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Temperaments of '''immunity family''' temper out the [[immunity comma]], 1638400/1594323 = {{monzo| 16-13 2 }} (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 ==
Subgroup: 2.3.5
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.  


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


{{Val list|legend=1| 5, 29, 34 }}
[[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 }}


[[Badness]]: 0.184738
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}


=== 7-limit ===
[[Badness]] (Sintel): 4.33
Subgroup: 2.3.5.7
 
=== 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


[[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 ===
=== 11-limit ===
Line 34: Line 54:
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:  
* WE: ~2 = 1202.6033{{c}}, ~7/4 = 954.1395{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1429{{c}}


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


Badness: 0.052661
Badness (Sintel): 1.74


==== 13-limit ====
==== 13-limit ====
Line 47: Line 69:
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:  
* WE: ~2 = 1202.5640{{c}}, ~7/4 = 954.0648{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1117{{c}}


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


Badness: 0.037078
Badness (Sintel): 1.53


=== Impunity ===
=== Impunity ===
Line 60: Line 84:
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:  
* WE: ~2 = 1201.6023{{c}}, ~7/4 = 953.4988{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3120{{c}}


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


Badness: 0.055243
Badness (Sintel): 1.83


==== 13-limit ====
==== 13-limit ====
Line 73: Line 99:
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:  
* WE: ~2 = 1200.5826{{c}}, ~7/4 = 952.4725{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3038{{c}}


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


Badness: 0.033196
Badness (Sintel): 1.37


== 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 }}


{{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 ===
=== 11-limit ===
Line 101: Line 131:
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 }}


POTE generator: ~64/55 = 246.603
Optimal tunings:  
* WE: ~2 = 1196.9902{{c}}, ~55/32 = 951.0056{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~55/32 = 953.3755{{c}}


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


Badness: 0.051515
Badness (Sintel): 1.70


== Immune ==
== Immune ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 781250/750141
[[Comma list]]: 225/224, 781250/750141


[[Mapping]]: [{{val|1 2 5 9}}, {{val|0 -2 -13 -30}}]
{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}
 
{{Multival|legend=1|2 13 30 16 42 33}}


[[POTE generator]]: ~567/500 = 247.583
[[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 }}


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


[[Badness]]: 0.144835
[[Badness]] (Sintel): 3.67


=== 11-limit ===
=== 11-limit ===
Line 129: Line 163:
Comma list: 100/99, 225/224, 1331/1323
Comma list: 100/99, 225/224, 1331/1323


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


POTE generator: ~140/121 = 247.589
Optimal tunings:  
* WE: ~2 = 1199.7081{{c}}, ~121/70 = 952.1793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/70 = 952.3983{{c}}


Optimal GPV sequence: {{Val list| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness: 0.050990
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 142: Line 178:
Comma list: 100/99, 169/168, 225/224, 275/273
Comma list: 100/99, 169/168, 225/224, 275/273


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


POTE generator: ~15/13 = 247.593
Optimal tunings:  
* WE: ~2 = 1198.8955{{c}}, ~26/15 = 952.0049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.7890{{c}}


Optimal GPV sequence: {{Val list| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}


Badness: 0.028521
Badness (Sintel): 0.858


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