Mint temperaments: Difference between revisions

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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 (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]]


POTE generator: ~25/24 = 140.995
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


Map: [<1 1 2|, <0 5 3|]
[[Subgroup]]: 2.3.5.7


EDOs: 8, 9, 17c
[[Comma list]]: 28/25, 36/35


Badness: 0.2005
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
: mapping generators: ~2, ~7/6


==7-limit==
[[Optimal tuning]]s:
Commas: 36/35, 392/375
* [[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 }}


POTE generator: ~15/14 = 140.927
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


Map: [<1 1 2 2|, <0 5 3 7|]
[[Badness]] (Sintel): 1.19


Wedgie: <<5 3 7 -7 -3 8||
== Progression ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''


EDOs: 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: 0.0484
[[Subgroup]]: 2.3.5.7


==11-limit==
[[Comma list]]: 36/35, 392/375
Commas: 36/35, 56/55, 77/75


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


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


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


Badness: 0.0261
[[Badness]] (Sintel): 1.22


==13-limit==
=== 11-limit ===
Commas: 26/25, 36/35, 56/55, 66/65
Subgroup: 2.3.5.7.11


[[POTE_tuning|POTE generator]]: ~15/14 = 140.751
Comma list: 36/35, 56/55, 77/75


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


EDOs: 8, 9, 17
Optimal tunings:  
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}


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


==17-limit==
Badness (Sintel): 0.861
Commas: 26/25, 36/35, 51/50, 56/55, 66/65


[[POTE_tuning|POTE generator]]: ~15/14 = 141.404
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


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


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


Badness: 0.0167
Optimal tunings:  
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}


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


[[POTE_tuning|POTE generator]]: ~15/14 = 140.479
Badness (Sintel): 0.750


Map: [&lt;1 1 2 2 3 3 4 4|, &lt;0 5 3 7 4 6 1 2|]
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


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


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


=Ripple=
Optimal tunings:
Comma: 6561/6250
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}


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


Map: [&lt;1 2 3|, &lt;0 -5 -8|]
Badness (Sintel): 0.853


EDOs: 12, 71b, 83b, 95b, 107bc, 119bc
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.1389
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65


==7-limit==
Mapping: {{mapping| 1 1 2 2 3 3 4 4 | 0 5 3 7 4 6 1 2 }}
Commas: 36/35, 2560/2401


[[POTE_tuning|POTE generator]]: ~21/20 = 99.483
Optimal tunings:  
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}


Map: [&lt;1 2 3 3|, &lt;0 -5 -8 -2|]
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}


Wedgie: &lt;&lt;5 8 2 1 -11 -18||
Badness (Sintel): 1.02


EDOs: [[12edo|12]], [[23edo|23]]
== Subklei ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


Badness: 0.0597
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]].  


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 36/35, 80/77, 126/121


POTE generator: ~21/20 = ~12/11 = 99.385
[[Comma list]]: 36/35, 1029/1000


Map: [&lt;1 2 3 3 4|, &lt;0 -5 -8 -2 -6|]
{{Mapping|legend=1| 1 -3 -3 -1 | 0 6 7 5 }}
: mapping generators: ~2, ~5/3


EDOs: 12
[[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 }}


Badness: 0.0388
{{Optimal ET sequence|legend=1| 4, 13cd, 17c, 21, 38c }}


==13-limit==
[[Badness]] (Sintel): 1.55
Commas: 36/35, 40/39, 66/65, 147/143


POTE generator: ~21/20 = ~12/11 = ~14/13 = 98.572
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 2 3 3 4 4|, &lt;0 -5 -8 -2 -6 -3|]
Comma list: 36/35, 77/75, 352/343


EDOs: 12f
Mapping: {{mapping| 1 -3 -3 -1 -8 | 0 6 7 5 15 }}


Badness: 0.0316
Optimal tunings:  
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}


