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|36/35]]. 36 is [http://en.wikipedia.org/wiki/Square_triangular_number both] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [http://en.wikipedia.org/wiki/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.


=Progression=
= Progression =
Comma: 3456/3125
== 5-limit ==
Comma list: 3456/3125


POTE generator: ~25/24 = 140.995
POTE generator: ~25/24 = 140.995


Map: [<1 1 2|, <0 5 3|]
Mapping: [{{val| 1 1 2 }}, {{val| 0 5 3 }}]


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


Badness: 0.2005
Badness: 0.2005


==7-limit==
== 7-limit ==
Commas: 36/35, 392/375
Comma list: 36/35, 392/375


POTE generator: ~15/14 = 140.927
POTE generator: ~15/14 = 140.927


Map: [<1 1 2 2|, <0 5 3 7|]
Mapping: [{{val| 1 1 2 2 }}, {{val| 0 5 3 7 }}]


Wedgie: <<5 3 7 -7 -3 8||
Wedgie: {{wedgie| 5 3 7 -7 -3 8 }}


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


Badness: 0.0484
Badness: 0.0484


==11-limit==
== 11-limit ==
Commas: 36/35, 56/55, 77/75
Comma list: 36/35, 56/55, 77/75


[[POTE_tuning|POTE generator]]: ~15/14 = 140.747
[[POTE generator]]: ~12/11 = 140.747


Map: [<1 1 2 2 3|, <0 5 3 7 4|]
Mapping: [{{val| 1 1 2 2 3 }}, {{val| 0 5 3 7 4 }}]


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


Badness: 0.0261
Badness: 0.0261


==13-limit==
== 13-limit ==
Commas: 26/25, 36/35, 56/55, 66/65
Comma list: 26/25, 36/35, 56/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 140.751
[[POTE generator]]: ~13/12 = 140.751


Map: Map: [<1 1 2 2 3 3|, <0 5 3 7 4 6|]
Mapping: [{{val| 1 1 2 2 3 3 }}, {{val| 0 5 3 7 4 6 }}]


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


Badness: 0.0182
Badness: 0.0182


==17-limit==
== 17-limit ==
Commas: 26/25, 36/35, 51/50, 56/55, 66/65
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 141.404
[[POTE generator]]: ~13/12 = 141.404


Map: [<1 1 2 2 3 3 4|, <0 5 3 7 4 6 1|]
Mapping: [{{val| 1 1 2 2 3 3 4 }}, {{val| 0 5 3 7 4 6 1 }}]


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


Badness: 0.0167
Badness: 0.0167


===19-limit===
== 19-limit ==
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 140.479
[[POTE generator]]: ~13/12 = 140.479


Map: [<1 1 2 2 3 3 4 4|, <0 5 3 7 4 6 1 2|]
Mapping: [{{val| 1 1 2 2 3 3 4 4 }}, {{val| 0 5 3 7 4 6 1 2 }}]


EDOs: 8, 9, 17, [[94edo|94]]
{{Val list|legend=1| 8d, 9, 17cg, 94… }}


Badness: 0.0168
Badness: 0.0168


=Ripple=
= Ripple =
Comma: 6561/6250
== 5-limit ==
Comma list: 6561/6250


POTE generator: ~27/25 = 100.838
POTE generator: ~27/25 = 100.838


Map: [<1 2 3|, <0 -5 -8|]
Mapping: [{{val| 1 2 3 }}, {{val| 0 -5 -8 }}]


EDOs: 12, 71b, 83b, 95b, 107bc, 119bc
{{Val list|legend=1| 11c, 12, 71b, 83b, 95b, 107bc, 119bc }}


Badness: 0.1389
Badness: 0.1389


==7-limit==
== 7-limit ==
Commas: 36/35, 2560/2401
{{see also| Slendro clan #Hemiripple }}
{{see also| Sensamagic clan #Cohemiripple }}


