Mint temperaments: Difference between revisions

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These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[Wikipedia: Triangular number|triangular number]], and this helps make 36/35 a septimal interval of considerable significance. Temperaments tempering it out include beep, dicot, dominant, father, diminished, august, hystrix, penta, hexadecimal, ripple and gorgo.
These are low complexity, high error temperaments tempering out the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[Wikipedia: Triangular number|triangular number]], and this helps make 36/35 a septimal interval of considerable significance. Temperaments tempering it out include [[Bug family #Beep|beep]], [[Dicot family|dicot]], [[Meantone family #Dominant|dominant]], [[Father family|father]], [[Dimipent family #Diminished|diminished]], [[Augmented family #August|august]], hystrix, penta, hexadecimal, ripple and gorgo.
 
= Progression =
== 5-limit (lafayette) ==


== Progression ==
=== 5-limit (lafayette) ===
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 16: Line 15:
[[Badness]]: 0.200483
[[Badness]]: 0.200483


== 7-limit ==
=== 7-limit ===
 
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 32: Line 30:
[[Badness]]: 0.048356
[[Badness]]: 0.048356


== 11-limit ==
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 42: Line 39:
POTE generators: ~12/11 = 140.747
POTE generators: ~12/11 = 140.747


{{Val list|legend=1| 8d, 9, 17c }}
Vals: {{Val list| 8d, 9, 17c }}


Badness: 0.026050
Badness: 0.026050


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 56: Line 52:
POTE generators: ~13/12 = 140.751
POTE generators: ~13/12 = 140.751


{{Val list|legend=1| 8d, 9, 17c }}
Vals: {{Val list| 8d, 9, 17c }}


Badness: 0.018158
Badness: 0.018158


== 17-limit ==
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 70: Line 65:
POTE generators: ~13/12 = 141.404
POTE generators: ~13/12 = 141.404


{{Val list|legend=1| 8d, 9, 17cg }}
Vals: {{Val list| 8d, 9, 17cg }}


Badness: 0.016743
Badness: 0.016743


== 19-limit ==
=== 19-limit ===
 
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 84: Line 78:
POTE generators: ~13/12 = 140.479
POTE generators: ~13/12 = 140.479


{{Val list|legend=1| 8d, 9, 17cg }}
Vals: {{Val list| 8d, 9, 17cg }}


Badness: 0.016841
Badness: 0.016841


= Ripple =
== Ripple ==
{{see also| Ripple family #Ripple }}
{{see also| Ripple family #Ripple }}


Line 105: Line 99:
[[Badness]]: 0.059735
[[Badness]]: 0.059735


== 11-limit ==
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 115: Line 108:
POTE generator: ~21/20 = 99.385
POTE generator: ~21/20 = 99.385


{{Val list|legend=1| 12 }}
Vals: {{Val list| 12 }}


Badness: 0.038811
Badness: 0.038811


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 129: Line 121:
POTE generator: ~21/20 = 98.572
POTE generator: ~21/20 = 98.572


{{Val list|legend=1| 12f }}
Vals: {{Val list| 12f }}


Badness: 0.031639
Badness: 0.031639


= Smate =
== Smate ==
{{see also| Smate family #Smate }}
{{see also| Smate family #Smate }}


Line 146: Line 138:
[[POTE generator]]: ~5/4 = 422.275
[[POTE generator]]: ~5/4 = 422.275


{{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }}
{{Val list|legend=1| 3d, 11d, 14, 17c, 37ccdd }}


[[Badness]]: 0.077871
[[Badness]]: 0.077871


== 11-limit ==
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 160: Line 151:
POTE generator: ~5/4 = 422.217
POTE generator: ~5/4 = 422.217


{{Val list|legend=1| 14, 17c, 37cde }}
Vals: {{Val list| 3de, 14, 17c, 37ccddee }}


Badness: 0.042518
Badness: 0.042518


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 174: Line 164:
POTE generator: ~5/4 = 423.020
POTE generator: ~5/4 = 423.020


{{Val list|legend=1| 14, 17c }}
Vals: {{Val list| 3de, 14, 17c }}


Badness: 0.036836
Badness: 0.036836


= Subklei =
== Subklei ==
== 5-limit (delorean) ==
=== 5-limit (delorean) ===
 
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 193: Line 182:
[[Badness]]: 0.355937
[[Badness]]: 0.355937


== 7-limit ==
=== 7-limit ===
 
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 209: Line 197:
[[Badness]]: 0.061101
[[Badness]]: 0.061101


== 11-limit ==
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 219: Line 206:
POTE generator: ~6/5 = 283.253
POTE generator: ~6/5 = 283.253


Vals: {{Val list| 17c, 55cd }}
Vals: {{Val list| 17c, 55ccd }}


Badness: 0.044651
Badness: 0.044651


=== 13-limit ===
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 239: Line 225:
Badness: 0.032324
Badness: 0.032324


