Subgroup monzos and vals: Difference between revisions
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Wikispaces>genewardsmith **Imported revision 201083430 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 201083594 - 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>2011-02-11 22: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-02-11 22:05:16 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>201083594</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> | ||
<h4>Original Wikitext content:</h4> | <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">Given a [[just intonation subgroup]], we can find a canonical form for its generators by means of the [[Normal lists|normal interval list]] which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short. | <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">Given a [[Just intonation subgroups|just intonation subgroup]], we can find a canonical form for its generators by means of the [[Normal lists|normal interval list]] which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short. | ||
For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the [[Tenney-Euclidean Tuning|pseudoinverse]] of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2>, and multiplying this by M` gives the smonzo |2 -3 2>. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.</pre></div> | For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the [[Tenney-Euclidean Tuning|pseudoinverse]] of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2>, and multiplying this by M` gives the smonzo |2 -3 2>. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.</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"><html><head><title>Smonzos and Svals</title></head><body>Given a <a class="wiki_link" href="/ | <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"><html><head><title>Smonzos and Svals</title></head><body>Given a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, we can find a canonical form for its generators by means of the <a class="wiki_link" href="/Normal%20lists">normal interval list</a> which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define <a class="wiki_link" href="/Vals%20and%20tuning%20space">vals</a> and Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short.<br /> | ||
<br /> | <br /> | ||
For example, consider the subgroup generated by the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a>, 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">pseudoinverse</a> of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2&gt;, and multiplying this by M` gives the smonzo |2 -3 2&gt;. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.</body></html></pre></div> | For example, consider the subgroup generated by the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a>, 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">pseudoinverse</a> of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2&gt;, and multiplying this by M` gives the smonzo |2 -3 2&gt;. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.</body></html></pre></div> | ||
Revision as of 22:05, 11 February 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 2011-02-11 22:05:16 UTC.
- The original revision id was 201083594.
- 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:
Given a [[Just intonation subgroups|just intonation subgroup]], we can find a canonical form for its generators by means of the [[Normal lists|normal interval list]] which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short. For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the [[Tenney-Euclidean Tuning|pseudoinverse]] of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2>, and multiplying this by M` gives the smonzo |2 -3 2>. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.
Original HTML content:
<html><head><title>Smonzos and Svals</title></head><body>Given a <a class="wiki_link" href="/Just%20intonation%20subgroups">just intonation subgroup</a>, we can find a canonical form for its generators by means of the <a class="wiki_link" href="/Normal%20lists">normal interval list</a> which may be computed from any finite set of generators. In the case of the full p-limit group for any prime p, this consists of the primes from 2 to p in ascending order. This is precisely the ordered list used to define <a class="wiki_link" href="/Vals%20and%20tuning%20space">vals</a> and Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to p. This generalization we may call the subgroup monzos and subgroup vals, or smonzos and svals for short.<br /> <br /> For example, consider the subgroup generated by the <a class="wiki_link" href="/The%20Archipelago">barbados triad</a>, 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning">pseudoinverse</a> of M, M`. If u is a monzo for an interval in the subgroup, then uM` gives the corresponding smonzo. For instance, the monzo for 676/675 is |2 -3 2 0 0 2>, and multiplying this by M` gives the smonzo |2 -3 2>. We may check this is the correct smonzo from 2^2 3^(-3) (13/5)^2, which is 676/675 as desired.</body></html>