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Temperaments of '''immunity family''' temper out the [[immunity comma]] (1638400/1594323, sasayoyo).
{{Technical data page}}
The '''immunity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[immunity comma]], 1638400/1594323 ({{monzo| 16 -13 2 }}).


== Immunity ==
== Immunity ==
Subgroup: 2.3.5
Immunity is naturally a 2.3.5.13 [[subgroup temperament]] as the generator serves as ~26/15 (or its [[octave complement]] ~15/13), tempering out [[676/675]].


[[Comma]]: 1638400/1594323
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val|1 0 -8}}, {{val|0 2 13}}]
[[Comma list]]: 1638400/1594323


[[POTE generator]]: ~729/640 = 247.070
{{Mapping|legend=1| 1 0 -8 | 0 2 13 }}


{{Val list|legend=1| 5, 29, 34 }}
: mapping generators: ~2, ~1280/729


[[Badness]]: 0.184738
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1280/729 = 952.706
: [[error map]]: {{val| 0.000 +3.456 -1.141 }}
* [[POTE]]: ~2 = 1200.000, ~1280/729 = 952.930
: error map: {{val| 0.000 +3.905 +1.775 }}


=== 7-limit ===
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
Subgroup: 2.3.5.7
 
[[Badness]] (Smith): 0.184738
 
=== 2.3.5.13 subgroup ===
Subgroup: 2.3.5.13
 
Comma list: 676/675, 3200/3159
 
Sval mapping: {{mapping| 1 0 -8 -9 | 0 2 13 16 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~26/15 = 952.643
* CWE: ~2 = 1200.000, ~26/15 = 952.789
 
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}
 
Badness (Smith): 0.257
 
== Septimal immunity ==
{{See also| Semaphoresmic clan }}
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 2240/2187
[[Comma list]]: 49/48, 2240/2187


[[Mapping]]: [{{val|1 0 -8 2}}, {{val|0 2 13 1}}]
{{Mapping|legend=1| 1 0 -8 2 | 0 2 13 1 }}
 
{{Multival|legend=1|2 13 1 16 -4 -34}}


[[POTE generator]]: ~7/6 = 247.555
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/4 = 952.767
: [[error map]]: {{val| 0.000 +3.580 -0.337 -16.058 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 952.445
: error map: {{val| 0.000 +2.936 -4.525 -16.381 }}


{{Val list|legend=1| 5, 29, 34, 63d }}
{{Optimal ET sequence|legend=1| 5, …, 29, 34, 63d }}


[[Badness]]: 0.077631
[[Badness]] (Smith): 0.077631


=== 11-limit ===
=== 11-limit ===
Line 34: Line 62:
Comma list: 49/48, 55/54, 896/891
Comma list: 49/48, 55/54, 896/891


Mapping: [{{val|1 0 -8 2 9}}, {{val|0 2 13 1 -7}}]
Mapping: {{mapping| 1 0 -8 2 9 | 0 2 13 1 -7 }}


POTE generator: ~7/6 = 247.926
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.442
* POTE: ~2 = 1200.000, ~7/4 = 952.074


Vals: {{Val list| 5, 24c, 29, 63de, 92cddee }}
{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}


Badness: 0.052661
Badness (Smith): 0.052661


==== 13-limit ====
==== 13-limit ====
Line 47: Line 77:
Comma list: 49/48, 55/54, 91/90, 896/891
Comma list: 49/48, 55/54, 91/90, 896/891


Mapping: [{{val|1 0 -8 2 9 -9}}, {{val|0 2 13 1 -7 16}}]
Mapping: {{mapping| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}


POTE generator: ~7/6 = 247.969
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.472
* POTE: ~2 = 1200.000, ~7/4 = 952.031


Vals: {{Val list| 5, 24cf, 29, 92cddee }}
{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}


Badness: 0.037078
Badness (Smith): 0.037078


=== Impunity ===
=== Impunity ===
Line 60: Line 92:
Comma list: 49/48, 100/99, 1232/1215
Comma list: 49/48, 100/99, 1232/1215


Mapping: [{{val|1 0 -8 2 -14}}, {{val|0 2 13 1 22}}]
Mapping: {{mapping| 1 0 -8 2 -14 | 0 2 13 1 22 }}


