Mint temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of low complexity, high error, temperaments which temper 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. These temperaments equate 6/5 with 7/6, 5/4 with 9/7, and 7/4 with 9/5, so minor and major thirds and sixths are intervals of 5 and 7 at the same time.

Temperaments discussed elsewhere include

• Father (+16/15) → Father family
• Dominant (+64/63) → Meantone family
• Armodue (+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) → Diminished family
• Smate (+2048/1875) → Smate family
• Darkstone (+1875/1792) → Magic family
• Ripple (+2560/2401) → Ripple family
• Whitewood (+2187/2048) → Whitewood family

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

Optimal tunings:

• 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 (Sintel): 1.1864

Progression

Not to be confused with Progress.

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

Optimal tunings:

• 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 (Sintel): 1.2237

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 (Sintel): 0.8612

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 (Sintel): 0.7503

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 (Sintel): 0.8529

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 (Sintel): 1.0244

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

Optimal tunings:

• 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 (Sintel): 1.5463

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 (Sintel): 1.4761

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 (Sintel): 1.3357

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 (Sintel): 1.5620

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 (Sintel): 1.5166

Naian

For the 5-limit version, see Miscellaneous 5-limit temperaments #Naian.

Subgroup: 2.3.5.7

Comma list: 36/35, 9604/9375

Mapping: [⟨1 5 5 7], ⟨0 -9 -7 -11]]

mapping generators: ~2, ~98/75

Optimal tunings:

• CTE: ~2 = 1200.000, ~98/75 = 456.494

error map: ⟨0.000, -10.401, +18.228, +9.740]

• CWE: ~2 = 1200.000, ~98/75 = 456.012

error map: ⟨0.000, -6.059, +21.606, +15.047]

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 2.8912

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 2541/2500

Mapping: [⟨1 5 5 7 5], ⟨0 -9 -7 -11 -4]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~77/60 = 456.625
• CWE: ~2 = 1200.000, ~77/60 = 455.798

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 1.9522

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 66/65, 507/500

Mapping: [⟨1 5 5 7 5 6], ⟨0 -9 -7 -11 -4 -6]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~13/10 = 456.767
• CWE: ~2 = 1200.000, ~13/10 = 455.730

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 1.5538

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 36/35, 51/50, 56/55, 66/65, 170/169

Mapping: [⟨1 5 5 7 5 6 6], ⟨0 -9 -7 -11 -4 -6 -5]]

Optimal tunings:

• CTE: ~2 = 1200.000, ~13/10 = 456.821
• CWE: ~2 = 1200.000, ~13/10 = 455.732

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 1.4332

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

Optimal tunings:

• 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 (Sintel): 2.9070

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 (Sintel): 1.6700

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 (Sintel): 1.3678

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

Optimal tunings:

• 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 (Sintel): 7.7857

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 (Sintel): 4.2942

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 (Sintel): 3.1927