48

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:

• WE: ~55/48 = 239.7335 ¢, ~11/8 = 554.3239 ¢ (~25/24 = 74.8569 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 554.8505 ¢ (~25/24 = 74.8505 ¢)

Optimal ET sequence: 15, 50d, 65d, 80

Badness (Sintel): 1.44

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:

• WE: ~55/48 = 239.7764 ¢, ~11/8 = 554.1429 ¢ (~25/24 = 74.5902 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 554.6627 ¢ (~25/24 = 74.6627 ¢)

Optimal ET sequence: 15f, 50df, 65d, 80, 145d

Badness (Sintel): 1.84

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 1040/1029, 1331/1323

Mapping: [⟨5 1 7 21 15 37 32], ⟨0 3 2 -3 1 -8 -5]]

Optimal tunings:

• CTE: ~55/48 = 239.7935 ¢, ~11/8 = 554.1209 ¢ (~25/24 = 74.5340 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 554.6141 ¢ (~25/24 = 74.6141 ¢)

Optimal ET sequence: 15f, 50dfg, 65d, 80, 145d

Badness (Sintel): 1.61

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 190/189, 256/255, 351/350, 1040/1029, 1331/1323

Mapping: [⟨5 1 7 21 15 37 32 12], ⟨0 3 2 -3 1 -8 -5 4]]

Optimal tunings:

• WE: ~55/48 = 239.8147 ¢, ~11/8 = 554.2353 ¢ (~25/24 = 74.6060 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 554.6675 ¢ (~25/24 = 74.6675 ¢)

Optimal ET sequence: 15f, 65d, 80

Badness (Sintel): 1.54

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 190/189, 253/252, 256/255, 351/350, 1040/1029, 1331/1323

Mapping: [⟨5 1 7 21 15 37 32 12 18], ⟨0 3 2 -3 1 -8 -5 4 2]]

Optimal tunings:

• WE: ~23/20 = 239.8299 ¢, ~11/8 = 554.2946 ¢ (~24/23 = 74.6347 ¢)
• CWE: ~23/20 = 240.0000 ¢, ~11/8 = 554.6878 ¢ (~24/23 = 74.6878 ¢)

Optimal ET sequence: 15f, 65d, 80

Badness (Sintel): 1.46

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 176/175, 190/189, 232/231, 253/252, 256/255, 351/350, 1040/1029, 1331/1323

Mapping: [⟨5 1 7 21 15 37 32 12 18 22], ⟨0 3 2 -3 1 -8 -5 4 2 1]]

Optimal tunings:

• WE: ~23/20 = 239.8259 ¢, ~11/8 = 554.2822 ¢ (~24/23 = 74.6304 ¢)
• CWE: ~23/20 = 240.0000 ¢, ~11/8 = 554.6835 ¢ (~24/23 = 74.6835 ¢)

Optimal ET sequence: 15f, 65d, 80

Badness (Sintel): 1.38

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:

• WE: ~55/48 = 239.7425 ¢, ~11/8 = 554.6648 ¢ (~25/24 = 75.1797 ¢)
• CWE: ~55/48 = 240.0000 ¢, ~11/8 = 555.1626 ¢ (~25/24 = 75.1626 ¢)

Optimal ET sequence: 15, 80, 175bcde, 255bcdde

Badness (Sintel): 1.57

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 blackwood.

Subgroup: 2.3.5.7

Comma list: 126/125, 16807/16384

Mapping: [⟨5 1 7 14], ⟨0 3 2 0]]

Optimal tunings:

• WE: ~8/7 = 240.2499 ¢, ~48/35 = 553.6203 ¢ (~21/20 = 73.1204 ¢)

error map: ⟨+1.250 -0.844 +2.676 -5.327]

• CWE: ~8/7 = 240.0000 ¢, ~48/35 = 553.2893 ¢ (~21/20 = 73.2893 ¢)

error map: ⟨0.000 -2.087 +0.265 -8.826]

Optimal ET sequence: 15, 35, 50, 65, 115d

Badness (Sintel): 2.71

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:

• WE: ~8/7 = 240.1524 ¢, ~11/8 = 553.6154 ¢ (~21/20 = 73.3106 ¢)
• CWE: ~8/7 = 240.0000 ¢, ~11/8 = 553.3989 ¢ (~21/20 = 73.3989 ¢)

Optimal ET sequence: 15, 35, 50, 65, 115d

Badness (Sintel): 1.49