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Temperaments of '''immunity family''' temper out 1638400/1594323 (immunity comma, 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 ==
[[Comma]]: 1638400/1594323
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]].


[[Mapping]]: [{{val|1 0 -8}}, {{val|0 2 13}}]
[[Subgroup]]: 2.3.5


[[POTE generator]]: ~729/640 = 247.070
[[Comma list]]: 1638400/1594323


{{Val list|legend=1| 5, 29, 34 }}
{{Mapping|legend=1| 1 0 -8 | 0 2 13 }}


[[Badness]]: 0.184738
: mapping generators: ~2, ~1280/729
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000¢, ~1280/729 = 952.7056¢
: [[error map]]: {{val| 0.000 +3.456 -1.141 }}
* [[CWE]]: ~2 = 1200.0000¢, ~1280/729 = 952.8424¢
: error map: {{val| 0.000 +3.730 +0.638 }}
 
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
 
[[Badness]] (Sintel): 4.334
 
=== 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 = 1200.0000¢, ~26/15 = 952.6431¢
* CWE: ~2 = 1200.0000¢, ~26/15 = 952.7890¢
 
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}
 
Badness (Sintel): 0.858
 
=== Overview to extensions ===
The second comma of the comma list defines which 7-limit family member we are looking at:
* Septimal immunity adds [[49/48]];
* Immunized adds [[64/63]];
* Immune adds [[225/224]].
 
== Septimal immunity ==
{{See also| Semaphoresmic clan }}
 
[[Subgroup]]: 2.3.5.7


== 7-limit ==
[[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}}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000¢, ~7/4 = 952.7674¢
: [[error map]]: {{val| 0.000 +3.580 -0.337 -16.058 }}
* [[CWE]]: ~2 = 1200.0000¢, ~7/4 = 952.5376¢
: error map: {{val| 0.000 +3.120 -3.325 -16.288 }}


[[POTE generator]]: ~7/6 = 247.555
{{Optimal ET sequence|legend=1| 5, …, 29, 34, 63d }}


{{Val list|legend=1| 5, 29, 34, 63d }}
[[Badness]] (Sintel): 1.965


[[Badness]]: 0.077631
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
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 = 1200.0000¢, ~7/4 = 952.4418¢
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1429¢


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


Badness: 0.052661
Badness (Sintel): 1.741
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
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 = 1200.0000¢, ~7/4 = 952.4723¢
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1117¢


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


Badness: 0.037078
Badness (Sintel): 1.532
 
=== Impunity ===
Subgroup: 2.3.5.7.11


== Impunity ==
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 = 1200.0000¢, ~7/4 = 952.5282¢
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3120¢


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


Badness: 0.055243
Badness (Sintel): 1.826
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
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 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.5292¢
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3038¢


POTE generator: ~7/6 = 247.783
{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}


Vals: {{Val list| 29, 63d, 92cdd }}
Badness (Sintel): 1.372


Badness: 0.033196
== Immunized ==
[[Subgroup]]: 2.3.5.7


= Immunized =
[[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 }}


{{Multival|legend=1|2 13 -4 16 -12 -46}}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000¢, ~140/81 = 953.0010¢
: [[error map]]: {{val| 0.000 +4.047 +2.669 +19.170 }}
* [[CWE]]: ~2 = 1200.0000¢, ~140/81 = 953.3954¢
: error map: {{val| 0.000 +4.836 +7.827 +17.592 }}


[[POTE generator]]: ~81/70 = 246.479
{{Optimal ET sequence|legend=1| 5, 29d, 34d, 39d, 73dd, 112cddd }}


{{Val list|legend=1| 5, 34d, 39d, 73dd }}
[[Badness]] (Sintel): 2.090


[[Badness]]: 0.082592
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 11-limit ==
Comma list: 64/63, 99/98, 1944/1925  
Comma list: 64/63, 99/98, 1944/1925  


Mapping: [{{val|1 0 -8 6 13}}, {{val|0 2 13 -4 -12}}]
Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000¢, ~55/32 = 953.2713¢
* CWE: ~2 = 1200.0000¢, ~55/32 = 953.3755¢
 
