Logarithmic approximants: Difference between revisions

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<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:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:05:26 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:21:56 UTC</tt>.

: The original revision id was <tt>~~541616090~~</tt>.

: The original revision id was <tt>541616972</tt>.

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

<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">WORK IN PROGRESS

=Introduction=

=Introduction=

The term //logarithmic approximant// (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

* Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?

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=**Bimodular approximants**=

==Definition==

==Definition==

The bimodular approximant of an interval with frequency ratio //r = n/d// is

[[math]]

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[[math]]

==Properties==

==Properties==

When //r// is small, //v// provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio //r// is

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Relationships of this sort can be identified in all equal temperaments.

==Bimodular commas==

==Bimodular commas==

As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals //J//1 and //J//2 (with //J//1 < //J//2) and their approximants //v//1 and //v//2, we define the //bimodular residue// as

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=**Padé approximants of order (1,2)**=

==Definition==

==Definition==

In the section on bimodular approximants it was shown than an interval of logarithmic size //J// (measured in dineper units) is related to its bimodular approximant by

[[math]]

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Another way to express this relationship is with a continued fraction:

[[math]]

\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))

\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))

[[math]]

The first convergent of this continued fraction is //v//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

[[math]]

~~~~\qquad y = \frac{3v}{3-v^2~~~~}

\qquad y = \frac{3v}{3-v^2}

[[math]]

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

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=**Quadratic approximants**=

==Definition==

==Definition==

The quadratic approximant //q// of an interval //J// with frequency ratio //r// = //n/////d// is

[[math]]

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The presence of a square root in the denominator of //q// (except where //J// is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

==Properties==

==Properties==

If //v//[//J//] and //q//[//J//] denote, respectively, the bimodular and quadratic approximants of an interval //J// with frequency ratio //r//, and //q//n denotes //q//[//J//n] , then

[[math]]

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<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>Logarithmic approximants</title></head><body>WORK IN PROGRESS

<h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

<h1 id="toc0"><a name="Introduction"></a>Introduction</h1>

The term logarithmic approximant (or approximant for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:

<ul><li>Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?</li><li>Why are certain commas small, and roughly how small are they?</li><li>Why does the 3-limit framework produce aesthetically pleasing scale structures?</li></ul>

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<ul><li>Bimodular approximants (first order rational approximants)</li><li>Padé approximants of order (1,2) (second order rational approximants)</li><li>Quadratic approximants</li></ul>

<h1 id="toc1"><a name="Bimodular approximants"></a>Bimodular approximants</h1>

<h2 id="toc2"><a name="Bimodular approximants-Definition"></a>Definition</h2>

<h2 id="toc2"><a name="Bimodular approximants-Definition"></a>Definition</h2>

The bimodular approximant of an interval with frequency ratio r = n/d is

<!-- ws:start:WikiTextMathRule:3:

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--><script type="math/tex">\qquad r = \frac{1+v}{1-v}</script>

<h2 id="toc3"><a name="Bimodular approximants-Properties"></a>Properties</h2>

<h2 id="toc3"><a name="Bimodular approximants-Properties"></a>Properties</h2>

When r is small, v provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.

Noting that the exact size (in dineper units) of the interval with frequency ratio r is

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Relationships of this sort can be identified in all equal temperaments.

<h2 id="toc5"><a name="Bimodular approximants-Bimodular commas"></a>Bimodular commas</h2>

<h2 id="toc5"><a name="Bimodular approximants-Bimodular commas"></a>Bimodular commas</h2>

As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.

Given two intervals J1 and J2 (with J1 &lt; J2) and their approximants v1 and v2, we define the bimodular residue as

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<h1 id="toc7"><a name="Padé approximants of order (1,2)"></a>Padé approximants of order (1,2)</h1>

<h2 id="toc8"><a name="Padé approximants of order (1,2)-Definition"></a>Definition</h2>

<h2 id="toc8"><a name="Padé approximants of order (1,2)-Definition"></a>Definition</h2>

In the section on bimodular approximants it was shown than an interval of logarithmic size J (measured in dineper units) is related to its bimodular approximant by

<!-- ws:start:WikiTextMathRule:22:

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<!-- ws:start:WikiTextMathRule:24:

[[math]]&lt;br/&gt;

\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;br/&gt;[[math]]

\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))</script>

--><script type="math/tex">\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))</script>

The first convergent of this continued fraction is v, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

<!-- ws:start:WikiTextMathRule:25:

[[math]]&lt;br/&gt;

~~&lt;span style=&quot;font-family: Georgia,serif; font-size: 110%;&quot;&gt;~~\qquad y = \frac{3v}{3-v^2~~&lt;/span&gt;~~}&lt;br/&gt;[[math]]

\qquad y = \frac{3v}{3-v^2}&lt;br/&gt;[[math]]

--><script type="math/tex~~">\qquad y = \frac{3v}{3-v^2~~}</script>

--><script type="math/tex">\qquad y = \frac{3v}{3-v^2}</script>

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

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<h1 id="toc9"><a name="Quadratic approximants"></a>Quadratic approximants</h1>

<h2 id="toc10"><a name="Quadratic approximants-Definition"></a>Definition</h2>

