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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]] ({{monzo|legend=1| 16 -13 2 }}, [[ratio]]: 1638400/1594323).


== Immunity ==
== Immunity ==
[[Comma]]: 1638400/1594323
The head of this family is immunity, which splits the perfect twelfth in halves, and [[5/4]] is found in 13 generator steps. Its [[ploidacot]] is alpha-dicot. It is a member of the [[syntonic–limmic equivalence continuum]] with equivalence number ''n'' = 2, meaning the [[Pythagorean limma]] is equated with a stack of two [[syntonic comma]]s.


[[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 }}
: mapping generators: ~2, ~1280/729


[[Badness]]: 0.184738
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9215{{c}}, ~1280/729 = 952.0734{{c}}
: [[error map]]: {{val| -1.079 +2.192 -0.731 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 952.8424{{c}}
: error map: {{val| 0.000 +3.730 +0.638 }}
 
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
 
[[Badness]] (Sintel): 4.33
 
=== 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]].
 
The immunity family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], discussed in [[#Subgroup extensions]].
 
== 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:
* [[WE]]: ~2 = 1201.4508{{c}}, ~7/4 = 953.5968{{c}}
: [[error map]]: {{val| +1.451 +5.239 -1.162 -12.327 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 952.5376{{c}}
: 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): 1.96


[[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|legend=0| 1 0 -8 2 9 | 0 2 13 1 -7 }}
 
Optimal tunings:
* WE: ~2 = 1202.6033{{c}}, ~7/4 = 954.1395{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1429{{c}}


POTE generator: ~7/6 = 247.926
{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}


Vals: {{Val list| 5, 24c, 29, 63de, 92cddee }}
Badness (Sintel): 1.74


Badness: 0.052661
==== 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|legend=0| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}
 
Optimal tunings:
* WE: ~2 = 1202.5640{{c}}, ~7/4 = 954.0648{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1117{{c}}


POTE generator: ~7/6 = 247.969
{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}


Vals: {{Val list| 5, 24cf, 29, 92cddee }}
Badness (Sintel): 1.53


Badness: 0.037078
=== 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|legend=0| 1 0 -8 2 -14 | 0 2 13 1 22 }}
 
Optimal tunings:
* WE: ~2 = 1201.6023{{c}}, ~7/4 = 953.4988{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3120{{c}}


POTE generator: ~7/6 = 247.773
{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}


Vals: {{Val list| 29, 63d, 92cdd }}
Badness (Sintel): 1.83


Badness: 0.055243
==== 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|legend=0| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}
 
Optimal tunings:
* WE: ~2 = 1200.5826{{c}}, ~7/4 = 952.4725{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3038{{c}}


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


Vals: {{Val list| 29, 63d, 92cdd }}
Badness (Sintel): 1.37


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:
* [[WE]]: ~2 = 1196.6463{{c}}, ~140/81 = 950.8564{{c}}
: [[error map]]: {{val| -3.354 -0.242 +1.649 +7.626 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 953.3954{{c}}
: 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): 2.09


[[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|legend=0| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
 
Optimal tunings:
* WE: ~2 = 1196.9902{{c}}, ~55/32 = 951.0056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/32 = 953.3755{{c}}
 
{{Optimal ET sequence|legend=0| 5, 29de, 34d, 39d, 73dd }}
 
Badness (Sintel): 1.70
 
== 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:
* [[WE]]: ~2 = 1199.7353{{c}}, ~1000/567 = 952.2068{{c}}
: [[error map]]: {{val| -0.265 +2.459 -5.507 +2.939 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1000/567 = 952.4061{{c}}
: error map: {{val| 0.000 +2.857 -5.034 +3.357 }}
 
{{Optimal ET sequence|legend=1| 29, 34d, 63 }}
 
[[Badness]] (Sintel): 3.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 1331/1323
 
{{Mapping|legend=0| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
 
Optimal tunings:
* WE: ~2 = 1199.7081{{c}}, ~121/70 = 952.1793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/70 = 952.3983{{c}}
 
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
 
Badness (Sintel): 1.69
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 169/168, 225/224, 275/273
 
{{Mapping|legend=0| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.8625{{c}}, ~26/15 = 952.2977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.4008{{c}}
 
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
 
Badness (Sintel): 1.18
 
== Subgroup extensions ==
=== Immunity (2.3.5.13) ===
Immunity is naturally a [[2.3.5.13 subgroup|2.3.5.13-subgroup]] temperament as the generator serves as ~[[26/15]] (or its [[octave complement]] ~[[15/13]]), tempering out [[676/675]].
 
Subgroup: 2.3.5.13
 
Comma list: 676/675, 3200/3159
 
{{Mapping|legend=2| 1 0 -8 -9 | 0 2 13 16 }}


POTE generator: ~64/55 = 246.603
Optimal tunings:  
* WE: ~2 = 1198.8955{{c}}, ~26/15 = 952.0049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.7890{{c}}


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


Badness: 0.051515
Badness (Sintel): 0.858


[[Category:Theory]]
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Temperament family]]
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Immunity]]
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Immunity_family"