Cataharry family: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 166579211 - Original comment: **
Tags: Mobile edit Mobile web edit
 
(28 intermediate revisions by 8 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>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-29 21:13:07 UTC</tt>.<br>
: The original revision id was <tt>166579211</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">The cataharry family of rank three temperaments tempers out the cataharry comma, 19683/19600.


Comma: 19683/19600
The '''cataharry family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the [[cataharry comma]], 19683/19600.


7 and 9 limit minimax
== Cataharry ==
[|1 0 0 0&gt;, |4/9 0 2/9 2/9&gt;,
[[Subgroup]]: 2.3.5.7
|0 0 1 0&gt;, |0 0 0 1&gt;]
Eigenmonzos: 2, 8/7, 5/4


Map: [&lt;1 0 0 -2|, &lt;0 2 0 9|, &lt;0 0 1 -1|]
[[Comma list]]: [[19683/19600]]
Generators: 2, 140/81, 5
Edos: 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717


==11 limit children==
{{Mapping|legend=1| 1 0 0 -2 | 0 2 0 9 | 0 0 1 -1 }}


===Albus===
: mapping generators: ~2, ~140/81, ~5
Commas: 540/539, 8019/8000


Minimax tuning
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3&gt;,
* [[7-odd-limit|7-]] and [[9-odd-limit]]:
|5 0 7/2 -2 -3/2&gt;, |5 0 5/2 -1 -3/2&gt;,
: {{monzo list| 1 0 0 0 | 4/9 0 2/9 2/9 | 0 0 1 0 | 0 0 0 1 }}
|5 0 5/2 -2 -1/2&gt;]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
Eigenmonzos: 2, 11/10, 7/5


Map: [&lt;1 0 0 -2 6|, &lt;0 2 0 9 -12|, &lt;0 0 1 -1 3|]
{{Optimal ET sequence|legend=1| 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587 }}
Generators: 2, 140/99, 5


13-limit
[[Badness]]: 0.301 × 10<sup>-3</sup>
Commas: 351/350, 540/539, 729/728


13 and 15 limit minimax
[[Projection pair]]s: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7
[|1 0 0 0 0 0&gt;, |8/3 0 4/3 -2/3 -2/3 0&gt;,
|5 0 7/2 -2 -3/2 0&gt;, |5 0 5/2 -1 -3/2 0&gt;,
|5 0 5/2 -2 -1/2 0&gt;, |8 0 11/2 -3 -5/2 0&gt;]
Eigenmonzos: 2, 11/10, 7/5


Map: [&lt;1 0 0 -2 6 -1|, &lt;0 2 0 9 -12 3|, &lt;0 0 1 -1 3 1|]
== Albus ==
Generators: 2, 26/15, 5
[[Subgroup]]: 2.3.5.7.11
Edos: 19, 53, 58, 72, 130, 183, 313, 385, 457


===Snape===
[[Comma list]]: 540/539, 8019/8000
Commas: 19683/19600, 441/440


13-limit
{{Mapping|legend=1| 1 0 0 -2 6 | 0 2 0 9 -12 | 0 0 1 -1 3 }}
Commas: 351/350, 441/440, 729/728


===Hagrid===
: mapping generators: ~2, ~140/81, ~5
Commas: 243/242, 9801/9800


13-limit
[[Minimax tuning]]:
Commas: 243/242, 351/350, 729/728
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 }}, {{monzo| 5 0 7/2 -2 -3/2 }}, {{monzo| 5 0 5/2 -1 -3/2 }}, {{monzo| 5 0 5/2 -2 -1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5


===Draco===
{{Optimal ET sequence|legend=1| 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640 }}
Commas: 19683/19600, 176/175


13-limit
[[Badness]]: 0.621 × 10<sup>-3</sup>
Commas: 176/175, 351/350, 676/675</pre></div>
 
