Hobbit: Difference between revisions

Revision as of 23:53, 3 October 2010

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This revision was by author genewardsmith and made on 2010-10-03 23:53:14 UTC.

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A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.

==Definition==

To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:

(1) If o = |1 0 0 ... 0> is the monzo for 2 in the [[Harmonic Limit|p-limit]] group.

(2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.

(3) Form the (i+1)**x**n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.

(4) Monzo weight N by multiplying on the right by a n**x**n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND.

(5) Now find Q = M`M, where M` is the [[RMS tuning|Moore-Penrose pseudoinverse]] of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M.

(6) Let P = I - Q, where I is the identity matrix.

(7) For a p-limit monzo or [[Fractional monzos|fractional monzo]] m we now define the seminorm

||m||_s = ||mDP||

where the norm on the right is the ordinary Euclidean norm.

(8) If v[1] is odd then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 <= <J|m> < 1 where J is the JI mapping <log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.

(9) If v[1] is even, we choose a monzo u such that ||u||_s > 0 and ||u||_s is minimal. Then for each integer j, 0 <= j < v[1], we choose a corresponding monzo mj such that <v|m> = j, 0 <= <J|m> < 1, and ||m - u/2||_s is minimal.

(10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale.

Note that

Original HTML content:

<html><head><title>Hobbits</title></head><body>A <em>hobbit scale</em> is a generalization of <a class="wiki_link" href="/MOSScales">MOS</a> for arbitrary regular temperaments which is a sort of cousin to <a class="wiki_link" href="/Dwarves">dwarf scales</a>. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h2>
<br />
To define the hobbit scale we first define a particular <a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow">seminorm</a> on interval space. This seminorm applies to <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzos</a> and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:<br />
<br />
(1) If o = |1 0 0 ... 0&gt; is the monzo for 2 in the <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group.<br />
<br />
(2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.<br />
<br />
(3) Form the (i+1)<strong>x</strong>n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.<br />
<br />
(4) Monzo weight N by multiplying on the right by a n<strong>x</strong>n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND.<br />
<br />
(5) Now find Q = M`M, where M` is the <a class="wiki_link" href="/RMS%20tuning">Moore-Penrose pseudoinverse</a> of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M.<br />
<br />
(6) Let P = I - Q, where I is the identity matrix.<br />
<br />
(7) For a p-limit monzo or <a class="wiki_link" href="/Fractional%20monzos">fractional monzo</a> m we now define the seminorm<br />
<br />


<table class="wiki_table">
    <tr>
        <td>m<br />
</td>
        <td>_s =<br />
</td>
        <td>mDP<br />
</td>
    </tr>
</table>

