Lesfip scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 291516939 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 291517247 - 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>2012-01-11 21: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-11 21:17:07 UTC</tt>.<br> | ||
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Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| < e, where Xs is an indeterminate corresponding to s. Let B be the sum ∑(Xs - Xt - c)^2 over all triples s∊S, t∊S, c∊C with s>t and |s-t-c| < e, where Xs and Xt are indeterminates corresponding to s and t. Now let Q = A+B. We define L(S, C, e) to be the set S' of values for the indeterminates minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the indeterminates, leading to n linear equations in n unknowns. If the system is underdetermined this might not lead to a unique solution; in this case L(S, C, e) is undefined. By adding more members to C, increasing the value of e, or both, it's possible to find a unique solution to a different, but related, problem. | Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| < e, where Xs is an indeterminate corresponding to s. Let B be the sum ∑(Xs - Xt - c)^2 over all triples s∊S, t∊S, c∊C with s>t and |s-t-c| < e, where Xs and Xt are indeterminates corresponding to s and t. Now let Q = A+B. We define L(S, C, e) to be the set S' of values for the indeterminates minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the indeterminates, leading to n linear equations in n unknowns. If the system is underdetermined this might not lead to a unique solution; in this case L(S, C, e) is undefined. By adding more members to C, increasing the value of e, or both, it's possible to find a unique solution to a different, but related, problem. | ||
A lesfip scale is a fixed point of L for a given | A lesfip scale is a fixed point of L for a given choice of C and some range a < e < b of values for e; that is, a set S such that L(S, C, e) = S, together with (in Scala format) 1200, representing the octave class. Lesfip are discrete points in the space of possible n-note octave repeating scales, surrounded by a basin of attraction. They can be found by iterating L, discarding the extra note when two notes converge to the same value, and stopping when a fixed point is reached. | ||
=Examples= | =Examples= | ||
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Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| &lt; e, where Xs is an indeterminate corresponding to s. Let B be the sum ∑(Xs - Xt - c)^2 over all triples s∊S, t∊S, c∊C with s&gt;t and |s-t-c| &lt; e, where Xs and Xt are indeterminates corresponding to s and t. Now let Q = A+B. We define L(S, C, e) to be the set S' of values for the indeterminates minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the indeterminates, leading to n linear equations in n unknowns. If the system is underdetermined this might not lead to a unique solution; in this case L(S, C, e) is undefined. By adding more members to C, increasing the value of e, or both, it's possible to find a unique solution to a different, but related, problem. <br /> | Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| &lt; e, where Xs is an indeterminate corresponding to s. Let B be the sum ∑(Xs - Xt - c)^2 over all triples s∊S, t∊S, c∊C with s&gt;t and |s-t-c| &lt; e, where Xs and Xt are indeterminates corresponding to s and t. Now let Q = A+B. We define L(S, C, e) to be the set S' of values for the indeterminates minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the indeterminates, leading to n linear equations in n unknowns. If the system is underdetermined this might not lead to a unique solution; in this case L(S, C, e) is undefined. By adding more members to C, increasing the value of e, or both, it's possible to find a unique solution to a different, but related, problem. <br /> | ||
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A lesfip scale is a fixed point of L for a given | A lesfip scale is a fixed point of L for a given choice of C and some range a &lt; e &lt; b of values for e; that is, a set S such that L(S, C, e) = S, together with (in Scala format) 1200, representing the octave class. Lesfip are discrete points in the space of possible n-note octave repeating scales, surrounded by a basin of attraction. They can be found by iterating L, discarding the extra note when two notes converge to the same value, and stopping when a fixed point is reached.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Examples"></a><!-- ws:end:WikiTextHeadingRule:4 -->Examples</h1> | ||
Examples of lesfip scales can be found at <a class="wiki_link" href="/Scalesmith">Scalesmith</a>.</body></html></pre></div> | Examples of lesfip scales can be found at <a class="wiki_link" href="/Scalesmith">Scalesmith</a>.</body></html></pre></div> | ||