Immunity family: Difference between revisions

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


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 10:


{{Mapping|legend=1| 1 0 -8 | 0 2 13 }}
{{Mapping|legend=1| 1 0 -8 | 0 2 13 }}
: mapping generators: ~2, ~1280/729
: mapping generators: ~2, ~1280/729


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1280/729 = 952.706
* [[WE]]: ~2 = 1198.9215{{c}}, ~1280/729 = 952.0734{{c}}
: [[error map]]: {{val| 0.000 +3.456 -1.141 }}
: [[error map]]: {{val| -1.079 +2.192 -0.731 }}
* [[POTE]]: ~2 = 1200.000, ~1280/729 = 952.930
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 952.8424{{c}}
: error map: {{val| 0.000 +3.905 +1.775 }}
: error map: {{val| 0.000 +3.730 +0.638 }}


{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}


[[Badness]] (Smith): 0.184738
[[Badness]] (Sintel): 4.33
 
=== 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 }}
=== 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]].


Optimal tunings:
The immunity family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], discussed in [[#Subgroup extensions]].  
* 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 ==
== Septimal immunity ==
Line 48: Line 40:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 952.767
* [[WE]]: ~2 = 1201.4508{{c}}, ~7/4 = 953.5968{{c}}
: [[error map]]: {{val| 0.000 +3.580 -0.337 -16.058 }}
: [[error map]]: {{val| +1.451 +5.239 -1.162 -12.327 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 952.445
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 952.5376{{c}}
: error map: {{val| 0.000 +2.936 -4.525 -16.381 }}
: error map: {{val| 0.000 +3.120 -3.325 -16.288 }}


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


[[Badness]] (Smith): 0.077631
[[Badness]] (Sintel): 1.96


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


Mapping: {{mapping| 1 0 -8 2 9 | 0 2 13 1 -7 }}
{{Mapping|legend=0| 1 0 -8 2 9 | 0 2 13 1 -7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.442
* WE: ~2 = 1202.6033{{c}}, ~7/4 = 954.1395{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 952.074
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1429{{c}}


{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}
{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}


Badness (Smith): 0.052661
Badness (Sintel): 1.74


==== 13-limit ====
==== 13-limit ====
Line 77: Line 69:
Comma list: 49/48, 55/54, 91/90, 896/891
Comma list: 49/48, 55/54, 91/90, 896/891


Mapping: {{mapping| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}
{{Mapping|legend=0| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.472
* WE: ~2 = 1202.5640{{c}}, ~7/4 = 954.0648{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 952.031
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1117{{c}}


{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}
{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}


Badness (Smith): 0.037078
Badness (Sintel): 1.53


=== Impunity ===
=== Impunity ===
Line 92: Line 84:
Comma list: 49/48, 100/99, 1232/1215
Comma list: 49/48, 100/99, 1232/1215


Mapping: {{mapping| 1 0 -8 2 -14 | 0 2 13 1 22 }}
{{Mapping|legend=0| 1 0 -8 2 -14 | 0 2 13 1 22 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.528
* WE: ~2 = 1201.6023{{c}}, ~7/4 = 953.4988{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 952.227
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3120{{c}}


{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}
{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}


Badness (Smith): 0.055243
Badness (Sintel): 1.83


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


Mapping: {{mapping| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}
{{Mapping|legend=0| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.529
* WE: ~2 = 1200.5826{{c}}, ~7/4 = 952.4725{{c}}
* POTE: ~2 = 1200.000, ~7/4 = 952.217
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3038{{c}}


{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}
{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}


Badness (Smith): 0.033196
Badness (Sintel): 1.37


== Immunized ==
== Immunized ==
Line 125: Line 117:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~140/81 = 953.001
* [[WE]]: ~2 = 1196.6463{{c}}, ~140/81 = 950.8564{{c}}
: [[error map]]: {{val| 0.000 +4.047 +2.669 +19.170 }}
: [[error map]]: {{val| -3.354 -0.242 +1.649 +7.626 }}
* [[POTE]]: ~2 = 1200.000, ~140/81 = 953.521
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 953.3954{{c}}
: error map: {{val| 0.000 +5.088 +9.463 +17.089 }}
: error map: {{val| 0.000 +4.836 +7.827 +17.592 }}


{{Optimal ET sequence|legend=1| 5, 29d, 34d, 39d, 73dd, 112cddd }}
{{Optimal ET sequence|legend=1| 5, 29d, 34d, 39d, 73dd, 112cddd }}


[[Badness]] (Smith): 0.082592
[[Badness]] (Sintel): 2.09


=== 11-limit ===
=== 11-limit ===
Line 139: Line 131:
Comma list: 64/63, 99/98, 1944/1925  
Comma list: 64/63, 99/98, 1944/1925  


Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
{{Mapping|legend=0| 1 0 -8 6 13 | 0 2 13 -4 -12 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~55/32 = 953.271
* WE: ~2 = 1196.9902{{c}}, ~55/32 = 951.0056{{c}}
* POTE: ~2 = 1200.000, ~55/32 = 953.397
* CWE: ~2 = 1200.0000{{c}}, ~55/32 = 953.3755{{c}}


