Octagar temperaments
This is a collection of rank-2 temperaments tempering out the octagar comma, [5 -4 3 -2⟩ = 4000/3969.
Temperaments considered below are slendi and slithy. Discussed elsewhere are:
- Ripple (+36/35) → Ripple family
- Nautilus (+49/48) → Porcupine family
- Injera (+50/49 or 81/80) → Meantone family
- Augene (+64/63 or 126/125) → Augmented family
- Garibaldi (+225/224) → Schismatic family
- Octacot (+245/243) → Tetracot family
- Roman (+525/512) → Wesley family
- Superkleismic (+875/864 or 1029/1024) → Shibboleth family and Gamelismic clan
- Quartonic (+1728/1715) → Quartonic family
- Bidia (+2048/2025 or 3136/3125) → Diaschismic family
- Hamity (+2430/2401) → Amity family
- Hemikleismic (+6144/6125) → Kleismic family
- Pluto (+10976/10935) → Mirkwai clan
- Quintupole (+458752/455625) → Quintaleap family
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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.181
Optimal ET sequence: 41, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.180
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]]
Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 29.125
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.282
Optimal ET sequence: 27, 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]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.291
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 89.286
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.9605
Optimal ET sequence: 26, 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]]
Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.7028
Optimal ET sequence: 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]]
Optimal tuning (POTE): ~21/20 = 1\13, ~3/2 = 701.3573
Optimal ET sequence: 26, 39df, 65d
Badness: 0.037851
Dhaivatic
Subgroup: 2.3.5.7
Comma list: 4000/3969, 2100875/2097152
Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 319.198
Optimal ET sequence: 15, 64c, 79, 94, 203, 297
Badness: 0.168373
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1331/1323, 4000/3969
Mapping: [⟨1 -4 -3 6 0], ⟨0 21 20 -12 13]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 319.197
Optimal ET sequence: 15, 64c, 79, 94, 203, 297c
Badness: 0.051216
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 275/273, 325/324, 385/384, 1331/1323
Mapping: [⟨1 -4 -3 6 0 -8], ⟨0 21 20 -12 13 44]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 319.165
Optimal ET sequence: 15, 49bceff, 64cf, 79, 94
Badness: 0.037468