Mint temperaments: Difference between revisions

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Removed redundant list at the top so it doesn't need that needs to be updated for every new temp
→Shallowtone: normalize the mapping; complete the data
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[[Comma list]]: [[Shallowtone comma|295245/262144]]
[[Comma list]]: [[Shallowtone comma|295245/262144]]


{{Mapping|legend=1|1 1 8|0 1 -10}}
{{Mapping|legend=1| 1 0 18 | 0 1 -10 }}


Mapping generators: ~2, ~3/2
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.801
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 681.8012
* [[CWE]]: ~2 = 1\1, ~3/2 = 682.6617


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b, 44b, 51b, 58bc, 65bbc, 72bbcc, 79bbcc}}
{{Optimal ET sequence|legend=1| 7, 30b, 37b, 44b, 51b, 58bc, 65bbc }}
 
[[Badness]]: 0.666


=== 7-limit ===
=== 7-limit ===
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[[Comma list]]: 36/35, 295245/262144
[[Comma list]]: 36/35, 295245/262144


{{Mapping|legend=1|1 1 8 -4|0 1 -10 12}}
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}


Mapping generators: ~2, ~3/2
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 681.2780
* [[CWE]]: ~2 = 1\1, ~3/2 = 681.2447


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.278
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
[[Badness]]: 0.308


=== 11-limit ===
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 36/35, 45/44, 295245/262144
Comma list: 36/35, 45/44, 72171/65536


{{Mapping|legend=1|1 1 8 -4 8|0 1 -10 12 -8}}
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


Mapping generators: ~2, ~3/2
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 681.2538
* CWE: ~2 = 1\1, ~3/2 = 681.3267


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.327
Optimal ET sequence: {{Optimal ET sequence| 7, 30b, 37b }}


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
Badness: 0.130


=== 13-limit ===
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 36/35, 45/44, 27/26, 34749/32768


{{Mapping|legend=1|1 1 8 -4 8 2|0 1 -10 12 -8 3}}
Comma list: 36/35, 45/44, 27/26, 16731/16384


Mapping generators: ~2, ~3/2
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.267
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 681.2374
* CWE: ~2 = 1\1, ~3/2 = 681.2669


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
Optimal ET sequence: {{Optimal ET sequence| 7, 30b, 37b }}


Badness: 0.0773


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Revision as of 08:54, 18 April 2024

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

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

See also: Ripple family § 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

Optimal ET sequence: 12

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

Optimal ET sequence: 12

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

Optimal ET sequence: 12f

Badness: 0.031639

Smate

See also: Smate family § 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

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 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

See also: 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

Subgroup: 2.3.5

Comma list: 295245/262144

Mapping: [⟨1 0 18], ⟨0 1 -10]]

mapping generators: ~2, ~3

Optimal tunings:

  • CTE: ~2 = 1\1, ~3/2 = 681.8012
  • CWE: ~2 = 1\1, ~3/2 = 682.6617

Optimal ET sequence: 7, 30b, 37b, 44b, 51b, 58bc, 65bbc

Badness: 0.666

7-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: 36/35, 45/44, 27/26, 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

  1. ↑ Yahoo! Tuning Group | Progression temperament
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