Logarithmic approximants
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WORK IN PROGRESS
=Introduction=
<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;">Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why are certain commas small, and roughly how small are they?</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why is the ratio of the perfect fifth to the perfect fourth close to √2?</span>
The exact size, in cents, of an interval with frequency ratio //r// is
[[math]]
\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}
[[math]]
where for just intervals r is rational and can be written as the ratio of two integers:
[[math]]
\qquad r = n/d
[[math]]
When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as
[[math]]
\qquad J = \tfrac{1}{2} \ln{r}
[[math]]
This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e<span style="vertical-align: super;">2</span> = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.
Comparing the two units of measurement we find
1 dineper = 2400/ln(2) = 3462.468 cents
which is about 1.4 semitones short of three octaves.
The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth.
Three types of approximants are described here:
* Bimodular approximants (first order rational approximants)
* Padé approximants of order (1,2) (second order rational approximants)
* Quadratic approximants
=**<span style="font-size: 20px;">Bimodular approximants</span>**=
==**<span style="font-size: 15px;">Definition</span>**==
The bimodular approximant of an interval with frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r = n/d</span>// is
[[math]]
\qquad v = \frac{r-1}{r+1}
[[math]]
//<span style="font-family: Georgia,serif; font-size: 110%;">v </span>//can thus be expressed as
[[math]]
\qquad v = \frac{n-d}{n+d} \\
[[math]]
<span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum)
<span style="color: #ffffff;">######</span> =½ (frequency difference) / (mean frequency)
<span style="font-family: Georgia,serif; font-size: 110%;">//r// </span>can be retrieved from <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> using the inverse relation
[[math]]
\qquad r = \frac{1+v}{1-v}
[[math]]
==**<span style="font-size: 15px;">Properties of bimodular approximants</span>**==
When <span style="font-family: Georgia,serif; font-size: 110%;">//r// </span>is small, <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> 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 <span style="font-family: Georgia,serif; font-size: 110%;">//r//</span> is
[[math]]
\qquad J = \tfrac{1}{2} \ln{r}
[[math]]
the relationship between <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span> can be expressed as
[[math]]
\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...
[[math]]
which shows that <span style="font-family: Georgia,serif; font-size: 110%;">//v// ≈ //J//</span> and provides an indication of the size and sign of the error involved in this approximation.
//<span style="font-family: Georgia;">J</span>// can be expressed in terms of <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> as
[[math]]
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...
[[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 //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]]
\qquad -\tfrac{1}{3}v^3 = -0.00267...
[[math]]
The approximants of superparticular intervals are reciprocals of odd integers:
<Figure>^^^
If <span style="font-family: Georgia,serif; font-size: 110%;">//v//[//J//] </span>denotes the bimodular approximant of an interval <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span> with frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//,
[[math]]
\qquad v[-J] = -v[J] \\
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}
[[math]]
This last result is equivalent to the identity expressing <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 + </span><span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif;">)</span> in terms of <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> and <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">).</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:
Two perfect fourths (//r// = 4/3, //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)
Three major thirds (//r// = 5/4, //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/9) or two __7/5__s (//<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/6) or five __8/7__s (//<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/15) approximate an octave (//r// = 2/1,//<span style="font-family: Georgia,serif; font-size: 110%;"> v</span>// = 1/3)
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.
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos alpha.
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos beta.
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos gamma. This temperament has high accuracy because it conforms to the policy noted above.
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.
Tuning the intervals __9/7__, __7/5__ and __5/3__ in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.
Tuning the intervals __11/9__, __9/7__, __3/2__ and __5/3__ in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.
Relationships of this sort can be identified in all equal temperaments.
==<span style="font-size: 15px;">Bimodular commas</span>==
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 <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> (with<span style="font-family: Georgia,serif; font-size: 110%;"> //J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> < <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>) and their approximants <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>, we define the //bimodular residue// as
[[math]]
\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}
[[math]]
and using the Taylor series expansion of <span style="font-family: Georgia,serif; font-size: 110%;">//J//(//v//)</span> we find
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)
[[math]]
The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> with integer coefficients sharing no common factor:
[[math]]
\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)
[[math]]
where
[[math]]
\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}
[[math]]
and (with rare exceptions)
[[math]]
\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}
[[math]]
The bimodular residue is accurately estimated by
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)
[[math]]
and therefore
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m
[[math]]
===Examples===
If the source intervals are the perfect fourth (//f = __4/3__)// and the perfect fifth (//F = __3/2__//), <span style="font-family: Georgia,serif; font-size: 110%;">//v//1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">//v//2 = 1/5</span>, and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// is the Pythagorean comma:
[[math]]
\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f
[[math]]
If the source intervals are the perfect fourth (//f// //= __4/3__//) and the minor seventh (//m<span style="vertical-align: sub;">7 = __9/5__</span>//), <span style="font-family: Georgia,serif; font-size: 110%;">//v//1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">//v//2 = 2/7</span>, //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>//r <span style="font-family: Georgia,serif; font-size: 110%;">= 2/7</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// is the syntonic comma:
[[math]]
\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f
[[math]]
Further examples of bimodular commas are provided in Reference 1^^^. See also __Don Page comma^^^__ (another name for this type of comma).
