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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
* ''[[Beep]]'' (+21/20) → [[Bug family #Beep|Bug family]]
* [[Father]] (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* [[Dominant (temperament)|Dominant]] (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Dominant]]'' (+64/63) → [[Meantone family #Dominant|Meantone family]]
* ''[[Armodue (temperament)|Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Father]]'' (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Mujannabic|Dicot family]]
* ''[[Diminished]]'' (+50/49) → [[Dimipent family #Diminished|Dimipent family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[August]]'' (+128/125) → [[Augmented family #August|Augmented 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]]
* ''[[Hystrix]]'' (+160/147) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Hexadecimal]]'' (+135/128) → [[Pelogic family|Pelogic family]]
* [[Diminished (temperament)|Diminished]] (+50/49) → [[Diminished family #Septimal diminished|Diminished family]]
* ''[[Gorgo]]'' (+1029/1024) → [[Gamelismic clan #Gorgo|Gamelismic clan]]
* ''[[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]]
* ''[[Rip]]'' (+2560/2401) → [[Ripple family #Rip|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


Considered below are progression, ripple, smate, subklei, slurpee, and penta.
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 28/25, 36/35
 
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
: mapping generators: ~2, ~7/6
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1186.8093{{c}}, ~7/6 = 237.3390{{c}}
: [[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 }}
 
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
 
[[Badness]] (Sintel): 1.19


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


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.  
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 25: Line 48:


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


{{Multival|legend=1| 5 3 7 -7 -3 8 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1193.9544{{c}}, ~15/14 = 140.2169{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 140.927
: [[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 ET sequence|legend=1| 8d, 9, 17c }}
{{Optimal ET sequence|legend=1| 8d, 9, 17c }}


[[Badness]]: 0.048356
[[Badness]] (Sintel): 1.22


=== 11-limit ===
=== 11-limit ===
Line 43: Line 67:
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}
Mapping: {{mapping| 1 1 2 2 3 | 0 5 3 7 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 140.747
Optimal tunings:
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}


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


Badness: 0.026050
Badness (Sintel): 0.861


=== 13-limit ===
=== 13-limit ===
Line 56: Line 82:
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}
Mapping: {{mapping| 1 1 2 2 3 3 | 0 5 3 7 4 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.751
Optimal tunings:
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}


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


Badness: 0.018158
Badness (Sintel): 0.750


=== 17-limit ===
=== 17-limit ===
Line 69: Line 97:
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}
Mapping: {{mapping| 1 1 2 2 3 3 4 | 0 5 3 7 4 6 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 141.404
Optimal tunings:
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}


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


Badness: 0.016743
Badness (Sintel): 0.853


=== 19-limit ===
=== 19-limit ===
Line 82: Line 112:
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 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 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 140.479
Optimal tunings:
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
Badness (Sintel): 1.02


Badness: 0.016841
== Subklei ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


== Ripple ==
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]].
{{see also| Ripple family #Ripple }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}
{{Mapping|legend=1| 1 -3 -3 -1 | 0 6 7 5 }}
: mapping generators: ~2, ~5/3


{{Multival|legend=1| 5 8 2 1 -11 -18 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.5285{{c}}, ~5/3 = 915.8950{{c}}
: [[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 99.483
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}


{{Optimal ET sequence|legend=1| 12 }}
[[Badness]] (Sintel): 1.55
 
[[Badness]]: 0.059735


=== 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: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
Mapping: {{mapping| 1 -3 -3 -1 -8 | 0 6 7 5 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385
Optimal tunings:
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}


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


Badness: 0.038811
Badness (Sintel): 1.48


=== 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: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}
Mapping: {{mapping| 1 3 4 4 7 7 | 0 6 7 5 15 14 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 98.572
Optimal tunings:
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}


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


Badness: 0.031639
Badness (Sintel): 1.34


== Smate ==
=== Subkla ===
{{See also| Smate family #Smate }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 2048/1875
 
{{Mapping|legend=1| 1 3 2 6 | 0 -4 1 -9 }}
 
{{Multival|legend=1| 4 -1 9 -11 3 24 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 422.275
 
{{Optimal ET sequence|legend=1| 3d, 11d, 14, 17c, 37ccdd }}
 
[[Badness]]: 0.077871
 
=== 11-limit ===
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: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}
Mapping: {{mapping| 1 -3 -3 -1 5 | 0 6 7 5 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217
Optimal tunings:
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}


{{Optimal ET sequence|legend=1| 3de, 14, 17c, 37ccddee }}
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}


Badness: 0.042518
Badness (Sintel): 1.56


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


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


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020
Optimal tunings:
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}


{{Optimal ET sequence|legend=1| 3de, 14, 17c }}
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


Badness: 0.036836
Badness (Sintel): 1.52


== Subklei ==
== Naian ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Delorean]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Naian]].''
 
