Immunity family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 488046868 - Original comment: **
Re-organize & + intro
 
(30 intermediate revisions by 13 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''immunity family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[immunity comma]] ({{monzo|legend=1| 16 -13 2 }}, [[ratio]]: 1638400/1594323).
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-02-07 17:33:08 UTC</tt>.<br>
: The original revision id was <tt>488046868</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc]]


=Immunity=
== Immunity ==
Comma: 1638400/1594323
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.


POTE generator: ~729/640 = 247.070
[[Subgroup]]: 2.3.5


Map: [&lt;1 0 -8|, &lt;0 2 13|]
[[Comma list]]: 1638400/1594323
EDOs: 5, 29, 34
Badness: 0.18473


==7-limit==
{{Mapping|legend=1| 1 0 -8 | 0 2 13 }}
Commas: 49/48, 2240/2187
: mapping generators: ~2, ~1280/729


POTE generator: ~8/7 = 247.555
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.9215{{c}}, ~1280/729 = 952.0734{{c}}
: [[error map]]: {{val| -1.079 +2.192 -0.731 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1280/729 = 952.8424{{c}}
: error map: {{val| 0.000 +3.730 +0.638 }}


Map: [&lt;1 0 -8 2|, &lt;0 2 13 1|]
{{Optimal ET sequence|legend=1| 5, …, 29, 34 }}
Wedgie: &lt;&lt;2 13 1 16 -4 -34||
EDOs: 5, 29, 34
Badness: 0.0776


==11-limit==
[[Badness]] (Sintel): 4.33
Commas: 49/48, 55/54, 896/891


POTE generator: ~8/7 = 247.926
=== 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]].


Map: [&lt;1 0 -8 2 9|, &lt;0 2 13 1 -7|]
The immunity family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], discussed in [[#Subgroup extensions]].  
EDOs: 5, 29, 63de, 92cde
Badness: 0.0527


==13-limit==
== Septimal immunity ==
Commas: 49/48, 55/54, 91/90, 896/891
{{See also| Semaphoresmic clan }}


POTE generator: ~8/7 = 247.97
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 0 -8 2 9 -9|, &lt;0 2 13 1 -7 16|]
[[Comma list]]: 49/48, 2240/2187
EDOs: 5, 29, 92cde
Badness: 0.0371


==Impunity==
{{Mapping|legend=1| 1 0 -8 2 | 0 2 13 1 }}
Commas: 49/48, 100/99, 1232/1215


POTE generator: ~8/7 = 247.773
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.4508{{c}}, ~7/4 = 953.5968{{c}}
: [[error map]]: {{val| +1.451 +5.239 -1.162 -12.327 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 952.5376{{c}}
: error map: {{val| 0.000 +3.120 -3.325 -16.288 }}


Map: [&lt;1 0 -8 2 -14|, &lt;0 2 13 1 22|]
{{Optimal ET sequence|legend=1| 5, …, 29, 34, 63d }}
EDOs: 29, 63d, 92cd
Badness: 0.0552


===13-limit===
[[Badness]] (Sintel): 1.96
Commas: 49/48, 91/90, 100/99, 352/351


POTE generator: ~8/7 = 247.783
=== 11-limit ===
Subgroup: 2.3.5.7.11


Map: [&lt;1 0 -8 2 -14 -9|, &lt;0 2 13 1 22 16|]
Comma list: 49/48, 55/54, 896/891
EDOs: 29, 63d, 92cd
Badness: 0.0332


=Immunized=
Mapping: {{mapping| 1 0 -8 2 9 | 0 2 13 1 -7 }}
Commas: 64/63, 8748/8575


POTE generator: ~81/70 = 246.479
Optimal tunings:  
* WE: ~2 = 1202.6033{{c}}, ~7/4 = 954.1395{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1429{{c}}


Map: [&lt;1 0 -8 6|, &lt;0 2 13 -4|]
{{Optimal ET sequence|legend=0| 5, 24c, 29, 63de }}
Wedgie: &lt;&lt;2 13 -4 16 -12 -46||
 
EDOs: 5, 34d, 39d, 73d
Badness (Sintel): 1.74
Badness: 0.0826</pre></div>
 
<h4>Original HTML content:</h4>
==== 13-limit ====
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Immunity family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:14:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:14 --&gt;&lt;!-- ws:start:WikiTextTocRule:15: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Immunity"&gt;Immunity&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:15 --&gt;&lt;!-- ws:start:WikiTextTocRule:16: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Immunity-7-limit"&gt;7-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:16 --&gt;&lt;!-- ws:start:WikiTextTocRule:17: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Immunity-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Comma list: 49/48, 55/54, 91/90, 896/891
&lt;!-- ws:end:WikiTextTocRule:17 --&gt;&lt;!-- ws:start:WikiTextTocRule:18: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Immunity-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:18 --&gt;&lt;!-- ws:start:WikiTextTocRule:19: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Immunity-Impunity"&gt;Impunity&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 -8 2 9 -9 | 0 2 13 1 -7 16 }}
&lt;!-- ws:end:WikiTextTocRule:19 --&gt;&lt;!-- ws:start:WikiTextTocRule:20: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Immunity-Impunity-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Immunized"&gt;Immunized&lt;/a&gt;&lt;/div&gt;
Optimal tunings:  
&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt;&lt;/div&gt;
* WE: ~2 = 1202.5640{{c}}, ~7/4 = 954.0648{{c}}
&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.1117{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Immunity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Immunity&lt;/h1&gt;
 
Comma: 1638400/1594323&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 24cf, 29 }}
&lt;br /&gt;
 
