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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]], 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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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


Optimal GPV sequence: {{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 }}


{{Multival|legend=1|2 13 -4 16 -12 -46}}
[[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 }}


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


=== 11-limit ===
=== 11-limit ===
Line 101: Line 139:
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:
* CTE: ~2 = 1200.000, ~55/32 = 953.271
* POTE: ~2 = 1200.000, ~55/32 = 953.397


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


Badness: 0.051515
Badness (Smith): 0.051515


== 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:
* [[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| 29, 34d, 63 }}
{{Optimal ET sequence|legend=1| 29, 34d, 63 }}


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


=== 11-limit ===
=== 11-limit ===
Line 129: Line 171:
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:
* CTE: ~2 = 1200.000, ~121/70 = 952.375
* POTE: ~2 = 1200.000, ~121/70 = 952.411


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


Badness: 0.050990
Badness (Smith): 0.050990


=== 13-limit ===
=== 13-limit ===
Line 142: Line 186:
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 }}


POTE generator: ~15/13 = 247.593
Optimal tunings:
* CTE: ~2 = 1200.000, ~26/15 = 952.389
* POTE: ~2 = 1200.000, ~26/15 = 952.407


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


Badness: 0.028521
Badness (Smith): 0.028521


[[Category:Theory]]
[[Category:Temperament]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Immunity family| ]] <!-- main article -->
[[Category:Pages with mostly numerical content]]
[[Category:Immunity]]
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Immunity_family"