Named by [[Xenllium]] in 2026, naian may be described as {{nowrap| 8d & 21 }} with a [[ploidacot]] signature of epsilon-enneacot.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 1029/1000
[[Comma list]]: 36/35, 9604/9375
 
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


{{Multival|legend=1| 6 7 5 -3 -9 -8 }}
{{Mapping|legend=1| 1 -4 -2 -4 | 0 9 7 11 }}
: mapping generators: ~2, ~75/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 284.219
[[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 }}


{{Optimal ET sequence|legend=1| 4, 17c, 21, 38c }}
{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}


[[Badness]]: 0.061101
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 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, 56/55, 2541/2500


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


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.253
Optimal tunings:
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}


{{Optimal ET sequence|legend=1| 17c, 55ccd }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}


Badness: 0.044651
Badness (Sintel): 1.95


==== 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, 56/55, 66/65, 507/500


Mapping: {{mapping| 1 3 4 4 7 7 | 0 -6 -7 -5 -15 -14 }}
Mapping: {{mapping| 1 -4 -2 -4 1 0 | 0 9 7 11 4 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 282.856
Optimal tunings:
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}


{{Optimal ET sequence|legend=1| 17c }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


Badness: 0.032324
Badness (Sintel): 1.55


=== Subkla ===
=== 17-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 36/35, 56/55, 1029/1000
Comma list: 36/35, 51/50, 56/55, 66/65, 170/169


Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}
Mapping: {{mapping| 1 -4 -2 -4 1 0 1 | 0 9 7 11 4 6 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.822
Optimal tunings:
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}


{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


Badness: 0.047247
Badness (Sintel): 1.43


==== 13-limit ====
== Slurpee ==
Subgroup: 2.3.5.7.11.13
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''
 
Comma list: 36/35, 56/55, 66/65, 640/637
 
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.882
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.  
 
{{Optimal ET sequence|legend=1| 4, 17c, 21, 38ce, 55ccde }}
 
Badness: 0.036701
 
== Slurpee ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Slurpee]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 251: Line 279:


{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 7 11 3 1 -15 -24 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 72.303
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
[[Badness]] (Sintel): 2.91
 
[[Badness]]: 0.114869


=== 11-limit ===
=== 11-limit ===
Line 267: Line 298:
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.297
Optimal tunings:
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}


{{Optimal ET sequence|legend=1| 16, 17c, 33, 50cd, 83bccd }}
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


Badness: 0.050514
Badness (Sintel): 1.67


=== 13-limit ===
=== 13-limit ===
Line 280: Line 313:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 72.409
Optimal tunings:
 
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
{{Optimal ET sequence|legend=1| 16, 17c, 33, 83bccd }}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}
 
Badness: 0.033102
 
== Penta ==
{{See also| Gamelismic clan #Gidorah }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 28/25, 36/35
 
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
 
{{Multival|legend=1| 3 2 4 -4 -2 4 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 239.977
 
{{Optimal ET sequence|legend=1| 1bd, 5 }}
 
[[Badness]]: 0.046882
 
== Shallowtone ==
[[Subgroup]]: 2.3.5


[[Comma list]]: [[Shallowtone comma|295245/262144]]
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}


{{Mapping|legend=1|1 1 8|0 1 -10}}
Badness (Sintel): 1.37


Mapping generators: ~2, ~3/2
== Shallowtone ==
{{Main| Shallowtone }}
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.801
[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b, 44b, 51b, 58bc, 65bbc, 72bbcc, 79bbcc}}
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 295245/262144
[[Comma list]]: 36/35, 295245/262144


{{Mapping|legend=1|1 1 8|0 1 -10 12}}
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
: mapping generators: ~2, ~3


Mapping generators: ~2, ~3/2
[[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 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.278
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
[[Badness]] (Sintel): 7.79


=== 11-limit ===
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 36/35, 45/44, 295245/262144
Comma list: 36/35, 45/44, 72171/65536


{{Mapping|legend=1|1 1 8|0 1 -10 12 -8}}
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


Mapping generators: ~2, ~3/2
Optimal tunings:
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.327
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
Badness (Sintel): 4.29


=== 13-limit ===
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 36/35, 45/44, 27/26, 34749/32768
Comma list: 27/26, 36/35, 45/44, 16731/16384


{{Mapping|legend=1|1 1 8|0 1 -10 12 -8}}
Mapping: {{Mapping| 1 0 18 -16 16 -1 | 0 1 -10 12 -8 3 }}


Mapping generators: ~2, ~3/2
Optimal tunings:
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 681.267
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


[[Support]]ing [[ET]]s: {{EDOs|7, 30b, 37b}}
Badness (Sintel): 3.19


== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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