Octagar temperaments

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This is a collection of temperaments tempering out the octagar comma, [5 -4 3 -2 = 4000/3969.

Temperaments discussed elsewhere include:

Considered below are slendi and slithy.

Slendi

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [1 2 3 3], 0 -17 -28 -8]]

Wedgie⟨⟨17 28 8 5 -35 -60]]

POTE generator: ~49/48 = 29.181

Optimal GPV sequence41, 206, 247bc, 288bcd, 329bcd

Badness: 0.142903

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 46656/46585

Mapping: [1 2 3 3 4], 0 -17 -28 -8 -22]]

POTE generator: ~49/48 = 29.180

Vals: 41, 206e

Badness: 0.063217

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 245/242, 343/338

Mapping: [1 2 3 3 4 4], 0 -17 -28 -8 -22 -1]]

POTE generator: ~49/48 = 29.125

Vals: 41, 124ef, 165ef

Badness: 0.043644

Slithy

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [1 3 5 4], 0 -19 -36 -16]]

Wedgie⟨⟨19 36 16 13 -28 -64]]

POTE generator: ~21/20 = 89.282

Optimal GPV sequence27, 94, 121, 215, 336d

Badness: 0.107482

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187, 4375/4356

Mapping: [1 3 5 4 8], 0 -19 -36 -16 -61]]

POTE generator: ~21/20 = 89.291

Vals: 94, 121, 215, 336de, 551bde

Badness: 0.045895

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539, 1575/1573

Mapping: [1 3 5 4 8 4], 0 -19 -36 -16 -61 -4]]

POTE generator: ~21/20 = 89.286

Vals: 94, 121, 215, 336def

Badness: 0.028492

Tridecatonic

See also: High badness temperaments #Tridecatonic

The tridecatonic temperament (26&39) has a period of 1/13 octave. It tempers out the tridecalimma, [-11 26 -13 in the 5-limit. There are some extensions for 7-limit, including tridecatonic (26&39d, tempering out 4000/3969), trideci (26&39, tempering out 4375/4374), and bosonic (26&130, tempering out 321489/320000).

Subgroup: 2.3.5.7

Comma list: 4000/3969, 8748/8575

Mapping: [13 21 31 37], 0 -1 -2 -1]]

POTE generator: ~3/2 = 701.9605

Optimal GPV sequence26, 39d, 65d

Badness: 0.139213

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 1944/1925, 2560/2541

Mapping: [13 21 31 37 45], 0 -1 -2 -1 0]]

POTE generator: ~3/2 = 701.7028

Vals: 26, 39d, 65d

Badness: 0.063257

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 275/273, 1040/1029

Mapping: [13 21 31 37 45 49], 0 -1 -2 -1 0 -2]]

POTE generator: ~3/2 = 701.3573

Vals: 26, 39df, 65d

Badness: 0.037851