Mint temperaments

(Redirected from Progression)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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]]

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

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

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