Mint temperaments
This is a collection of low complexity, high error temperaments tempering out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance.
Temperaments discussed elsewhere include
- Beep (+21/20) → Bug family
- Dicot (+25/24) → Dicot family
- Dominant (+64/63) → Meantone family
- Father (+16/15) → Father family
- Diminished (+50/49) → Dimipent family
- August (+128/125) → Augmented family
- Hystrix (+160/147) → Porcupine family
- Hexadecimal (+135/128) → Pelogic family
- Gorgo (+1029/1024) → Gamelismic clan
- Darkstone (+1875/1792) → Magic family
Progression
- For the 5-limit version of this temperament, see High badness temperaments #Lafayette.
Progression, named by Gene Ward Smith in 2011[1] can be described as the 8d & 9 temperament, and 17edo is an obvious tuning for it.
Subgroup: 2.3.5.7
Comma list: 36/35, 392/375
Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]
- mapping generators: ~2, ~15/14
Wedgie: ⟨⟨ 5 3 7 -7 -3 8 ]]
Optimal tuning (POTE): ~2 = 1\1, ~15/14 = 140.927
Optimal ET sequence: 8d, 9, 17c
Badness: 0.048356
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 77/75
Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]
Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 140.747
Optimal ET sequence: 8d, 9, 17c
Badness: 0.026050
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 66/65
Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.751
Optimal ET sequence: 8d, 9, 17c
Badness: 0.018158
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65
Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 141.404
Optimal ET sequence: 8d, 9, 17cg
Badness: 0.016743
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.479
Optimal ET sequence: 8d, 9, 17cg
Badness: 0.016841
Ripple
Subgroup: 2.3.5.7
Comma list: 36/35, 2560/2401
Mapping: [⟨1 2 3 3], ⟨0 -5 -8 -2]]
Wedgie: ⟨⟨ 5 8 2 1 -11 -18 ]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.483
Badness: 0.059735
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 80/77, 126/121
Mapping: [⟨1 2 3 3 4], ⟨0 -5 -8 -2 -6]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385
Badness: 0.038811
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 40/39, 66/65, 147/143
Mapping: [⟨1 2 3 3 4 4], ⟨0 -5 -8 -2 -6 -3]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 98.572
Badness: 0.031639
Smate
Subgroup: 2.3.5.7
Comma list: 36/35, 2048/1875
Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]
Wedgie: ⟨⟨ 4 -1 9 -11 3 24 ]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.275
Optimal ET sequence: 3d, 11d, 14, 17c, 37ccdd
Badness: 0.077871
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 243/242
Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217
Optimal ET sequence: 3de, 14, 17c, 37ccddee
Badness: 0.042518
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 243/242
Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020
Optimal ET sequence: 3de, 14, 17c
Badness: 0.036836
Subklei
- For the 5-limit version of this temperament, see High badness temperaments #Delorean.
Subgroup: 2.3.5.7
Comma list: 36/35, 1029/1000
Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]
Wedgie: ⟨⟨ 6 7 5 -3 -9 -8 ]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 284.219
Optimal ET sequence: 4, 17c, 21, 38c
Badness: 0.061101
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 77/75, 352/343
Mapping: [⟨1 3 4 4 7], ⟨0 -6 -7 -5 -15]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.253
Optimal ET sequence: 17c, 55ccd
Badness: 0.044651
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 66/65, 352/343
Mapping: [⟨1 3 4 4 7 7], ⟨0 -6 -7 -5 -15 -14]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 282.856
Badness: 0.032324
Subkla
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 1029/1000
Mapping: [⟨1 3 4 4 3], ⟨0 -6 -7 -5 2]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.822
Optimal ET sequence: 4, 17c, 21, 38ce, 55ccde
Badness: 0.047247
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 66/65, 640/637
Mapping: [⟨1 3 4 4 3 3], ⟨0 -6 -7 -5 2 3]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.882
Optimal ET sequence: 4, 17c, 21, 38ce, 55ccde
Badness: 0.036701
Slurpee
- For the 5-limit version of this temperament, see High badness temperaments #Slurpee.
Subgroup: 2.3.5.7
Comma list: 36/35, 51200/50421
Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]
Wedgie: ⟨⟨ 7 11 3 1 -15 -24 ]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.303
Optimal ET sequence: 16, 17c, 33, 50cd, 83bccd
Badness: 0.114869
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 121/120, 352/343
Mapping: [⟨1 2 3 3 4], ⟨0 -7 -11 -3 -9]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.297
Optimal ET sequence: 16, 17c, 33, 50cd, 83bccd
Badness: 0.050514
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 66/65, 143/140, 352/343
Mapping: [⟨1 2 3 3 4 4], ⟨0 -7 -11 -3 -9 -5]]
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.409
Optimal ET sequence: 16, 17c, 33, 83bccd
Badness: 0.033102
Penta
Subgroup: 2.3.5.7
Comma list: 28/25, 36/35
Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]
Wedgie: ⟨⟨ 3 2 4 -4 -2 4 ]]
Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 239.977
Badness: 0.046882
Shallowtone
- For the 5-limit version of this temperament, see Syntonic–chromatic equivalence continuum.
Subgroup: 2.3.5.7
Comma list: 36/35, 295245/262144
Mapping: [⟨1 0 18 -16], ⟨0 1 -10 12]]
Optimal ET sequence: 7, 30b, 37b
Badness: 0.308
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 72171/65536
Mapping: [⟨1 0 18 -16 16], ⟨0 1 -10 12 -8]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 681.2538
- CWE: ~2 = 1\1, ~3/2 = 681.3267
Optimal ET sequence: 7, 30b, 37b
Badness: 0.130
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 16731/16384
Mapping: [⟨1 0 18 -16 16 -1], ⟨0 1 -10 12 -8 3]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 681.2374
- CWE: ~2 = 1\1, ~3/2 = 681.2669
Optimal ET sequence: 7, 30b, 37b
Badness: 0.0773