Porwell family: Difference between revisions

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<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 '''porwell family''' of [[rank-3 temperament]]s tempers out the porwell comma, {{monzo| 11 1 -3 -2 }} = [[6144/6125]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-12 19:02:16 UTC</tt>.<br>
: The original revision id was <tt>187562083</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 porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.


Comma: 6144/6125
== Porwell a.k.a. hewuermity ==
[[Subgroup]]: 2.3.5.7


7 and 9 limit minimax
[[Comma list]]: [[6144/6125]]
[|1 0 0 0&gt;, |0 1 0 0&gt;, |11/5 1/5 2/5 -2/5&gt;,
|11/5 1/5 -3/5 3/5&gt;]
Eigenmonzos: 4/3, 7/5


Map: [&lt;1 0 1 4|, &lt;0 1 1 -1|, &lt;0 0 -2 3|]
{{Mapping|legend=1| 1 0 1 4 | 0 1 1 -1 | 0 0 -2 3 }}
Generators: 2, 3, 35/32
Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130, 229, 412


==11 limit children==
: mapping generators: ~2, ~3, ~35/32


===Zeus===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.3482, ~35/32 = 157.4677
Commas: 121/120, 176/175


Minimax tuning
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0&gt;,
* 7- and [[9-odd-limit]]
|22/9 2/9 1/3 -4/9 0&gt;, |22/9 2/9 -2/3 5/9 0&gt;,
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 11/5 1/5 2/5 -2/5 | 11/5 1/5 -3/5 3/5 }}
|10/3 2/3 0 -1/3 0&gt;]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
Eigenmonzos: 2, 9/7, 7/5


Lattice basis: 11/10, 11/8
{{Optimal ET sequence|legend=1| 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381 }}
Map to lattice: [&lt;0 1 -1 2 0|, &lt;0 1 1 -1 1|]
Angle(11/10, 11/8) = 87.464 degrees


Map: [&lt;1 0 1 4 2|, &lt;0 1 1 -1 1|, &lt;0 0 -2 3 1|]
[[Badness]]: 0.142 × 10<sup>-3</sup>
Generators: 2, 3, 11/10


Zeus11[22] [[Hobbits|hobbit]] [[transversal]]
[[Projection pair]]s: 3 6125/2048 to 2.5.7
33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
11/6, 15/8, 64/33, 2
Zeus11[24] hobbit transversal
33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
11/6, 15/8, 64/33, 2


13-limit
== Zeus ==
{{Main| Zeus }}


====Tridecimal Zeus====
Zeus tempers out [[121/120]], [[176/175]], and [[385/384]]. See [[Biyatismic clan #Zeus]].
Commas: 121/120, 176/175, 351/350


13-limit minimax
== Jupiter ==
[|1 0 0 0 0 0&gt;, |11/9 10/9 -1/3 -2/9 0 0&gt;,
[[Subgroup]]: 2.3.5.7.11
|22/9 2/9 1/3 -4/9 0 0&gt;, |22/9 2/9 -2/3 5/9 0 0&gt;,
|10/3 2/3 0 -1/3 0 0&gt;, |14/3 -8/3 1 1/3 0 0&gt;]
Eigenmonzos: 2, 9/7, 7/5


15-limit minimax
[[Comma list]]: 540/539, 5632/5625
[|1 0 0 0 0 0&gt;, |0 1 0 0 0 0&gt;,
|11/5 1/5 2/5 -2/5 0 0&gt;,|11/5 1/5 -3/5 3/5 0 0&gt;,
|13/5 3/5 1/5 -1/5 0 0&gt;, |38/5 -12/5 1/5 -1/5 0 0&gt;]
Eigenmonzos: 2, 4/3, 7/5


Lattice basis: 11/10 length 0.7898 11/8 length 1.002
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 1 -1 6 | 0 0 -2 3 -8 }}
Angle(11/10, 11/8) = 106.7439 degrees
Map to lattice: [&lt;0 1 -1 2 0 -3|, &lt;0 1 1 -1 1 -2|]


Map: [&lt;1 0 1 4 2 7|, &lt;0 1 1 -1 1 -2|, &lt;0 0 -2 3 -1  -1|]
{{Optimal ET sequence|legend=1| 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde }}
Generators: 2, 3, 11/10
Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130, 229, 282, 381, 412


Zeus13[22] hobbit transversal
[[Badness]]: 0.562 × 10<sup>-3</sup>
260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2


====Tinia====
== Hera ==
Commas: 66/65, 121/120, 176/175</pre></div>
[[Subgroup]]: 2.3.5.7.11
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Porwell family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The porwell family of rank three temperaments tempers out the porwell comma, 6144/6125.&lt;br /&gt;
[[Comma list]]: 6144/6125, 8019/8000
&lt;br /&gt;
 