<h4>Original HTML content:</h4>
=== Madagascar ===
<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;Cataharry family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The cataharry family of rank three temperaments tempers out the cataharry comma, 19683/19600.&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Comma: 19683/19600&lt;br /&gt;
Comma list: 351/350, 540/539, 676/675
&lt;br /&gt;
 
7 and 9 limit minimax&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -2 6 -1 | 0 2 0 9 -12 3 | 0 0 1 -1 3 1 }}
[|1 0 0 0&amp;gt;, |4/9 0 2/9 2/9&amp;gt;, &lt;br /&gt;
 
|0 0 1 0&amp;gt;, |0 0 0 1&amp;gt;]&lt;br /&gt;
[[Mapping to lattice]]: [{{val| 0 -2 -3 -6 3 -6 }}, {{val| 0 0 -1 1 -3 -1 }}]
Eigenmonzos: 2, 8/7, 5/4&lt;br /&gt;
 
&lt;br /&gt;
Lattice basis:
Map: [&amp;lt;1 0 0 -2|, &amp;lt;0 2 0 9|, &amp;lt;0 0 1 -1|]&lt;br /&gt;
: 15/13 length = 0.3019, 26/25 length = 0.9477
Generators: 2, 140/81, 5&lt;br /&gt;
: Angle (15/13, 26/25) = 81.6476
Edos: 19, 24, 48, 53, 58, 72, 130, 202, 457, 587, 659, 717&lt;br /&gt;
 
&lt;br /&gt;
Minimax tuning:  
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-11 limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;11 limit children&lt;/h2&gt;
* 13- and 15-odd-limit
&lt;br /&gt;
: [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 8/3 0 4/3 -2/3 -2/3 0 }}, {{monzo| 5 0 7/2 -2 -3/2 0 }}, {{monzo| 5 0 5/2 -1 -3/2 0 }}, {{monzo| 5 0 5/2 -2 -1/2 0 }}, {{monzo| 8 0 11/2 -3 -5/2 0 }}]
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-11 limit children-Albus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Albus&lt;/h3&gt;
: unchanged-interval (eigenmonzo) basis: 2.7/5.11/5
Commas: 540/539, 8019/8000&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313 }}
Minimax tuning&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |8/3 0 4/3 -2/3 -2/3&amp;gt;, &lt;br /&gt;
Badness: 0.560 × 10<sup>-3</sup>
|5 0 7/2 -2 -3/2&amp;gt;, |5 0 5/2 -1 -3/2&amp;gt;, &lt;br /&gt;
 
|5 0 5/2 -2 -1/2&amp;gt;]&lt;br /&gt;
Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11
Eigenmonzos: 2, 11/10, 7/5&lt;br /&gt;
 
&lt;br /&gt;
Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3
Map: [&amp;lt;1 0 0 -2 6|, &amp;lt;0 2 0 9 -12|, &amp;lt;0 0 1 -1 3|]&lt;br /&gt;
 
Generators: 2, 140/99, 5&lt;br /&gt;
Scales: [[madagascar19]]
&lt;br /&gt;
 
13-limit &lt;br /&gt;
== Snape ==
Commas: 351/350, 540/539, 729/728&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
13 and 15 limit minimax&lt;br /&gt;
[[Comma list]]: 441/440, 19683/19600
[|1 0 0 0 0 0&amp;gt;, |8/3 0 4/3 -2/3 -2/3 0&amp;gt;, &lt;br /&gt;
 
|5 0 7/2 -2 -3/2 0&amp;gt;, |5 0 5/2 -1 -3/2 0&amp;gt;, &lt;br /&gt;
{{Mapping|legend=1| 1 0 0 -2 -7 | 0 2 0 9 22 | 0 0 1 -1 -3 }}
|5 0 5/2 -2 -1/2 0&amp;gt;, |8 0 11/2 -3 -5/2 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 11/10, 7/5&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 0 -2 6 -1|, &amp;lt;0 2 0 9 -12 3|, &amp;lt;0 0 1 -1 3 1|]&lt;br /&gt;
[[Badness]]: 0.988 × 10<sup>-3</sup>
Generators: 2, 26/15, 5&lt;br /&gt;
 