[[POTE_tuning|POTE generator]]: ~21/20 = 99.483
Comma list: 36/35, 2560/2401


Map: [<1 2 3 3|, <0 -5 -8 -2|]
[[POTE generator]]: ~21/20 = 99.483


Wedgie: <<5 8 2 1 -11 -18||
Mapping: [{{val| 1 2 3 3 }}, {{val| 0 -5 -8 -2 }}]


EDOs: [[12edo|12]], [[23edo|23]]
Wedgie: {{wedgie| 5 8 2 1 -11 -18 }}
 
{{Val list|legend=1| 11c, 12 }}


Badness: 0.0597
Badness: 0.0597


==11-limit==
== 11-limit ==
Commas: 36/35, 80/77, 126/121
Comma list: 36/35, 80/77, 126/121


POTE generator: ~21/20 = ~12/11 = 99.385
POTE generator: ~21/20 = 99.385


Map: [<1 2 3 3 4|, <0 -5 -8 -2 -6|]
Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -5 -8 -2 -6 }}]


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


Badness: 0.0388
Badness: 0.0388


==13-limit==
== 13-limit ==
Commas: 36/35, 40/39, 66/65, 147/143
Comma list: 36/35, 40/39, 66/65, 147/143


POTE generator: ~21/20 = ~12/11 = ~14/13 = 98.572
POTE generator: ~21/20 = 98.572


Map: [<1 2 3 3 4 4|, <0 -5 -8 -2 -6 -3|]
Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -5 -8 -2 -6 -3 }}]


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


Badness: 0.0316
Badness: 0.0316


=Smate=
= Smate =
Comma: 2048/1875
Comma list: 2048/1875


POTE generator: ~5/4 = 420.855
POTE generator: ~5/4 = 420.855


Map: [<1 3 2|, <0 -4 1|]
Mapping: [{{val| 1 3 2 }}, {{val| 0 -4 1 }}]


EDOs: 11, 14, 17c, 20c, 37c
{{Val list|legend=1| 11, 14, 17c, 20c, 37c }}


Badness: 0.179
Badness: 0.179


==7-limit==
== 7-limit ==
Commas: 36/35, 2048/1875
Comma list: 36/35, 2048/1875


POTE generator: ~5/4 = 422.275
POTE generator: ~5/4 = 422.275


Map: [<1 3 2 6|, <0 -4 1 -9|]
Mapping: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]


Wedgie: <<4 -1 9 -11 3 24||
Wedgie: {{wedgie| 4 -1 9 -11 3 24 }}


EDOs: 14, 17c, 37cd
{{Val list|legend=1| 14, 17c, 37cd }}


Badness: 0.0779
Badness: 0.0779


==11-limit==
=== 11-limit ===
Commas: 36/35 56/55 243/242
Comma list: 36/35 56/55 243/242


POTE generator: ~5/4 = 422.217
POTE generator: ~5/4 = 422.217


Map: [<1 3 2 6 7|, <0 -4 1 -9 -10|]
Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]


EDOs: 14, 17c, 37cde
{{Val list|legend=1| 14, 17c, 37cde }}


Badness: 0.0425
Badness: 0.0425


==13-limit==
=== 13-limit ===
Commas: 26/25, 36/35, 56/55, 243/242  
Comma list: 26/25, 36/35, 56/55, 243/242  


POTE generator: ~5/4 = 423.020
POTE generator: ~5/4 = 423.020


Map: [<1 3 2 6 7 3|, <0 -4 1 -9 -10 2|]
Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]


EDOs: 14, 17c
{{Val list|legend=1| 14, 17c }}


Badness: 0.0368
Badness: 0.0368


=Hemismate=
== Hemismate ==
commas: 256/245, 392/375  
Comma list: 256/245, 392/375  


POTE generator: ~8/7 = 210.452
POTE generator: ~8/7 = 210.452


Map: [<1 3 2 3|, <0 -8 2 -1|]
Mapping: [{{val| 1 3 2 3 }}, {{val| 0 -8 2 -1 }}]