Comma: 6144/6125&lt;br /&gt;
{{Mapping|legend=1| 1 0 1 4 9 | 0 1 1 -1 -3 | 0 0 -2 3 -6 }}
&lt;br /&gt;
 
7 and 9 limit minimax&lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.8035, ~35/32 = 157.1649
[|1 0 0 0&amp;gt;, |0 1 0 0&amp;gt;, |11/5 1/5 2/5 -2/5&amp;gt;, &lt;br /&gt;
 
|11/5 1/5 -3/5 3/5&amp;gt;]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c }}
Eigenmonzos: 4/3, 7/5&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]]: 2.23 × 10<sup>-3</sup>
Map: [&amp;lt;1 0 1 4|, &amp;lt;0 1 1 -1|, &amp;lt;0 0 -2 3|]&lt;br /&gt;
 
Generators: 2, 3, 35/32&lt;br /&gt;
=== 13-limit ===
Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130, 229, 412&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
&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;
Comma list: 351/350, 2080/2079, 4096/4095
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-11 limit children-Zeus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Zeus&lt;/h3&gt;
Mapping: {{mapping| 1 0 1 4 9 7 | 0 1 1 -1 -3 -2 | 0 0 -2 3 -6 -1 }}
Commas: 121/120, 176/175&lt;br /&gt;
 
&lt;br /&gt;
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.6641, ~35/32 = 157.1899
Minimax tuning&lt;br /&gt;
 
[|1 0 0 0 0&amp;gt;, |11/9 10/9 -1/3 -2/9 0&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 46, 53, 84, 99, 130, 183, 313 }}
|22/9 2/9 1/3 -4/9 0&amp;gt;, |22/9 2/9 -2/3 5/9 0&amp;gt;, &lt;br /&gt;
 
|10/3 2/3 0 -1/3 0&amp;gt;]&lt;br /&gt;
Badness: 1.30 × 10<sup>-3</sup>
Eigenmonzos: 2, 9/7, 7/5&lt;br /&gt;
 
&lt;br /&gt;
== Semiporwell ==
Lattice basis: 11/10, 11/8&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Map to lattice: [&amp;lt;0 1 -1 2 0|, &amp;lt;0 1 1 -1 1|]&lt;br /&gt;
 
Angle(11/10, 11/8) = 87.464 degrees&lt;br /&gt;
[[Comma list]]: 6144/6125, 9801/9800
&lt;br /&gt;
 
Map: [&amp;lt;1 0 1 4 2|, &amp;lt;0 1 1 -1 1|, &amp;lt;0 0 -2 3 1|]&lt;br /&gt;
{{Mapping|legend=1| 2 0 0 11 14 | 0 1 1 -1 -2 | 0 0 2 -3 -1 }}
Generators: 2, 3, 11/10&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~99/70, ~3, ~128/99
Zeus11[22] &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit&lt;/a&gt; &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt;&lt;br /&gt;
 
33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3, &lt;br /&gt;
[[Optimal tuning]] ([[POTE]]): ~99/70 = 1\2, ~3/2 = 702.4613, ~128/99 = 442.4903
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4, &lt;br /&gt;
 
11/6, 15/8, 64/33, 2&lt;br /&gt;
{{Optimal ET sequence|legend=1| 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee }}
Zeus11[24] hobbit transversal&lt;br /&gt;
 
33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,&lt;br /&gt;
[[Badness]]: 1.25 × 10<sup>-3</sup>
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9, &lt;br /&gt;
 
11/6, 15/8, 64/33, 2&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
13-limit&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 1716/1715, 2080/2079, 3584/3575
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc2"&gt;&lt;a name="x-11 limit children-Zeus-Tridecimal Zeus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Tridecimal Zeus&lt;/h4&gt;
 
Commas: 121/120, 176/175, 351/350&lt;br /&gt;
Mapping: {{mapping| 2 0 0 11 14 15 | 0 1 1 -1 -2 -1 | 0 0 2 -3 -1 -6 }}
&lt;br /&gt;
 
13-limit minimax&lt;br /&gt;
Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.3964, ~84/65 = 442.5902
[|1 0 0 0 0 0&amp;gt;, |11/9 10/9 -1/3 -2/9 0 0&amp;gt;, &lt;br /&gt;
 
|22/9 2/9 1/3 -4/9 0 0&amp;gt;, |22/9 2/9 -2/3 5/9 0 0&amp;gt;, &lt;br /&gt;
{{Optimal ET sequence|legend=1| 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff }}
|10/3 2/3 0 -1/3 0 0&amp;gt;, |14/3 -8/3 1 1/3 0 0&amp;gt;]&lt;br /&gt;
 
Eigenmonzos: 2, 9/7, 7/5&lt;br /&gt;
Badness: 1.22 × 10<sup>-3</sup>
&lt;br /&gt;
 