=**<span style="font-size: 21.33px;">Padé approximants of order (1,2)</span>**=
In the section on bimodular approximants it was shown than an interval of logarithmic size //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// (measured in dineper units) is related to its bimodular approximant by
[[math]]
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...
[[math]]
where
[[math]]
\qquad v = \frac{r-1}{r+1}
[[math]]
and //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>// is the interval’s frequency ratio.
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 - ...)))
[[math]]
The first convergent of this continued fraction is //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is<span style="font-family: Georgia,serif; font-size: 110%;"> 3//v///(3-//v//</span><span style="vertical-align: super;">2</span>).
Values of this rational approximant for some simple 5-limit intervals are shown in the table below.
|| //Interval// || //(1,2) Padé approximant// ||
|| Perfect twelfth __3/1__ || 6/11 ||
|| Octave 2/1 || 9/26 ||
|| Major sixth 5/3 || 12 15/ ||
|| Perfect fifth 3/2 || 74/47 ||
|| Perfect fourth 4/3 || 21/146 ||
|| Major third 5/4 || 27/242 ||
^^^Original HTML content:
<html><head><title>Logarithmic approximants</title></head><body>WORK IN PROGRESS<br />
<!-- ws:start:WikiTextHeadingRule:25:<h1> --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:25 -->Introduction</h1>
<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 />
<ul><li><span style="font-family: Arial,Helvetica,sans-serif;">Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why are certain commas small, and roughly how small are they?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why is the ratio of the perfect fifth to the perfect fourth close to √2?</span></li></ul><br />
The exact size, in cents, of an interval with frequency ratio <em>r</em> is<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]<br/>
\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}<br />
<br/>[[math]]
--><script type="math/tex">\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}
</script><!-- ws:end:WikiTextMathRule:0 --><br />
where for just intervals r is rational and can be written as the ratio of two integers:<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]<br/>
\qquad r = n/d<br />
<br/>[[math]]
--><script type="math/tex">\qquad r = n/d
</script><!-- ws:end:WikiTextMathRule:1 --><br />
When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]<br/>
\qquad J = \tfrac{1}{2} \ln{r}<br />
<br/>[[math]]
--><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{r}
</script><!-- ws:end:WikiTextMathRule:2 --><br />
This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e<span style="vertical-align: super;">2</span> = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.<br />
Comparing the two units of measurement we find<br />
1 dineper = 2400/ln(2) = 3462.468 cents<br />
which is about 1.4 semitones short of three octaves.<br />
<br />
The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus <u>3/2</u> is the perfect fifth.<br />
<br />
Three types of approximants are described here:<br />
<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><br />
<!-- ws:start:WikiTextHeadingRule:27:<h1> --><h1 id="toc1"><a name="Bimodular approximants"></a><!-- ws:end:WikiTextHeadingRule:27 --><strong><span style="font-size: 20px;">Bimodular approximants</span></strong></h1>
<!-- ws:start:WikiTextHeadingRule:29:<h2> --><h2 id="toc2"><a name="Bimodular approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:29 --><strong><span style="font-size: 15px;">Definition</span></strong></h2>
The bimodular approximant of an interval with frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r = n/d</span></em> is<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]<br/>
\qquad v = \frac{r-1}{r+1}<br />
<br/>[[math]]
--><script type="math/tex">\qquad v = \frac{r-1}{r+1}
</script><!-- ws:end:WikiTextMathRule:3 --><br />
<em><span style="font-family: Georgia,serif; font-size: 110%;">v </span></em>can thus be expressed as<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]<br/>
\qquad v = \frac{n-d}{n+d} \\<br />
<br/>[[math]]
--><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> =½ (frequency difference) / (mean frequency)<br />
<span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> </span>can be retrieved from <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> using the inverse relation<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]<br/>
\qquad r = \frac{1+v}{1-v}<br/>[[math]]
--><script type="math/tex">\qquad r = \frac{1+v}{1-v}</script><!-- ws:end:WikiTextMathRule:5 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:31:<h2> --><h2 id="toc3"><a name="Bimodular approximants-Properties of bimodular approximants"></a><!-- ws:end:WikiTextHeadingRule:31 --><strong><span style="font-size: 15px;">Properties of bimodular approximants</span></strong></h2>
When <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> </span>is small, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> 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.<br />
Noting that the exact size (in dineper units) of the interval with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em></span> is<br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]<br/>
\qquad J = \tfrac{1}{2} \ln{r}<br/>[[math]]
--><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{r}</script><!-- ws:end:WikiTextMathRule:6 --><br />