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


Badness (Smith): 0.051515
Badness (Sintel): 1.70


== Immune ==
== Immune ==
Line 157: Line 149:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1000/567 = 952.386
* [[WE]]: ~2 = 1199.7353{{c}}, ~1000/567 = 952.2068{{c}}
: [[error map]]: {{val| 0.000 +2.818 -5.292 +2.763 }}
: [[error map]]: {{val| -0.265 +2.459 -5.507 +2.939 }}
* [[POTE]]: ~2 = 1200.000, ~1000/567 = 952.417
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1000/567 = 952.4061{{c}}
: error map: {{val| 0.000 +2.879 -4.893 +3.683 }}
: error map: {{val| 0.000 +2.857 -5.034 +3.357 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 171: Line 163:
Comma list: 100/99, 225/224, 1331/1323
Comma list: 100/99, 225/224, 1331/1323


Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
{{Mapping|legend=0| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~121/70 = 952.375
* WE: ~2 = 1199.7081{{c}}, ~121/70 = 952.1793{{c}}
* POTE: ~2 = 1200.000, ~121/70 = 952.411
* CWE: ~2 = 1200.0000{{c}}, ~121/70 = 952.3983{{c}}


{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness (Smith): 0.050990
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 186: Line 178:
Comma list: 100/99, 169/168, 225/224, 275/273
Comma list: 100/99, 169/168, 225/224, 275/273


Mapping: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}
{{Mapping|legend=0| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~26/15 = 952.389
* WE: ~2 = 1199.8625{{c}}, ~26/15 = 952.2977{{c}}
* POTE: ~2 = 1200.000, ~26/15 = 952.407
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.4008{{c}}


{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness (Smith): 0.028521
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 }}
 
Optimal tunings:
* WE: ~2 = 1198.8955{{c}}, ~26/15 = 952.0049{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.7890{{c}}
 
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}
 
Badness (Sintel): 0.858


[[Category:Temperament families]]
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Immunity family| ]] <!-- Main article -->
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Immunity| ]] <!-- Key article -->
[[Category:Rank 2]]
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 17:44, 30 September 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 (monzo: [16 -13 2⟩, ratio: 1638400/1594323).

Immunity

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 commas.

Subgroup: 2.3.5

Comma list: 1638400/1594323

Mapping: [⟨1 0 -8], ⟨0 2 13]]

mapping generators: ~2, ~1280/729

Optimal tunings:

  • WE: ~2 = 1198.9215 ¢, ~1280/729 = 952.0734 ¢
error map: ⟨-1.079 +2.192 -0.731]
  • CWE: ~2 = 1200.0000 ¢, ~1280/729 = 952.8424 ¢
error map: ⟨0.000 +3.730 +0.638]

Optimal ET sequence: 5, …, 29, 34

Badness (Sintel): 4.33

Overview to extensions

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

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

Mapping: [⟨1 0 -8 2], ⟨0 2 13 1]]

Optimal tunings:

  • WE: ~2 = 1201.4508 ¢, ~7/4 = 953.5968 ¢
error map: ⟨+1.451 +5.239 -1.162 -12.327]
  • 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): 1.96

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:

  • WE: ~2 = 1202.6033 ¢, ~7/4 = 954.1395 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 952.1429 ¢

Optimal ET sequence: 5, 24c, 29, 63de

Badness (Sintel): 1.74

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:

  • WE: ~2 = 1202.5640 ¢, ~7/4 = 954.0648 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 952.1117 ¢

Optimal ET sequence: 5, 24cf, 29

Badness (Sintel): 1.53

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:

  • WE: ~2 = 1201.6023 ¢, ~7/4 = 953.4988 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 952.3120 ¢

Optimal ET sequence: 5e, 24cee, 29

Badness (Sintel): 1.83

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:

  • WE: ~2 = 1200.5826 ¢, ~7/4 = 952.4725 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 952.3038 ¢

Optimal ET sequence: 5e, 24ceef, 29

Badness (Sintel): 1.37

Immunized

Subgroup: 2.3.5.7

Comma list: 64/63, 8748/8575

Mapping: [⟨1 0 -8 6], ⟨0 2 13 -4]]

Optimal tunings:

  • WE: ~2 = 1196.6463 ¢, ~140/81 = 950.8564 ¢
error map: ⟨-3.354 -0.242 +1.649 +7.626]
  • 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): 2.09

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:

  • WE: ~2 = 1196.9902 ¢, ~55/32 = 951.0056 ¢
  • CWE: ~2 = 1200.0000 ¢, ~55/32 = 953.3755 ¢

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

Badness (Sintel): 1.70

Immune

Subgroup: 2.3.5.7

Comma list: 225/224, 781250/750141

Mapping: [⟨1 2 5 9], ⟨0 -2 -13 -30]]

Optimal tunings:

  • WE: ~2 = 1199.7353 ¢, ~1000/567 = 952.2068 ¢
error map: ⟨-0.265 +2.459 -5.507 +2.939]
  • 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): 3.67

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:

  • WE: ~2 = 1199.7081 ¢, ~121/70 = 952.1793 ¢
  • CWE: ~2 = 1200.0000 ¢, ~121/70 = 952.3983 ¢

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

Optimal tunings:

  • WE: ~2 = 1199.8625 ¢, ~26/15 = 952.2977 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 952.4008 ¢

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 1.18

Subgroup extensions

Immunity (2.3.5.13)

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: 2.3.5.13

Comma list: 676/675, 3200/3159

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

Optimal tunings:

  • WE: ~2 = 1198.8955 ¢, ~26/15 = 952.0049 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 952.7890 ¢

Optimal ET sequence: 5, …, 29, 34

Badness (Sintel): 0.858

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