Octagar temperaments

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

Temperaments discussed elsewhere include:

• Ripple, {36/35, 2560/2401} → Ripple family
• Nautilus, {49/48, 250/243} → Porcupine family
• Injera, {50/49, 81/80} → Meantone family
• Augene, {64/63, 126/125} → Augmented family
• Garibaldi, {225/224, 3125/3087} → Schismatic family
• Octacot, {245/243, 2401/2400} → Tetracot family
• Roman, {525/512, 3125/3024} → Wesley family and Avicennmic temperaments
• Superkleismic, {875/864, 1029/1024} → Shibboleth family and Gamelismic clan
• Quartonic, {1728/1715, 4000/3969} → Orwellismic temperaments
• Bidia, {2048/2025, 3136/3125} → Diaschismic family
• Hamity, {2430/2401, 4000/3969} → Amity family
• Hemikleismic, {4000/3969, 6144/6125} → Kleismic family
• Pluto, {4000/3969, 10976/10935} → Mirkwai clan
• Quintupole, {4000/3969, 458752/455625} → Quintaleap family

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

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: Template:Val list

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: Template:Val list

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

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: Template:Val list

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: Template:Val list

Badness: 0.028492

Tridecatonic

The tridecatonic temperament (26&39d) has a period of 1/13 octave. It tempers out the tridecalimma, [-11 26 -13⟩ = 2541865828329/2500000000000 in the 5-limit.

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

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: Template:Val list

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: Template:Val list

Badness: 0.037851