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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 (temperament)|Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Mujannabic|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]]
* ''[[Rip]]'' (+2560/2401) → [[Ripple family #Rip|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


Period: 1\1
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


Optimal ([[POTE]]) generators: ~25/24 = 140.995
[[Subgroup]]: 2.3.5.7


EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
[[Comma list]]: 28/25, 36/35


Scales:  
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
: mapping generators: ~2, ~7/6


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1186.8093{{c}}, ~7/6 = 237.3390{{c}}
<div class="mw-collapsible-content">
: [[error map]]: {{val| -13.191 -3.129 +61.983 -45.851 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 231.845{{c}}
: error map: {{val| 0.000 +6.291 +85.850 -24.498 }}


Comma list: 3456/3125
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


POTE generator: ~25/24 = 140.995
[[Badness]] (Sintel): 1.19


Mapping: [{{val| 1 1 2 }}, {{val| 0 5 3 }}]
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


{{Val list|legend=1| 8, 9, 17c }}
Named by [[Gene Ward Smith]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18891.html Yahoo! Tuning Group | ''Progression temperament'']</ref>, progression can be described as the {{nowrap| 8d & 9 }} temperament. It has a generator that is a somewhat flat neutral second, three make [[5/4]], five make [[3/2]], and seven make [[7/4]], with a [[ploidacot]] signature of pentacot. [[17edo]] is an obvious tuning for it.


Badness: 0.2005
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 36/35, 392/375


== 7-limit ==
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
: mapping generators: ~2, ~15/14


Period: 1\1
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1193.9544{{c}}, ~15/14 = 140.2169{{c}}
: [[error map]]: {{val| -6.046 -6.916 +22.246 +0.601 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}
: error map: {{val| 0.000 -2.460 +33.384 +10.468 }}


Optimal ([[POTE]]) generators: ~15/14 = 140.927
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
[[Badness]] (Sintel): 1.22


Scales:
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 392/375
 
Mapping: [{{val| 1 1 2 2 }}, {{val| 0 5 3 7 }}]
 
Wedgie: {{wedgie| 5 3 7 -7 -3 8 }}
 
{{Val list|legend=1| 8d, 9, 17c }}
 
Badness: 0.0484
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~12/11 = 140.747
 
EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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 }}


{{Val list|legend=1| 8d, 9, 17c }}
Optimal tunings:
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}


Badness: 0.0261
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


</div></div>
Badness (Sintel): 0.861


== 13-limit ==
=== 13-limit ===
 
Subgroup: 2.3.5.7.11.13
Period: 1\1
 
Optimal ([[POTE]]) generators: ~13/12 = 140.751
 
EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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 }}
 
{{Val list|legend=1| 8d, 9, 17c }}
 
Badness: 0.0182
 
</div></div>
 
== 17-limit ==


Period: 1\1
Optimal tunings:  
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}


Optimal ([[POTE]]) generators: ~13/12 = 141.404
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
Badness (Sintel): 0.750


Scales:
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


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 }}
 
{{Val list|legend=1| 8d, 9, 17cg }}
 
Badness: 0.0167
 
</div></div>
 
== 19-limit ==
 
Period: 1\1


Optimal ([[POTE]]) generators: ~13/12 = 140.479
Optimal tunings:  
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}


EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Scales:  
Badness (Sintel): 0.853


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
=== 19-limit ===
<div style="line-height:1.6;">Technical data</div>
Subgroup: 2.3.5.7.11.13.17.19
<div class="mw-collapsible-content">


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 }}


{{Val list|legend=1| 8d, 9, 17cg, 94… }}
Optimal tunings:
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}


Badness: 0.0168
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


</div></div>
Badness (Sintel): 1.02


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


Period: 1\1
Subklei is likely named for its flatter minor third generator than [[kleismic]]. It tempers out [[1029/1000]] as well as [[2401/2400]] and can be described as {{nowrap| 17c & 21 }}. Its [[ploidacot]] is delta-hexacot. Note that in the data below, the generator is the [[5/3]][[~]][[12/7]] major sixth, so that six generators minus four octaves give the [[3/2|perfect fifth]].


