Lesfip scales: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-12 02:26:48 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-12 13:49:17 UTC</tt>.

: The original revision id was <tt>~~291555967~~</tt>.

: The original revision id was <tt>291735689</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 15:

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 a variable 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 variables 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 variables minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the variables, 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 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.

Line 31:

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.

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.

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 a variable 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 variables 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 variables minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the variables, 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 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.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-01-12 02:26:48 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-01-12 13:49:17 UTC</tt>.<br>
-: The original revision id was <tt>291555967</tt>.<br>
+: The original revision id was <tt>291735689</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>
@@ Line 15: / Line 15: @@
 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 [[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| &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.
+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 a variable 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 variables 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 variables minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the variables, 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 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.
@@ Line 31: / Line 31: @@
 Suppose S are the notes of a scale, expressed in terms of a set of notes 0 &amp;lt; s &amp;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 &amp;lt; c &amp;lt; 1200 for c∊C of consonance targets, for example the values in cents of all members of a q-limit &lt;a class="wiki_link" href="/tonality%20diamond"&gt;tonality diamond&lt;/a&gt;. 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.&lt;br /&gt;
 &lt;br /&gt;
-Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| &amp;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&amp;gt;t and |s-t-c| &amp;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. &lt;br /&gt;
+Now form two sums: let A be the sum ∑(Xs - c)^2 over all pairs s∊S, c∊C with |s-c| &amp;lt; e, where Xs is a variable corresponding to s. Let B be the sum ∑(Xs - Xt - c)^2 over all triples s∊S, t∊S, c∊C with s&amp;gt;t and |s-t-c| &amp;lt; e, where Xs and Xt are variables 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 variables minimizing Q, if a unique minimum exists. This can be found by differentiating Q with respect to each of the variables, 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. &lt;br /&gt;
 &lt;br /&gt;
 A lesfip scale is a fixed point of L for a given choice of C and some range a &amp;lt; e &amp;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.&lt;br /&gt;