Wedgie: <<8 -2 1 -22 -21 8||
Wedgie: {{wedgie| 8 -2 1 -22 -21 8 }}


EDOs: 6, 11, 17c, 40bcd
{{Val list|legend=1| 6, 11, 17c, 40bcd }}


Badness: 0.1543
Badness: 0.1543


==11-limit==
=== 11-limit ===
Commas: 56/55, 77/75, 256/245
Comma list: 56/55, 77/75, 256/245


POTE generator: ~8/7 =  210.481
POTE generator: ~8/7 =  210.481


Map: [<1 3 2 3 4|, <0 -8 2 -1 -3|]
Mapping: [{{val| 1 3 2 3 4 }}, {{val| 0 -8 2 -1 -3 }}]


EDOs: 6, 11, 17c, 40bcde
{{Val list|legend=1| 6, 11, 17c, 40bcde }}


Badness: 0.0655
Badness: 0.0655


==13-limit==
=== 13-limit ===
Commas: 26/25, 56/55, 77/75, 256/245
Commas: 26/25, 56/55, 77/75, 256/245


POTE generator: ~8/7 = 210.974
POTE generator: ~8/7 = 210.974


Map: [<1 3 2 3 4 3|, <0 -8 2 -1 -3 4|]
Mapping: [{{val| 1 3 2 3 4 3 }}, {{val| 0 -8 2 -1 -3 4 }}]


EDOs: 6, 11, 17c
{{Val list|legend=1| 6, 11, 17c }}


Badness: 0.0505
Badness: 0.0505

Revision as of 10:11, 28 February 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

Comma list: 3456/3125

POTE generator: ~25/24 = 140.995

Mapping: [⟨1 1 2], ⟨0 5 3]]

Template:Val list

Badness: 0.2005

7-limit

Comma list: 36/35, 392/375

POTE generator: ~15/14 = 140.927

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

Wedgie: ⟨⟨5 3 7 -7 -3 8]]

Template:Val list

Badness: 0.0484

11-limit

Comma list: 36/35, 56/55, 77/75

POTE generator: ~12/11 = 140.747

Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]

Template:Val list

Badness: 0.0261

13-limit

Comma list: 26/25, 36/35, 56/55, 66/65

POTE generator: ~13/12 = 140.751

Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]

Template:Val list

Badness: 0.0182

17-limit

Comma list: 26/25, 36/35, 51/50, 56/55, 66/65

POTE generator: ~13/12 = 141.404

Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]

Template:Val list

Badness: 0.0167

19-limit

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

POTE generator: ~13/12 = 140.479

Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]

Template:Val list

Badness: 0.0168

Ripple

5-limit

Comma list: 6561/6250

POTE generator: ~27/25 = 100.838

Mapping: [⟨1 2 3], ⟨0 -5 -8]]

Template:Val list

Badness: 0.1389

7-limit

See also: Slendro clan § Hemiripple
See also: Sensamagic clan § Cohemiripple

Comma list: 36/35, 2560/2401

POTE generator: ~21/20 = 99.483

Mapping: [⟨1 2 3 3], ⟨0 -5 -8 -2]]

Wedgie: ⟨⟨5 8 2 1 -11 -18]]

Template:Val list

Badness: 0.0597

11-limit

Comma list: 36/35, 80/77, 126/121

POTE generator: ~21/20 = 99.385

Mapping: [⟨1 2 3 3 4], ⟨0 -5 -8 -2 -6]]

Template:Val list

Badness: 0.0388

13-limit

Comma list: 36/35, 40/39, 66/65, 147/143

POTE generator: ~21/20 = 98.572

Mapping: [⟨1 2 3 3 4 4], ⟨0 -5 -8 -2 -6 -3]]

Template:Val list

Badness: 0.0316

Smate

Comma list: 2048/1875

POTE generator: ~5/4 = 420.855

Mapping: [⟨1 3 2], ⟨0 -4 1]]

Template:Val list

Badness: 0.179

7-limit

Comma list: 36/35, 2048/1875

POTE generator: ~5/4 = 422.275

Mapping: [⟨1 3 2 6], ⟨0 -4 1 -9]]

Wedgie: ⟨⟨4 -1 9 -11 3 24]]