POTE generator: ~729/640 = 247.070&lt;br /&gt;
Badness (Sintel): 1.53
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -8|, &amp;lt;0 2 13|]&lt;br /&gt;
=== Impunity ===
EDOs: 5, 29, 34&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.18473&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 49/48, 100/99, 1232/1215
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Immunity-7-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit&lt;/h2&gt;
 
Commas: 49/48, 2240/2187&lt;br /&gt;
Mapping: {{mapping| 1 0 -8 2 -14 | 0 2 13 1 22 }}
&lt;br /&gt;
 
POTE generator: ~8/7 = 247.555&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~2 = 1201.6023{{c}}, ~7/4 = 953.4988{{c}}
Map: [&amp;lt;1 0 -8 2|, &amp;lt;0 2 13 1|]&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3120{{c}}
Wedgie: &amp;lt;&amp;lt;2 13 1 16 -4 -34||&lt;br /&gt;
 
EDOs: 5, 29, 34&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5e, 24cee, 29 }}
Badness: 0.0776&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.83
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Immunity-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 49/48, 55/54, 896/891&lt;br /&gt;
==== 13-limit ====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~8/7 = 247.926&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 49/48, 91/90, 100/99, 352/351
Map: [&amp;lt;1 0 -8 2 9|, &amp;lt;0 2 13 1 -7|]&lt;br /&gt;
 
EDOs: 5, 29, 63de, 92cde&lt;br /&gt;
Mapping: {{mapping| 1 0 -8 2 -14 -9 | 0 2 13 1 22 16 }}
Badness: 0.0527&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Immunity-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
* WE: ~2 = 1200.5826{{c}}, ~7/4 = 952.4725{{c}}
Commas: 49/48, 55/54, 91/90, 896/891&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 952.3038{{c}}
&lt;br /&gt;
 
POTE generator: ~8/7 = 247.97&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5e, 24ceef, 29 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -8 2 9 -9|, &amp;lt;0 2 13 1 -7 16|]&lt;br /&gt;
Badness (Sintel): 1.37
EDOs: 5, 29, 92cde&lt;br /&gt;
 
Badness: 0.0371&lt;br /&gt;
== Immunized ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Immunity-Impunity"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Impunity&lt;/h2&gt;
 
Commas: 49/48, 100/99, 1232/1215&lt;br /&gt;
[[Comma list]]: 64/63, 8748/8575
&lt;br /&gt;
 
POTE generator: ~8/7 = 247.773&lt;br /&gt;
{{Mapping|legend=1| 1 0 -8 6 | 0 2 13 -4 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 -8 2 -14|, &amp;lt;0 2 13 1 22|]&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 29, 63d, 92cd&lt;br /&gt;
* [[WE]]: ~2 = 1196.6463{{c}}, ~140/81 = 950.8564{{c}}
Badness: 0.0552&lt;br /&gt;
: [[error map]]: {{val| -3.354 -0.242 +1.649 +7.626 }}
&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 953.3954{{c}}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Immunity-Impunity-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;13-limit&lt;/h3&gt;
: error map: {{val| 0.000 +4.836 +7.827 +17.592 }}
Commas: 49/48, 91/90, 100/99, 352/351&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 5, 29d, 34d, 39d, 73dd, 112cddd }}
POTE generator: ~8/7 = 247.783&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Sintel): 2.09
Map: [&amp;lt;1 0 -8 2 -14 -9|, &amp;lt;0 2 13 1 22 16|]&lt;br /&gt;
 
EDOs: 29, 63d, 92cd&lt;br /&gt;
=== 11-limit ===
Badness: 0.0332&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Immunized"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Immunized&lt;/h1&gt;
Comma list: 64/63, 99/98, 1944/1925
Commas: 64/63, 8748/8575&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 -8 6 13 | 0 2 13 -4 -12 }}
POTE generator: ~81/70 = 246.479&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:
Map: [&amp;lt;1 0 -8 6|, &amp;lt;0 2 13 -4|]&lt;br /&gt;
* WE: ~2 = 1196.9902{{c}}, ~55/32 = 951.0056{{c}}
Wedgie: &amp;lt;&amp;lt;2 13 -4 16 -12 -46||&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~55/32 = 953.3755{{c}}
EDOs: 5, 34d, 39d, 73d&lt;br /&gt;
 
Badness: 0.0826&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Optimal ET sequence|legend=0| 5, 29de, 34d, 39d, 73dd }}
 
Badness (Sintel): 1.70
 
== Immune ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 781250/750141
 
{{Mapping|legend=1| 1 2 5 9 | 0 -2 -13 -30 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7353{{c}}, ~1000/567 = 952.2068{{c}}
: [[error map]]: {{val| -0.265 +2.459 -5.507 +2.939 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1000/567 = 952.4061{{c}}
: error map: {{val| 0.000 +2.857 -5.034 +3.357 }}
 
{{Optimal ET sequence|legend=1| 29, 34d, 63 }}
 
[[Badness]] (Sintel): 3.67
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 225/224, 1331/1323
 
Mapping: {{mapping| 1 2 5 9 8 | 0 -2 -13 -30 -22 }}
 
Optimal tunings:  
* WE: ~2 = 1199.7081{{c}}, ~121/70 = 952.1793{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~121/70 = 952.3983{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 5 9 8 7 | 0 -2 -13 -30 -22 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.8625{{c}}, ~26/15 = 952.2977{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 952.4008{{c}}
 
{{Optimal ET sequence|legend=0| 29, 34d, 63 }}
 
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| ]] <!-- Key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

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

Retrieved from "https://en.xen.wiki/index.php?title=Immunity_family&oldid=236304"