Logarithmic approximants: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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<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//[[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

* Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?

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[[math]]

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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.

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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)

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* Quadratic approximants

**Bimodular approximants**

=**Bimodular approximants**=

**Definition**

==**Definition**==

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

[[math]]

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[[math]]

**Properties of bimodular approximants**

==**Properties of bimodular approximants**==

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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the relationship between //v// and //J// can be expressed as

[[math]]

\qquad v = \frac{r-1}{r+1} = \frac{~~\exp~~{2J}-1}{~~\exp~~{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...

\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 //v// ≈ //J// and provides an indication of the size and sign of the error involved in this approximation.

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[[math]]

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

The approximants of superparticular intervals are reciprocals of odd integers:

<Figure>^^^

If //v[J]// denotes the bimodular approximant of an interval //J// with frequency ratio //r//,

If //v//[//J//] denotes the bimodular approximant of an interval //J// with frequency ratio //r//,

[[math]]

\qquad v[-J] = -v[J] \\

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This last result is equivalent to the identity expressing tanh(//J//1 + //J//1) in terms of tanh(//J//1) and tanh(//J//2).

**Bimodular approximants and equal temperaments**

=**Bimodular approximants and equal temperaments**=

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, = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)

Two perfect fourths (//r// = 4/3, //v// = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)

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)

Three major thirds (//r// = 5/4, //v// = 1/9) or two __7/5__s (//v// = 1/6) or five __8/7__s (//v// = 1/15) approximate an octave (//r// = 2/1,// v// = 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.

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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 J1 and J2 (with J1<J2) and their approximants v1 and v2, we define the //bimodular residue//as

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

[[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 J(v) we find

and using the Taylor series expansion of //J//(//v//) 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 J1 and J2 with integer coefficients sharing no common factor:

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 //J//1 and //J//2 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)

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and (with rare exceptions)

[[math]]

\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}

\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

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[[math]]

Examples:

===Examples===

If the source intervals are the perfect fourth (f) and the perfect fifth (F),

If the source intervals are the perfect fourth (//f = __4/3__)// and the perfect fifth (//F = __3/2__//), //v//1 = 1/7, //v//2 = 1/5, and //b// is the Pythagorean comma:

v1 = 1/7, v2 = 1/5, and 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) and the minor seventh (m7),

If the source intervals are the perfect fourth (//f// //= __4/3__//) and the minor seventh (//m7 = __9/5__//), //v//1 = 1/7, //v//2 = 2/7, //b//r = 2/7 and //b// is the syntonic comma:

v1 = 1/7, v2 = 2/7, ~~br(J1~~,~~J2)~~ = 2/7 and b~~(J1,J2)~~ 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).</pre></div>

Further examples of bimodular commas are provided in Reference 1^^^. See also __Don Page comma^^^__ (another name for this type of comma).

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

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]]

\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 //r// 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 //v//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is 3//v///(3-//v//2).

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 ||

^^^</pre></div>

<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

<~~br /~~>

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

<~~strong~~><~~span style~~="~~font-size: 15px;~~">~~Introduction~~</~~span~~><~~/strong~~>

The term logarithmic approximant<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a> (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 reasonably accurate approximation to 5-limit just intonation?</li><li>Why are certain commas small, and roughly how small are they?</li><li>Why is the ratio of the perfect fifth to the perfect fourth close to √2?</li></ul>

The exact size, in cents, of an interval with frequency ratio r is

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--><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{r}

</script>

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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.

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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:

<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>

Bimodular approximants

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

Definition

<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:

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

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

Properties of bimodular approximants

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

<!-- ws:start:WikiTextMathRule:6:

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<!-- ws:start:WikiTextMathRule:7:

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

\qquad v = \frac{r-1}{r+1} = \frac{~~\exp~~{2J}-1}{~~\exp~~{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&lt;br/&gt;[[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 - ...&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad v = \frac{r-1}{r+1} = \frac{~~\exp~~{2J}-1}{~~\exp~~{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</script>

--><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>

which shows that v ≈ J and provides an indication of the size and sign of the error involved in this approximation.