<h2 id="toc10"><a name="Quadratic approximants-Definition"></a>Definition</h2>

The quadratic approximant q of an interval J with frequency ratio r = n/d is

<!-- ws:start:WikiTextMathRule:26:

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The presence of a square root in the denominator of q (except where J is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.

<h2 id="toc11"><a name="Quadratic approximants-Properties"></a>Properties</h2>

<h2 id="toc11"><a name="Quadratic approximants-Properties"></a>Properties</h2>

If v[J] and q[J] denote, respectively, the bimodular and quadratic approximants of an interval J with frequency ratio r, and qn denotes q[Jn] , then

<!-- ws:start:WikiTextMathRule:31:

@@ 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:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:05:26 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:21:56 UTC</tt>.<br>
-: The original revision id was <tt>541616090</tt>.<br>
+: The original revision id was <tt>541616972</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">WORK IN PROGRESS
-=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
+=&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant// (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;
@@ Line 41: / Line 41: @@
 =**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**=
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 The bimodular approximant of an interval with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;// is
 [[math]]
@@ Line 59: / Line 59: @@
 [[math]]
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
 When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r//&lt;/span&gt; is
@@ Line 114: / Line 114: @@
 Relationships of this sort can be identified in all equal temperaments.
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;==
 As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the //bimodular residue// as
@@ Line 158: / Line 158: @@
 =**&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;**=
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 In the section on bimodular approximants it was shown than an interval of logarithmic size //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// (measured in dineper units) is related to its bimodular approximant by
 [[math]]
@@ Line 170: / Line 170: @@
 Another way to express this relationship is with a continued fraction:
 [[math]]
-\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))
+\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))
 [[math]]
 The first convergent of this continued fraction is //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
 [[math]]
-&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;\qquad y = \frac{3v}{3-v^2&lt;/span&gt;}
+\qquad y = \frac{3v}{3-v^2}
 [[math]]
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.
@@ Line 197: / Line 197: @@
 =**&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;**=
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = //n/////d//&lt;/span&gt; is
 [[math]]
@@ Line 245: / Line 245: @@
 The presence of a square root in the denominator of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// (except where //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
 If //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[//J//]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//n]&lt;/span&gt; , then
 [[math]]
@@ Line 380: / Line 380: @@
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;WORK IN PROGRESS&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
   &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
@@ Line 414: / Line 414: @@
 &lt;ul&gt;&lt;li&gt;Bimodular approximants (first order rational approximants)&lt;/li&gt;&lt;li&gt;Padé approximants of order (1,2) (second order rational approximants)&lt;/li&gt;&lt;li&gt;Quadratic approximants&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The bimodular approximant of an interval with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;&lt;/em&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
@@ Line 437: / Line 437: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad r = \frac{1+v}{1-v}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:5 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
   When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.&lt;br /&gt;
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
@@ Line 500: / Line 500: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
   As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.&lt;br /&gt;
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &amp;lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the &lt;em&gt;bimodular residue&lt;/em&gt; as&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   In the section on bimodular approximants it was shown than an interval of logarithmic size &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; (measured in dineper units) is related to its bimodular approximant by&lt;br /&gt;
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 [[math]]&amp;lt;br/&amp;gt;
-\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&amp;lt;br/&amp;gt;[[math]]
+\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:24 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad J = \tanh^{-1}{v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:24 --&gt;&lt;br /&gt;
 The first convergent of this continued fraction is &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is&lt;br /&gt;
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 [[math]]&amp;lt;br/&amp;gt;
-&amp;lt;span style=&amp;quot;font-family: Georgia,serif; font-size: 110%;&amp;quot;&amp;gt;\qquad y = \frac{3v}{3-v^2&amp;lt;/span&amp;gt;}&amp;lt;br/&amp;gt;[[math]]
+\qquad y = \frac{3v}{3-v^2}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;\qquad y = \frac{3v}{3-v^2&lt;/span&gt;}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:25 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad y = \frac{3v}{3-v^2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:25 --&gt;&lt;br /&gt;
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.&lt;br /&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = &lt;em&gt;n&lt;/em&gt;&lt;em&gt;/d&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
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 The presence of a square root in the denominator of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; (except where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
   If &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;, and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;n]&lt;/span&gt; , then&lt;br /&gt;
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