POTE generator: ~7/6 = 247.773
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.528
* POTE: ~2 = 1200.000, ~7/4 = 952.227


Vals: {{Val list| 29, 63d, 92cdd }}
{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}


Badness: 0.055243
Badness (Smith): 0.055243


==== 13-limit ====
==== 13-limit ====
Line 73: Line 107:
Comma list: 49/48, 91/90, 100/99, 352/351
Comma list: 49/48, 91/90, 100/99, 352/351


Mapping: [{{val|1 0 -8 2 -14 -9}}, {{val|0 2 13 1 22 16}}]
Mapping: {{mapping| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}


POTE generator: ~7/6 = 247.783
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.529
* POTE: ~2 = 1200.000, ~7/4 = 952.217


Vals: {{Val list| 29, 63d, 92cdd }}
{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}


Badness: 0.033196
Badness (Smith): 0.033196


== Immunized ==
== Immunized ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64/63, 8748/8575
[[Comma list]]: 64/63, 8748/8575


[[Mapping]]: [{{val|1 0 -8 6}}, {{val|0 2 13 -4}}]
{{Mapping|legend=1| 1 0 -8 6 | 0 2 13 -4 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~140/81 = 953.001
: [[error map]]: {{val| 0.000 +4.047 +2.669 +19.170 }}
* [[POTE]]: ~2 = 1200.000, ~140/81 = 953.521
: error map: {{val| 0.000 +5.088 +9.463 +17.089 }}
 
{{Optimal ET sequence|legend=1| 5, 29d, 34d, 39d, 73dd, 112cddd }}
 
[[Badness]] (Smith): 0.082592
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 99/98, 1944/1925
 
Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~55/32 = 953.271
* POTE: ~2 = 1200.000, ~55/32 = 953.397
 
{{Optimal ET sequence|legend=0| 5, 29de, 34d, 39d, 73dd }}
 
Badness (Smith): 0.051515
 
== Immune ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250/750141


{{Multival|legend=1|2 13 -4 16 -12 -46}}
{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}


[[POTE generator]]: ~81/70 = 246.479
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~1000/567 = 952.386
: [[error map]]: {{val| 0.000 +2.818 -5.292 +2.763 }}
* [[POTE]]: ~2 = 1200.000, ~1000/567 = 952.417
: error map: {{val| 0.000 +2.879 -4.893 +3.683 }}


{{Val list|legend=1| 5, 34d, 39d, 73dd }}
{{Optimal ET sequence|legend=1| 29, 34d, 63 }}


[[Badness]]: 0.082592
[[Badness]] (Smith): 0.144835


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 64/63, 99/98, 1944/1925
Comma list: 100/99, 225/224, 1331/1323
 
Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~121/70 = 952.375
* POTE: ~2 = 1200.000, ~121/70 = 952.411
 
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
 
Badness (Smith): 0.050990
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 225/224, 275/273


Mapping: [{{val|1 0 -8 6 13}}, {{val|0 2 13 -4 -12}}]
Mapping: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}


POTE generator: ~64/55 = 246.603
Optimal tunings:
* CTE: ~2 = 1200.000, ~26/15 = 952.389
* POTE: ~2 = 1200.000, ~26/15 = 952.407


Vals: {{Val list| 5, 34d, 39d, 73dd }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness: 0.051515
Badness (Smith): 0.028521


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament]]
[[Category:Pages with mostly numerical content]]
[[Category:Temperament family]]
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Immunity family| ]] <!-- main article -->
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Immunity]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:39, 24 June 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.

The immunity family of temperaments tempers out the immunity comma, 1638400/1594323 ([16 -13 2⟩).

Immunity

Immunity is naturally a 2.3.5.13 subgroup temperament as the generator serves as ~26/15 (or its octave complement ~15/13), tempering out 676/675.

