Mint temperaments: Difference between revisions

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Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* ''[[Beep]]'' (+21/20) → [[Bug family #Beep|Bug family]]
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Hexadecimal]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Dominant (temperament)|Dominant]]'' (+64/63) → [[Meantone family #Dominant|Meantone family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[Father]]'' (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Diminished (temperament)|Diminished]]'' (+50/49) → [[Dimipent family #Diminished|Dimipent family]]
* ''[[August]]'' (+128/125) → [[Augmented family #August|Augmented family]]
* ''[[August]]'' (+128/125) → [[Augmented family #August|Augmented family]]
* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hexadecimal]]'' (+135/128) → [[Pelogic family|Pelogic family]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Dimipent family #Diminished|Dimipent family]]
* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]
* ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]
* ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]
* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]


== Progression ==
== Progression ==
Line 85: Line 87:


Badness: 0.016841
Badness: 0.016841
== Ripple ==
{{see also| Ripple family #Ripple }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 36/35, 2560/2401
{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}
{{Multival|legend=1| 5 8 2 1 -11 -18 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 99.483
{{Optimal ET sequence|legend=1| 12 }}
[[Badness]]: 0.059735
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 36/35, 80/77, 126/121
Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385
{{Optimal ET sequence|legend=1| 12 }}
Badness: 0.038811
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 40/39, 66/65, 147/143
Mapping: {{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|legend=1| 12f }}
Badness: 0.031639
== Smate ==
{{See also| Smate family #Smate }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 36/35, 2048/1875
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}
{{Multival|legend=1| 4 -1 9 -11 3 24 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 422.275
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217
{{Optimal ET sequence|legend=1| 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: {{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|legend=1| 3de, 14, 17c }}
Badness: 0.036836


== Subklei ==
== Subklei ==

Revision as of 09:27, 1 January 2025

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

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

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]]

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

  1. ↑ Yahoo! Tuning Group | Progression temperament
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