<br />
where the norm on the right is the ordinary Euclidean norm.<br />
<br />
(8) If v[1] is odd then for each integer j, 0 &lt;= j &lt; v[1], we choose a corresponding monzo mj such that &lt;v|m&gt;  <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="j, 0"></a><!-- ws:end:WikiTextHeadingRule:2 --> j, 0 &lt;</h1>
 &lt;J|m&gt; &lt; 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.<br />
<br />
(9) If v[1] is even, we choose a monzo u such that ||u||_s &gt; 0 and ||u||_s is minimal. Then for each integer j, 0 &lt;= j &lt; v[1], we choose a corresponding monzo mj such that &lt;v|m&gt;  <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="j, 0"></a><!-- ws:end:WikiTextHeadingRule:4 --> j, 0 &lt;</h1>
 &lt;J|m&gt; &lt; 1, and ||m - u/2||_s is minimal.<br />
<br />
(10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale.<br />
<br />
Note that</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>2010-10-03 22:05:41 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-03 23:53:14 UTC</tt>.<br>
-: The original revision id was <tt>167459105</tt>.<br>
+: The original revision id was <tt>167479385</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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]. Given a regular temperament and a val v which supports (or belongs to) the temperament, there is a unique scale for the temperament which can be tuned to any tuning of the temperament with v[1] notes to the octave.</pre></div>
+<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">A //hobbit scale// is a generalization of [[MOSScales|MOS]] for arbitrary regular temperaments which is a sort of cousin to [[Dwarves|dwarf scales]]. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.
+==Definition==
+To define the hobbit scale we first define a particular [[http://mathworld.wolfram.com/Seminorm.html|seminorm]] on interval space. This seminorm applies to [[Monzos and Interval Space|monzos]] and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:
+(1) If o = |1 0 0 ... 0&gt; is the monzo for 2 in the [[Harmonic Limit|p-limit]] group.
+(2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.
+(3) Form the (i+1)**x**n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.
+(4) Monzo weight N by multiplying on the right by a n**x**n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND.
+(5) Now find Q = M`M, where M` is the [[RMS tuning|Moore-Penrose pseudoinverse]] of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M.
+(6) Let P = I - Q, where I is the identity matrix.
+(7) For a p-limit monzo or [[Fractional monzos|fractional monzo]] m we now define the seminorm
+||m||_s = ||mDP||
+where the norm on the right is the ordinary Euclidean norm.
+(8) If v[1] is odd then for each integer j, 0 &lt;= j &lt; v[1], we choose a corresponding monzo mj such that &lt;v|m&gt; = j, 0 &lt;= &lt;J|m&gt; &lt; 1 where J is the JI mapping &lt;log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.
+(9) If v[1] is even, we choose a monzo u such that ||u||_s &gt; 0 and ||u||_s is minimal. Then for each integer j, 0 &lt;= j &lt; v[1], we choose a corresponding monzo mj such that &lt;v|m&gt; = j, 0 &lt;= &lt;J|m&gt; &lt; 1, and ||m - u/2||_s is minimal.
+(10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale.
+Note that </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;Hobbits&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;hobbit scale&lt;/em&gt; is a generalization of &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; for arbitrary regular temperaments which is a sort of cousin to &lt;a class="wiki_link" href="/Dwarves"&gt;dwarf scales&lt;/a&gt;. Given a regular temperament and a val v which supports (or belongs to) the temperament, there is a unique scale for the temperament which can be tuned to any tuning of the temperament with v[1] notes to the octave.&lt;/body&gt;&lt;/html&gt;</pre></div>
+<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;Hobbits&lt;/title&gt;&lt;/head&gt;&lt;body&gt;A &lt;em&gt;hobbit scale&lt;/em&gt; is a generalization of &lt;a class="wiki_link" href="/MOSScales"&gt;MOS&lt;/a&gt; for arbitrary regular temperaments which is a sort of cousin to &lt;a class="wiki_link" href="/Dwarves"&gt;dwarf scales&lt;/a&gt;. Given a regular temperament and an equal temperament val v which supports (or belongs to) the temperament, there is a unique scale for the temperament, which can be tuned to any tuning of the temperament, containing v[1] notes to the octave.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Definition&lt;/h2&gt;
+&lt;br /&gt;
+To define the hobbit scale we first define a particular &lt;a class="wiki_link_ext" href="http://mathworld.wolfram.com/Seminorm.html" rel="nofollow"&gt;seminorm&lt;/a&gt; on interval space. This seminorm applies to &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;monzos&lt;/a&gt; and has the property that the seminorm of a comma of the temperament, or of the unison, the octave and any power of two is 0. It may be defined as follows:&lt;br /&gt;
+&lt;br /&gt;
+(1) If o = |1 0 0 ... 0&amp;gt; is the monzo for 2 in the &lt;a class="wiki_link" href="/Harmonic%20Limit"&gt;p-limit&lt;/a&gt; group.&lt;br /&gt;
+&lt;br /&gt;
+(2) c1, c2, ..., ci are monzos for a basis for the commas of the temperament.&lt;br /&gt;
+&lt;br /&gt;
+(3) Form the (i+1)&lt;strong&gt;x&lt;/strong&gt;n matrix N = [o, c1, c2, ..., ci] whose rows consist of o and the commas ck.&lt;br /&gt;
+&lt;br /&gt;
+(4) Monzo weight N by multiplying on the right by a n&lt;strong&gt;x&lt;/strong&gt;n diagonal matrix D consisting of log2(qk) along the diagonal, where qk are the primes from 2 to p, obtaining M = ND.&lt;br /&gt;
+&lt;br /&gt;
+(5) Now find Q = M`M, where M` is the &lt;a class="wiki_link" href="/RMS%20tuning"&gt;Moore-Penrose pseudoinverse&lt;/a&gt; of M. On the assumption that the ck form a basis for the commas, then M has linearly independent rows, and by a property of the pseudoinverse, M` = M*(MM*)^(-1), where M* is the transpose of M, so that Q = M*(MM*)^(1)M.&lt;br /&gt;
+&lt;br /&gt;
+(6) Let P = I - Q, where I is the identity matrix.&lt;br /&gt;
+&lt;br /&gt;
+(7) For a p-limit monzo or &lt;a class="wiki_link" href="/Fractional%20monzos"&gt;fractional monzo&lt;/a&gt; m we now define the seminorm&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;m&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;_s =&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;mDP&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+where the norm on the right is the ordinary Euclidean norm.&lt;br /&gt;
+&lt;br /&gt;
+(8) If v[1] is odd then for each integer j, 0 &amp;lt;= j &amp;lt; v[1], we choose a corresponding monzo mj such that &amp;lt;v|m&amp;gt;  &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="j, 0"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt; j, 0 &amp;lt;&lt;/h1&gt;
+ &amp;lt;J|m&amp;gt; &amp;lt; 1 where J is the JI mapping &amp;lt;log2(2) log2(3) ... log2(p)|, and ||m||_s is minimal.&lt;br /&gt;
+&lt;br /&gt;
+(9) If v[1] is even, we choose a monzo u such that ||u||_s &amp;gt; 0 and ||u||_s is minimal. Then for each integer j, 0 &amp;lt;= j &amp;lt; v[1], we choose a corresponding monzo mj such that &amp;lt;v|m&amp;gt;  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="j, 0"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt; j, 0 &amp;lt;&lt;/h1&gt;
+ &amp;lt;J|m&amp;gt; &amp;lt; 1, and ||m - u/2||_s is minimal.&lt;br /&gt;
+&lt;br /&gt;
+(10) We now apply the chosen tuning to the monzos mj, obtaining values (in cents or fractional monzos) defining a scale. The monzos mj are defined only modulo the commas and the octave o, but since the commas are tempered out and mj is in the octave range from 0 to 1200 cents, this does not affect the definition of the scale.&lt;br /&gt;
+&lt;br /&gt;
+Note that&lt;/body&gt;&lt;/html&gt;</pre></div>

Hobbit: Difference between revisions

Revision as of 23:53, 3 October 2010

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