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. This equates very different intervals with each other—6/5 with 7/6, 5/4 with 9/7, and 7/4 with 9/5. In a sense, what mint is all about is pretending that minor and major thirds and sixths are 5-limit and 7-limit at the same time, and like any temperament that seems to involve "pretending", mint is an exotemperament as it is not far from the edge of what can be sensibly called a temperament at all.
Temperaments discussed elsewhere include
- Father (+16/15) → Father family
- Dominant (+64/63) → Meantone family
- Hexadecimal (+135/128) → Mavila family
- Dicot (+25/24) → Dicot family
- Beep (+21/20) → Bug family
- August (+128/125) → Augmented family
- Gorgo (+1029/1024) → Gamelismic clan
- Hystrix (+160/147) → Porcupine family
- Diminished (+50/49) → Dimipent family
- Smate (+2048/1875) → Smate family
- Darkstone (+1875/1792) → Magic family
- Ripple (+2560/2401) → Ripple family
Progression
- For the 5-limit version, see Miscellaneous 5-limit 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 ]]
- CTE: ~2 = 1200.000, ~15/14 = 138.611
- error map: ⟨0.000 -8.900 +29.519 +1.451]
- POTE: ~2 = 1200.000, ~15/14 = 140.927
- error map: ⟨0.000 +2.680 +36.467 +17.663]
Optimal ET sequence: 8d, 9, 17c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~12/11 = 138.556
- POTE: ~2 = 1200.000, ~12/11 = 140.747
Optimal ET sequence: 8d, 9, 17c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~13/12 = 138.741
- POTE: ~2 = 1200.000, ~13/12 = 140.751
Optimal ET sequence: 8d, 9, 17c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~13/12 = 138.649
- POTE: ~2 = 1200.000, ~13/12 = 141.404
Optimal ET sequence: 8d, 9, 17cg
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~13/12 = 138.750
- POTE: ~2 = 1200.000, ~13/12 = 140.479
Optimal ET sequence: 8d, 9, 17cg
Badness (Smith): 0.016841
Subklei
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Delorean.
Subgroup: 2.3.5.7
Comma list: 36/35, 1029/1000
Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]
- mapping generators: ~2, ~7/6
Wedgie: ⟨⟨ 6 7 5 -3 -9 -8 ]]
- CTE: ~2 = 1200.000, ~7/6 = 284.986
- error map: ⟨0.000 -11.870 +18.785 +6.245]
- POTE: ~2 = 1200.000, ~7/6 = 284.219
- error map: ⟨0.000 -7.268 +24.154 +10.080]
Optimal ET sequence: 4, 13cd, 17c, 21, 38c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~7/6 = 284.265
- POTE: ~2 = 1200.000, ~7/6 = 283.253
Optimal ET sequence: 4e, …, 13cdee, 17c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~7/6 = 283.918
- POTE: ~2 = 1200.000, ~7/6 = 282.856
Optimal ET sequence: 4ef, …, 13cdeef, 17c
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~7/6 = 284.870
- POTE: ~2 = 1200.000, ~7/6 = 283.822
Optimal ET sequence: 4, 17c, 21, 38ce
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~7/6 = 284.758
- POTE: ~2 = 1200.000, ~7/6 = 283.882
Optimal ET sequence: 4, 17c, 38ce
Badness (Smith): 0.036701
Slurpee
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Slurpee.
Subgroup: 2.3.5.7
Comma list: 36/35, 51200/50421
Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]
- mapping generators: ~2, ~21/20
Wedgie: ⟨⟨ 7 11 3 1 -15 -24 ]]
- CTE: ~2 = 1200.000, ~21/20 = 72.776
- error map: ⟨0.000 -11.385 +13.153 +12.847]
- POTE: ~2 = 1200.000, ~21/20 = 72.303
- error map: ⟨0.000 -8.075 +18.355 +14.265]
Optimal ET sequence: 16, 17c, 33
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~21/20 = 72.681
- POTE: ~2 = 1200.000, ~21/20 = 72.297
Optimal ET sequence: 16, 17c, 33
Badness (Smith): 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 tunings:
- CTE: ~2 = 1200.000, ~21/20 = 72.653
- POTE: ~2 = 1200.000, ~21/20 = 72.409
Optimal ET sequence: 16, 17c, 33
Badness (Smith): 0.033102
Penta
- For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.
Subgroup: 2.3.5.7
Comma list: 28/25, 36/35
Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]
- mapping generators: ~2, ~7/6
Wedgie: ⟨⟨ 3 2 4 -4 -2 4 ]]
- CTE: ~2 = 1200.000, ~7/6 = 231.845
- error map: ⟨0.000 -6.421 +77.376 -41.447]
- POTE: ~2 = 1200.000, ~7/6 = 239.977
- error map: ⟨0.000 +17.976 +93.640 -8.918]
Optimal ET sequence: 1bd, …, 4bcd, 5
Badness (Smith): 0.046882
Shallowtone
- For the 5-limit version, see Syntonic–chromatic equivalence continuum #Shallowtone (5-limit).
Subgroup: 2.3.5.7
Comma list: 36/35, 295245/262144
Mapping: [⟨1 0 18 -16], ⟨0 1 -10 12]]
- mapping generators: ~2, ~3
- CTE: ~2 = 1200.000, ~3/2 = 681.278
- error map: ⟨0.000 -20.677 +0.906 +6.510]
- CWE: ~2 = 1200.000, ~3/2 = 681.245
- error map: ⟨0.000 -20.710 +1.239 +6.110]
Optimal ET sequence: 7, 30b, 37b
Badness (Smith): 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 = 1200.000, ~3/2 = 681.254
- CWE: ~2 = 1200.000, ~3/2 = 681.327
Optimal ET sequence: 7, 30b, 37b
Badness (Smith): 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 = 1200.000, ~3/2 = 681.237
- CWE: ~2 = 1200.000, ~3/2 = 681.267
Optimal ET sequence: 7, 30b, 37b
Badness (Smith): 0.0773