Immunity family: Difference between revisions

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+CTE tunings. +error maps. Spell the periods in cents. Note the badness metric being used. Misc. cleanup
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Temperaments of '''immunity family''' temper out the [[immunity comma]], 1638400/1594323 = {{monzo| 16 -13 2 }} (sasayoyo).
The '''immunity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[immunity comma]], 1638400/1594323 ({{monzo| 16 -13 2 }}).


== Immunity ==
== Immunity ==
Line 12: Line 12:
: mapping generators: ~2, ~1280/729
: mapping generators: ~2, ~1280/729


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1280/729 = 952.930
[[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 }}


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


[[Badness]]: 0.184738
[[Badness]] (Smith): 0.184738


=== 2.3.5.13 subgroup ===
=== 2.3.5.13 subgroup ===
Line 25: Line 29:
Sval mapping: {{mapping| 1 0 -8 -9 | 0 2 13 16 }}
Sval mapping: {{mapping| 1 0 -8 -9 | 0 2 13 16 }}


Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 952.6431
Optimal tunings:
* CTE: ~2 = 1200.000, ~26/15 = 952.643
* CWE: ~2 = 1200.000, ~26/15 = 952.789


{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
{{Optimal ET sequence|legend=0| 5, …, 29, 34 }}


Badness: 0.257
Badness (Smith): 0.257


== Septimal immunity ==
== Septimal immunity ==
{{See also| Slendro clan }}
{{See also| Semaphoresmic clan }}


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Multival|legend=1| 2 13 1 16 -4 -34 }}
{{Multival|legend=1| 2 13 1 16 -4 -34 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/4 = 952.445
[[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 }}


{{Optimal ET sequence|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 55: Line 65:
Mapping: {{mapping| 1 0 -8 2 9 | 0 2 13 1 -7 }}
Mapping: {{mapping| 1 0 -8 2 9 | 0 2 13 1 -7 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 952.074
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.442
* POTE: ~2 = 1200.000, ~7/4 = 952.074


{{Optimal ET sequence|legend=1| 5, 24c, 29, 63de }}
{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}


Badness: 0.052661
Badness (Smith): 0.052661


==== 13-limit ====
==== 13-limit ====
Line 68: Line 80:
Mapping: {{mapping| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}
Mapping: {{mapping| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 952.031
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.472
* POTE: ~2 = 1200.000, ~7/4 = 952.031


{{Optimal ET sequence|legend=1| 5, 24cf, 29 }}
{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}


Badness: 0.037078
Badness (Smith): 0.037078


=== Impunity ===
=== Impunity ===
Line 81: Line 95:
Mapping: {{mapping| 1 0 -8 2 -14 | 0 2 13 1 22 }}
Mapping: {{mapping| 1 0 -8 2 -14 | 0 2 13 1 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 952.227
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.528
* POTE: ~2 = 1200.000, ~7/4 = 952.227


{{Optimal ET sequence|legend=1| 5e, 24cee, 29 }}
{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}


Badness: 0.055243
Badness (Smith): 0.055243


==== 13-limit ====
==== 13-limit ====
Line 94: Line 110:
Mapping: {{mapping| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}
Mapping: {{mapping| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~7/4 = 952.217
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/4 = 952.529
* POTE: ~2 = 1200.000, ~7/4 = 952.217


{{Optimal ET sequence|legend=1| 5e, 24ceef, 29 }}
{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}


Badness: 0.033196
Badness (Smith): 0.033196


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~140/81 = 953.521
[[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 122: Line 144:
Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/32 = 953.397
Optimal tunings:
* CTE: ~2 = 1200.000, ~55/32 = 953.271
* POTE: ~2 = 1200.000, ~55/32 = 953.397


{{Optimal ET sequence|legend=1| 5, 29de, 34d, 39d, 73dd }}
{{Optimal ET sequence|legend=0| 5, 29de, 34d, 39d, 73dd }}


Badness: 0.051515
Badness (Smith): 0.051515


== Immune ==
== Immune ==
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{{Multival|legend=1| 2 13 30 16 42 33 }}
{{Multival|legend=1| 2 13 30 16 42 33 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1000/567 = 952.417
[[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 }}


{{Optimal ET sequence|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 150: Line 178:
Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}