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


POTE generator: ~5/4 = 420.855
Badness (Sintel): 1.48


Map: [&lt;1 3 2|, &lt;0 -4 1|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 11, 14, 17c, 20c, 37c
Comma list: 26/25, 36/35, 66/65, 352/343


Badness: 0.179
Mapping: {{mapping| 1 3 4 4 7 7 | 0 6 7 5 15 14 }}


==7-limit==
Optimal tunings:
Commas: 36/35, 2048/1875
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}


POTE generator: ~5/4 = 422.275
{{Optimal ET sequence|legend=0| 4ef, …, 13cdeef, 17c }}


Map: [&lt;1 3 2 6|, &lt;0 -4 1 -9|]
Badness (Sintel): 1.34


Wedgie: &lt;&lt;4 -1 9 -11 3 24||
=== Subkla ===
Subgroup: 2.3.5.7.11


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


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


==11-limit==
Optimal tunings:
Commas: 36/35 56/55 243/242
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}


POTE generator: ~5/4 = 422.217
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}


Map: [&lt;1 3 2 6 7|, &lt;0 -4 1 -9 -10|]
Badness (Sintel): 1.56


EDOs: 14, 17c, 37cde
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Badness: 0.0425
Comma list: 36/35, 56/55, 66/65, 640/637


==13-limit==
Mapping: {{mapping| 1 -3 -3 -1 5 6 | 0 6 7 5 -2 -3 }}
Commas: 26/25, 36/35, 56/55, 243/242


POTE generator: ~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}}


Map: [&lt;1 3 2 6 7 3|, &lt;0 -4 1 -9 -10 2|]
{{Optimal ET sequence|legend=0| 4, 17c, 38ce }}


EDOs: 14, 17c
Badness (Sintel): 1.52


Badness: 0.0368
== Naian ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Naian]].''


=Hemismate=
Named by [[Xenllium]] in 2026, naian may be described as {{nowrap| 8d & 21 }} with a [[ploidacot]] signature of epsilon-enneacot.
commas: 256/245, 392/375


POTE generator: ~8/7 = 210.452
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 3 2 3|, &lt;0 -8 2 -1|]
[[Comma list]]: 36/35, 9604/9375


Wedgie: &lt;&lt;8 -2 1 -22 -21 8||
{{Mapping|legend=1| 1 -4 -2 -4 | 0 9 7 11 }}
: mapping generators: ~2, ~75/49


EDOs: 6, 11, 17c, 40bcd
[[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 }}


Badness: 0.1543
{{Optimal ET sequence|legend=1| 8d, 21, 29cd }}


==11-limit==
[[Badness]] (Sintel): 2.89
Commas: 56/55, 77/75, 256/245


POTE generator: ~8/7 =  210.481
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 3 2 3 4|, &lt;0 -8 2 -1 -3|]
Comma list: 36/35, 56/55, 2541/2500


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


Badness: 0.0655
Optimal tunings:  
* WE: ~2 = 1194.3937{{c}}, ~75/49 = 741.2117{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 744.2018{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 8d, 21, 29cde }}
Commas: 26/25, 56/55, 77/75, 256/245


POTE generator: ~8/7 = 210.974
Badness (Sintel): 1.95


Map: [&lt;1 3 2 3 4 3|, &lt;0 -8 2 -1 -3 4|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 6, 11, 17c
Comma list: 36/35, 56/55, 66/65, 507/500


Badness: 0.0505
Mapping: {{mapping| 1 -4 -2 -4 1 0 | 0 9 7 11 4 6 }}


=Subklei=
Optimal tunings:
Comma: 17496/15625
* WE: ~2 = 1193.8564{{c}}, ~20/13 = 740.9635{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2703{{c}}


POTE generator: ~6/5 = 284.293
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


Map: [&lt;1 3 4|, &lt;0 -6 -7|]
Badness (Sintel): 1.55


EDOs: 4, 17c, 21, 38c
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Badness: 0.3559
Comma list: 36/35, 51/50, 56/55, 66/65, 170/169