J can be expressed in terms of v as

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--><script type="math/tex">\qquad -\tfrac{1}{3}v^3 = -0.00267...</script>

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

The approximants of superparticular intervals are reciprocals of odd integers:

&lt;Figure&gt;

&lt;Figure&gt;^^^

If v[J] denotes the bimodular approximant of an interval J with frequency ratio r,

If v[J] denotes the bimodular approximant of an interval J with frequency ratio r,

<!-- ws:start:WikiTextMathRule:12:

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

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This last result is equivalent to the identity expressing tanh(J1 + J1) in terms of tanh(J1) and tanh(J2).

Bimodular approximants and equal temperaments

<h1 id="toc4"><a name="Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</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:

Two perfect fourths (r = 4/3, = 1/7) approximate a minor seventh (r = 9/5, = 2/7)

Two perfect fourths (r = 4/3, v = 1/7) approximate a minor seventh (r = 9/5, = 2/7)

Three major thirds (r = 5/4, = 1/9) or two 7/5s ( = 1/6) or five 8/7s ( = 1/15) approximate an octave (r = 2/1, = 1/3)

Three major thirds (r = 5/4, v = 1/9) or two 7/5s (v = 1/6) or five 8/7s (v = 1/15) approximate an octave (r = 2/1, v = 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.

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

Bimodular commas

<h2 id="toc5"><a name="Bimodular approximants and equal temperaments-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 residueas

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

<!-- ws:start:WikiTextMathRule:13:

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

\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}</script>

and using the Taylor series expansion of J(v) we find

and using the Taylor series expansion of J(v) we find

<!-- ws:start:WikiTextMathRule:14:

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

\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)&lt;br/&gt;[[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>

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 J1 and J2 with integer coefficients sharing no common factor:

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 J1 and J2 with integer coefficients sharing no common factor:

<!-- ws:start:WikiTextMathRule:15:

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

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<!-- ws:start:WikiTextMathRule:17:

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

\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}&lt;br/&gt;[[math]]

\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}</script>

--><script type="math/tex">\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}</script>

The bimodular residue is accurately estimated by

<!-- ws:start:WikiTextMathRule:18:

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--><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>

Examples:

<h3 id="toc6"><a name="Bimodular approximants and equal temperaments-Bimodular commas-Examples"></a>Examples</h3>

If the source intervals are the perfect fourth (f) and the perfect fifth (F),

If the source intervals are the perfect fourth (f = 4/3) and the perfect fifth (F = 3/2), v1 = 1/7, v2 = 1/5, and b is the Pythagorean comma:

v1 = 1/7, v2 = 1/5, and is the Pythagorean comma:

<!-- ws:start:WikiTextMathRule:20:

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

\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&lt;br/&gt;[[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>

If the source intervals are the perfect fourth (f) and the minor seventh (m7),

If the source intervals are the perfect fourth (f = 4/3) and the minor seventh (m7 = 9/5), v1 = 1/7, v2 = 2/7, br = 2/7 and b is the syntonic comma:

v1 = 1/7, v2 = 2/7, ~~br(J1~~,~~J2)~~ = 2/7 and b~~(J1,J2)~~ is the syntonic comma:

<!-- ws:start:WikiTextMathRule:21:

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

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--><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>

Further examples of bimodular commas are provided in Reference 1. See also Don Page comma (another name for this type of comma).</body></html></pre></div>

Further examples of bimodular commas are provided in Reference 1^^^. See also Don Page comma^^^ (another name for this type of comma).

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

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:

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

\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script>

where

<!-- ws:start:WikiTextMathRule:23:

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

\qquad v = \frac{r-1}{r+1}&lt;br/&gt;[[math]]

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

and r is the interval’s frequency ratio.

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

<!-- 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]]

--><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 3v/(3-v2).

