Tp tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 350573252 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 350573386 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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>2012-07-06 00:28: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-07-06 00:28:53 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>350573386</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> | ||
| Line 25: | Line 25: | ||
=L2 tuning= | =L2 tuning= | ||
In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately | In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament+measures#TE error|TE error]]. Starting from a [[gencom]] for the temperament, we may find the tuning and the L2 error by the following proceedure: | ||
# Convert the gencom into monzo form. | # Convert the gencom into monzo form. | ||
| Line 58: | Line 58: | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc2"><a name="L2 tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->L2 tuning</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc2"><a name="L2 tuning"></a><!-- ws:end:WikiTextHeadingRule:6 -->L2 tuning</h1> | ||
In the special case where p = 2, the Lp norm 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 L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately | In the special case where p = 2, the Lp norm 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 L2 tuning, or TE tuning, extended to arbitrary JI groups, and the L2 error, which is E2(S) for the temperament S, and which is approximately proportional to [[Tenney-Euclidean temperament+measures#TE error|TE error]]. Starting from a <a class="wiki_link" href="/gencom">gencom</a> for the temperament, we may find the tuning and the L2 error by the following proceedure:<br /> | ||
<br /> | <br /> | ||
<ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.</li><li>Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.</li><li>Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> closest to the JIP using the distance function defined by Q. As usual the JIP is &lt;1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods.</li><li>Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</li><li>Given the tuning T, we may find the L2 error as PE2s(T) by constrained optimization of Err(T) under the constraint M∙M = 1.</li></ol></body></html></pre></div> | <ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.</li><li>Take the dot product of M with a vector &lt;x1 x2 ... xn| consisting of n new indeterminates x1, x2, ..., xn; call that X.</li><li>Maximize X^2 subject to the constraint that the M∙M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> closest to the JIP using the distance function defined by Q. As usual the JIP is &lt;1 1 1 ... 1|. Since Q is a quadratic polynomial, this minimal distance is easily found by calculus methods.</li><li>Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</li><li>Given the tuning T, we may find the L2 error as PE2s(T) by constrained optimization of Err(T) under the constraint M∙M = 1.</li></ol></body></html></pre></div> | ||