Optimal ([[POTE]]) generators: ~27/25 = 100.838
[[Subgroup]]: 2.3.5.7


EDO generators: [[8edo|1\8]], [[9edo|1\9]], [[17edo|2\17]]
[[Comma list]]: 36/35, 1029/1000


Scales:  
{{Mapping|legend=1| 1 -3 -3 -1 | 0 6 7 5 }}
: mapping generators: ~2, ~5/3


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Optimal tuning]]s:
<div style="line-height:1.6;">Technical data</div>
* [[WE]]: ~2 = 1197.5285{{c}}, ~5/3 = 915.8950{{c}}
<div class="mw-collapsible-content">
: [[error map]]: {{val| -2.471 -11.171 +18.366 +3.121 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}
: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}


Comma list: 6561/6250
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}


Mapping: [{{val| 1 2 3 }}, {{val| 0 -5 -8 }}]
[[Badness]] (Sintel): 1.55


{{Val list|legend=1| 11c, 12, 71b, 83b, 95b, 107bc, 119bc }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.1389
Comma list: 36/35, 77/75, 352/343


</div></div>
Mapping: {{mapping| 1 -3 -3 -1 -8 | 0 6 7 5 15 }}


== 7-limit ==
Optimal tunings:
{{see also| Slendro clan #Hemiripple }}
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
{{see also| Sensamagic clan #Cohemiripple }}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}


Optimal ([[POTE]]) generators: ~21/20 = 99.483
Badness (Sintel): 1.48


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 3 4 4 7 7 | 0 6 7 5 15 14 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35, 2560/2401
Optimal tunings:  
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}


Mapping: [{{val| 1 2 3 3 }}, {{val| 0 -5 -8 -2 }}]
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}


Wedgie: {{wedgie| 5 8 2 1 -11 -18 }}
Badness (Sintel): 1.34


{{Val list|legend=1| 11c, 12 }}
=== Subkla ===
Subgroup: 2.3.5.7.11


Badness: 0.0597
Comma list: 36/35, 56/55, 1029/1000


</div></div>
Mapping: {{mapping| 1 -3 -3 -1 5 | 0 6 7 5 -2 }}


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}


Period: 1\1
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}


Optimal ([[POTE]]) generators: ~21/20 = 99.385
Badness (Sintel): 1.56


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Scales:  
Comma list: 36/35, 56/55, 66/65, 640/637


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Mapping: {{mapping| 1 -3 -3 -1 5 6 | 0 6 7 5 -2 -3 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35, 80/77, 126/121
Optimal tunings:  
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}


Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -5 -8 -2 -6 }}]
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


{{Val list|legend=1| 12 }}
Badness (Sintel): 1.52


Badness: 0.0388
== Naian ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Naian]].''


</div></div>
Named by [[Xenllium]] in 2026, naian may be described as {{nowrap| 8d & 21 }} with a [[ploidacot]] signature of epsilon-enneacot.


== 13-limit ==
[[Subgroup]]: 2.3.5.7


Period: 1\1
[[Comma list]]: 36/35, 9604/9375


Optimal ([[POTE]]) generators: ~21/20 = 98.572
{{Mapping|legend=1| 1 -4 -2 -4 | 0 9 7 11 }}
: mapping generators: ~2, ~75/49


EDO generators: [[11edo|1\11]], [[12edo|1\12]], [[13edo|1\13]]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.8807{{c}}, ~75/49 = 741.8522{{c}}
: [[error map]]: {{val| -4.119 -8.808 +14.890 +8.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 743.9885{{c}}
: error map: {{val| 0.000 -6.059 +21.606 +15.047 }}


Scales:
{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
[[Badness]] (Sintel): 2.89
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 40/39, 66/65, 147/143
 
Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -5 -8 -2 -6 -3 }}]
 
{{Val list|legend=1| 12f }}
 
Badness: 0.0316
 
</div></div>
 
= Smate =
== 5-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~5/4 = 420.855
 
EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[20edo|7\20]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 2048/1875
 
Mapping: [{{val| 1 3 2 }}, {{val| 0 -4 1 }}]
 
{{Val list|legend=1| 11, 14, 17c, 20c, 37c }}
 
Badness: 0.179
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~5/4 = 422.275
 
EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[20edo|7\20]]
 
Scales:  
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 2048/1875
 
Mapping: [{{val| 1 3 2 6 }}, {{val| 0 -4 1 -9 }}]
 
Wedgie: {{wedgie| 4 -1 9 -11 3 24 }}
 
{{Val list|legend=1| 14, 17c, 37cd }}
 
Badness: 0.0779
 
</div></div>


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


Period: 1\1
Comma list: 36/35, 56/55, 2541/2500


Optimal ([[POTE]]) generators: ~5/4 = 422.217
Mapping: {{mapping| 1 -4 -2 -4 1 | 0 9 7 11 4 }}


EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[20edo|7\20]]
Optimal tunings:  
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}