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

<tr>

<td>Interval

</td>

<td>(1,2) Padé approximant

</td>

</tr>

<tr>

<td>Perfect twelfth 3/1

</td>

<td>6/11

</td>

</tr>

<tr>

<td>Octave 2/1

</td>

<td>9/26

</td>

</tr>

<tr>

<td>Major sixth 5/3

</td>

<td>12 15/

</td>

</tr>

<tr>

<td>Perfect fifth 3/2

</td>

<td>74/47

</td>

</tr>

<tr>

<td>Perfect fourth 4/3

</td>

<td>21/146

</td>

</tr>

<tr>

<td>Major third 5/4

</td>

<td>27/242

</td>

</tr>

</table>

^^^</body></html></pre></div>

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 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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-: The original revision id was <tt>541434588</tt>.<br>
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 <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=
-**&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 do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?&lt;/span&gt;
@@ Line 29: / Line 28: @@
 [[math]]
-This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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.
+This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; = 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
 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)
@@ Line 38: / Line 40: @@
 * Quadratic approximants
-**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**
+=**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**=
-**&lt;span style="font-size: 15px;"&gt;Definition&lt;/span&gt;**
+==**&lt;span style="font-size: 15px;"&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 57: / Line 59: @@
 [[math]]
-**&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;**
+==**&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
@@ Line 65: / Line 67: @@
 the relationship between &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt; can be expressed as
 [[math]]
-\qquad v = \frac{r-1}{r+1} = \frac{\exp{2J}-1}{\exp{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...
+\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 &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.
@@ Line 87: / Line 89: @@
 [[math]]
-The figure shows the approximants of the first 31 superparticular intervals, which are reciprocals of odd integers.
+The approximants of superparticular intervals are reciprocals of odd integers:
-&lt;Figure&gt;
+&lt;Figure&gt;^^^
-If &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v[J]// &lt;/span&gt;denotes the bimodular approximant 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;//,
+If &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//[//J//] &lt;/span&gt;denotes the bimodular approximant 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;//,
 [[math]]
 \qquad v[-J] = -v[J] \\
@@ Line 97: / Line 99: @@
 This last result is equivalent to the identity expressing &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1 + &lt;/span&gt;&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;span style="font-family: Georgia,serif;"&gt;)&lt;/span&gt; in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;).&lt;/span&gt;
-**&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;**
+=**&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;**=
 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, = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)
+Two perfect fourths (//r// = 4/3, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)
-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)
+Three major thirds (//r// = 5/4, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// = 1/9) or two __7/5__s (//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// = 1/6) or five __8/7__s (//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// = 1/15) approximate an octave (//r// = 2/1,//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; v&lt;/span&gt;// = 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.
@@ Line 112: / Line 114: @@
 Relationships of this sort can be identified in all equal temperaments.
-&lt;span style="font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;
+==&lt;span style="font-size: 15px;"&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 J1 and J2 (with J1&lt;J2) and their approximants v1 and v2, we define the //bimodular residue//as
+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
 [[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 J(v) we find
+and using the Taylor series expansion of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//(//v//)&lt;/span&gt; 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 J1 and J2 with integer coefficients sharing no common factor:
+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 &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 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)
@@ Line 132: / Line 134: @@
 and (with rare exceptions)
 [[math]]
-\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}
+\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
@@ Line 143: / Line 145: @@
 [[math]]
-Examples:
+===Examples===
-If the source intervals are the perfect fourth (f) and the perfect fifth (F),
+If the source intervals are the perfect fourth (//f = __4/3__)// and the perfect fifth (//F = __3/2__//), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 1/5&lt;/span&gt;, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the Pythagorean comma:
-v1 = 1/7, v2 = 1/5, and 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) and the minor seventh (m7),
+If the source intervals are the perfect fourth (//f// //= __4/3__//) and the minor seventh (//m&lt;span style="vertical-align: sub;"&gt;7 = __9/5__&lt;/span&gt;//), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 2/7&lt;/span&gt;, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the syntonic comma:
-v1 = 1/7, v2 = 2/7, br(J1,J2) = 2/7 and b(J1,J2) 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).</pre></div>
+Further examples of bimodular commas are provided in Reference 1^^^. See also __Don Page comma^^^__ (another name for this type of comma).
+=**&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&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]]
+\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 //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;// 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 //&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&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; 3//v///(3-//v//&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;).
