K*N subgroups: Difference between revisions

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**Imported revision 232592028 - Original comment: **
Wikispaces>genewardsmith
**Imported revision 232643896 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>2011-05-28 12:50:44 UTC</tt>.<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-05-28 22:10:33 UTC</tt>.<br>
: The original revision id was <tt>232592028</tt>.<br>
: The original revision id was <tt>232643896</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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<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">For any [[Harmonic limit|prime limit]] p, EDO N and positive integer k, the p-limit k*N subgroup is the largest [[Just intonation subgroups|just intonation subgroup]] of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas.  
<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">For any [[Harmonic limit|prime limit]] p, EDO N and positive integer k, the p-limit k*N subgroup is the largest [[Just intonation subgroups|just intonation subgroup]] of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas.  


A procedure for finding the k*N subgroup is to take enough of the intervals of the p-limit which are mapped to a value divisible by k by the k*N patent val, and add to this set 2 and a basis for the commas of the k*N patent val, and then reduce this to a [[Normal lists|normal interval list]], giving the canonical list of generators for the subgroup. To get "enough" intervals, take the [[Diamonds|diamond]] of the primes from 2 to p, which is the tonality diamond, then the diamond of the diamond, and so forth, until the iterated diamond construction includes a basis for the commas of the k*N patent val.
A procedure for finding the k*N subgroup is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists|normal interval list]]. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.


For example, the 5-limit diamond is 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2. The intervals mapped to an even integer by &lt;12 19 28| are 5/4, 8/5 and 2, and a basis for the 5-limit commas is 81/80 and 128/125. If we add 81/80 and 128/125 to the diamond and reduce to the normal list, we get 2.81.5 for the 2*6 subgroup, which is not correct; however the diamond of the diamond (or the diamond of the diamond of the diamond, etc.) gives the correct subgroup, 2.9.5. The diamond of the diamond of the diamond contains a basis for the commas, and therefore the procedure described above would give the correct answer.</pre></div>
</pre></div>
<h4>Original HTML content:</h4>
<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;k*N subgroups&lt;/title&gt;&lt;/head&gt;&lt;body&gt;For any &lt;a class="wiki_link" href="/Harmonic%20limit"&gt;prime limit&lt;/a&gt; p, EDO N and positive integer k, the p-limit k*N subgroup is the largest &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; for k*N-edo. This also means they temper out the same commas. &lt;br /&gt;
<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;k*N subgroups&lt;/title&gt;&lt;/head&gt;&lt;body&gt;For any &lt;a class="wiki_link" href="/Harmonic%20limit"&gt;prime limit&lt;/a&gt; p, EDO N and positive integer k, the p-limit k*N subgroup is the largest &lt;a class="wiki_link" href="/Just%20intonation%20subgroups"&gt;just intonation subgroup&lt;/a&gt; of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the &lt;a class="wiki_link" href="/patent%20val"&gt;patent val&lt;/a&gt; for k*N-edo. This also means they temper out the same commas. &lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A procedure for finding the k*N subgroup is to take enough of the intervals of the p-limit which are mapped to a value divisible by k by the k*N patent val, and add to this set 2 and a basis for the commas of the k*N patent val, and then reduce this to a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal interval list&lt;/a&gt;, giving the canonical list of generators for the subgroup. To get &amp;quot;enough&amp;quot; intervals, take the &lt;a class="wiki_link" href="/Diamonds"&gt;diamond&lt;/a&gt; of the primes from 2 to p, which is the tonality diamond, then the diamond of the diamond, and so forth, until the iterated diamond construction includes a basis for the commas of the k*N patent val.&lt;br /&gt;
A procedure for finding the k*N subgroup is to take the product m of the odd primes less than or equal to p, and then find the &lt;a class="wiki_link" href="/Euler%20genera"&gt;Euler genus&lt;/a&gt; Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal interval list&lt;/a&gt;. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;br /&gt;
For example, the 5-limit diamond is 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 2. The intervals mapped to an even integer by &amp;lt;12 19 28| are 5/4, 8/5 and 2, and a basis for the 5-limit commas is 81/80 and 128/125. If we add 81/80 and 128/125 to the diamond and reduce to the normal list, we get 2.81.5 for the 2*6 subgroup, which is not correct; however the diamond of the diamond (or the diamond of the diamond of the diamond, etc.) gives the correct subgroup, 2.9.5. The diamond of the diamond of the diamond contains a basis for the commas, and therefore the procedure described above would give the correct answer.&lt;/body&gt;&lt;/html&gt;</pre></div>

Revision as of 22:10, 28 May 2011

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 2011-05-28 22:10:33 UTC.
The original revision id was 232643896.
The revision comment was:

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

Original Wikitext content:

For any [[Harmonic limit|prime limit]] p, EDO N and positive integer k, the p-limit k*N subgroup is the largest [[Just intonation subgroups|just intonation subgroup]] of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the [[patent val]] for k*N-edo. This also means they temper out the same commas. 

A procedure for finding the k*N subgroup is to take the product m of the odd primes less than or equal to p, and then find the [[Euler genera|Euler genus]] Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding [[Normal lists|normal interval list]]. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.

Original HTML content:

<html><head><title>k*N subgroups</title></head><body>For any <a class="wiki_link" href="/Harmonic%20limit">prime limit</a> p, EDO N and positive integer k, the p-limit k*N subgroup is the largest <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a> of the p-limit on which N-edo and k*N-edo approximate intervals to the same values using the mapping supplied by the <a class="wiki_link" href="/patent%20val">patent val</a> for k*N-edo. This also means they temper out the same commas. <br />
<br />
A procedure for finding the k*N subgroup is to take the product m of the odd primes less than or equal to p, and then find the <a class="wiki_link" href="/Euler%20genera">Euler genus</a> Euler(m^i) for integers 1, 2, and so forth. Here Euler(d) for an odd integer d is the set of all divisors of d reduced to an octave, and including 2. For each such genus, select those intervals such that the k*N patent val maps the interval to a number divisible by k, and then find the corresponding <a class="wiki_link" href="/Normal%20lists">normal interval list</a>. When two successive values i and i+1 lead to the same normal list, return that as the canonical list of generators for the k*N subgroup.</body></html>
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