Logarithmic approximants: Difference between revisions
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-19 08:27: | : This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-19 08:27:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>541426660</tt>.<br> | ||
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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.<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"> | <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 | ||
WORK IN PROGRESS | |||
**<span style="font-size: 15px;">Introduction</span>** | **<span style="font-size: 15px;">Introduction</span>** | ||
<span style="font-family: Arial,Helvetica,sans-serif;">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</span> | <span style="font-family: Arial,Helvetica,sans-serif;">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</span> | ||
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The exact size, in cents, of an interval with frequency ratio //r// is | The exact size, in cents, of an interval with frequency ratio //r// is | ||
[[math]] | [[math]] | ||
\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2} | \qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2} | ||
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where for just intervals r is rational and can be written as the ratio of two integers: | where for just intervals r is rational and can be written as the ratio of two integers: | ||
[[math]] | [[math]] | ||
\qquad r = n/d | \qquad r = n/d | ||
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[[math]] | [[math]] | ||
\qquad v = \frac{n-d}{n+d} \\ | \qquad v = \frac{n-d}{n+d} \\ | ||
[[math]] | [[math]] | ||
<span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum) | <span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum) | ||
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[[math]] | [[math]] | ||
\qquad J = \tanh^(-1){v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ... | \qquad J = \tanh^(-1){v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ... | ||
[[math]] | [[math]] | ||
The function <span style="font-family: Georgia,serif; font-size: 110%;">//v(r)//</span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> in the region of // | The function <span style="font-family: Georgia,serif; font-size: 110%;">//v(r)//</span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> 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. | ||
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[[math]] | [[math]] | ||
\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273... | \qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273... | ||
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[[math]] | [[math]] | ||
\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2 | \qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2 | ||
[[math]] | [[math]] | ||
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and the Taylor series indicates that the error in this value is about | and the Taylor series indicates that the error in this value is about | ||
[[math]] | [[math]] | ||
\qquad -\tfrac{1}{3}v^3 = -0.00267... | \qquad -\tfrac{1}{3}v^3 = -0.00267... | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
\qquad v[-J] = -v[J] | \qquad v[-J] = -v[J] | ||
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2} | \qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2} | ||
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**<span style="font-size: 20px;">Bimodular approximants and equal temperaments</span>** | **<span style="font-size: 20px;">Bimodular approximants and equal temperaments</span>** | ||
While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example: | While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example: | ||
Two perfect fourths (// | Two perfect fourths (//r// = 4/3, = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7) | ||
Three major thirds (// | Three major thirds (//r// = 5/4, = 1/9) or two __7/5__s ( = 1/6) or five __8/7__s ( = 1/15) approximate an octave (//r// = 2/1, = 1/3) | ||
Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments. | Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments. | ||
Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval. | Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval. | ||
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Further examples of bimodular commas are provided in Reference 1. See also __Don Page comma__ (another name for this type of comma).</pre></div> | Further examples of bimodular commas are provided in Reference 1. See also __Don Page comma__ (another name for this type of comma).</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>Logarithmic approximants</title></head><body> | <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<br /> | ||
WORK IN PROGRESS<br /> | |||
<strong><span style="font-size: 15px;">Introduction</span></strong><br /> | <strong><span style="font-size: 15px;">Introduction</span></strong><br /> | ||
<span style="font-family: Arial,Helvetica,sans-serif;">The term <em>logarithmic approximant</em><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a> (or <em>approximant</em> for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as</span><br /> | <span style="font-family: Arial,Helvetica,sans-serif;">The term <em>logarithmic approximant</em><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a> (or <em>approximant</em> for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as</span><br /> | ||
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[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
\qquad v = \frac{n-d}{n+d} \\&lt;br /&gt; | \qquad v = \frac{n-d}{n+d} \\&lt;br /&gt; | ||
&lt;br/&gt;[[math]] | |||
--><script type="math/tex">\qquad v = \frac{n-d}{n+d} \\ | --><script type="math/tex">\qquad v = \frac{n-d}{n+d} \\ | ||
</script><!-- ws:end:WikiTextMathRule:4 --><br /> | |||
<span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum)<br /> | <span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum)<br /> | ||
<br /> | <br /> | ||
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--><script type="math/tex">\qquad J = \tanh^(-1){v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:8 --><br /> | --><script type="math/tex">\qquad J = \tanh^(-1){v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:8 --><br /> | ||
<br /> | <br /> | ||
The function <span style="font-family: Georgia,serif; font-size: 110%;"><em>v(r)</em></span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> <em>J(r) =</em>½ ln <em>r</em> </span> in the region of <em | The function <span style="font-family: Georgia,serif; font-size: 110%;"><em>v(r)</em></span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> <em>J(r) =</em>½ ln <em>r</em> </span> in the region of <em>r</em> = 1, which has the property of matching the function value and its first and second derivatives at this value of <em>r</em>. The bimodular approximant function is thus accurate to second order in <em>r</em> – 1.<br /> | ||
<br /> | <br /> | ||
<br /> | <br /> | ||
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<strong><span style="font-size: 20px;">Bimodular approximants and equal temperaments</span></strong><br /> | <strong><span style="font-size: 20px;">Bimodular approximants and equal temperaments</span></strong><br /> | ||
While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:<br /> | While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:<br /> | ||
Two perfect fourths (<em | Two perfect fourths (<em>r</em> = 4/3, = 1/7) approximate a minor seventh (<em>r</em> = 9/5, = 2/7)<br /> | ||
Three major thirds (<em | Three major thirds (<em>r</em> = 5/4, = 1/9) or two <u>7/5</u>s ( = 1/6) or five <u>8/7</u>s ( = 1/15) approximate an octave (<em>r</em> = 2/1, = 1/3)<br /> | ||
Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.<br /> | Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.<br /> | ||
Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.<br /> | Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.<br /> | ||