+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 ||
+^^^</pre></div>
 <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;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Introduction&lt;/h1&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;
-&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;
 &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 reasonably accurate 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 is the ratio of the perfect fifth to the perfect fourth close to √2?&lt;/span&gt;&lt;/li&gt;&lt;/ul&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;
@@ Line 183: / Line 209: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad J = \tfrac{1}{2} \ln{r}
 &lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:2 --&gt;&lt;br /&gt;
-This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 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.&lt;br /&gt;
+This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt; = 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.&lt;br /&gt;
 Comparing the two units of measurement we find&lt;br /&gt;
 dineper = 2400/ln(2) = 3462.468 cents&lt;br /&gt;
 which is about 1.4 semitones short of three octaves.&lt;br /&gt;
+&lt;br /&gt;
+The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus &lt;u&gt;3/2&lt;/u&gt; is the perfect fifth.&lt;br /&gt;
+&lt;br /&gt;
 Three types of approximants are described here:&lt;br /&gt;
 &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;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Definition&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:29:&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:29 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Definition&lt;/span&gt;&lt;/strong&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;
+ 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:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 213: / Line 242: @@
   --&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;strong&gt;&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:31:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties of bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:31 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;&lt;/strong&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;
+ 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;
 &lt;!-- ws:start:WikiTextMathRule:6:
@@ Line 223: / Line 252: @@
 &lt;!-- ws:start:WikiTextMathRule:7:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad v = \frac{r-1}{r+1} = \frac{\exp{2J}-1}{\exp{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&amp;lt;br/&amp;gt;[[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 - ...&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad v = \frac{r-1}{r+1} = \frac{\exp{2J}-1}{\exp{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\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 - ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:7 --&gt;&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;
@@ Line 249: / Line 278: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad -\tfrac{1}{3}v^3 = -0.00267...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-The figure shows the approximants of the first 31 superparticular intervals, which are reciprocals of odd integers.&lt;br /&gt;
+The approximants of superparticular intervals are reciprocals of odd integers:&lt;br /&gt;
-&amp;lt;Figure&amp;gt;&lt;br /&gt;
+&amp;lt;Figure&amp;gt;^^^&lt;br /&gt;
 &lt;br /&gt;
-If &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v[J]&lt;/em&gt; &lt;/span&gt;denotes the bimodular approximant of an interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&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;,&lt;br /&gt;
+If &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;denotes the bimodular approximant of an interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&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;,&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:12:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 261: / Line 290: @@
 This last result is equivalent to the identity expressing &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&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;&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;&lt;span style="font-family: Georgia,serif;"&gt;)&lt;/span&gt; in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&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;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&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;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;).&lt;/span&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;
+&lt;!-- ws:start:WikiTextHeadingRule:33:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:33 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-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:&lt;br /&gt;
+ 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:&lt;br /&gt;
-Two perfect fourths (&lt;em&gt;r&lt;/em&gt; = 4/3, = 1/7) approximate a minor seventh (&lt;em&gt;r&lt;/em&gt; = 9/5, = 2/7)&lt;br /&gt;
+Two perfect fourths (&lt;em&gt;r&lt;/em&gt; = 4/3, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; = 1/7) approximate a minor seventh (&lt;em&gt;r&lt;/em&gt; = 9/5, = 2/7)&lt;br /&gt;
-Three major thirds (&lt;em&gt;r&lt;/em&gt; = 5/4, = 1/9) or two &lt;u&gt;7/5&lt;/u&gt;s ( = 1/6) or five &lt;u&gt;8/7&lt;/u&gt;s ( = 1/15) approximate an octave (&lt;em&gt;r&lt;/em&gt; = 2/1, = 1/3)&lt;br /&gt;
+Three major thirds (&lt;em&gt;r&lt;/em&gt; = 5/4, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; = 1/9) or two &lt;u&gt;7/5&lt;/u&gt;s (&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; = 1/6) or five &lt;u&gt;8/7&lt;/u&gt;s (&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; = 1/15) approximate an octave (&lt;em&gt;r&lt;/em&gt; = 2/1,&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; v&lt;/span&gt;&lt;/em&gt; = 1/3)&lt;br /&gt;
 Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.&lt;br /&gt;
 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.&lt;br /&gt;
@@ Line 276: / Line 305: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;span style="font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:35:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants and equal temperaments-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:35 --&gt;&lt;span style="font-size: 15px;"&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;
