Mint temperaments: Difference between revisions

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This is a collection of low complexity, high error [[regular temperament|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. 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.  
{{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]]'' (+135/128) → [[Mavila family #Armodue|Mavila family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* ''[[Dicot]]'' (+25/24) → [[Dicot family|Dicot family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
* [[Beep]] (+21/20) → [[Bug family #Beep|Bug family]]
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* ''[[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]]
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== 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]].''


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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: 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:  
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: mapping generators: ~2, ~7/6
: mapping generators: ~2, ~7/6
{{Multival|legend=1| 3 2 4 -4 -2 4 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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[[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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