==7-limit==
Mapping: {{mapping| 1 -4 -2 -4 1 0 1 | 0 9 7 11 4 6 5 }}
Commas: 36/35, 1029/1000


POTE generator: ~6/5 = 284.219
Optimal tunings:  
* WE: ~2 = 1194.2330{{c}}, ~20/13 = 741.1242{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.2679{{c}}


Map: [&lt;1 3 4 4|, &lt;0 -6 -7 -5|]
{{Optimal ET sequence|legend=0| 8d, 21, 29cdef }}


Wedgie: &lt;&lt;6 7 5 -3 -9 -8||
Badness (Sintel): 1.43


EDOs: 4, 17c, 21, 38c
== Slurpee ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''


Badness: 0.0611
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.  


==11-limit==
[[Subgroup]]: 2.3.5.7
Commas: 36/35, 77/75, 352/343


POTE generator: ~6/5 = 283.253
[[Comma list]]: 36/35, 51200/50421


Map: [&lt;1 3 4 4 7|, &lt;0 -6 -7 -5 -15|]
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
: mapping generators: ~2, ~21/20


EDOs: 17c, 55cd
[[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 }}


Badness: 0.0447
{{Optimal ET sequence|legend=1| 16, 17c, 33 }}


==Subkla==
[[Badness]] (Sintel): 2.91
Commas: 36/35, 56/55, 1029/1000


POTE generator: ~6/5 = 283.822
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 3 4 4 3|, &lt;0 -6 -7 -5 2|]
Comma list: 36/35, 121/120, 352/343


EDOs: 4, 17c, 21, 38ce, 55cde
Mapping: {{mapping| 1 2 3 3 4 | 0 -7 -11 -3 -9 }}


Badness: 0.0472
Optimal tunings:  
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 16, 17c, 33 }}
Commas: 36/35, 56/55, 66/65, 640/637


POTE generator: ~6/5 = 283.882
Badness (Sintel): 1.67


Map: [&lt;1 3 4 4 3 3|, &lt;0 -6 -7 -5 2 3|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 4, 17c, 21, 38ce, 55cde
Comma list: 36/35, 66/65, 143/140, 352/343


Badness: 0.0367
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -7 -11 -3 -9 -5 }}


==13-limit==
Optimal tunings:
Commas: 26/25, 36/35, 66/65, 352/343
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}


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


Map: [&lt;1 3 4 4 7 7|, &lt;0 -6 -7 -5 -15 -14|]
Badness (Sintel): 1.37


EDOs: 17c
== Shallowtone ==
{{Main| Shallowtone }}
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


Badness: 0.0323
[[Subgroup]]: 2.3.5.7


=Slurpee=
[[Comma list]]: 36/35, 295245/262144
Comma: 177147/156250


POTE generator: ~27/25 = 72.742
{{Mapping|legend=1| 1 0 18 -16 | 0 1 -10 12 }}
: mapping generators: ~2, ~3


Map: [&lt;1 2 3|, &lt;0 -7 -11|]
[[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 }}


EDOs: 16, 17c, 33
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}


Badness: 0.6893
[[Badness]] (Sintel): 7.79


==7-limit==
=== 11-limit ===
Commas: 36/35, 51200/50421
Subgroup: 2.3.5.7.11


POTE generator: ~21/20 = 72.303
Comma list: 36/35, 45/44, 72171/65536


Map: [&lt;1 2 3 3|, &lt;0 -7 -11 -3|]
Mapping: {{Mapping| 1 0 18 -16 16 | 0 1 -10 12 -8 }}


Wedgie: &lt;&lt;7 11 3 1 -15 -24||
Optimal tunings:
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}


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


Badness: 0.1149
Badness (Sintel): 4.29


==11-limit==
=== 13-limit ===
Commas: 36/35, 121/120, 352/343
Subgroup: 2.3.5.7.11.13


POTE generator: ~21/20 = 72.297
Comma list: 27/26, 36/35, 45/44, 16731/16384


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


EDOs: 16, 17c, 33, 50cd, 83bcd
Optimal tunings:
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}


