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Mint temperaments: Difference between revisions

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This is a collection of 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 [[triangular number]], and this helps make 36/35 a septimal interval of considerable significance.  
{{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.  


Temperaments discussed elsewhere include
Temperaments discussed elsewhere include
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Hexadecimal]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
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* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Dimipent family #Diminished|Dimipent family]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]
* ''[[Smate]]'' (+2048/1875) → [[Smate family #Septimal smate|Smate family]]
* ''[[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]]
* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


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


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: mapping generators: ~2, ~15/14
: mapping generators: ~2, ~15/14
{{Multival|legend=1| 5 3 7 -7 -3 8 }}


[[Optimal tuning]]s:  
[[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
* [[POTE]]: ~2 = 1200.000, ~15/14 = 140.927
: error map: {{val| 0.000 +2.680 +36.467 +17.663 }}


{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747
* POTE: ~2 = 1200.000, ~12/11 = 140.747


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


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


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


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{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


{{Multival|legend=1| 6 7 5 -3 -9 -8 }}
: mapping generators: ~2, ~7/6


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~6/5 = 284.219
* [[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 }}


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


[[Badness]] (Smith): 0.061101
[[Badness]] (Smith): 0.061101
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~6/5 = 283.253
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* POTE: ~2 = 1200.000, ~7/6 = 283.253


{{Optimal ET sequence|legend=0| 17c, 55ccd }}
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}


Badness (Smith): 0.044651
Badness (Smith): 0.044651
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~6/5 = 282.856
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* POTE: ~2 = 1200.000, ~7/6 = 282.856


{{Optimal ET sequence|legend=0| 17c }}
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}


Badness (Smith): 0.032324
Badness (Smith): 0.032324
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~6/5 = 283.822
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* POTE: ~2 = 1200.000, ~7/6 = 283.822


{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce, 55ccde }}
{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce }}


Badness (Smith): 0.047247
Badness (Smith): 0.047247
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Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000, ~6/5 = 283.882
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* POTE: ~2 = 1200.000, ~7/6 = 283.882


{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce, 55ccde }}
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


Badness (Smith): 0.036701
Badness (Smith): 0.036701
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{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}


{{Multival|legend=1| 7 11 3 1 -15 -24 }}
: mapping generators: ~2, ~21/20


[[Optimal tuning]]s:  
[[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
* [[POTE]]: ~2 = 1200.000, ~21/20 = 72.303
: error map: {{val| 0.000 -8.075 +18.355 +14.265 }}


{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


[[Badness]] (Smith): 0.114869
[[Badness]] (Smith): 0.114869
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* POTE: ~2 = 1200.000, ~21/20 = 72.297
* POTE: ~2 = 1200.000, ~21/20 = 72.297


{{Optimal ET sequence|legend=0| 16, 17c, 33, 50cd, 83bccd }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


Badness (Smith): 0.050514
Badness (Smith): 0.050514
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* POTE: ~2 = 1200.000, ~21/20 = 72.409
* POTE: ~2 = 1200.000, ~21/20 = 72.409


{{Optimal ET sequence|legend=0| 16, 17c, 33, 83bccd }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


Badness (Smith): 0.033102
Badness (Smith): 0.033102
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{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}


{{Multival|legend=1| 3 2 4 -4 -2 4 }}
: mapping generators: ~2, ~7/6


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000, ~6/5 = 239.977
* [[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 }}


{{Optimal ET sequence|legend=1| 1bd, 5 }}
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


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


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
: mapping generators: ~2, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.2780
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.278
* [[CWE]]: ~2 = 1200.000, ~3/2 = 681.2447
: [[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 }}


{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.2538
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* CWE: ~2 = 1200.000, ~3/2 = 681.3267
* CWE: ~2 = 1200.000, ~3/2 = 681.327


{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
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Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.2374
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* CWE: ~2 = 1200.000, ~3/2 = 681.2669
* CWE: ~2 = 1200.000, ~3/2 = 681.267


{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}
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[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mint_temperaments"