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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
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