Lesfip scales - Revision history

Fredg999: Categories, markup (header levels and more)

2021-08-14T04:04:54Z

Categories, markup (header levels and more)

← Older revision		Revision as of 04:04, 14 August 2021
Line 1:		Line 1:
	=Introduction=		== Introduction ==
	A ''lesfip scale'' (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [~~http~~:~~//en.wikipedia.org/wiki/Least_squares~~ least squares algorithm] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		A '''lesfip scale''' (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[Wikipedia:Least squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[~~Periodic_scale~~\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[~~Circulating_Temperaments\|circulating temperaments~~]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[~~val-werck\|~~val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.		A lesfip scale may be found by choosing a [[Periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[well temperament]]s Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.

	=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\|~~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 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.		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.
Line 11:		Line 11:
	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.		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 ==
	Examples of lesfip scales can be found at [[Scalesmith\|Scalesmith]].		Examples of lesfip scales can be found at [[Scalesmith\|Scalesmith]].
	~~[[Category:lesfip]]~~
	[[Category:scales]]		[[Category:Lesfip scales\| ]] <!-- main article -->

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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 13:49:17 UTC</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>~~
	~~<h4>Original Wikitext content:</h4>~~
	<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">
	=Introduction=		=Introduction=
	A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		A ''lesfip scale'' (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [http://en.wikipedia.org/wiki/Least_squares least squares algorithm] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[~~periodic scale~~\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.		A lesfip scale may be found by choosing a [[Periodic_scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[Circulating_Temperaments\|circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck\|val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.

	=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\|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 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.		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.
Line 20:		Line 12:

	=Examples=		=Examples=
	Examples of lesfip scales can be found at [[Scalesmith]].~~</pre></div>~~		Examples of lesfip scales can be found at [[Scalesmith\|Scalesmith]].
	~~<h4>Original HTML content:</h4>~~		[[Category:lesfip]]
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body><br />		[[Category:scales]]
	~~<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1>~~
	~~A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http~~://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
	~~<br />~~
	A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a> Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, <a class="wiki_link" href="/val-werck">val-werck</a>. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.<br />
	~~<br />~~
	~~<!-- ws~~:~~start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1>~~
	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 />
	~~<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 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. <br />
	~~<br />~~
	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 />~~
	~~<br />~~
	~~<!-- 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>~~

Wikispaces>genewardsmith: Imported revision 291735689 - Original comment:

2012-01-12T13:49:17Z

**Imported revision 291735689 - Original comment: **

Wikispaces>genewardsmith: Imported revision 291555967 - Original comment:

2012-01-12T02:26:48Z

**Imported revision 291555967 - Original comment: **

← Older revision		Revision as of 02:26, 12 January 2012
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>2012-01-~~11 21~~:17:07 UTC</tt>.<br>		: 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>~~291517247~~</tt>.<br>		: 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>
Line 13:		Line 13:

	=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.
Line 29:		Line 29:
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1>		<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1>
	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 />
	<br />		<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 />		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 />

Wikispaces>genewardsmith: Imported revision 291517247 - Original comment:

2012-01-11T21:17:07Z

**Imported revision 291517247 - Original comment: **

← Older revision		Revision as of 21:17, 11 January 2012
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>2012-01-11 21:15:31 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-01-11 21:17:07 UTC</tt>.<br>
	: The original revision id was <tt>~~291516939~~</tt>.<br>		: The original revision id was <tt>291517247</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 17:		Line 17:
	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 ~~vhoice~~ 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.		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=
Line 33:		Line 33:
	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 />
	<br />		<br />
	A lesfip scale is a fixed point of L for a given ~~vhoice~~ 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 />		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 />
	<br />		<br />
	<!-- 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>

Wikispaces>genewardsmith: Imported revision 291516939 - Original comment:

2012-01-11T21:15:31Z

**Imported revision 291516939 - Original comment: **

← Older revision		Revision as of 21:15, 11 January 2012
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>~~2011~~-01-~~10 17~~:15:13 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-01-11 21:15:31 UTC</tt>.<br>
	: The original revision id was <tt>~~192279720~~</tt>.<br>		: The original revision id was <tt>291516939</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>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		<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">
			=Introduction=
			A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.		A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.

			=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.