Optimal tuning (POTE): ~2 = 1\1, ~121/70 = 952.411
Optimal tunings:
* CTE: ~2 = 1200.000, ~121/70 = 952.375
* POTE: ~2 = 1200.000, ~121/70 = 952.411


{{Optimal ET sequence|legend=1| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness: 0.050990
Badness (Smith): 0.050990


=== 13-limit ===
=== 13-limit ===
Line 163: Line 193:
Mapping: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}
Mapping: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 952.407
Optimal tunings:
* CTE: ~2 = 1200.000, ~26/15 = 952.389
* POTE: ~2 = 1200.000, ~26/15 = 952.407


{{Optimal ET sequence|legend=1| 29, 34d, 63 }}
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}


Badness: 0.028521
Badness (Smith): 0.028521


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

Revision as of 10:47, 13 March 2025

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 = 1200.000, ~1280/729 = 952.706
error map: ⟨0.000 +3.456 -1.141]
  • POTE: ~2 = 1200.000, ~1280/729 = 952.930
error map: ⟨0.000 +3.905 +1.775]

Optimal ET sequence: 5, …, 29, 34

Badness (Smith): 0.184738

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 = 1200.000, ~26/15 = 952.643
  • CWE: ~2 = 1200.000, ~26/15 = 952.789

Optimal ET sequence: 5, …, 29, 34

Badness (Smith): 0.257

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]]

Wedgie: ⟨⟨ 2 13 1 16 -4 -34 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.767
error map: ⟨0.000 +3.580 -0.337 -16.058]
  • POTE: ~2 = 1200.000, ~7/4 = 952.445
error map: ⟨0.000 +2.936 -4.525 -16.381]

Optimal ET sequence: 5, …, 29, 34, 63d

Badness (Smith): 0.077631

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 = 1200.000, ~7/4 = 952.442
  • POTE: ~2 = 1200.000, ~7/4 = 952.074

Optimal ET sequence: 5, 24c, 29, 63de

Badness (Smith): 0.052661

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 = 1200.000, ~7/4 = 952.472
  • POTE: ~2 = 1200.000, ~7/4 = 952.031

Optimal ET sequence: 5, 24cf, 29

Badness (Smith): 0.037078

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 = 1200.000, ~7/4 = 952.528
  • POTE: ~2 = 1200.000, ~7/4 = 952.227

Optimal ET sequence: 5e, 24cee, 29

Badness (Smith): 0.055243

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 = 1200.000, ~7/4 = 952.529
  • POTE: ~2 = 1200.000, ~7/4 = 952.217

Optimal ET sequence: 5e, 24ceef, 29

Badness (Smith): 0.033196

Immunized

Subgroup: 2.3.5.7

Comma list: 64/63, 8748/8575

Mapping: [⟨1 0 -8 6], ⟨0 2 13 -4]]

Wedgie: ⟨⟨ 2 13 -4 16 -12 -46 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~140/81 = 953.001
error map: ⟨0.000 +4.047 +2.669 +19.170]
  • POTE: ~2 = 1200.000, ~140/81 = 953.521
error map: ⟨0.000 +5.088 +9.463 +17.089]

Optimal ET sequence: 5, 29d, 34d, 39d, 73dd, 112cddd

Badness (Smith): 0.082592

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 = 1200.000, ~55/32 = 953.271
  • POTE: ~2 = 1200.000, ~55/32 = 953.397

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

Badness (Smith): 0.051515

Immune

Subgroup: 2.3.5.7

Comma list: 225/224, 781250/750141

Mapping: [⟨1 2 5 9], ⟨0 -2 -13 -30]]

Wedgie: ⟨⟨ 2 13 30 16 42 33 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~1000/567 = 952.386
error map: ⟨0.000 +2.818 -5.292 +2.763]
  • POTE: ~2 = 1200.000, ~1000/567 = 952.417
error map: ⟨0.000 +2.879 -4.893 +3.683]

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.144835

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 = 1200.000, ~121/70 = 952.375
  • POTE: ~2 = 1200.000, ~121/70 = 952.411

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.050990

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 = 1200.000, ~26/15 = 952.389
  • POTE: ~2 = 1200.000, ~26/15 = 952.407

Optimal ET sequence: 29, 34d, 63

Badness (Smith): 0.028521

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