Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 357975966 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 472472284 - 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> | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-11-26 13:21:08 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>472472284</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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**Tp tuning** is a generalzation of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. If p ≥ 1, define the Tp norm, which we may also call the Tp complexity, of any monzo in weighted coordinates b as | **Tp tuning** is a generalzation of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. If p ≥ 1, define the Tp norm, which we may also call the Tp complexity, of any monzo in weighted coordinates b as | ||
[[math]] | [[math]] | ||
|| |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p} | ||\ |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p} | ||
[[math]] | [[math]] | ||
where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m, | where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m, | ||
[[math]] | [[math]] | ||
|| |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p} | ||\ |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p} | ||
[[math]] | [[math]] | ||
If q is any positive rational number, ||q||_p is the Tp norm defined by the monzo. | If q is any positive rational number, ||q||_p is the Tp norm defined by the monzo. | ||
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=T2 tuning= | =T2 tuning= | ||
In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the | In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which when divided by the square root of the number n of primes in the prime limit, is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the RMS error, which for a tuning map T is ||(T - J)/n ||_2 = ||T - J||. | ||
For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives | For an example, consider [[Chromatic pairs#Indium|indium temperament]], with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the [[Tenney-Euclidean tuning|usual methods]], in particular the pseudoinverse, we find that the T2 tuning map is <1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting <1200 1200 1200 1200 1200| gives <-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</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>Tp tuning</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Dual norm">Dual norm</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Applying the Hahn-Banach theorem">Applying the Hahn-Banach theorem</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#T2 tuning">T2 tuning</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | <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>Tp tuning</title></head><body><!-- ws:start:WikiTextTocRule:10:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:10 --><!-- ws:start:WikiTextTocRule:11: --><a href="#Definition">Definition</a><!-- ws:end:WikiTextTocRule:11 --><!-- ws:start:WikiTextTocRule:12: --> | <a href="#Dual norm">Dual norm</a><!-- ws:end:WikiTextTocRule:12 --><!-- ws:start:WikiTextTocRule:13: --> | <a href="#Applying the Hahn-Banach theorem">Applying the Hahn-Banach theorem</a><!-- ws:end:WikiTextTocRule:13 --><!-- ws:start:WikiTextTocRule:14: --> | <a href="#T2 tuning">T2 tuning</a><!-- ws:end:WikiTextTocRule:14 --><!-- ws:start:WikiTextTocRule:15: --> | ||
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<!-- ws:start:WikiTextMathRule:0: | <!-- ws:start:WikiTextMathRule:0: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
|| |b_2 \ b_3 \ ... \ b_k&gt; ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}&lt;br/&gt;[[math]] | ||\ |b_2 \ b_3 \ ... \ b_k&gt; ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">|| |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">||\ |b_2 \ b_3 \ ... \ b_k> ||_p = (|b_2|^p + |b_3|^p + ... + |b_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m,<br /> | where 2, 3, ... k are the primes up to k in order. In unweighted coordinates, this would be, for unweighted monzo m,<br /> | ||
<!-- ws:start:WikiTextMathRule:1: | <!-- ws:start:WikiTextMathRule:1: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
|| |m_2 \ m_3 \ ... \ m_k&gt; ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}&lt;br/&gt;[[math]] | ||\ |m_2 \ m_3 \ ... \ m_k&gt; ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">|| |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | --><script type="math/tex">||\ |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) m_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
If q is any positive rational number, ||q||_p is the Tp norm defined by the monzo.<br /> | If q is any positive rational number, ||q||_p is the Tp norm defined by the monzo.<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="T2 tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->T2 tuning</h1> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc3"><a name="T2 tuning"></a><!-- ws:end:WikiTextHeadingRule:8 -->T2 tuning</h1> | ||
In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the | In the special case where p = 2, the Tp norm for the full prime limit becomes the L2 norm, which when divided by the square root of the number n of primes in the prime limit, is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is T2 tuning extended to arbitrary JI groups, and the RMS error, which for a tuning map T is ||(T - J)/n ||_2 = ||T - J||.<br /> | ||
<br /> | <br /> | ||
For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the T2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, gives | For an example, consider <a class="wiki_link" href="/Chromatic%20pairs#Indium">indium temperament</a>, with group 2.5/3.7/3.11/3 and comma basis 3025/3024 and 3125/3087. The corresponding full 11-limit temperament is of rank three, and using the <a class="wiki_link" href="/Tenney-Euclidean%20tuning">usual methods</a>, in particular the pseudoinverse, we find that the T2 tuning map is &lt;1199.552 1901.846 2783.579 3371.401 4153.996|. Applying that to 12/11 gives a generator of 146.995, and multiplying that by 1200.0/1199.552 gives a POT2 tuning, or extended POTE tuning, of 147.010. Converting the tuning map to weighted coordinates and subtracting &lt;1200 1200 1200 1200 1200| gives &lt;-0.4475 -.0685 -1.1778 0.9172 0.7741|. The ordinary Euclidean norm of this, ie the square root of the dot product, is 1.7414, and dividing by sqrt(5) gives the RMS error, 0.77879 cents.</body></html></pre></div> | ||