			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 vhoice 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 of lesfip scales can be found at [[Scalesmith]].</pre></div>		Examples of lesfip scales can be found at [[Scalesmith]].</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />		<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body><br />
			<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h1>
			A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
	<br />		<br />
	A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a> Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, <a class="wiki_link" href="/val-werck">val-werck</a>. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.<br />		A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a> Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, <a class="wiki_link" href="/val-werck">val-werck</a>. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.<br />
	<br />		<br />
			<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:2 -->Definition</h1>
			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 />
			<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 />
			<br />
			A lesfip scale is a fixed point of L for a given vhoice 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 />
			<br />
			<!-- 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>

Wikispaces>genewardsmith: Imported revision 192279720 - Original comment:

2011-01-10T17:15:13Z

**Imported revision 192279720 - Original comment: **

Wikispaces>genewardsmith: Imported revision 192272166 - Original comment:

2011-01-10T16:44:48Z

**Imported revision 192272166 - Original comment: **

← Older revision		Revision as of 16:44, 10 January 2011
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>2011-01-10 16:37:30 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-01-10 16:44:48 UTC</tt>.<br>
	: The original revision id was <tt>~~192270434~~</tt>.<br>		: The original revision id was <tt>192272166</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 8:		Line 8:
	<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</pre></div>		A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.

			Examples of lesfip scales can be found at [[Scalesmith]].</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />		<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
	<br />		<br />
	A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</body></html></pre></div>		A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.<br />
			<br />
			Examples of lesfip scales can be found at <a class="wiki_link" href="/Scalesmith">Scalesmith</a>.</body></html></pre></div>

Wikispaces>genewardsmith: Imported revision 192270434 - Original comment:

2011-01-10T16:37:30Z

**Imported revision 192270434 - Original comment: **

Wikispaces>genewardsmith: Imported revision 192264102 - Original comment:

2011-01-10T16:14:15Z

**Imported revision 192264102 - Original comment: **

← Older revision		Revision as of 13:49, 12 January 2012
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>2012-01-12 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 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 15:		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.		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.		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:		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.<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 />
	<br />		<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 />		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. <br />
	<br />		<br />
	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 />		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 />

← Older revision		Revision as of 17:15, 10 January 2011
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>2011-01-10 16:44:48 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-01-10 17:15:13 UTC</tt>.<br>
	: The original revision id was <tt>~~192272166~~</tt>.<br>		: The original revision id was <tt>192279720</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 8:		Line 8:
	<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.		A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous [[circulating temperaments]] Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, [[val-werck]]. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.

	Examples of lesfip scales can be found at [[Scalesmith]].</pre></div>		Examples of lesfip scales can be found at [[Scalesmith]].</pre></div>
Line 14:		Line 14:
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />		<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
	<br />		<br />
	A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.<br />		A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration. Two distinct scales can easily converge on one lesfip scale; an example is provided by the famous <a class="wiki_link" href="/circulating%20temperaments">circulating temperaments</a> Vallotti-Young and Werckmeister III, which converge onto the same temperament extraordinaire, <a class="wiki_link" href="/val-werck">val-werck</a>. It can also happen that during the iteration two notes merge into one, decreasing the size of the scale.<br />
	<br />		<br />
	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>

← Older revision		Revision as of 16:37, 10 January 2011
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>2011-01-10 16:14:15 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-01-10 16:37:30 UTC</tt>.<br>
	: The original revision id was <tt>~~192264102~~</tt>.<br>		: The original revision id was <tt>192270434</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 8:		Line 8:
	<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.		<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

	A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, an an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</pre></div>		A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />		<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
	<br />		<br />
	A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, an an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</body></html></pre></div>		A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, and an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</body></html></pre></div>

← Older revision		Revision as of 16:14, 10 January 2011
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>2011-01-10 14:~~52:07~~ UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2011-01-10 16:14:15 UTC</tt>.<br>
	: The original revision id was <tt>~~192238344~~</tt>.<br>		: The original revision id was <tt>192264102</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>
	<h4>Original Wikitext content:</h4>		<h4>Original Wikitext content:</h4>
	<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the least squares algorithm minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.</pre></div>		<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">A //lesfip scale// (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the [[http://en.wikipedia.org/wiki/Least_squares\|least squares algorithm]] minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.

			A lesfip scale may be found by choosing a [[periodic scale\|scale]], a target set of intervals, an an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML content:</h4>
	<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the least squares algorithm minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.</body></html></pre></div>		<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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Lesfip scales</title></head><body>A <em>lesfip scale</em> (LEast Squares FIxed Point) is a scale which for some range of values for error tolerance, is a fixed point for the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Least_squares" rel="nofollow">least squares algorithm</a> minimizing the sum of squared error relative to some target list of intervals. That is, when we minimize the sum of the squares of all of the errors less than some fixed value such as seven cents, we obtain the same scale back again.<br />
			<br />
			A lesfip scale may be found by choosing a <a class="wiki_link" href="/periodic%20scale">scale</a>, a target set of intervals, an an error tolerance. The least squares optimization is iterated until a fixed point is arrived at. Occasionally there may be a problem when the system becomes underdetermined, which can be fixed by increasing the error tolerance by some amount for one iteration.</body></html></pre></div>

Lesfip scales - Revision history

Fredg999: Categories, markup (header levels and more)

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>genewardsmith: **Imported revision 291735689 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 291555967 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 291517247 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 291516939 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 192279720 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 192272166 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 192270434 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 192264102 - Original comment: **