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 = 1200.0000¢, ~1280/729 = 952.7056¢
* [[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 }}
* [[CWE]]: ~2 = 1200.0000¢, ~1280/729 = 952.8424¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 952.8424{{c}}
: error map: {{val| 0.000 +3.730 +0.638 }}
: 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]] (Sintel): 4.334
[[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 }}
 
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 ===
=== Overview to extensions ===
Line 43: Line 27:
* Immunized adds [[64/63]];  
* Immunized adds [[64/63]];  
* Immune adds [[225/224]].
* 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 ==
== Septimal immunity ==
Line 54: Line 40:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000¢, ~7/4 = 952.7674¢
* [[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 }}
* [[CWE]]: ~2 = 1200.0000¢, ~7/4 = 952.5376¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 952.5376{{c}}
: error map: {{val| 0.000 +3.120 -3.325 -16.288 }}
: 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]] (Sintel): 1.965
[[Badness]] (Sintel): 1.96


=== 11-limit ===
=== 11-limit ===
Line 71: Line 57:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.4418¢
* WE: ~2 = 1202.6033{{c}}, ~7/4 = 954.1395{{c}}
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1429¢
* 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 (Sintel): 1.741
Badness (Sintel): 1.74


==== 13-limit ====
==== 13-limit ====
Line 86: Line 72:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.4723¢
* WE: ~2 = 1202.5640{{c}}, ~7/4 = 954.0648{{c}}
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1117¢
* 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 (Sintel): 1.532
Badness (Sintel): 1.53


=== Impunity ===
=== Impunity ===
Line 101: Line 87:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.5282¢
* WE: ~2 = 1201.6023{{c}}, ~7/4 = 953.4988{{c}}
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3120¢
* 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 (Sintel): 1.826
Badness (Sintel): 1.83


==== 13-limit ====
==== 13-limit ====
Line 116: Line 102:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.5292¢
* WE: ~2 = 1200.5826{{c}}, ~7/4 = 952.4725{{c}}
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3038¢
* 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 (Sintel): 1.372
Badness (Sintel): 1.37


== Immunized ==
== Immunized ==
Line 131: Line 117:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000¢, ~140/81 = 953.0010¢
* [[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 }}
* [[CWE]]: ~2 = 1200.0000¢, ~140/81 = 953.3954¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 953.3954{{c}}
: error map: {{val| 0.000 +4.836 +7.827 +17.592 }}
: 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]] (Sintel): 2.090
[[Badness]] (Sintel): 2.09


=== 11-limit ===
=== 11-limit ===
Line 148: Line 134:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~55/32 = 953.2713¢
* WE: ~2 = 1196.9902{{c}}, ~55/32 = 951.0056{{c}}
* CWE: ~2 = 1200.0000¢, ~55/32 = 953.3755¢
* 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 (Sintel): 1.703
Badness (Sintel): 1.70


== Immune ==
== Immune ==
Line 163: Line 149:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000¢, ~1000/567 = 952.3863¢
* [[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 }}
* [[CWE]]: ~2 = 1200.0000¢, ~1000/567 = 952.4061¢
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1000/567 = 952.4061{{c}}
: error map: {{val| 0.000 +2.857 -5.034 +3.357 }}
: 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]] (Sintel): 3.665
[[Badness]] (Sintel): 3.67


=== 11-limit ===
=== 11-limit ===
Line 180: Line 166:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~121/70 = 952.3747¢
* WE: ~2 = 1199.7081{{c}}, ~121/70 = 952.1793{{c}}
* CWE: ~2 = 1200.0000¢, ~121/70 = 952.3983¢
* 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 (Sintel): 1.686
Badness (Sintel): 1.69


=== 13-limit ===
=== 13-limit ===
Line 195: Line 181:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000¢, ~26/15 = 952.3891¢
* WE: ~2 = 1199.8625{{c}}, ~26/15 = 952.2977{{c}}
* CWE: ~2 = 1200.0000¢, ~26/15 = 952.4008¢
* 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 (Sintel): 1.179
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 }}
 
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:Immunity family| ]] <!-- Main article -->
[[Category:Immunity family| ]] <!-- Main article -->

Latest revision as of 03:13, 27 August 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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