Immunity family: Difference between revisions

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{{Technical data page}}<br><br>
{{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]], 1638400/1594323 ({{monzo| 16 -13 2 }}).


== 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]].  
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
[[Subgroup]]: 2.3.5
Line 14: Line 14:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1280/729 = 952.706
* [[CTE]]: ~2 = 1200.0000¢, ~1280/729 = 952.7056¢
: [[error map]]: {{val| 0.000 +3.456 -1.141 }}
: [[error map]]: {{val| 0.000 +3.456 -1.141 }}
* [[POTE]]: ~2 = 1200.000, ~1280/729 = 952.930
* [[CWE]]: ~2 = 1200.0000¢, ~1280/729 = 952.8424¢
: 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): 4.334


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
Line 31: Line 31:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~26/15 = 952.643
* CTE: ~2 = 1200.0000¢, ~26/15 = 952.6431¢
* CWE: ~2 = 1200.000, ~26/15 = 952.789
* CWE: ~2 = 1200.0000¢, ~26/15 = 952.7890¢


{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}


Badness (Smith): 0.257
Badness (Sintel): 0.858
 
=== 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]].


== Septimal immunity ==
== Septimal immunity ==
Line 46: Line 52:


{{Mapping|legend=1| 1 0 -8 2 | 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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~7/4 = 952.767
* [[CTE]]: ~2 = 1200.0000¢, ~7/4 = 952.7674¢
: [[error map]]: {{val| 0.000 +3.580 -0.337 -16.058 }}
: [[error map]]: {{val| 0.000 +3.580 -0.337 -16.058 }}
* [[POTE]]: ~2 = 1200.000, ~7/4 = 952.445
* [[CWE]]: ~2 = 1200.0000¢, ~7/4 = 952.5376¢
: 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): 1.965


=== 11-limit ===
=== 11-limit ===
Line 67: Line 71:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.442
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.4418¢
* POTE: ~2 = 1200.000, ~7/4 = 952.074
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1429¢


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


Badness (Smith): 0.052661
Badness (Sintel): 1.741


==== 13-limit ====
==== 13-limit ====
Line 82: Line 86:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.472
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.4723¢
* POTE: ~2 = 1200.000, ~7/4 = 952.031
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.1117¢


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


Badness (Smith): 0.037078
Badness (Sintel): 1.532


=== Impunity ===
=== Impunity ===
Line 97: Line 101:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.528
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.5282¢
* POTE: ~2 = 1200.000, ~7/4 = 952.227
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3120¢


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


Badness (Smith): 0.055243
Badness (Sintel): 1.826


==== 13-limit ====
==== 13-limit ====
Line 112: Line 116:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~7/4 = 952.529
* CTE: ~2 = 1200.0000¢, ~7/4 = 952.5292¢
* POTE: ~2 = 1200.000, ~7/4 = 952.217
* CWE: ~2 = 1200.0000¢, ~7/4 = 952.3038¢


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


Badness (Smith): 0.033196
Badness (Sintel): 1.372


== Immunized ==
== Immunized ==
Line 125: Line 129:


{{Mapping|legend=1| 1 0 -8 6 | 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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~140/81 = 953.001
* [[CTE]]: ~2 = 1200.0000¢, ~140/81 = 953.0010¢
: [[error map]]: {{val| 0.000 +4.047 +2.669 +19.170 }}
: [[error map]]: {{val| 0.000 +4.047 +2.669 +19.170 }}
* [[POTE]]: ~2 = 1200.000, ~140/81 = 953.521
* [[CWE]]: ~2 = 1200.0000¢, ~140/81 = 953.3954¢
: 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): 2.090


=== 11-limit ===
=== 11-limit ===
Line 146: Line 148:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~55/32 = 953.271
* CTE: ~2 = 1200.0000¢, ~55/32 = 953.2713¢
* POTE: ~2 = 1200.000, ~55/32 = 953.397
* CWE: ~2 = 1200.0000¢, ~55/32 = 953.3755¢


{{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): 1.703


== Immune ==
== Immune ==
Line 159: Line 161:


{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}
{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}
{{Multival|legend=1| 2 13 30 16 42 33 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~1000/567 = 952.386
* [[CTE]]: ~2 = 1200.0000¢, ~1000/567 = 952.3863¢
: [[error map]]: {{val| 0.000 +2.818 -5.292 +2.763 }}
: [[error map]]: {{val| 0.000 +2.818 -5.292 +2.763 }}
* [[POTE]]: ~2 = 1200.000, ~1000/567 = 952.417
* [[CWE]]: ~2 = 1200.0000¢, ~1000/567 = 952.4061¢
: 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): 3.665


