Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 347566980 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 347567292 - 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>2012-06-23 23: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-23 23:24:26 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>347567292</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">**Lp tuning** is a generalzation of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as | <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">**Lp tuning** is a generalzation of [[TOP tuning|TOP]] and [[Tenney-Euclidean tuning|TE]] tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as | ||
[[math]] | [[math]] | ||
|| |b_2 \ b_3 \ ... \ b_k> | || |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> | || |m_2 \ m_3 \ ... \ m_k> ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) b_k|^p)^{1/p} | ||
[[math]] | [[math]] | ||
If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. | If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. | ||
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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; | || |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> | --><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; | || |m_2 \ m_3 \ ... \ m_k&gt; ||_p = (|\log_2(2) m_2|^p + |\log_2(3)m_3|^p + ... + |\log_2(k) b_k|^p)^{1/p}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">|| |m_2 \ m_3 \ ... \ m_k> | --><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) b_k|^p)^{1/p}</script><!-- ws:end:WikiTextMathRule:1 --><br /> | ||
If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. <br /> | If q is any positive rational number, ||q||_p is the Lp norm defined by the monzo. <br /> | ||
<br /> | <br /> | ||
For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We derfine the error of the tuning on q, Err(q), as |T(q) - cents(q)}, and if q ≠ 1, the <em>Lp proportional error</em> as PEp(q) = Err(q)/||q||_p.</body></html></pre></div> | For some just intonation group G, which is to say some finitely generated group of positive rational numbers which can be either a full prime-limit group or some subgroup of such a group, a regular temperament tuning T is defined by a linear map from monzos belonging to G to a value in cents, such that T(c) = 0 for any comma c of the temperament. We derfine the error of the tuning on q, Err(q), as |T(q) - cents(q)}, and if q ≠ 1, the <em>Lp proportional error</em> as PEp(q) = Err(q)/||q||_p.</body></html></pre></div> | ||