Badness: 0.0505
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


==13-limit==
Badness (Sintel): 3.19
Commas: 36/35. 66/65, 143/140, 352/343


POTE generator: ~21/20 = 72.409
== References ==


Map: [&lt;1 2 3 3 4 4|, &lt;0 -7 -11 -3 -9 -5|]
[[Category:Temperament collections]]
 
[[Category:Mint temperaments| ]] <!-- main article -->
EDOs: 16, 17c, 33, 83bcd
[[Category:Rank 2]]
 
Badness: 0.0331
 
[[Category:definition]]
[[Category:overview]]
[[Category:quartonic]]
[[Category:temperament]]

Latest revision as of 10:15, 29 May 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of low complexity, high error, temperaments which temper out the septimal quarter-tone, 36/35. 36 is both a 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

Penta

For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.

Subgroup: 2.3.5.7

Comma list: 28/25, 36/35

Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • WE: ~2 = 1186.8093 ¢, ~7/6 = 237.3390 ¢
error map: ⟨-13.191 -3.129 +61.983 -45.851]
  • CWE: ~2 = 1200.0000 ¢, ~7/6 = 231.845 ¢
error map: ⟨0.000 +6.291 +85.850 -24.498]

Optimal ET sequence: 1bd, …, 4bcd, 5

Badness (Sintel): 1.19

Progression

Not to be confused with Progress.
For the 5-limit version, see Miscellaneous 5-limit temperaments #Lafayette.

Named by Gene Ward Smith in 2011[1], progression can be described as the 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

Comma list: 36/35, 392/375

Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]

mapping generators: ~2, ~15/14

Optimal tunings:

  • WE: ~2 = 1193.9544 ¢, ~15/14 = 140.2169 ¢
error map: ⟨-6.046 -6.916 +22.246 +0.601]
  • CWE: ~2 = 1200.000 ¢, ~15/14 = 139.8991 ¢
error map: ⟨0.000 -2.460 +33.384 +10.468]

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 1.22

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]

Optimal tunings:

  • WE: ~2 = 1194.7089 ¢, ~12/11 = 140.1262 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.8776 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.861

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]

Optimal tunings:

  • WE: ~2 = 1195.3694 ¢, ~13/12 = 140.2080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.9749 ¢

Optimal ET sequence: 8d, 9, 17c

Badness (Sintel): 0.750

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]

Optimal tunings:

  • WE: ~2 = 1193.8350 ¢, ~13/12 = 140.6779 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.5885 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 0.853

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65

Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]

Optimal tunings:

  • WE: ~2 = 1196.1446 ¢, ~13/12 = 140.0276 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.0301 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 1.02

Subklei

For the 5-limit version, see Miscellaneous 5-limit temperaments #Delorean.

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 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 perfect fifth.

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 -3 -3 -1], ⟨0 6 7 5]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • WE: ~2 = 1197.5285 ¢, ~5/3 = 915.8950 ¢
error map: ⟨-2.471 -11.171 +18.366 +3.121]
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.4228 ¢
error map: ⟨0.000 -9.418 +21.646 +8.288]

Optimal ET sequence: 4, 13cd, 17c, 21, 38c

Badness (Sintel): 1.55

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 -3 -1 -8], ⟨0 6 7 5 15]]

Optimal tunings:

  • WE: ~2 = 1196.3345 ¢, ~5/3 = 913.9469 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.2970 ¢

Optimal ET sequence: 4e, …, 13cdee, 17c

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 3 4 4 7 7], ⟨0 6 7 5 15 14]]

Optimal tunings:

  • WE: ~2 = 1196.0784 ¢, ~5/3 = 914.1466 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.6712 ¢

Optimal ET sequence: 4ef, …, 13cdeef, 17c

Badness (Sintel): 1.34

Subkla

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 -3 -1 5], ⟨0 6 7 5 -2]]

Optimal tunings:

  • WE: ~2 = 1196.3809 ¢, ~5/3 = 913.4153 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.8804 ¢

Optimal ET sequence: 4, 13cd, 17c, 38ce

Badness (Sintel): 1.56

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 -3 -1 5 6], ⟨0 6 7 5 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1196.5477 ¢, ~5/3 = 913.4822 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.9416 ¢

Optimal ET sequence: 4, 17c, 38ce

Badness (Sintel): 1.52

Naian

For the 5-limit version, see Miscellaneous 5-limit temperaments #Naian.