== Subkla ==
=== Subkla ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 249: Line 234:
POTE generator: ~6/5 = 283.822
POTE generator: ~6/5 = 283.822


Vals: {{Val list| 4, 17c, 21, 38ce, 55cde }}
Vals: {{Val list| 4, 17c, 21, 38ce, 55ccde }}


Badness: 0.047247
Badness: 0.047247


=== 13-limit ===
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 263: Line 247:
POTE generator: ~6/5 = 283.882
POTE generator: ~6/5 = 283.882


Vals: {{Val list| 4, 17c, 21, 38ce, 55cde }}
Vals: {{Val list| 4, 17c, 21, 38ce, 55ccde }}


Badness: 0.036701
Badness: 0.036701


= Slurpee =
== Slurpee ==
== 5-limit ==
 
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 282: Line 264:
[[Badness]]: 0.689274
[[Badness]]: 0.689274


== 7-limit ==
=== 7-limit ===
 
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 294: Line 275:
[[POTE generator]]: ~21/20 = 72.303
[[POTE generator]]: ~21/20 = 72.303


{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
{{Val list|legend=1| 16, 17c, 33, 50cd, 83bccd }}


[[Badness]]: 0.114869
[[Badness]]: 0.114869


== 11-limit ==
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 308: Line 288:
POTE generator: ~21/20 = 72.297
POTE generator: ~21/20 = 72.297


{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
Vals: {{Val list| 16, 17c, 33, 50cd, 83bccd }}


Badness: 0.050514
Badness: 0.050514


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 322: Line 301:
POTE generator: ~21/20 = 72.409
POTE generator: ~21/20 = 72.409


{{Val list|legend=1| 16, 17c, 33, 83bcd }}
Vals: {{Val list| 16, 17c, 33, 83bccd }}


Badness: 0.033102
Badness: 0.033102


= Penta =
== Penta ==
{{see also| Gamelismic clan #Gidorah }}
{{see also| Gamelismic clan #Gidorah }}


Revision as of 11:55, 8 June 2021

These are 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 tempering it out include beep, dicot, dominant, father, diminished, august, hystrix, penta, hexadecimal, ripple and gorgo.

Progression

5-limit (lafayette)

Subgroup: 2.3.5

Comma list: 3456/3125

Mapping: [⟨1 1 2], ⟨0 5 3]]

POTE generator: ~25/24 = 140.995

Template:Val list

Badness: 0.200483

7-limit

Subgroup: 2.3.5.7

Comma list: 36/35, 392/375

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

Wedgie: ⟨⟨ 5 3 7 -7 -3 8 ]]

POTE generator: ~15/14 = 140.927

Template:Val list

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]]

POTE generators: ~12/11 = 140.747

Vals: Template:Val list

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]]

POTE generators: ~13/12 = 140.751

Vals: Template:Val list

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]]

POTE generators: ~13/12 = 141.404

Vals: Template:Val list

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]]

POTE generators: ~13/12 = 140.479

Vals: Template:Val list

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 ]]

POTE generator: ~21/20 = 99.483

Template:Val list

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]]

POTE generator: ~21/20 = 99.385

Vals: Template:Val list

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]]

POTE generator: ~21/20 = 98.572

Vals: Template:Val list

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 ]]

POTE generator: ~5/4 = 422.275

Template:Val list

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]]

POTE generator: ~5/4 = 422.217

Vals: Template:Val list

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]]

POTE generator: ~5/4 = 423.020

Vals: Template:Val list

Badness: 0.036836

Subklei

5-limit (delorean)

Subgroup: 2.3.5

Comma list: 17496/15625

Mapping: [⟨1 3 4], ⟨0 -6 -7]]

POTE generator: ~6/5 = 284.293

Template:Val list

Badness: 0.355937

7-limit

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 ]]

POTE generator: ~6/5 = 284.219

Template:Val list

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]]

POTE generator: ~6/5 = 283.253

Vals: Template:Val list

Badness: 0.044651

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 66/65, 352/343

POTE generator: ~6/5 = 282.856

Mapping: [⟨1 3 4 4 7 7], ⟨0 -6 -7 -5 -15 -14]]

POTE generator: ~6/5 = 282.856

Vals: Template:Val list

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]]

POTE generator: ~6/5 = 283.822

Vals: Template:Val list

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]]

POTE generator: ~6/5 = 283.882

Vals: Template:Val list

Badness: 0.036701

Slurpee

Subgroup: 2.3.5

Comma list: 177147/156250

Mapping: [⟨1 2 3], ⟨0 -7 -11]]

POTE generator: ~27/25 = 72.742

Template:Val list

Badness: 0.689274

7-limit

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 ]]

POTE generator: ~21/20 = 72.303

Template:Val list

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]]

POTE generator: ~21/20 = 72.297

Vals: Template:Val list

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]]

POTE generator: ~21/20 = 72.409

Vals: Template:Val list

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 ]]

POTE generator: ~6/5 = 239.977

Template:Val list

Badness: 0.046882

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