{{Optimal ET sequence|legend=0| 5, 29de, 34d, 39d, 73dd }}
 
Badness (Sintel): 1.703
 
== Immune ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250/750141
 
{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000¢, ~1000/567 = 952.3863¢
: [[error map]]: {{val| 0.000 +2.818 -5.292 +2.763 }}
* [[CWE]]: ~2 = 1200.0000¢, ~1000/567 = 952.4061¢
: error map: {{val| 0.000 +2.857 -5.034 +3.357 }}
 
{{Optimal ET sequence|legend=1| 29, 34d, 63 }}
 
[[Badness]] (Sintel): 3.665
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 1331/1323
 
Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
 
Optimal tunings:
* CTE: ~2 = 1200.0000¢, ~121/70 = 952.3747¢
* CWE: ~2 = 1200.0000¢, ~121/70 = 952.3983¢
 
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
 
Badness (Sintel): 1.686
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 225/224, 275/273
 
Mapping: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}


POTE generator: ~64/55 = 246.603
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~26/15 = 952.3891¢
* CWE: ~2 = 1200.0000¢, ~26/15 = 952.4008¢


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


Badness: 0.051515
Badness (Sintel): 1.179


[[Category:Theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Immunity]]
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 07:50, 27 February 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.

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 = 1200.0000¢, ~1280/729 = 952.7056¢
error map: ⟨0.000 +3.456 -1.141]
  • CWE: ~2 = 1200.0000¢, ~1280/729 = 952.8424¢
error map: ⟨0.000 +3.730 +0.638]

Optimal ET sequence: 5, …, 29, 34

Badness (Sintel): 4.334

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 = 1200.0000¢, ~26/15 = 952.6431¢
  • CWE: ~2 = 1200.0000¢, ~26/15 = 952.7890¢

Optimal ET sequence: 5, …, 29, 34

Badness (Sintel): 0.858

Overview to extensions

The second comma of the comma list defines which 7-limit family member we are looking at:

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 = 1200.0000¢, ~7/4 = 952.7674¢
error map: ⟨0.000 +3.580 -0.337 -16.058]
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.5376¢
error map: ⟨0.000 +3.120 -3.325 -16.288]

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

Badness (Sintel): 1.965

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 = 1200.0000¢, ~7/4 = 952.4418¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.1429¢

Optimal ET sequence: 5, 24c, 29, 63de

Badness (Sintel): 1.741

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 = 1200.0000¢, ~7/4 = 952.4723¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.1117¢

Optimal ET sequence: 5, 24cf, 29

Badness (Sintel): 1.532

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 = 1200.0000¢, ~7/4 = 952.5282¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.3120¢

Optimal ET sequence: 5e, 24cee, 29

Badness (Sintel): 1.826

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 = 1200.0000¢, ~7/4 = 952.5292¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.3038¢

Optimal ET sequence: 5e, 24ceef, 29

Badness (Sintel): 1.372

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 = 1200.0000¢, ~140/81 = 953.0010¢
error map: ⟨0.000 +4.047 +2.669 +19.170]
  • CWE: ~2 = 1200.0000¢, ~140/81 = 953.3954¢
error map: ⟨0.000 +4.836 +7.827 +17.592]

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

Badness (Sintel): 2.090

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 = 1200.0000¢, ~55/32 = 953.2713¢
  • CWE: ~2 = 1200.0000¢, ~55/32 = 953.3755¢

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

Badness (Sintel): 1.703

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 = 1200.0000¢, ~1000/567 = 952.3863¢
error map: ⟨0.000 +2.818 -5.292 +2.763]
  • CWE: ~2 = 1200.0000¢, ~1000/567 = 952.4061¢
error map: ⟨0.000 +2.857 -5.034 +3.357]

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 3.665

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 = 1200.0000¢, ~121/70 = 952.3747¢
  • CWE: ~2 = 1200.0000¢, ~121/70 = 952.3983¢

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 1.686

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 = 1200.0000¢, ~26/15 = 952.3891¢
  • CWE: ~2 = 1200.0000¢, ~26/15 = 952.4008¢

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 1.179

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