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

• 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 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

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]]

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

Optimal ET sequence: 17c

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, 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]]

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

For the 5-limit version, see Gamelismic clan #Gidorah.

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

Optimal ET sequence: 1bd, 5

Badness: 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]]

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 681.2780
• CWE: ~2 = 1\1, ~3/2 = 681.2447

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