Mint temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
m No septimal dicot anymore!
 
(22 intermediate revisions by 6 users not shown)
Line 1: Line 1:
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. This equates very different intervals with each other—[[6/5]] with [[7/6]], [[5/4]] with [[9/7]], and [[7/4]] with [[9/5]]. In a sense, what mint is all about is pretending that minor and major thirds and sixths are simultaneously 5-limit and 7-limit, and like any temperament that seems to involve "pretending", mint is not far from the edge of what can be sensibly called a temperament at all. In other words, it is an [[exotemperament]].
{{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 (temperament)|Armodue]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Mujannabic|Dicot family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug 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]]
* ''[[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]]
* ''[[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]].''
 
[[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 ==
{{Distinguish| Progress }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lafayette]].''
: ''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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~15/14 = 138.611
* [[WE]]: ~2 = 1193.9544{{c}}, ~15/14 = 140.2169{{c}}
: [[error map]]: {{val| 0.000 -8.900 +29.519 +1.451 }}
: [[error map]]: {{val| -6.046 -6.916 +22.246 +0.601 }}
* [[POTE]]: ~2 = 1200.000, ~15/14 = 140.927
* [[CWE]]: ~2 = 1200.000{{c}}, ~15/14 = 139.8991{{c}}
: error map: {{val| 0.000 +2.680 +36.467 +17.663 }}
: 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]] (Smith): 0.048356
[[Badness]] (Sintel): 1.22


=== 11-limit ===
=== 11-limit ===
Line 48: Line 68:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* WE: ~2 = 1194.7089{{c}}, ~12/11 = 140.1262{{c}}
* POTE: ~2 = 1200.000, ~12/11 = 140.747
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.8776{{c}}


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


Badness (Smith): 0.026050
Badness (Sintel): 0.861


=== 13-limit ===
=== 13-limit ===
Line 63: Line 83:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.741
* WE: ~2 = 1195.3694{{c}}, ~13/12 = 140.2080{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 140.751
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.9749{{c}}


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


Badness (Smith): 0.018158
Badness (Sintel): 0.750


=== 17-limit ===
=== 17-limit ===
Line 78: Line 98:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.649
* WE: ~2 = 1193.8350{{c}}, ~13/12 = 140.6779{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 141.404
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.5885{{c}}


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


Badness (Smith): 0.016743
Badness (Sintel): 0.853


=== 19-limit ===
=== 19-limit ===
Line 93: Line 113:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~13/12 = 138.750
* WE: ~2 = 1196.1446{{c}}, ~13/12 = 140.0276{{c}}
* POTE: ~2 = 1200.000, ~13/12 = 140.479
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.0301{{c}}


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


Badness (Smith): 0.016841
Badness (Sintel): 1.02


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 107: Line 129:
[[Comma list]]: 36/35, 1029/1000
[[Comma list]]: 36/35, 1029/1000


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/6 = 284.986
* [[WE]]: ~2 = 1197.5285{{c}}, ~5/3 = 915.8950{{c}}
: [[error map]]: {{val| 0.000 -11.870 +18.785 +6.245 }}
: [[error map]]: {{val| -2.471 -11.171 +18.366 +3.121 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 284.219
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 915.4228{{c}}
: error map: {{val| 0.000 -7.268 +24.154 +10.080 }}
: error map: {{val| 0.000 -9.418 +21.646 +8.288 }}


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


[[Badness]] (Smith): 0.061101
[[Badness]] (Sintel): 1.55


=== 11-limit ===
=== 11-limit ===
Line 128: Line 147:
Comma list: 36/35, 77/75, 352/343
Comma list: 36/35, 77/75, 352/343


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* WE: ~2 = 1196.3345{{c}}, ~5/3 = 913.9469{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 283.253
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.2970{{c}}


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


Badness (Smith): 0.044651
Badness (Sintel): 1.48


==== 13-limit ====
==== 13-limit ====
Line 143: Line 162:
Comma list: 26/25, 36/35, 66/65, 352/343  
Comma list: 26/25, 36/35, 66/65, 352/343  


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* WE: ~2 = 1196.0784{{c}}, ~5/3 = 914.1466{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 282.856
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 916.6712{{c}}


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


Badness (Smith): 0.032324
Badness (Sintel): 1.34


=== Subkla ===
=== Subkla ===
Line 158: Line 177:
Comma list: 36/35, 56/55, 1029/1000
Comma list: 36/35, 56/55, 1029/1000


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* WE: ~2 = 1196.3809{{c}}, ~5/3 = 913.4153{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 283.822
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.8804{{c}}


{{Optimal ET sequence|legend=0| 4, 17c, 21, 38ce }}
{{Optimal ET sequence|legend=0| 4, 13cd, 17c, 38ce }}