15-limit minimax&lt;br /&gt;
[[Category:Temperament families]]
[|1 0 0 0 0 0&amp;gt;, |0 1 0 0 0 0&amp;gt;, &lt;br /&gt;
[[Category:Pages with mostly numerical content]]
|11/5 1/5 2/5 -2/5 0 0&amp;gt;,|11/5 1/5 -3/5 3/5 0 0&amp;gt;, &lt;br /&gt;
[[Category:Porwell family| ]] <!-- main article -->
|13/5 3/5 1/5 -1/5 0 0&amp;gt;, |38/5 -12/5 1/5 -1/5 0 0&amp;gt;]&lt;br /&gt;
[[Category:Porwell| ]] <!-- key article -->
Eigenmonzos: 2, 4/3, 7/5&lt;br /&gt;
[[Category:Rank 3]]
&lt;br /&gt;
Lattice basis: 11/10 length 0.7898 11/8 length 1.002&lt;br /&gt;
Angle(11/10, 11/8) = 106.7439 degrees&lt;br /&gt;
Map to lattice: [&amp;lt;0 1 -1 2 0 -3|, &amp;lt;0 1 1 -1 1 -2|]&lt;br /&gt;
&lt;br /&gt;
Map: [&amp;lt;1 0 1 4 2 7|, &amp;lt;0 1 1 -1 1 -2|, &amp;lt;0 0 -2 3 -1 -1|]&lt;br /&gt;
Generators: 2, 3, 11/10&lt;br /&gt;
Edos: 6, 7, 8, 9, 15, 16, 22, 23, 24, 31, 37, 38, 39, 40, 46, 47, 53, 55, 59, 68, 75, 77, 84, 99, 130, 229, 282, 381, 412&lt;br /&gt;
&lt;br /&gt;
Zeus13[22] hobbit transversal&lt;br /&gt;
260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h4&amp;gt; --&gt;&lt;h4 id="toc3"&gt;&lt;a name="x-11 limit children-Zeus-Tinia"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Tinia&lt;/h4&gt;
Commas: 66/65, 121/120, 176/175&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 00:38, 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 porwell family of rank-3 temperaments tempers out the porwell comma, [11 1 -3 -2⟩ = 6144/6125.

Porwell a.k.a. hewuermity

Subgroup: 2.3.5.7

Comma list: 6144/6125

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

mapping generators: ~2, ~3, ~35/32

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.3482, ~35/32 = 157.4677

Minimax tuning:

[[1 0 0 0⟩, [0 1 0 0⟩, [11/5 1/5 2/5 -2/5⟩, [11/5 1/5 -3/5 3/5⟩]
unchanged-interval (eigenmonzo) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 84, 99, 229, 251, 282, 381

Badness: 0.142 × 10-3

Projection pairs: 3 6125/2048 to 2.5.7

Zeus

Main article: Zeus

Zeus tempers out 121/120, 176/175, and 385/384. See Biyatismic clan #Zeus.

Jupiter

Subgroup: 2.3.5.7.11

Comma list: 540/539, 5632/5625

Mapping: [⟨1 0 1 4 -5], ⟨0 1 1 -1 6], ⟨0 0 -2 3 -8]]

Optimal ET sequence: 22, 31, 53, 84e, 99e, 121, 130, 152, 282, 434de, 465d, 617de, 747cde, 899cdde

Badness: 0.562 × 10-3

Hera

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 8019/8000

Mapping: [⟨1 0 1 4 9], ⟨0 1 1 -1 -3], ⟨0 0 -2 3 -6]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8035, ~35/32 = 157.1649

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313, 359, 542c, 672c

Badness: 2.23 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 4096/4095

Mapping: [⟨1 0 1 4 9 7], ⟨0 1 1 -1 -3 -2], ⟨0 0 -2 3 -6 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.6641, ~35/32 = 157.1899

Optimal ET sequence: 46, 53, 84, 99, 130, 183, 313

Badness: 1.30 × 10-3

Semiporwell

Subgroup: 2.3.5.7.11

Comma list: 6144/6125, 9801/9800

Mapping: [⟨2 0 0 11 14], ⟨0 1 1 -1 -2], ⟨0 0 2 -3 -1]]

mapping generators: ~99/70, ~3, ~128/99

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.4613, ~128/99 = 442.4903

Optimal ET sequence: 22, 46, 68, 84, 106, 130, 152, 282, 328, 480, 1112bccddee, 1242ccddee, 1722bcccdddeee

Badness: 1.25 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3584/3575

Mapping: [⟨2 0 0 11 14 15], ⟨0 1 1 -1 -2 -1], ⟨0 0 2 -3 -1 -6]]

Optimal tuning (POTE): ~99/70 = 1\2, ~3/2 = 702.3964, ~84/65 = 442.5902

Optimal ET sequence: 22f, 38e, 46, 68, 84, 130, 198, 282, 328, 480f, 610cdef, 808cdeff, 1090ccdeefff, 1418bcccddeeefff

Badness: 1.22 × 10-3

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