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~121/70 = 952.375
* CTE: ~2 = 1200.0000¢, ~121/70 = 952.3747¢
* POTE: ~2 = 1200.000, ~121/70 = 952.411
* CWE: ~2 = 1200.0000¢, ~121/70 = 952.3983¢


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


Badness (Smith): 0.050990
Badness (Sintel): 1.686


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~26/15 = 952.389
* CTE: ~2 = 1200.0000¢, ~26/15 = 952.3891¢
* POTE: ~2 = 1200.000, ~26/15 = 952.407
* CWE: ~2 = 1200.0000¢, ~26/15 = 952.4008¢


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


Badness (Smith): 0.028521
Badness (Sintel): 1.179


[[Category:Temperament families]]
[[Category:Temperament families]]

Latest revision as of 07:50, 27 February 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, 1638400/1594323 ([16 -13 2⟩).

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.

Subgroup: 2.3.5

Comma list: 1638400/1594323

Mapping: [⟨1 0 -8], ⟨0 2 13]]

mapping generators: ~2, ~1280/729

Optimal tunings:

  • CTE: ~2 = 1200.0000¢, ~1280/729 = 952.7056¢
error map: ⟨0.000 +3.456 -1.141]
  • CWE: ~2 = 1200.0000¢, ~1280/729 = 952.8424¢
error map: ⟨0.000 +3.730 +0.638]

Optimal ET sequence: 5, …, 29, 34

Badness (Sintel): 4.334

2.3.5.13 subgroup

Subgroup: 2.3.5.13

Comma list: 676/675, 3200/3159

Sval 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: 5, …, 29, 34

Badness (Sintel): 0.858

Overview to extensions

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

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:

  • CTE: ~2 = 1200.0000¢, ~7/4 = 952.7674¢
error map: ⟨0.000 +3.580 -0.337 -16.058]
  • 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): 1.965

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:

  • CTE: ~2 = 1200.0000¢, ~7/4 = 952.4418¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.1429¢

Optimal ET sequence: 5, 24c, 29, 63de

Badness (Sintel): 1.741

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:

  • CTE: ~2 = 1200.0000¢, ~7/4 = 952.4723¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.1117¢

Optimal ET sequence: 5, 24cf, 29

Badness (Sintel): 1.532

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:

  • CTE: ~2 = 1200.0000¢, ~7/4 = 952.5282¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.3120¢

Optimal ET sequence: 5e, 24cee, 29

Badness (Sintel): 1.826

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:

  • CTE: ~2 = 1200.0000¢, ~7/4 = 952.5292¢
  • CWE: ~2 = 1200.0000¢, ~7/4 = 952.3038¢

Optimal ET sequence: 5e, 24ceef, 29

Badness (Sintel): 1.372

Immunized

Subgroup: 2.3.5.7

Comma list: 64/63, 8748/8575

Mapping: [⟨1 0 -8 6], ⟨0 2 13 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.0000¢, ~140/81 = 953.0010¢
error map: ⟨0.000 +4.047 +2.669 +19.170]
  • 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): 2.090

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:

  • CTE: ~2 = 1200.0000¢, ~55/32 = 953.2713¢
  • CWE: ~2 = 1200.0000¢, ~55/32 = 953.3755¢

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

Badness (Sintel): 1.703

Immune

Subgroup: 2.3.5.7

Comma list: 225/224, 781250/750141

Mapping: [⟨1 2 5 9], ⟨0 -2 -13 -30]]

Optimal tunings:

  • CTE: ~2 = 1200.0000¢, ~1000/567 = 952.3863¢
error map: ⟨0.000 +2.818 -5.292 +2.763]
  • 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): 3.665

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:

  • CTE: ~2 = 1200.0000¢, ~121/70 = 952.3747¢
  • CWE: ~2 = 1200.0000¢, ~121/70 = 952.3983¢

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 1.686

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:

  • CTE: ~2 = 1200.0000¢, ~26/15 = 952.3891¢
  • CWE: ~2 = 1200.0000¢, ~26/15 = 952.4008¢

Optimal ET sequence: 29, 34d, 63

Badness (Sintel): 1.179

Retrieved from "https://en.xen.wiki/index.php?title=Immunity_family&oldid=224852"