Regular temperament: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-18 02:13:16 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-18 02:14:18 UTC</tt>.

: The original revision id was <tt>~~302943820~~</tt>.

: The original revision id was <tt>302943850</tt>.

: The revision comment was: <tt></tt>

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

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For example, using "∨" to represent the interior product, we have mir = <<6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0> is <0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1> which is <1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0> which is also <1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.

As explained on the [[Interior product#Applications|interior ~~proiduct~~]] page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.

As explained on the [[Interior product#Applications|interior product]] page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.

* **[[Normal lists|Normal val lists]]**

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For example, using &quot;∨&quot; to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.

As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior ~~proiduct~~</a> page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.

As explained on the <a class="wiki_link" href="/Interior%20product#Applications">interior product</a> page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.

<ul><li><a class="wiki_link" href="/Normal%20lists">Normal val lists</a></li></ul>Given a list of vals, we may <a class="wiki_link" href="/Saturation">saturate</a> it and reduce it using the <a class="wiki_link" href="/Normal%20lists">Hermite normal form</a> to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&lt;1 1 3 3|, &lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].

@@ 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>2012-02-18 02:13:16 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-02-18 02:14:18 UTC</tt>.<br>
-: The original revision id was <tt>302943820</tt>.<br>
+: The original revision id was <tt>302943850</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 15: / Line 15: @@
 For example, using "∨" to represent the interior product, we have mir = &lt;&lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0&gt; is &lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&gt; which is &lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&gt; which is also &lt;1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.
-As explained on the [[Interior product#Applications|interior proiduct]] page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.
+As explained on the [[Interior product#Applications|interior product]] page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.
 * **[[Normal lists|Normal val lists]]**
@@ Line 70: / Line 70: @@
 For example, using &amp;quot;∨&amp;quot; to represent the interior product, we have mir = &amp;lt;&amp;lt;6 -7 -2 -25 -20 15|| for the wedgie of 7-limit miracle. Then the interior product mir ∨ |1 0 0 0&amp;gt; is &amp;lt;0 -6 7 2|, with 15/14 we get mir ∨ |-1 1 1 -1&amp;gt; which is &amp;lt;1 1 3 3|, and with 16/15 we get mir ∨ |4 -1 -1 0&amp;gt; which is also &amp;lt;1 1 3 3|; this val tempers out the commas of miracle and 15/14 (or 16/15), sending all of them to the unison.&lt;br /&gt;
 &lt;br /&gt;
-As explained on the &lt;a class="wiki_link" href="/Interior%20product#Applications"&gt;interior proiduct&lt;/a&gt; page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.&lt;br /&gt;
+As explained on the &lt;a class="wiki_link" href="/Interior%20product#Applications"&gt;interior product&lt;/a&gt; page, if W is the wedgie, then the tuning of map for the temperament can be defined via a multimonzo V which has the property that for every JI interval q, the tempered value of q is given by the dot product (W∨q).V.&lt;br /&gt;
 &lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;strong&gt;&lt;a class="wiki_link" href="/Normal%20lists"&gt;Normal val lists&lt;/a&gt;&lt;/strong&gt;&lt;/li&gt;&lt;/ul&gt;Given a list of vals, we may &lt;a class="wiki_link" href="/Saturation"&gt;saturate&lt;/a&gt; it and reduce it using the &lt;a class="wiki_link" href="/Normal%20lists"&gt;Hermite normal form&lt;/a&gt; to a normal val list, which canonically represents the abstract temperament. Applying the vals successively (an operation we may regard as a matrix multiplication if we like) to a rational interval gives an element in an abelian group representing the notes of the temperament. For example, the normal val list for 7-limit miracle is [&amp;lt;1 1 3 3|, &amp;lt;0 6 -7 -2|] and applying this to the monzo for either 16/15 or 15/14 leads to [0 1].&lt;br /&gt;