Named by Xenllium in 2026, naian may be described as 8d & 21 with a ploidacot signature of epsilon-enneacot.

Subgroup: 2.3.5.7

Comma list: 36/35, 9604/9375

Mapping: [⟨1 -4 -2 -4], ⟨0 9 7 11]]

mapping generators: ~2, ~75/49

Optimal tunings:

  • WE: ~2 = 1195.8807 ¢, ~75/49 = 741.8522 ¢
error map: ⟨-4.119 -8.808 +14.890 +8.026]
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 743.9885 ¢
error map: ⟨0.000 -6.059 +21.606 +15.047]

Optimal ET sequence: 8d, 21, 29cd

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 2541/2500

Mapping: [⟨1 -4 -2 -4 1], ⟨0 9 7 11 4]]

Optimal tunings:

  • WE: ~2 = 1194.3937 ¢, ~75/49 = 741.2117 ¢
  • CWE: ~2 = 1200.0000 ¢, ~75/49 = 744.2018 ¢

Optimal ET sequence: 8d, 21, 29cde

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 -2 -4 1 0], ⟨0 9 7 11 4 6]]

Optimal tunings:

  • WE: ~2 = 1193.8564 ¢, ~20/13 = 740.9635 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2703 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.55

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -4 -2 -4 1 0 1], ⟨0 9 7 11 4 6 5]]

Optimal tunings:

  • WE: ~2 = 1194.2330 ¢, ~20/13 = 741.1242 ¢
  • CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2679 ¢

Optimal ET sequence: 8d, 21, 29cdef

Badness (Sintel): 1.43

Slurpee

For the 5-limit version, see Miscellaneous 5-limit temperaments #Slurpee.

Slurpee may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 36/35, 51200/50421

Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1197.9281 ¢, ~21/20 = 72.1780 ¢
error map: ⟨-2.072 -11.345 +13.512 +8.424]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4941 ¢
error map: ⟨0.000 -9.414 +16.251 +13.692]

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 2.91

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -7 -11 -3 -9]]

Optimal tunings:

  • WE: ~2 = 1198.4220 ¢, ~21/20 = 72.2015 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4470 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.67

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -7 -11 -3 -9 -5]]

Optimal tunings:

  • WE: ~2 = 1199.0238 ¢, ~21/20 = 72.3506 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4956 ¢

Optimal ET sequence: 16, 17c, 33

Badness (Sintel): 1.37

Shallowtone

Main article: Shallowtone
For the 5-limit version, see Syntonic–chromatic equivalence continuum #Shallowtone (5-limit).

Subgroup: 2.3.5.7

Comma list: 36/35, 295245/262144

Mapping: [⟨1 0 18 -16], ⟨0 1 -10 12]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1202.4397 ¢, ~3/2 = 682.6219 ¢
error map: ⟨+2.440 -16.893 +6.986 +12.877]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2447 ¢
error map: ⟨0.000 -20.710 +1.239 +6.110]

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 7.79

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 18 -16 16], ⟨0 1 -10 12 -8]]

Optimal tunings:

  • WE: ~2 = 1202.0285 ¢, ~3/2 = 682.4922 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.3267 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 4.29

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 18 -16 16 -1], ⟨0 1 -10 12 -8 3]]

Optimal tunings:

  • WE: ~2 = 1201.4928 ¢, ~3/2 = 682.1188 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2669 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 3.19

References

  1. ↑ Yahoo! Tuning Group | Progression temperament
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