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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.
{{Technical data page}}
This is a collection of low [[complexity]], high [[error]], [[regular temperament|temperaments]] which [[temper out]] the septimal quarter-tone, [[36/35]]. 36 is [[Wikipedia: Square triangular number|both]] a [http://mathworld.wolfram.com/SquareNumber.html square] and a [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate [[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]], so minor and major thirds and sixths are intervals of 5 and 7 at the same time.  


= Progression =
Temperaments discussed elsewhere include
== 5-limit ==
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug 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]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]
* ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]
* ''[[Darkstone]]'' (+1875/1792) → [[Magic family #Darkstone|Magic family]]
* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


Subgroup: 2.3.5
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


[[Comma list]]: 3456/3125
Progression, named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18891.html Yahoo! Tuning Group | ''Progression temperament'']</ref> can be described as the 8d & 9 temperament, and [[17edo]] is an obvious tuning for it.


[[Mapping]]: [{{val| 1 1 2 }}, {{val| 0 5 3 }}]
[[Subgroup]]: 2.3.5.7
 
[[POTE generator]]: ~25/24 = 140.995
 
{{Val list|legend=1| 8, 9, 17c }}
 
[[Badness]]: 0.200483
 
== 7-limit ==
 
Subgroup: 2.3.5.7


[[Comma list]]: 36/35, 392/375
[[Comma list]]: 36/35, 392/375


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 5 3 7 }}]
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
 
{{Multival|legend=1| 5 3 7 -7 -3 8 }}


[[POTE generator]]: ~15/14 = 140.927
: mapping generators: ~2, ~15/14


{{Val list|legend=1| 8d, 9, 17c }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~15/14 = 138.611
: [[error map]]: {{val| 0.000 -8.900 +29.519 +1.451 }}
* [[POTE]]: ~2 = 1200.000, ~15/14 = 140.927
: error map: {{val| 0.000 +2.680 +36.467 +17.663 }}


[[Badness]]: 0.048356
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


== 11-limit ==
[[Badness]] (Smith): 0.048356


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


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


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


POTE generators: ~12/11 = 140.747
Optimal tunings:
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747


{{Val list|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Badness: 0.026050
Badness (Smith): 0.026050
 
== 13-limit ==


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


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


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


POTE generators: ~13/12 = 140.751
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 138.741
* POTE: ~2 = 1200.000, ~13/12 = 140.751


{{Val list|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


Badness: 0.018158
Badness (Smith): 0.018158
 
== 17-limit ==


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


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


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


POTE generators: ~13/12 = 141.404
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 138.649
* POTE: ~2 = 1200.000, ~13/12 = 141.404


{{Val list|legend=1| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Badness: 0.016743
Badness (Smith): 0.016743
 
== 19-limit ==


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


Comma list: 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


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


POTE generators: ~13/12 = 140.479
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 138.750
* POTE: ~2 = 1200.000, ~13/12 = 140.479


{{Val list|legend=1| 8d, 9, 17cg }}
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Badness: 0.016841
Badness (Smith): 0.016841


= Ripple =
== Subklei ==
{{see also| Ripple family #Ripple }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2560/2401
[[Comma list]]: 36/35, 1029/1000


[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -5 -8 -2 }}]
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


{{Multival|legend=1| 5 8 2 1 -11 -18 }}
: mapping generators: ~2, ~7/6


[[POTE generator]]: ~21/20 = 99.483
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/6 = 284.986
: [[error map]]: {{val| 0.000 -11.870 +18.785 +6.245 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 284.219
: error map: {{val| 0.000 -7.268 +24.154 +10.080 }}


{{Val list|legend=1| 12 }}
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}


[[Badness]]: 0.059735
[[Badness]] (Smith): 0.061101
 
== 11-limit ==


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


Comma list: 36/35, 80/77, 126/121
Comma list: 36/35, 77/75, 352/343


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -5 -8 -2 -6 }}]
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}


POTE generator: ~21/20 = 99.385
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* POTE: ~2 = 1200.000, ~7/6 = 283.253


{{Val list|legend=1| 12 }}
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}


Badness: 0.038811
Badness (Smith): 0.044651
 
== 13-limit ==


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


Comma list: 36/35, 40/39, 66/65, 147/143
Comma list: 26/25, 36/35, 66/65, 352/343


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -5 -8 -2 -6 -3 }}]
Mapping: {{mapping| 1 3 4 4 7 7 | 0 -6 -7 -5 -15 -14 }}


POTE generator: ~21/20 = 98.572
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* POTE: ~2 = 1200.000, ~7/6 = 282.856


{{Val list|legend=1| 12f }}
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}


Badness: 0.031639
Badness (Smith): 0.032324
 
= Smate =
{{see also| Smate family #Smate }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 36/35, 2048/1875
 
[[Mapping]]: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]
 
{{Multival|legend=1| 4 -1 9 -11 3 24 }}
 
[[POTE generator]]: ~5/4 = 422.275
 
{{Val list|legend=1| 3d, 11d, 14, 17c, 37cd }}
 
[[Badness]]: 0.077871
 
== 11-limit ==


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


Comma list: 36/35, 56/55, 243/242
Comma list: 36/35, 56/55, 1029/1000
 
Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]


POTE generator: ~5/4 = 422.217
Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}


{{Val list|legend=1| 14, 17c, 37cde }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* POTE: ~2 = 1200.000, ~7/6 = 283.822


Badness: 0.042518
{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce }}


== 13-limit ==
Badness (Smith): 0.047247


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


Comma list: 26/25, 36/35, 56/55, 243/242
Comma list: 36/35, 56/55, 66/65, 640/637


Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}


POTE generator: ~5/4 = 423.020
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* POTE: ~2 = 1200.000, ~7/6 = 283.882


{{Val list|legend=1| 14, 17c }}
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


Badness: 0.036836
Badness (Smith): 0.036701


= Subklei =
== Slurpee ==
== 5-limit ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 17496/15625
[[Comma list]]: 36/35, 51200/50421
 
[[Mapping]]: [{{val| 1 3 4 }}, {{val| 0 -6 -7 }}]
 
[[POTE generator]]: ~6/5 = 284.293
 
{{Val list|legend=1| 4, 17c, 21, 38c }}
 
[[Badness]]: 0.355937
 
== 7-limit ==
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 36/35, 1029/1000
 
[[Mapping]]: [{{val| 1 3 4 4 }}, {{val| 0 -6 -7 -5 }}]


{{Multival|legend=1| 6 7 5 -3 -9 -8 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}


[[POTE generator]]: ~6/5 = 284.219
: mapping generators: ~2, ~21/20


{{Val list|legend=1| 4, 17c, 21, 38c }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~21/20 = 72.776
: [[error map]]: {{val| 0.000 -11.385 +13.153 +12.847 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 72.303
: error map: {{val| 0.000 -8.075 +18.355 +14.265 }}


[[Badness]]: 0.061101
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


== 11-limit ==
[[Badness]] (Smith): 0.114869


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


Comma list: 36/35, 77/75, 352/343
Comma list: 36/35, 121/120, 352/343  


Mapping: [{{val| 1 3 4 4 7 }}, {{val| 0 -6 -7 -5 -15 }}]
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}


POTE generator: ~6/5 = 283.253
Optimal tunings:
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* POTE: ~2 = 1200.000, ~21/20 = 72.297


{{Val list|legend=1| 17c, 55cd }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


Badness: 0.044651
Badness (Smith): 0.050514


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


Comma list: 26/25, 36/35, 66/65, 352/343  
Comma list: 36/35, 66/65, 143/140, 352/343


POTE generator: ~6/5 = 282.856
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}


Mapping: [{{val| 1 3 4 4 7 7 }}, {{val| 0 -6 -7 -5 -15 -14 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* POTE: ~2 = 1200.000, ~21/20 = 72.409


POTE generator: ~6/5 = 282.856
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


{{Val list|legend=1| 17c }}
Badness (Smith): 0.033102


Badness: 0.032324
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


== Subkla ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11
[[Comma list]]: 28/25, 36/35


Comma list: 36/35, 56/55, 1029/1000
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}


Mapping: [{{val| 1 3 4 4 3 }}, {{val| 0 -6 -7 -5 2 }}]
: mapping generators: ~2, ~7/6


POTE generator: ~6/5 = 283.822
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/6 = 231.845
: [[error map]]: {{val| 0.000 -6.421 +77.376 -41.447 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 239.977
: error map: {{val| 0.000 +17.976 +93.640 -8.918 }}


{{Val list|legend=1| 4, 17c, 21, 38ce, 55cde }}
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


Badness: 0.047247
[[Badness]] (Smith): 0.046882


=== 13-limit ===
== Shallowtone ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


Comma list: 36/35, 56/55, 66/65, 640/637
[[Comma list]]: 36/35, 295245/262144


Mapping: [{{val| 1 3 4 4 3 3 }}, {{val| 0 -6 -7 -5 2 3 }}]
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}


POTE generator: ~6/5 = 283.882
: mapping generators: ~2, ~3


{{Val list|legend=1| 4, 17c, 21, 38ce, 55cde }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.278
: [[error map]]: {{val| 0.000 -20.677 +0.906 +6.510 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 681.245
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}


Badness: 0.036701
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}


= Slurpee =
[[Badness]] (Smith): 0.308
== 5-limit ==
 
Subgroup: 2.3.5
 
[[Comma list]]: 177147/156250
 
[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -7 -11 }}]
 
[[POTE generator]]: ~27/25 = 72.742
 
{{Val list|legend=1| 16, 17c, 33 }}
 
[[Badness]]: 0.689274
 
== 7-limit ==
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 36/35, 51200/50421
 
[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -7 -11 -3 }}]
 
{{Multival|legend=1| 7 11 3 1 -15 -24 }}
 
[[POTE generator]]: ~21/20 = 72.303
 
{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
 
[[Badness]]: 0.114869
 
== 11-limit ==


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


Comma list: 36/35, 121/120, 352/343
Comma list: 36/35, 45/44, 72171/65536


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -7 -11 -3 -9 }}]
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


POTE generator: ~21/20 = 72.297
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* CWE: ~2 = 1200.000, ~3/2 = 681.327


{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


Badness: 0.050514
Badness (Smith): 0.130
 
== 13-limit ==


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


Comma list: 36/35. 66/65, 143/140, 352/343
Comma list: 27/26, 36/35, 45/44, 16731/16384
 
Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -7 -11 -3 -9 -5 }}]
 
POTE generator: ~21/20 = 72.409
 
{{Val list|legend=1| 16, 17c, 33, 83bcd }}
 
Badness: 0.033102
 
= Penta =
{{see also| Gamelismic clan #Gidorah }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 28/25, 36/35


[[Mapping]]: [{{val| 1 1 2 2 }}, {{val| 0 3 2 4 }}]
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}


{{Multival|legend=1| 3 2 4 -4 -2 4 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* CWE: ~2 = 1200.000, ~3/2 = 681.267


[[POTE generator]]: ~6/5 = 239.977
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


{{Val list|legend=1| 1bd, 5 }}
Badness (Smith): 0.0773


[[Badness]]: 0.046882
== Notes ==


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Pages with mostly numerical content]]
[[Category:Mint]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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