Scales:
{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.95
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 56/55, 243/242
 
Mapping: [{{val| 1 3 2 6 7 }}, {{val| 0 -4 1 -9 -10 }}]
 
{{Val list|legend=1| 14, 17c, 37cde }}
 
Badness: 0.0425
 
</div></div>


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


Period: 1\1
Comma list: 36/35, 56/55, 66/65, 507/500


Optimal ([[POTE]]) generators: ~5/4 = 423.020
Mapping: {{mapping| 1 -4 -2 -4 1 0 | 0 9 7 11 4 6 }}


EDO generators: [[14edo|5\14]], [[17edo|6\17]], [[20edo|7\20]]
Optimal tunings:  
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}


Scales:
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.55
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 26/25, 36/35, 56/55, 243/242
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 3 2 6 7 3 }}, {{val| 0 -4 1 -9 -10 2 }}]
Comma list: 36/35, 51/50, 56/55, 66/65, 170/169


{{Val list|legend=1| 14, 17c }}
Mapping: {{mapping| 1 -4 -2 -4 1 0 1 | 0 9 7 11 4 6 5 }}


Badness: 0.0368
Optimal tunings:  
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


== Hemismate ==
Badness (Sintel): 1.43


Period: 1\1
== Slurpee ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''


Optimal ([[POTE]]) generators: ~8/7 = 210.452
Slurpee may be described as the {{nowrap| 16 & 17c }} temperament. It has a generator that is a somewhat flat semitone of [[~]][[21/20]], three make [[8/7]], seven make [[4/3]], and eleven make [[8/5]], with a [[ploidacot]] signature of omega-heptacot.  


EDO generators: [[11edo|2\11]], [[17edo|3\17]], [[23edo|4\23]]
[[Subgroup]]: 2.3.5.7


Scales:  
[[Comma list]]: 36/35, 51200/50421


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
<div style="line-height:1.6;">Technical data</div>
: mapping generators: ~2, ~21/20
<div class="mw-collapsible-content">


Comma list: 256/245, 392/375
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.9281{{c}}, ~21/20 = 72.1780{{c}}
: [[error map]]: {{val| -2.072 -11.345 +13.512 +8.424 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 72.4941{{c}}
: error map: {{val| 0.000 -9.414 +16.251 +13.692 }}


Mapping: [{{val| 1 3 2 3 }}, {{val| 0 -8 2 -1 }}]
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


Wedgie: {{wedgie| 8 -2 1 -22 -21 8 }}
[[Badness]] (Sintel): 2.91
 
{{Val list|legend=1| 6, 11, 17c, 40bcd }}
 
Badness: 0.1543
 
</div></div>


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


Period: 1\1
Comma list: 36/35, 121/120, 352/343
 
Optimal ([[POTE]]) generators: ~8/7 = 210.481


EDO generators: [[11edo|2\11]], [[17edo|3\17]], [[23edo|4\23]]
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}


Scales:  
Optimal tunings:  
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 56/55, 77/75, 256/245
Badness (Sintel): 1.67
 
Mapping: [{{val| 1 3 2 3 4 }}, {{val| 0 -8 2 -1 -3 }}]
 
{{Val list|legend=1| 6, 11, 17c, 40bcde }}
 
Badness: 0.0655
 
</div></div>


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


Period: 1\1
Comma list: 36/35, 66/65, 143/140, 352/343


Optimal ([[POTE]]) generators: ~8/7 = 210.974
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}


EDO generators: [[11edo|2\11]], [[17edo|3\17]], [[23edo|4\23]]
Optimal tunings:  
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}


Scales:
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Badness (Sintel): 1.37
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 26/25, 56/55, 77/75, 256/245
== Shallowtone ==
{{Main| Shallowtone }}
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


Mapping: [{{val| 1 3 2 3 4 3 }}, {{val| 0 -8 2 -1 -3 4 }}]
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 6, 11, 17c }}
[[Comma list]]: 36/35, 295245/262144


Badness: 0.0505
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
: mapping generators: ~2, ~3


</div></div>
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.4397{{c}}, ~3/2 = 682.6219{{c}}
: [[error map]]: {{val| +2.440 -16.893 +6.986 +12.877 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 681.2447{{c}}
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}


= Subklei =
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
== 5-limit ==


Period: 1\1
[[Badness]] (Sintel): 7.79


Optimal ([[POTE]]) generators: ~6/5 = 284.293
=== 11-limit ===
 
Subgroup: 2.3.5.7.11
EDO generators: [[17edo|4\17]], [[21edo|5\21]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 17496/15625
 