the relationship between <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span> can be expressed as<br />
<!-- ws:start:WikiTextMathRule:7:
[[math]]<br/>
\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...<br/>[[math]]
--><script type="math/tex">\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</script><!-- ws:end:WikiTextMathRule:7 --><br />
which shows that <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em> ≈ <em>J</em></span> and provides an indication of the size and sign of the error involved in this approximation.<br />
<em><span style="font-family: Georgia;">J</span></em> can be expressed in terms of <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> as<br />
<!-- ws:start:WikiTextMathRule:8:
[[math]]<br/>
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...<br/>[[math]]
--><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 />
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 />
As an example, the size of the perfect fifth (in dNp units) is<br />
<!-- ws:start:WikiTextMathRule:9:
[[math]]<br/>
\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...<br/>[[math]]
--><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...</script><!-- ws:end:WikiTextMathRule:9 --><br />
The bimodular approximant for this interval (r = 3/2) is<br />
<!-- ws:start:WikiTextMathRule:10:
[[math]]<br/>
\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2<br/>[[math]]
--><script type="math/tex">\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2</script><!-- ws:end:WikiTextMathRule:10 --><br />
and the Taylor series indicates that the error in this value is about<br />
<!-- ws:start:WikiTextMathRule:11:
[[math]]<br/>
\qquad -\tfrac{1}{3}v^3 = -0.00267...<br/>[[math]]
--><script type="math/tex">\qquad -\tfrac{1}{3}v^3 = -0.00267...</script><!-- ws:end:WikiTextMathRule:11 --><br />
<br />
The approximants of superparticular intervals are reciprocals of odd integers:<br />
<Figure>^^^<br />
<br />
If <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>[<em>J</em>] </span>denotes the bimodular approximant of an interval <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span> with frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>,<br />
<!-- ws:start:WikiTextMathRule:12:
[[math]]<br/>
\qquad v[-J] = -v[J] \\<br />
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}<br/>[[math]]
--><script type="math/tex">\qquad v[-J] = -v[J] \\
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}</script><!-- ws:end:WikiTextMathRule:12 --><br />
This last result is equivalent to the identity expressing <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 + </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif;">)</span> in terms of <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> and <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">).</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:33:<h1> --><h1 id="toc4"><a name="Bimodular approximants and equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:33 --><strong><span style="font-size: 20px;">Bimodular approximants and equal temperaments</span></strong></h1>
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>r</em> = 4/3, <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/7) approximate a minor seventh (<em>r</em> = 9/5, = 2/7)<br />
Three major thirds (<em>r</em> = 5/4, <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/9) or two <u>7/5</u>s (<em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/6) or five <u>8/7</u>s (<em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/15) approximate an octave (<em>r</em> = 2/1,<em><span style="font-family: Georgia,serif; font-size: 110%;"> v</span></em> = 1/3)<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 />
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.<br />
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos alpha.<br />
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos beta.<br />
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos gamma. This temperament has high accuracy because it conforms to the policy noted above.<br />
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.<br />
Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.<br />
Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.<br />
Relationships of this sort can be identified in all equal temperaments.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:35:<h2> --><h2 id="toc5"><a name="Bimodular approximants and equal temperaments-Bimodular commas"></a><!-- ws:end:WikiTextHeadingRule:35 --><span style="font-size: 15px;">Bimodular commas</span></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.<br />
Given two intervals <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> (with<span style="font-family: Georgia,serif; font-size: 110%;"> <em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> < <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>) and their approximants <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>, we define the <em>bimodular residue</em> as<br />
<!-- ws:start:WikiTextMathRule:13:
[[math]]<br/>
\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}<br/>[[math]]
--><script type="math/tex">\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}</script><!-- ws:end:WikiTextMathRule:13 --><br />
and using the Taylor series expansion of <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>(<em>v</em>)</span> we find<br />
<!-- ws:start:WikiTextMathRule:14:
[[math]]<br/>
