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{{Technical data page}}
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.
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
Comma: 3456/3125
* [[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]]


POTE generator: ~25/24 = 140.995
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


Map: [<1 1 2|, <0 5 3|]
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.


EDOs: 8, 9, 17c
[[Subgroup]]: 2.3.5.7


Badness: 0.2005
[[Comma list]]: 36/35, 392/375


==7-limit==
{{Mapping|legend=1| 1 1 2 2 | 0 5 3 7 }}
Commas: 36/35, 392/375


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


Map: [&lt;1 1 2 2|, &lt;0 5 3 7|]
[[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 }}


Wedgie: &lt;&lt;5 3 7 -7 -3 8||
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


EDOs: 9, 17c
[[Badness]] (Smith): 0.048356


Badness: 0.0484
=== 11-limit ===
Subgroup: 2.3.5.7.11


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


[[POTE_tuning|POTE generator]]: ~15/14 = 140.747
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}


Map: [&lt;1 1 2 2 3|, &lt;0 5 3 7 4|]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747


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


Badness: 0.0261
Badness (Smith): 0.026050


==13-limit==
=== 13-limit ===
Commas: 26/25, 36/35, 56/55, 66/65
Subgroup: 2.3.5.7.11.13


[[POTE_tuning|POTE generator]]: ~15/14 = 140.751
Comma list: 26/25, 36/35, 56/55, 66/65


Map: Map: [&lt;1 1 2 2 3 3|, &lt;0 5 3 7 4 6|]
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}


EDOs: 8, 9, 17
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.741
* POTE: ~2 = 1200.000, ~13/12 = 140.751


Badness: 0.0182
{{Optimal ET sequence|legend=0| 8d, 9, 17c }}


==17-limit==
Badness (Smith): 0.018158
Commas: 26/25, 36/35, 51/50, 56/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 141.404
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;1 1 2 2 3 3 4|, &lt;0 5 3 7 4 6 1|]
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65


EDOs: 8, 9, 17
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}


Badness: 0.0167
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.649
* POTE: ~2 = 1200.000, ~13/12 = 141.404


===19-limit===
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
Commas: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 140.479
Badness (Smith): 0.016743


Map: [&lt;1 1 2 2 3 3 4 4|, &lt;0 5 3 7 4 6 1 2|]
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


EDOs: 8, 9, 17, [[94edo|94]]
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


Badness: 0.0168
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}


=Ripple=
Optimal tunings:
Comma: 6561/6250
* CTE: ~2 = 1200.000, ~13/12 = 138.750
* POTE: ~2 = 1200.000, ~13/12 = 140.479


POTE generator: ~27/25 = 100.838
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Map: [&lt;1 2 3|, &lt;0 -5 -8|]
Badness (Smith): 0.016841


EDOs: 12, 71b, 83b, 95b, 107bc, 119bc
== Subklei ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


Badness: 0.1389
[[Subgroup]]: 2.3.5.7


==7-limit==
[[Comma list]]: 36/35, 1029/1000
Commas: 36/35, 2560/2401


[[POTE_tuning|POTE generator]]: ~21/20 = 99.483
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


Map: [&lt;1 2 3 3|, &lt;0 -5 -8 -2|]
: mapping generators: ~2, ~7/6


Wedgie: &lt;&lt;5 8 2 1 -11 -18||
[[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 }}


EDOs: [[12edo|12]], [[23edo|23]]
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}


Badness: 0.0597
[[Badness]] (Smith): 0.061101


==11-limit==
=== 11-limit ===
Commas: 36/35, 80/77, 126/121
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = ~12/11 = 99.385
Comma list: 36/35, 77/75, 352/343


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


EDOs: 12
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* POTE: ~2 = 1200.000, ~7/6 = 283.253


Badness: 0.0388
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}


==13-limit==
Badness (Smith): 0.044651
Commas: 36/35, 40/39, 66/65, 147/143


POTE generator: ~21/20 = ~12/11 = ~14/13 = 98.572
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 3 4 4|, &lt;0 -5 -8 -2 -6 -3|]
Comma list: 26/25, 36/35, 66/65, 352/343


EDOs: 12f
Mapping: {{mapping| 1 3 4 4 7 7 | 0 -6 -7 -5 -15 -14 }}


Badness: 0.0316
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* POTE: ~2 = 1200.000, ~7/6 = 282.856


=Smate=
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}
Comma: 2048/1875


POTE generator: ~5/4 = 420.855
Badness (Smith): 0.032324


Map: [&lt;1 3 2|, &lt;0 -4 1|]
=== Subkla ===
Subgroup: 2.3.5.7.11


EDOs: 11, 14, 17c, 20c, 37c
Comma list: 36/35, 56/55, 1029/1000


Badness: 0.179
Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}


==7-limit==
Optimal tunings:
Commas: 36/35, 2048/1875
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* POTE: ~2 = 1200.000, ~7/6 = 283.822


POTE generator: ~5/4 = 422.275
{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce }}


Map: [&lt;1 3 2 6|, &lt;0 -4 1 -9|]
Badness (Smith): 0.047247


Wedgie: &lt;&lt;4 -1 9 -11 3 24||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 14, 17c, 37cd
Comma list: 36/35, 56/55, 66/65, 640/637


Badness: 0.0779
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}


==11-limit==
Optimal tunings:
Commas: 36/35 56/55 243/242
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* POTE: ~2 = 1200.000, ~7/6 = 283.882


POTE generator: ~5/4 = 422.217
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