+ 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 J1 and J2 (with J1&amp;lt;J2) and their approximants v1 and v2, we define the &lt;em&gt;bimodular residue&lt;/em&gt;as&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;
 &lt;!-- ws:start:WikiTextMathRule:13:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:13 --&gt;&lt;br /&gt;
-and using the Taylor series expansion of J(v) we find&lt;br /&gt;
+and using the Taylor series expansion of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;(&lt;em&gt;v&lt;/em&gt;)&lt;/span&gt; we find&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:14:
 [[math]]&amp;lt;br/&amp;gt;
 \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)&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\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)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:14 --&gt;&lt;br /&gt;
-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 J1 and J2 with integer coefficients sharing no common factor:&lt;br /&gt;
+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 &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;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; with integer coefficients sharing no common factor:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:15:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 301: / Line 330: @@
 &lt;!-- ws:start:WikiTextMathRule:17:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}&amp;lt;br/&amp;gt;[[math]]
+\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2}{GCD(g_1,g_2)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:17 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:17 --&gt;&lt;br /&gt;
 The bimodular residue is accurately estimated by&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:18:
@@ Line 314: / Line 343: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:19 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-Examples:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:37:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Bimodular approximants and equal temperaments-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:37 --&gt;Examples&lt;/h3&gt;
-If the source intervals are the perfect fourth (f) and the perfect fifth (F),&lt;br /&gt;
+ If the source intervals are the perfect fourth (&lt;em&gt;f = &lt;u&gt;4/3&lt;/u&gt;)&lt;/em&gt; and the perfect fifth (&lt;em&gt;F = &lt;u&gt;3/2&lt;/u&gt;&lt;/em&gt;), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;2 = 1/5&lt;/span&gt;, and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; is the Pythagorean comma:&lt;br /&gt;
-v1 = 1/7, v2 = 1/5, and is the Pythagorean comma:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:20:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:20 --&gt;&lt;br /&gt;
-If the source intervals are the perfect fourth (f) and the minor seventh (m7),&lt;br /&gt;
+If the source intervals are the perfect fourth (&lt;em&gt;f&lt;/em&gt; &lt;em&gt;= &lt;u&gt;4/3&lt;/u&gt;&lt;/em&gt;) and the minor seventh (&lt;em&gt;m&lt;span style="vertical-align: sub;"&gt;7 = &lt;u&gt;9/5&lt;/u&gt;&lt;/span&gt;&lt;/em&gt;), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;2 = 2/7&lt;/span&gt;, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; is the syntonic comma:&lt;br /&gt;
-v1 = 1/7, v2 = 2/7, br(J1,J2) = 2/7 and b(J1,J2) is the syntonic comma:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:21:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 328: / Line 355: @@
   --&gt;&lt;script type="math/tex"&gt;\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&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:21 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-Further examples of bimodular commas are provided in Reference 1. See also &lt;u&gt;Don Page comma&lt;/u&gt; (another name for this type of comma).&lt;/body&gt;&lt;/html&gt;</pre></div>
+Further examples of bimodular commas are provided in Reference 1^^^. See also &lt;u&gt;Don Page comma^^^&lt;/u&gt; (another name for this type of comma).&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:39:&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:39 --&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;
+ 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;
+&lt;!-- ws:start:WikiTextMathRule:22:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:22 --&gt;&lt;br /&gt;
+where&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:23:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad v = \frac{r-1}{r+1}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad v = \frac{r-1}{r+1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:23 --&gt;&lt;br /&gt;
+and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt; is the interval’s frequency ratio.&lt;br /&gt;
+Another way to express this relationship is with a continued fraction:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:24:
+[[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]]
+ --&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;span style="font-family: Georgia,serif; font-size: 110%;"&gt; 3&lt;em&gt;v&lt;/em&gt;/(3-&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;).&lt;br /&gt;
+Values of this rational approximant for some simple 5-limit intervals are shown in the table below.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;em&gt;Interval&lt;/em&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;em&gt;(1,2) Padé approximant&lt;/em&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Perfect twelfth &lt;u&gt;3/1&lt;/u&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6/11&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Octave 2/1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9/26&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Major sixth 5/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12 15/&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Perfect fifth 3/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;74/47&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Perfect fourth 4/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;21/146&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Major third 5/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;27/242&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+&lt;br /&gt;
+^^^&lt;/body&gt;&lt;/html&gt;</pre></div>