Mint temperaments: Difference between revisions

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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]]
* [[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]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Dominant (temperament)|Dominant]]'' (+64/63) → [[Meantone family #Dominant|Meantone family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* ''[[Father]]'' (+16/15) → [[Father family #Septimal father|Father family]]
* ''[[Diminished (temperament)|Diminished]]'' (+50/49) → [[Dimipent family #Diminished|Dimipent 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]]
* ''[[Ripple]]'' (+2560/2401) → [[Ripple family #Septimal ripple|Ripple family]]
* ''[[Whitewood]]'' (+2187/2048) → [[Whitewood family #Septimal whitewood|Whitewood family]]


== 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.  
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.  
Line 26: Line 31:
: mapping generators: ~2, ~15/14
: mapping generators: ~2, ~15/14


{{Multival|legend=1| 5 3 7 -7 -3 8 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~15/14 = 138.611
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/14 = 140.927
: [[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 }}


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


[[Badness]]: 0.048356
[[Badness]] (Smith): 0.048356


=== 11-limit ===
=== 11-limit ===
Line 41: Line 48:
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:
* CTE: ~2 = 1200.000, ~12/11 = 138.556
* POTE: ~2 = 1200.000, ~12/11 = 140.747


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


Badness: 0.026050
Badness (Smith): 0.026050


=== 13-limit ===
=== 13-limit ===
Line 54: Line 63:
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:
* CTE: ~2 = 1200.000, ~13/12 = 138.741
* POTE: ~2 = 1200.000, ~13/12 = 140.751


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


Badness: 0.018158
Badness (Smith): 0.018158


=== 17-limit ===
=== 17-limit ===
Line 67: Line 78:
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:
* CTE: ~2 = 1200.000, ~13/12 = 138.649
* POTE: ~2 = 1200.000, ~13/12 = 141.404


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


Badness: 0.016743
Badness (Smith): 0.016743


=== 19-limit ===
=== 19-limit ===
Line 80: Line 93:
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:
 
* CTE: ~2 = 1200.000, ~13/12 = 138.750
{{Optimal ET sequence|legend=1| 8d, 9, 17cg }}
* POTE: ~2 = 1200.000, ~13/12 = 140.479
 
Badness: 0.016841


== Ripple ==
{{Optimal ET sequence|legend=0| 8d, 9, 17cg }}
{{see also| Ripple family #Ripple }}


[[Subgroup]]: 2.3.5.7
Badness (Smith): 0.016841
 
[[Comma list]]: 36/35, 2560/2401
 
{{Mapping|legend=1| 1 2 3 3 | 0 -5 -8 -2 }}
 
{{Multival|legend=1| 5 8 2 1 -11 -18 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 99.483
 
{{Optimal ET sequence|legend=1| 12 }}
 
[[Badness]]: 0.059735
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 80/77, 126/121
 
Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 99.385
 
{{Optimal ET sequence|legend=1| 12 }}
 
Badness: 0.038811
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 36/35, 40/39, 66/65, 147/143
 
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 98.572
 
{{Optimal ET sequence|legend=1| 12f }}
 
Badness: 0.031639
 
== Smate ==
{{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
 
Comma list: 36/35, 56/55, 243/242
 
Mapping: {{mapping| 1 3 2 6 7 | 0 -4 1 -9 -10 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 422.217
 
{{Optimal ET sequence|legend=1| 3de, 14, 17c, 37ccddee }}
 
Badness: 0.042518
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 26/25, 36/35, 56/55, 243/242
 
Mapping: {{mapping| 1 3 2 6 7 3 | 0 -4 1 -9 -10 2 }}
 
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 423.020
 
{{Optimal ET sequence|legend=1| 3de, 14, 17c }}
 
Badness: 0.036836


== Subklei ==
== Subklei ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Delorean]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Delorean]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 181: Line 110:
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}
{{Mapping|legend=1| 1 3 4 4 | 0 -6 -7 -5 }}


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


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


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


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


=== 11-limit ===
=== 11-limit ===
Line 196: Line 129:
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}
Mapping: {{mapping| 1 3 4 4 7 | 0 -6 -7 -5 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.253
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.265
* POTE: ~2 = 1200.000, ~7/6 = 283.253


