# 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]]

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

#### 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

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