Edos: 19, 53, 58, 72, 130, 183, 313, 385, 457&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc2"&gt;&lt;a name="x-11 limit children-Snape"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Snape&lt;/h3&gt;
 
Commas: 19683/19600, 441/440&lt;br /&gt;
Comma list: 351/350, 441/440, 676/675
&lt;br /&gt;
 
13-limit&lt;br /&gt;
Mapping: {{mapping| 1 0 0 -2 -7 -1 | 0 2 0 9 22 3 | 0 0 1 -1 -3 1 }}
Commas: 351/350, 441/440, 729/728&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409 }}
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x-11 limit children-Hagrid"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Hagrid&lt;/h3&gt;
 
Commas: 243/242, 9801/9800&lt;br /&gt;
Badness: 0.786 × 10<sup>-3</sup>
&lt;br /&gt;
 
13-limit&lt;br /&gt;
== Hagrid ==
Commas: 243/242, 351/350, 729/728&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x-11 limit children-Draco"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Draco&lt;/h3&gt;
[[Comma list]]: 243/242, 9801/9800
Commas: 19683/19600, 176/175&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 2 0 0 -4 -1 | 0 2 0 9 5 | 0 0 1 -1 0 }}
13-limit&lt;br /&gt;
 
Commas: 176/175, 351/350, 676/675&lt;/body&gt;&lt;/html&gt;</pre></div>
: mapping generators: ~99/70, ~140/81, ~5
 
{{Optimal ET sequence|legend=1| 24, 34d, 48, 58, 72, 130, 202 }}
 
[[Badness]]: 1.116 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 364/363
 
Mapping: {{mapping| 2 0 0 -4 -1 -2 | 0 2 0 9 5 3 | 0 0 1 -1 0 1 }}
 
: mapping generators: ~99/70, ~26/15, ~5
 
{{Optimal ET sequence|legend=1| 24, 34d, 58, 72, 130 }}
 
Badness: 0.876 × 10<sup>-3</sup>
 
== Draco ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 176/175, 19683/19600
 
{{Mapping|legend=1| 1 0 0 -2 -6 | 0 2 0 9 9 | 0 0 1 -1 1 }}
 
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cd }}
 
[[Badness]]: 1.934 × 10<sup>-3</sup>
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 676/675
 
Mapping: {{mapping| 1 0 0 -2 -6 -1 | 0 2 0 9 9 3 | 0 0 1 -1 1 1 }}
 
{{Optimal ET sequence|legend=1| 19e, 24, 34d, 53, 58, 111, 280cdf }}
 
Badness: 0.999 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Cataharry family| ]] <!-- main article -->
[[Category:Cataharry| ]] <!-- key article -->
[[Category:Rank 3]]

Latest revision as of 00:39, 24 June 2025

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 cataharry family of rank-3 temperaments tempers out the cataharry comma, 19683/19600.

Cataharry

Subgroup: 2.3.5.7

Comma list: 19683/19600

Mapping: [⟨1 0 0 -2], ⟨0 2 0 9], ⟨0 0 1 -1]]

mapping generators: ~2, ~140/81, ~5

Minimax tuning:

[[1 0 0 0⟩, [4/9 0 2/9 2/9⟩, [0 0 1 0⟩, [0 0 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7

Optimal ET sequence: 19, 34d, 39d, 53, 72, 130, 183, 202, 385, 587

Badness: 0.301 × 10-3

Projection pairs: 3 19600/6561 7 1033052339200000000/150094635296999121 to 2.5.81/7

Albus

Subgroup: 2.3.5.7.11

Comma list: 540/539, 8019/8000

Mapping: [⟨1 0 0 -2 6], ⟨0 2 0 9 -12], ⟨0 0 1 -1 3]]

mapping generators: ~2, ~140/81, ~5

Minimax tuning:

[[1 0 0 0 0⟩, [8/3 0 4/3 -2/3 -2/3⟩, [5 0 7/2 -2 -3/2⟩, [5 0 5/2 -1 -3/2⟩, [5 0 5/2 -2 -1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 19, 33cd, 39d, 53, 58, 72, 130, 183, 255, 385, 640

Badness: 0.621 × 10-3

Madagascar

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675

Mapping: [⟨1 0 0 -2 6 -1], ⟨0 2 0 9 -12 3], ⟨0 0 1 -1 3 1]]

Mapping to lattice: [⟨0 -2 -3 -6 3 -6], ⟨0 0 -1 1 -3 -1]]

Lattice basis:

15/13 length = 0.3019, 26/25 length = 0.9477
Angle (15/13, 26/25) = 81.6476

Minimax tuning:

  • 13- and 15-odd-limit
[[1 0 0 0 0 0⟩, [8/3 0 4/3 -2/3 -2/3 0⟩, [5 0 7/2 -2 -3/2 0⟩, [5 0 5/2 -1 -3/2 0⟩, [5 0 5/2 -2 -1/2 0⟩, [8 0 11/2 -3 -5/2 0⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 19, 33cdff, 39df, 53, 58, 72, 111, 130, 183, 313

Badness: 0.560 × 10-3

Complexity spectrum: 15/13, 4/3, 13/10, 10/9, 6/5, 9/7, 18/13, 9/8, 5/4, 7/6, 13/12, 15/14, 16/15, 14/13, 8/7, 7/5, 16/13, 11/10, 15/11, 11/8, 12/11, 13/11, 11/9, 14/11

Projection pairs: 3 676/225 7 1357375919744/192216796875 11 16218292236328125/1491077447838784 13 8788/675 to 2.5.13/3

Scales: madagascar19

Snape

Subgroup: 2.3.5.7.11

Comma list: 441/440, 19683/19600

Mapping: [⟨1 0 0 -2 -7], ⟨0 2 0 9 22], ⟨0 0 1 -1 -3]]

Optimal ET sequence: 19e, 33cdee, 39dee, 53e, 58, 72, 130, 202, 351, 423d, 481, 553

Badness: 0.988 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440, 676/675

Mapping: [⟨1 0 0 -2 -7 -1], ⟨0 2 0 9 22 3], ⟨0 0 1 -1 -3 1]]

Optimal ET sequence: 19e, 33cdeeff, 39deef, 53e, 58, 72, 130, 279, 409

Badness: 0.786 × 10-3

Hagrid

Subgroup: 2.3.5.7.11

Comma list: 243/242, 9801/9800

Mapping: [⟨2 0 0 -4 -1], ⟨0 2 0 9 5], ⟨0 0 1 -1 0]]

mapping generators: ~99/70, ~140/81, ~5

Optimal ET sequence: 24, 34d, 48, 58, 72, 130, 202

Badness: 1.116 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363

Mapping: [⟨2 0 0 -4 -1 -2], ⟨0 2 0 9 5 3], ⟨0 0 1 -1 0 1]]

mapping generators: ~99/70, ~26/15, ~5

Optimal ET sequence: 24, 34d, 58, 72, 130

Badness: 0.876 × 10-3

Draco

Subgroup: 2.3.5.7.11

Comma list: 176/175, 19683/19600

Mapping: [⟨1 0 0 -2 -6], ⟨0 2 0 9 9], ⟨0 0 1 -1 1]]

Optimal ET sequence: 19e, 24, 34d, 53, 58, 111, 280cd

Badness: 1.934 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 676/675

Mapping: [⟨1 0 0 -2 -6 -1], ⟨0 2 0 9 9 3], ⟨0 0 1 -1 1 1]]

Optimal ET sequence: 19e, 24, 34d, 53, 58, 111, 280cdf

Badness: 0.999 × 10-3

Retrieved from "https://en.xen.wiki/index.php?title=Cataharry_family&oldid=201190"