Schismatic family: Difference between revisions

Revision as of 04:03, 14 September 2010

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author genewardsmith and made on 2010-09-14 04:03:40 UTC.

The original revision id was 162439025.

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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. 

The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.

==Seven limit children==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.

===Garibaldi===

7-limit
Commas: {225/224, 3125/3087}

Minimax tuning:
7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,
|5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]
9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, 
|5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]

Edos: 94, 135

Original HTML content:

<html><head><title>Schismatic family</title></head><body>The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br />
<br />
The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2>
The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-Seven limit children-Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:2 -->Garibaldi</h3>
<br />
7-limit<br />
Commas: {225/224, 3125/3087}<br />
<br />
Minimax tuning:<br />
7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,<br />
|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]<br />
9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;, <br />
|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]<br />
<br />
Edos: 94, 135</body></html>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-09-12 15:35:47 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-14 04:03:40 UTC</tt>.<br>
-: The original revision id was <tt>161998295</tt>.<br>
+: The original revision id was <tt>162439025</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>
@@ Line 11: / Line 11: @@
 ==Seven limit children==
-The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.</pre></div>
+The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.
+===Garibaldi===
+-limit
+Commas: {225/224, 3125/3087}
+Minimax tuning:
+-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,
+|5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]
+-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,
+|5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]
+Edos: 94, 135
+</pre></div>
 <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;Schismatic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its &lt;a class="wiki_link" href="/monzo"&gt;monzo&lt;/a&gt; is |-15 8 1&amp;gt;, and flipping that yields &amp;lt;&amp;lt;1 -8 -15|| for the &lt;a class="wiki_link" href="/Wedgies%20and%20Multivals"&gt;wedgie&lt;/a&gt;. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. &lt;br /&gt;
@@ Line 18: / Line 32: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Seven limit children&lt;/h2&gt;
-The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&amp;gt; gives garibaldi, |-44 26 0 1&amp;gt; grackle, |6 -2 0 -1&amp;gt; schism and |-59 39 0 -1&amp;gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&amp;gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&amp;gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&amp;gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&amp;gt; and slices the fifth in four.&lt;/body&gt;&lt;/html&gt;</pre></div>
+The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding |25 -14 0 -1&amp;gt; gives garibaldi, |-44 26 0 1&amp;gt; grackle, |6 -2 0 -1&amp;gt; schism and |-59 39 0 -1&amp;gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&amp;gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&amp;gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&amp;gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&amp;gt; and slices the fifth in four.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x-Seven limit children-Garibaldi"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Garibaldi&lt;/h3&gt;
+&lt;br /&gt;
+-limit&lt;br /&gt;
+Commas: {225/224, 3125/3087}&lt;br /&gt;
+&lt;br /&gt;
+Minimax tuning:&lt;br /&gt;
+-limit: [|1 0 0 0&amp;gt;, |5/3 1/15 0 -1/15&amp;gt;,&lt;br /&gt;
+|5/3 -8/15 0 8/15&amp;gt;, |5/3 -14/15 0 14/15&amp;gt;]&lt;br /&gt;
+-limit: [|1 0 0 0&amp;gt;, |25/16 1/8 0 -1/16&amp;gt;, &lt;br /&gt;
+|5/2 -1 0 1/2&amp;gt;, |25/8 -7/4 0 7/8&amp;gt;]&lt;br /&gt;
+&lt;br /&gt;
+Edos: 94, 135&lt;/body&gt;&lt;/html&gt;</pre></div>

Schismatic family: Difference between revisions

Revision as of 04:03, 14 September 2010

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

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