Map: [&lt;1 3 2 6 7|, &lt;0 -4 1 -9 -10|]
Badness (Smith): 0.036701


EDOs: 14, 17c, 37cde
== Slurpee ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''


Badness: 0.0425
[[Subgroup]]: 2.3.5.7


==13-limit==
[[Comma list]]: 36/35, 51200/50421
Commas: 26/25, 36/35, 56/55, 243/242


POTE generator: ~5/4 = 423.020
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}


Map: [&lt;1 3 2 6 7 3|, &lt;0 -4 1 -9 -10 2|]
: mapping generators: ~2, ~21/20


EDOs: 14, 17c
[[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: 0.0368
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


=Hemismate=
[[Badness]] (Smith): 0.114869
commas: 256/245, 392/375


POTE generator: ~8/7 = 210.452
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 3 2 3|, &lt;0 -8 2 -1|]
Comma list: 36/35, 121/120, 352/343


Wedgie: &lt;&lt;8 -2 1 -22 -21 8||
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}


EDOs: 6, 11, 17c, 40bcd
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* POTE: ~2 = 1200.000, ~21/20 = 72.297


Badness: 0.1543
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


==11-limit==
Badness (Smith): 0.050514
Commas: 56/55, 77/75, 256/245


POTE generator: ~8/7 =  210.481
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 3 2 3 4|, &lt;0 -8 2 -1 -3|]
Comma list: 36/35, 66/65, 143/140, 352/343


EDOs: 6, 11, 17c, 40bcde
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}


Badness: 0.0655
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* POTE: ~2 = 1200.000, ~21/20 = 72.409


==13-limit==
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
Commas: 26/25, 56/55, 77/75, 256/245


POTE generator: ~8/7 = 210.974
Badness (Smith): 0.033102


Map: [&lt;1 3 2 3 4 3|, &lt;0 -8 2 -1 -3 4|]
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


EDOs: 6, 11, 17c
[[Subgroup]]: 2.3.5.7


Badness: 0.0505
[[Comma list]]: 28/25, 36/35


=Subklei=
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
Comma: 17496/15625


POTE generator: ~6/5 = 284.293
: mapping generators: ~2, ~7/6


Map: [&lt;1 3 4|, &lt;0 -6 -7|]
[[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 }}


EDOs: 4, 17c, 21, 38c
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


Badness: 0.3559
[[Badness]] (Smith): 0.046882


==7-limit==
== Shallowtone ==
Commas: 36/35, 1029/1000
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


POTE generator: ~6/5 = 284.219
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 3 4 4|, &lt;0 -6 -7 -5|]
[[Comma list]]: 36/35, 295245/262144


Wedgie: &lt;&lt;6 7 5 -3 -9 -8||
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}


EDOs: 4, 17c, 21, 38c
: mapping generators: ~2, ~3


Badness: 0.0611
[[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 }}


==11-limit==
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
Commas: 36/35, 77/75, 352/343


POTE generator: ~6/5 = 283.253
[[Badness]] (Smith): 0.308


Map: [&lt;1 3 4 4 7|, &lt;0 -6 -7 -5 -15|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


EDOs: 17c, 55cd
Comma list: 36/35, 45/44, 72171/65536


Badness: 0.0447
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


==Subkla==
Optimal tunings:
Commas: 36/35, 56/55, 1029/1000
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* CWE: ~2 = 1200.000, ~3/2 = 681.327


POTE generator: ~6/5 = 283.822
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


Map: [&lt;1 3 4 4 3|, &lt;0 -6 -7 -5 2|]
Badness (Smith): 0.130


EDOs: 4, 17c, 21, 38ce, 55cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0472
Comma list: 27/26, 36/35, 45/44, 16731/16384


===13-limit===
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}
Commas: 36/35, 56/55, 66/65, 640/637


POTE generator: ~6/5 = 283.882
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* CWE: ~2 = 1200.000, ~3/2 = 681.267


Map: [&lt;1 3 4 4 3 3|, &lt;0 -6 -7 -5 2 3|]
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


EDOs: 4, 17c, 21, 38ce, 55cde
Badness (Smith): 0.0773


Badness: 0.0367
== Notes ==


==13-limit==
[[Category:Temperament collections]]
Commas: 26/25, 36/35, 66/65, 352/343
[[Category:Pages with mostly numerical content]]
 
[[Category:Mint temperaments| ]] <!-- main article -->
POTE generator: ~6/5 = 282.856
[[Category:Mint| ]] <!-- key article -->
 
[[Category:Rank 2]]
Map: [&lt;1 3 4 4 7 7|, &lt;0 -6 -7 -5 -15 -14|]
 
EDOs: 17c
 
Badness: 0.0323
 
=Slurpee=
Comma: 177147/156250
 
POTE generator: ~27/25 = 72.742
 
Map: [&lt;1 2 3|, &lt;0 -7 -11|]
 
EDOs: 16, 17c, 33
 
Badness: 0.6893
 
==7-limit==
Commas: 36/35, 51200/50421
 
POTE generator: ~21/20 = 72.303
 
Map: [&lt;1 2 3 3|, &lt;0 -7 -11 -3|]
 
Wedgie: &lt;&lt;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: [&lt;1 2 3 3 4|, &lt;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: [&lt;1 2 3 3 4 4|, &lt;0 -7 -11 -3 -9 -5|]
 
EDOs: 16, 17c, 33, 83bcd
 
Badness: 0.0331
[[Category:definition]]
[[Category:overview]]
[[Category:quartonic]]
[[Category:temperaments]]
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