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)<br/>[[math]]
--><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)</script><!-- ws:end:WikiTextMathRule:14 --><br />
The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> with integer coefficients sharing no common factor:<br />
<!-- ws:start:WikiTextMathRule:15:
[[math]]<br/>
\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)<br/>[[math]]
--><script type="math/tex">\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)</script><!-- ws:end:WikiTextMathRule:15 --><br />
where<br />
<!-- ws:start:WikiTextMathRule:16:
[[math]]<br/>
\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}<br/>[[math]]
--><script type="math/tex">\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}</script><!-- ws:end:WikiTextMathRule:16 --><br />
and (with rare exceptions)<br />
<!-- ws:start:WikiTextMathRule:17:
[[math]]<br/>
\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}<br/>[[math]]
--><script type="math/tex">\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}</script><!-- ws:end:WikiTextMathRule:17 --><br />
The bimodular residue is accurately estimated by<br />
<!-- ws:start:WikiTextMathRule:18:
[[math]]<br/>
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)<br/>[[math]]
--><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)</script><!-- ws:end:WikiTextMathRule:18 --><br />
and therefore<br />
<!-- ws:start:WikiTextMathRule:19:
[[math]]<br/>
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m<br/>[[math]]
--><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m</script><!-- ws:end:WikiTextMathRule:19 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:37:<h3> --><h3 id="toc6"><a name="Bimodular approximants and equal temperaments-Bimodular commas-Examples"></a><!-- ws:end:WikiTextHeadingRule:37 -->Examples</h3>
If the source intervals are the perfect fourth (<em>f = <u>4/3</u>)</em> and the perfect fifth (<em>F = <u>3/2</u></em>), <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>2 = 1/5</span>, and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> is the Pythagorean comma:<br />
<!-- ws:start:WikiTextMathRule:20:
[[math]]<br/>
\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f<br/>[[math]]
--><script type="math/tex">\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f</script><!-- ws:end:WikiTextMathRule:20 --><br />
If the source intervals are the perfect fourth (<em>f</em> <em>= <u>4/3</u></em>) and the minor seventh (<em>m<span style="vertical-align: sub;">7 = <u>9/5</u></span></em>), <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>2 = 2/7</span>, <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em>r <span style="font-family: Georgia,serif; font-size: 110%;">= 2/7</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> is the syntonic comma:<br />
<!-- ws:start:WikiTextMathRule:21:
[[math]]<br/>
\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f<br/>[[math]]
--><script type="math/tex">\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f</script><!-- ws:end:WikiTextMathRule:21 --><br />
<br />
Further examples of bimodular commas are provided in Reference 1^^^. See also <u>Don Page comma^^^</u> (another name for this type of comma).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:39:<h1> --><h1 id="toc7"><a name="Padé approximants of order (1,2)"></a><!-- ws:end:WikiTextHeadingRule:39 --><strong><span style="font-size: 21.33px;">Padé approximants of order (1,2)</span></strong></h1>
In the section on bimodular approximants it was shown than an interval of logarithmic size <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> (measured in dineper units) is related to its bimodular approximant by<br />
<!-- ws:start:WikiTextMathRule:22:
[[math]]<br/>
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...<br/>[[math]]
--><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:22 --><br />
where<br />
<!-- ws:start:WikiTextMathRule:23:
[[math]]<br/>
\qquad v = \frac{r-1}{r+1}<br/>[[math]]
--><script type="math/tex">\qquad v = \frac{r-1}{r+1}</script><!-- ws:end:WikiTextMathRule:23 --><br />
and <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em> is the interval’s frequency ratio.<br />
Another way to express this relationship is with a continued fraction:<br />
<!-- ws:start:WikiTextMathRule:24:
[[math]]<br/>
\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))<br/>[[math]]
--><script type="math/tex">\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))</script><!-- ws:end:WikiTextMathRule:24 --><br />
The first convergent of this continued fraction is <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em>, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is<span style="font-family: Georgia,serif; font-size: 110%;"> 3<em>v</em>/(3-<em>v</em></span><span style="vertical-align: super;">2</span>).<br />
Values of this rational approximant for some simple 5-limit intervals are shown in the table below.<br />
<table class="wiki_table">
<tr>
<td><em>Interval</em><br />
</td>
<td><em>(1,2) Padé approximant</em><br />
</td>
</tr>
<tr>
<td>Perfect twelfth <u>3/1</u><br />
</td>
<td>6/11<br />
</td>
</tr>
<tr>
<td>Octave 2/1<br />
</td>
<td>9/26<br />
</td>
</tr>
<tr>
<td>Major sixth 5/3<br />
</td>
<td>12 15/<br />
</td>
</tr>
<tr>
<td>Perfect fifth 3/2<br />
</td>
<td>74/47<br />
</td>
</tr>
<tr>
<td>Perfect fourth 4/3<br />
</td>
<td>21/146<br />
</td>
</tr>
<tr>
<td>Major third 5/4<br />
</td>
<td>27/242<br />
</td>
</tr>
</table>
<br />
<br />
^^^</body></html>