Trisedodge family

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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 sequence15, 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 sequence15, 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):

  • POTE: ~25/24 = 74.9480
  • CTE: ~25/24 = 74.515

Optimal ET sequence15, 50d, 65d, 80

Badness (Dirichlet): 3.485

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 sequence15, 50d, 65d, 80

Badness (Dirichlet): 1.438

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 sequence15, 80, 175bcde, 255bcdde

Badness (Dirichlet): 1.570

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.

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 sequence15, 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 sequence15, 35, 50, 65, 115d

Badness: 0.045070