{{Optimal ET sequence|legend=1| 17c, 55ccd }}
{{Optimal ET sequence|legend=0| 4e, …, 13cdee, 17c }}


Badness: 0.044651
Badness (Smith): 0.044651


==== 13-limit ====
==== 13-limit ====
Line 209: Line 144:
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 tuning (POTE): ~2 = 1\1, ~6/5 = 282.856
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 283.918
* POTE: ~2 = 1200.000, ~7/6 = 282.856


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


Badness: 0.032324
Badness (Smith): 0.032324


=== Subkla ===
=== Subkla ===
Line 222: Line 159:
Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}
Mapping: {{mapping| 1 3 4 4 3 | 0 -6 -7 -5 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.822
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.870
* POTE: ~2 = 1200.000, ~7/6 = 283.822


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


Badness: 0.047247
Badness (Smith): 0.047247


==== 13-limit ====
==== 13-limit ====
Line 235: Line 174:
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}
Mapping: {{mapping| 1 3 4 4 3 3 | 0 -6 -7 -5 2 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 283.882
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 284.758
* POTE: ~2 = 1200.000, ~7/6 = 283.882


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


Badness: 0.036701
Badness (Smith): 0.036701


== Slurpee ==
== Slurpee ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Slurpee]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Slurpee]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 250: Line 191:
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -7 -11 -3 }}


{{Multival|legend=1| 7 11 3 1 -15 -24 }}
: mapping generators: ~2, ~21/20


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 72.303
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 265: Line 210:
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:
* CTE: ~2 = 1200.000, ~21/20 = 72.681
* POTE: ~2 = 1200.000, ~21/20 = 72.297


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


Badness: 0.050514
Badness (Smith): 0.050514


=== 13-limit ===
=== 13-limit ===
Line 278: Line 225:
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:
* CTE: ~2 = 1200.000, ~21/20 = 72.653
* POTE: ~2 = 1200.000, ~21/20 = 72.409


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


Badness: 0.033102
Badness (Smith): 0.033102


== Penta ==
== Penta ==
{{See also| Gamelismic clan #Gidorah }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 293: Line 242:
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}
{{Mapping|legend=1| 1 1 2 2 | 0 3 2 4 }}


{{Multival|legend=1| 3 2 4 -4 -2 4 }}
: mapping generators: ~2, ~7/6


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 239.977
[[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, 5 }}
{{Optimal ET sequence|legend=1| 1bd, …, 4bcd, 5 }}


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


== Shallowtone ==
== Shallowtone ==
:''For the 5-limit version of this temperament, see [[Syntonic-chromatic equivalence continuum]].''
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Shallowtone (5-limit)]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 309: Line 262:


{{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


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~3/2 = 681.2780
* [[CTE]]: ~2 = 1200.000, ~3/2 = 681.278
* [[CWE]]: ~2 = 1\1, ~3/2 = 681.2447
: [[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 }}


{{Optimal ET sequence|legend=1| 7, 30b, 37b }}
{{Optimal ET sequence|legend=1| 7, 30b, 37b }}


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


=== 11-limit ===
=== 11-limit ===
Line 326: Line 283:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 681.2538
* CTE: ~2 = 1200.000, ~3/2 = 681.254
* CWE: ~2 = 1\1, ~3/2 = 681.3267
* CWE: ~2 = 1200.000, ~3/2 = 681.327


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


Badness: 0.130
Badness (Smith): 0.130


=== 13-limit ===
=== 13-limit ===
Line 341: Line 298:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 681.2374
* CTE: ~2 = 1200.000, ~3/2 = 681.237
* CWE: ~2 = 1\1, ~3/2 = 681.2669
* CWE: ~2 = 1200.000, ~3/2 = 681.267
 
{{Optimal ET sequence|legend=0| 7, 30b, 37b }}


Optimal ET sequence: {{Optimal ET sequence| 7, 30b, 37b }}
Badness (Smith): 0.0773


Badness: 0.0773
== Notes ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint temperaments| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 12:39, 21 August 2025

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

Progression

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

Progression, named by Gene Ward Smith in 2011[1] can be described as the 8d & 9 temperament, and 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:

