Lesfip scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 291517247 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 291555967 - 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- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-12 02:26:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>291555967</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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=Definition= | =Definition= | ||
Suppose S are the notes of a scale, expressed in terms of a set of notes 0 < s < 1200, for s∊S. This is the set of pitch classes of the scale, excepting the unison/octave class, expressed in terms of cents of a periodic scale. Let C likewise be a set 0 < c < 1200 for c∊C of consonance targets, for example the values in cents of all members of a q-limit [[tonality diamond]]. Let e be a positive real number denoting the maximum allowed error in an approximation of an interval of the scale to an element of the set of target consonances, expressed in cents. | Suppose S are the notes of a scale, expressed in terms of a set of notes 0 < s < 1200, for s∊S. This is the set of pitch classes of the scale, excepting the unison/octave class, expressed in terms of cents, of a periodic scale. Let C likewise be a set 0 < c < 1200 for c∊C of consonance targets, for example the values in cents of all members of a q-limit [[tonality diamond]]. Let e be a positive real number denoting the maximum allowed error in an approximation of an interval of the scale to an element of the set of target consonances, expressed in cents. | ||
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. | ||
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Suppose S are the notes of a scale, expressed in terms of a set of notes 0 &lt; s &lt; 1200, for s∊S. This is the set of pitch classes of the scale, excepting the unison/octave class, expressed in terms of cents of a periodic scale. Let C likewise be a set 0 &lt; c &lt; 1200 for c∊C of consonance targets, for example the values in cents of all members of a q-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>. Let e be a positive real number denoting the maximum allowed error in an approximation of an interval of the scale to an element of the set of target consonances, expressed in cents.<br /> | Suppose S are the notes of a scale, expressed in terms of a set of notes 0 &lt; s &lt; 1200, for s∊S. This is the set of pitch classes of the scale, excepting the unison/octave class, expressed in terms of cents, of a periodic scale. Let C likewise be a set 0 &lt; c &lt; 1200 for c∊C of consonance targets, for example the values in cents of all members of a q-limit <a class="wiki_link" href="/tonality%20diamond">tonality diamond</a>. Let e be a positive real number denoting the maximum allowed error in an approximation of an interval of the scale to an element of the set of target consonances, expressed in cents.<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| &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 /> | ||