Logarithmic approximants: Difference between revisions

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<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-size: 15px;"&gt;Introduction&lt;/span&gt;**

&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant//[[xenharmonic/Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]] (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 are certain commas small, and roughly how small are they?&lt;/span&gt;

* &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why is the ratio of the perfect fifth to the perfect fourth close to √2?&lt;/span&gt;

[[math]]

&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt; = (frequency difference) / (frequency sum)

&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt; =½ (frequency difference) / (mean frequency)

&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// &lt;/span&gt;can be retrieved from &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; using the inverse relation

**&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&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

which shows that &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v// ≈ //J//&lt;/span&gt; and provides an indication of the size and sign of the error involved in this approximation.

//&lt;span style="font-family: Georgia;"&gt;J&lt;/span&gt;// can be expressed in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; as

[[math]]

[[math]]

The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v(r)//&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J(r) =//½ ln //r// &lt;/span&gt; in the region of //r// = 1, which has the property of matching the function value and its first and second derivatives at this value of //r//. The bimodular approximant function is thus accurate to second order in //r// – 1.

As an example, the size of the perfect fifth (in dNp units) is

[[math]]

\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...

[[math]]

The bimodular approximant for this interval (r = 3/2) is

[[math]]

\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2

[[math]]

and the Taylor series indicates that the error in this value is about

[[math]]

The figure shows the approximants of the first 31 superparticular intervals, which are reciprocals of odd integers.

&lt;Figure&gt;

[[math]]

\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}

[[math]]

**&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;**

<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;strong&gt;&lt;span style="font-size: 15px;"&gt;Introduction&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;

&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1"&gt;{1}&lt;/a&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;

The exact size, in cents, of an interval with frequency ratio &lt;em&gt;r&lt;/em&gt; is&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:0:

&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:4 --&gt;&lt;br /&gt;

&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt; = (frequency difference) / (frequency sum)&lt;br /&gt;

&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt; =½ (frequency difference) / (mean frequency)&lt;br /&gt;

&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;can be retrieved from &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; using the inverse relation&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;br /&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;

which shows that &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt; ≈ &lt;em&gt;J&lt;/em&gt;&lt;/span&gt; and provides an indication of the size and sign of the error involved in this approximation.&lt;br /&gt;

&lt;em&gt;&lt;span style="font-family: Georgia;"&gt;J&lt;/span&gt;&lt;/em&gt; can be expressed in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; as&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:8:

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

The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v(r)&lt;/em&gt;&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J(r) =&lt;/em&gt;½ ln &lt;em&gt;r&lt;/em&gt; &lt;/span&gt; in the region of &lt;em&gt;r&lt;/em&gt; = 1, which has the property of matching the function value and its first and second derivatives at this value of &lt;em&gt;r&lt;/em&gt;. The bimodular approximant function is thus accurate to second order in &lt;em&gt;r&lt;/em&gt; – 1.&lt;br /&gt;

&lt;br /&gt;

As an example, the size of the perfect fifth (in dNp units) is&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:9:

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

The bimodular approximant for this interval (r = 3/2) is&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:10:

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

\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2&amp;lt;br/&amp;gt;[[math]]

  --&gt;&lt;script type="math/tex"&gt;\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:10 --&gt;&lt;br /&gt;

and the Taylor series indicates that the error in this value is about&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:11:

The figure shows the approximants of the first 31 superparticular intervals, which are reciprocals of odd integers.&lt;br /&gt;

&amp;lt;Figure&amp;gt;&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextMathRule:12:

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

\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}&amp;lt;br/&amp;gt;[[math]]

\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:12 --&gt;&lt;br /&gt;

&lt;br /&gt;

&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;