Badness (Smith): 0.047247
Badness (Sintel): 1.56


==== 13-limit ====
==== 13-limit ====
Line 173: Line 192:
Comma list: 36/35, 56/55, 66/65, 640/637
Comma list: 36/35, 56/55, 66/65, 640/637


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* WE: ~2 = 1196.5477{{c}}, ~5/3 = 913.4822{{c}}
* POTE: ~2 = 1200.000, ~7/6 = 283.882
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 915.9416{{c}}


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


Badness (Smith): 0.036701
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 {{nowrap| 8d & 21 }} with a [[ploidacot]] signature of epsilon-enneacot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 36/35, 9604/9375
 
{{Mapping|legend=1| 1 -4 -2 -4 | 0 9 7 11 }}
: mapping generators: ~2, ~75/49
 
[[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| 8d, 21, 29cd }}
 
[[Badness]] (Sintel): 2.89
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 56/55, 2541/2500
 
Mapping: {{mapping| 1 -4 -2 -4 1 | 0 9 7 11 4 }}
 
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=0| 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: {{mapping| 1 -4 -2 -4 1 0 | 0 9 7 11 4 6 }}
 
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=0| 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: {{mapping| 1 -4 -2 -4 1 0 1 | 0 9 7 11 4 6 5 }}
 
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=0| 8d, 21, 29cdef }}
 
Badness (Sintel): 1.43


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 191: 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
: mapping generators: ~2, ~21/20
{{Multival|legend=1| 7 11 3 1 -15 -24 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~21/20 = 72.776
* [[WE]]: ~2 = 1197.9281{{c}}, ~21/20 = 72.1780{{c}}
: [[error map]]: {{val| 0.000 -11.385 +13.153 +12.847 }}
: [[error map]]: {{val| -2.072 -11.345 +13.512 +8.424 }}
* [[POTE]]: ~2 = 1200.000, ~21/20 = 72.303
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 72.4941{{c}}
: error map: {{val| 0.000 -8.075 +18.355 +14.265 }}
: error map: {{val| 0.000 -9.414 +16.251 +13.692 }}


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


[[Badness]] (Smith): 0.114869
[[Badness]] (Sintel): 2.91


=== 11-limit ===
=== 11-limit ===
Line 214: Line 299:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* WE: ~2 = 1198.4220{{c}}, ~21/20 = 72.2015{{c}}
* POTE: ~2 = 1200.000, ~21/20 = 72.297
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4470{{c}}


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


Badness (Smith): 0.050514
Badness (Sintel): 1.67


=== 13-limit ===
=== 13-limit ===
Line 229: Line 314:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* WE: ~2 = 1199.0238{{c}}, ~21/20 = 72.3506{{c}}
* POTE: ~2 = 1200.000, ~21/20 = 72.409
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 72.4956{{c}}


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


Badness (Smith): 0.033102
Badness (Sintel): 1.37
 
== Penta ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''
 
[[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
 
{{Multival|legend=1| 3 2 4 -4 -2 4 }}
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}
 
[[Badness]] (Smith): 0.046882


== Shallowtone ==
== Shallowtone ==
{{Main| 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)]].''


Line 267: Line 330:


{{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
: mapping generators: ~2, ~3


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.278
* [[WE]]: ~2 = 1202.4397{{c}}, ~3/2 = 682.6219{{c}}
: [[error map]]: {{val| 0.000 -20.677 +0.906 +6.510 }}
: [[error map]]: {{val| +2.440 -16.893 +6.986 +12.877 }}
* [[CWE]]: ~2 = 1200.000, ~3/2 = 681.245
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 681.2447{{c}}
: error map: {{val| 0.000 -20.710 +1.239 +6.110 }}
: 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 }}


[[Badness]] (Smith): 0.308
[[Badness]] (Sintel): 7.79


=== 11-limit ===
=== 11-limit ===
Line 288: Line 350:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* WE: ~2 = 1202.0285{{c}}, ~3/2 = 682.4922{{c}}
* CWE: ~2 = 1200.000, ~3/2 = 681.327
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.3267{{c}}


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


Badness (Smith): 0.130
Badness (Sintel): 4.29


=== 13-limit ===
=== 13-limit ===
Line 303: Line 365:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* WE: ~2 = 1201.4928{{c}}, ~3/2 = 682.1188{{c}}
* CWE: ~2 = 1200.000, ~3/2 = 681.267
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 681.2669{{c}}


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


Badness (Smith): 0.0773
Badness (Sintel): 3.19


== Notes ==
== 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]]

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
Retrieved from "https://en.xen.wiki/index.php?title=Mint_temperaments&oldid=231277"