  • CTE: ~2 = 1200.000, ~15/14 = 138.611
error map: ⟨0.000 -8.900 +29.519 +1.451]
  • POTE: ~2 = 1200.000, ~15/14 = 140.927
error map: ⟨0.000 +2.680 +36.467 +17.663]

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.048356

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:

  • CTE: ~2 = 1200.000, ~12/11 = 138.556
  • POTE: ~2 = 1200.000, ~12/11 = 140.747

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.026050

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:

  • CTE: ~2 = 1200.000, ~13/12 = 138.741
  • POTE: ~2 = 1200.000, ~13/12 = 140.751

Optimal ET sequence: 8d, 9, 17c

Badness (Smith): 0.018158

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:

  • CTE: ~2 = 1200.000, ~13/12 = 138.649
  • POTE: ~2 = 1200.000, ~13/12 = 141.404

Optimal ET sequence: 8d, 9, 17cg

Badness (Smith): 0.016743

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:

  • CTE: ~2 = 1200.000, ~13/12 = 138.750
  • POTE: ~2 = 1200.000, ~13/12 = 140.479

Optimal ET sequence: 8d, 9, 17cg

Badness (Smith): 0.016841

Subklei

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

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 284.986
error map: ⟨0.000 -11.870 +18.785 +6.245]
  • POTE: ~2 = 1200.000, ~7/6 = 284.219
error map: ⟨0.000 -7.268 +24.154 +10.080]

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

Badness (Smith): 0.061101

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 284.265
  • POTE: ~2 = 1200.000, ~7/6 = 283.253

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

Badness (Smith): 0.044651

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:

  • CTE: ~2 = 1200.000, ~7/6 = 283.918
  • POTE: ~2 = 1200.000, ~7/6 = 282.856

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

Badness (Smith): 0.032324

Subkla

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 284.870
  • POTE: ~2 = 1200.000, ~7/6 = 283.822

Optimal ET sequence: 4, 17c, 21, 38ce

Badness (Smith): 0.047247

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 284.758
  • POTE: ~2 = 1200.000, ~7/6 = 283.882

Optimal ET sequence: 4, 17c, 38ce

Badness (Smith): 0.036701

Slurpee

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

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:

  • CTE: ~2 = 1200.000, ~21/20 = 72.776
error map: ⟨0.000 -11.385 +13.153 +12.847]
  • POTE: ~2 = 1200.000, ~21/20 = 72.303
error map: ⟨0.000 -8.075 +18.355 +14.265]

Optimal ET sequence: 16, 17c, 33

Badness (Smith): 0.114869

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:

  • CTE: ~2 = 1200.000, ~21/20 = 72.681
  • POTE: ~2 = 1200.000, ~21/20 = 72.297

Optimal ET sequence: 16, 17c, 33

Badness (Smith): 0.050514

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:

  • CTE: ~2 = 1200.000, ~21/20 = 72.653
  • POTE: ~2 = 1200.000, ~21/20 = 72.409

Optimal ET sequence: 16, 17c, 33

Badness (Smith): 0.033102

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:

  • CTE: ~2 = 1200.000, ~7/6 = 231.845
error map: ⟨0.000 -6.421 +77.376 -41.447]
  • POTE: ~2 = 1200.000, ~7/6 = 239.977
error map: ⟨0.000 +17.976 +93.640 -8.918]

Optimal ET sequence: 1bd, …, 4bcd, 5

Badness (Smith): 0.046882

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:

  • CTE: ~2 = 1200.000, ~3/2 = 681.278
error map: ⟨0.000 -20.677 +0.906 +6.510]
  • CWE: ~2 = 1200.000, ~3/2 = 681.245
error map: ⟨0.000 -20.710 +1.239 +6.110]

Optimal ET sequence: 7, 30b, 37b

Badness (Smith): 0.308

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:

  • CTE: ~2 = 1200.000, ~3/2 = 681.254
  • CWE: ~2 = 1200.000, ~3/2 = 681.327

Optimal ET sequence: 7, 30b, 37b

Badness (Smith): 0.130

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:

  • CTE: ~2 = 1200.000, ~3/2 = 681.237
  • CWE: ~2 = 1200.000, ~3/2 = 681.267

Optimal ET sequence: 7, 30b, 37b

Badness (Smith): 0.0773

Notes

  1. ↑ Yahoo! Tuning Group | Progression temperament
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