Subgroup: 2.3.5

Comma list: 1638400/1594323

Mapping: [⟨1 0 -8], ⟨0 2 13]]

mapping generators: ~2, ~1280/729

Optimal tunings:

  • CTE: ~2 = 1200.000, ~1280/729 = 952.706
error map: ⟨0.000 +3.456 -1.141]
  • POTE: ~2 = 1200.000, ~1280/729 = 952.930
error map: ⟨0.000 +3.905 +1.775]

Optimal ET sequence: 5, …, 29, 34

Badness (Smith): 0.184738

2.3.5.13 subgroup

Subgroup: 2.3.5.13

Comma list: 676/675, 3200/3159

Sval mapping: [⟨1 0 -8 -9], ⟨0 2 13 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~26/15 = 952.643
  • CWE: ~2 = 1200.000, ~26/15 = 952.789

Optimal ET sequence: 5, …, 29, 34

Badness (Smith): 0.257

Septimal immunity

See also: Semaphoresmic clan

Subgroup: 2.3.5.7

Comma list: 49/48, 2240/2187

Mapping: [⟨1 0 -8 2], ⟨0 2 13 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.767
error map: ⟨0.000 +3.580 -0.337 -16.058]
  • POTE: ~2 = 1200.000, ~7/4 = 952.445
error map: ⟨0.000 +2.936 -4.525 -16.381]

Optimal ET sequence: 5, …, 29, 34, 63d

Badness (Smith): 0.077631

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 896/891

Mapping: [⟨1 0 -8 2 9], ⟨0 2 13 1 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.442
  • POTE: ~2 = 1200.000, ~7/4 = 952.074

Optimal ET sequence: 5, 24c, 29, 63de

Badness (Smith): 0.052661

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 91/90, 896/891

Mapping: [⟨1 0 -8 2 9 -9], ⟨0 2 13 1 -7 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.472
  • POTE: ~2 = 1200.000, ~7/4 = 952.031

Optimal ET sequence: 5, 24cf, 29

Badness (Smith): 0.037078

Impunity

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 1232/1215

Mapping: [⟨1 0 -8 2 -14], ⟨0 2 13 1 22]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.528
  • POTE: ~2 = 1200.000, ~7/4 = 952.227

Optimal ET sequence: 5e, 24cee, 29

Badness (Smith): 0.055243

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 91/90, 100/99, 352/351

Mapping: [⟨1 0 -8 2 -14 -9], ⟨0 2 13 1 22 16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.529
  • POTE: ~2 = 1200.000, ~7/4 = 952.217

Optimal ET sequence: 5e, 24ceef, 29

Badness (Smith): 0.033196

Immunized

Subgroup: 2.3.5.7

Comma list: 64/63, 8748/8575

Mapping: [⟨1 0 -8 6], ⟨0 2 13 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~140/81 = 953.001
error map: ⟨0.000 +4.047 +2.669 +19.170]
  • POTE: ~2 = 1200.000, ~140/81 = 953.521
error map: ⟨0.000 +5.088 +9.463 +17.089]

Optimal ET sequence: 5, 29d, 34d, 39d, 73dd, 112cddd

Badness (Smith): 0.082592

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 99/98, 1944/1925

Mapping: [⟨1 0 -8 6 13], ⟨0 2 13 -4 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~55/32 = 953.271
  • POTE: ~2 = 1200.000, ~55/32 = 953.397

Optimal ET sequence: 5, 29de, 34d, 39d, 73dd

Badness (Smith): 0.051515

Immune

Subgroup: 2.3.5.7

Comma list: 225/224, 781250/750141

Mapping: [⟨1 2 5 9], ⟨0 -2 -13 -30]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~1000/567 = 952.386
error map: ⟨0.000 +2.818 -5.292 +2.763]
  • POTE: ~2 = 1200.000, ~1000/567 = 952.417
error map: ⟨0.000 +2.879 -4.893 +3.683]

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.144835

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 1331/1323

Mapping: [⟨1 2 5 9 8], ⟨0 -2 -13 -30 -22]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~121/70 = 952.375
  • POTE: ~2 = 1200.000, ~121/70 = 952.411

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.050990

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 225/224, 275/273

Mapping: [⟨1 2 5 9 8 7], ⟨0 -2 -13 -30 -22 -16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~26/15 = 952.389
  • POTE: ~2 = 1200.000, ~26/15 = 952.407

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.028521

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