Mapping: [{{val| 1 3 4 }}, {{val| 0 -6 -7 }}]
 
{{Val list|legend=1| 4, 17c, 21, 38c }}
 
Badness: 0.3559
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~6/5 = 284.219
 
EDO generators: [[17edo|4\17]], [[21edo|5\21]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 1029/1000
 
Mapping: [{{val| 1 3 4 4 }}, {{val| 0 -6 -7 -5 }}]
 
Wedgie: {{wedgie| 6 7 5 -3 -9 -8 }}
 
{{Val list|legend=1| 4, 17c, 21, 38c }}
 
Badness: 0.0611
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~6/5 = 283.253
 
EDO generators: [[17edo|4\17]], [[21edo|5\21]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 77/75, 352/343
 
Mapping: [{{val| 1 3 4 4 7 }}, {{val| 0 -6 -7 -5 -15 }}]
 
{{Val list|legend=1| 17c, 55cd }}
 
Badness: 0.0447
 
</div></div>
 
=== 13-limit ===
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~6/5 = 282.856
 
EDO generators: [[17edo|4\17]], [[21edo|5\21]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 26/25, 36/35, 66/65, 352/343
 
POTE generator: ~6/5 = 282.856
 
Mapping: [{{val| 1 3 4 4 7 7 }}, {{val| 0 -6 -7 -5 -15 -14 }}]
 
{{Val list|legend=1| 17c }}
 
Badness: 0.0323
 
</div></div>


== Subkla ==
Comma list: 36/35, 45/44, 72171/65536


Period: 1\1
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


Optimal ([[POTE]]) generators: ~6/5 = 283.822
Optimal tunings:
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}


EDO generators: [[17edo|4\17]], [[21edo|5\21]]
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


Scales:
Badness (Sintel): 4.29
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 56/55, 1029/1000
 
Mapping: [{{val| 1 3 4 4 3 }}, {{val| 0 -6 -7 -5 2 }}]
 
{{Val list|legend=1| 4, 17c, 21, 38ce, 55cde }}
 
Badness: 0.0472
 
</div></div>


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


Period: 1\1
Comma list: 27/26, 36/35, 45/44, 16731/16384
 
Optimal ([[POTE]]) generators: ~6/5 = 283.882
 
EDO generators: [[17edo|4\17]], [[21edo|5\21]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 56/55, 66/65, 640/637
 
Mapping: [{{val| 1 3 4 4 3 3 }}, {{val| 0 -6 -7 -5 2 3 }}]
 
{{Val list|legend=1| 4, 17c, 21, 38ce, 55cde }}
 
Badness: 0.0367
 
</div></div>
 
= Slurpee =
== 5-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~27/25 = 72.742
 
EDO generators: [[16edo|1\16]], [[17edo|1\17]], [[33edo|2\33]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 177147/156250
 
Mapping: [{{val| 1 2 3 }}, {{val| 0 -7 -11 }}]
 
{{Val list|legend=1| 16, 17c, 33 }}
 
Badness: 0.6893
 
</div></div>
 
== 7-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~21/20 = 72.303
 
EDO generators: [[16edo|1\16]], [[17edo|1\17]], [[33edo|2\33]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 51200/50421
 
Mapping: [{{val| 1 2 3 3 }}, {{val| 0 -7 -11 -3 }}]
 
Wedgie: {{wedgie| 7 11 3 1 -15 -24 }}
 
{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
 
Badness: 0.1149
 
</div></div>
 
== 11-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~21/20 = 72.297
 
EDO generators: [[16edo|1\16]], [[17edo|1\17]], [[33edo|2\33]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
 
Comma list: 36/35, 121/120, 352/343
 
Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -7 -11 -3 -9 }}]
 
{{Val list|legend=1| 16, 17c, 33, 50cd, 83bcd }}
 
Badness: 0.0505
 
</div></div>
 
== 13-limit ==
 
Period: 1\1
 
Optimal ([[POTE]]) generators: ~21/20 = 72.409
 
EDO generators: [[16edo|1\16]], [[17edo|1\17]], [[33edo|2\33]]
 
Scales:
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 36/35. 66/65, 143/140, 352/343
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}


Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -7 -11 -3 -9 -5 }}]
Optimal tunings:
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}


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


Badness: 0.0331
Badness (Sintel): 3.19


</div></div>
== References ==


[[Category:Theory]]
[[Category:Temperament collections]]
[[Category:Temperament]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Mint_temperaments"