Template:Val list

Badness: 0.0779

11-limit

Comma list: 36/35 56/55 243/242

POTE generator: ~5/4 = 422.217

Mapping: [⟨1 3 2 6 7], ⟨0 -4 1 -9 -10]]

Template:Val list

Badness: 0.0425

13-limit

Comma list: 26/25, 36/35, 56/55, 243/242

POTE generator: ~5/4 = 423.020

Mapping: [⟨1 3 2 6 7 3], ⟨0 -4 1 -9 -10 2]]

Template:Val list

Badness: 0.0368

Hemismate

Comma list: 256/245, 392/375

POTE generator: ~8/7 = 210.452

Mapping: [⟨1 3 2 3], ⟨0 -8 2 -1]]

Wedgie: ⟨⟨8 -2 1 -22 -21 8]]

Template:Val list

Badness: 0.1543

11-limit

Comma list: 56/55, 77/75, 256/245

POTE generator: ~8/7 = 210.481

Mapping: [⟨1 3 2 3 4], ⟨0 -8 2 -1 -3]]

Template:Val list

Badness: 0.0655

13-limit

Commas: 26/25, 56/55, 77/75, 256/245

POTE generator: ~8/7 = 210.974

Mapping: [⟨1 3 2 3 4 3], ⟨0 -8 2 -1 -3 4]]

Template:Val list

Badness: 0.0505

Subklei

Comma: 17496/15625

POTE generator: ~6/5 = 284.293

Map: [<1 3 4|, <0 -6 -7|]

EDOs: 4, 17c, 21, 38c

Badness: 0.3559

7-limit

Commas: 36/35, 1029/1000

POTE generator: ~6/5 = 284.219

Map: [<1 3 4 4|, <0 -6 -7 -5|]

Wedgie: <<6 7 5 -3 -9 -8||

EDOs: 4, 17c, 21, 38c

Badness: 0.0611

11-limit

Commas: 36/35, 77/75, 352/343

POTE generator: ~6/5 = 283.253

Map: [<1 3 4 4 7|, <0 -6 -7 -5 -15|]

EDOs: 17c, 55cd

Badness: 0.0447

Subkla

Commas: 36/35, 56/55, 1029/1000

POTE generator: ~6/5 = 283.822

Map: [<1 3 4 4 3|, <0 -6 -7 -5 2|]

EDOs: 4, 17c, 21, 38ce, 55cde

Badness: 0.0472

13-limit

Commas: 36/35, 56/55, 66/65, 640/637

POTE generator: ~6/5 = 283.882

Map: [<1 3 4 4 3 3|, <0 -6 -7 -5 2 3|]

EDOs: 4, 17c, 21, 38ce, 55cde

Badness: 0.0367

13-limit

Commas: 26/25, 36/35, 66/65, 352/343

POTE generator: ~6/5 = 282.856

Map: [<1 3 4 4 7 7|, <0 -6 -7 -5 -15 -14|]

EDOs: 17c

Badness: 0.0323

Slurpee

Comma: 177147/156250

POTE generator: ~27/25 = 72.742

Map: [<1 2 3|, <0 -7 -11|]

EDOs: 16, 17c, 33

Badness: 0.6893

7-limit

Commas: 36/35, 51200/50421

POTE generator: ~21/20 = 72.303

Map: [<1 2 3 3|, <0 -7 -11 -3|]

Wedgie: <<7 11 3 1 -15 -24||

EDOs: 16, 17c, 33, 50cd, 83bcd

Badness: 0.1149

11-limit

Commas: 36/35, 121/120, 352/343

POTE generator: ~21/20 = 72.297

Map: [<1 2 3 3 4|, <0 -7 -11 -3 -9|]

EDOs: 16, 17c, 33, 50cd, 83bcd

Badness: 0.0505

13-limit

Commas: 36/35. 66/65, 143/140, 352/343

POTE generator: ~21/20 = 72.409

Map: [<1 2 3 3 4 4|, <0 -7 -11 -3 -9 -5|]

EDOs: 16, 17c, 33, 83bcd

Badness: 0.0331

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