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// (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

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

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

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

* Why are certain commas small, and roughly how small are they?

* Why does the 3-limit framework produce aesthetically pleasing scale structures?

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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

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=**Bimodular approximants**=

==Definition==

==Definition==

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

[[math]]

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[[math]]

Properties

==Properties==

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

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

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

==Bimodular commas==

==Bimodular commas==

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

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

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

==Definition==

==Definition==

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

[[math]]

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=**Quadratic approximants**=

==Definition==

==Definition==

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

[[math]]

\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\

\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\

\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\

\qquad = \sinh{J} \\

\qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...

[[math]]

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

==Properties==

==Properties==

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

[[math]]

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The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity

[[math]]

\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}

\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}

[[math]]

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\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6

[[math]]

~~but~~ this can also be derived from bimodular approximants. Using

where //octave// = __2/1__, //large tone// = __9/8__.

However, this can also be derived from bimodular approximants. Using

[[math]]

\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}

[[math]]

with //J//2 = F =__3/2__ and //J//1 = //f// = __4/3__ this gives

with //J//2 = //F// =__3/2__ and //J//1 = //f// = __4/3__ this gives

[[math]]

\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\

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The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

== ==

==Relative sizes of intervals between 3 frequencies in arithmetic progression==

==Relative sizes of intervals between 3 frequencies in arithmetic progression==

===**Theorem**===

If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

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//T/////t// = 203.910/182.404 = 1.11790,

√5/2 = 1.11803.

===**Silver temperament**===

As the first example above ~~shows~~, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.

==**Silver temperament**==

As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

###Perfect fifth = __3/2__ = 702.944 cents

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\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}

[[math]]

and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows ~~(the~~ names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):

and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone) followed by tempered and just sizes in cents:

|| Octave

1200.00

(1200.00) || Perfect fourth

(1200.00) || Perfect fourth##

497.06

(498.04) || Tone

205.89

(203.91) || Limma

85.28

(90.22) || Pythag comma

35.32

(23.46) ||

|| Perfect 11th

|| Perfect 11th##

1697.06

(1698.04) || Perfect fifth

702.94

(701.96) || Minor third

(701.96) || Minor third##

291.17

(294.13) || Apotome

(294.13) || Apotome##

120.61

(113.69) || 17-tone comma

(113.69) || 17-tone comma##

49.96

(66.76) ||

~~where the limma is the Pythagorean semitone.~~

Thus for example:

###octave = ~~2×fourth~~ + tone

###octave = 2×fourth + tone

###fourth = ~~2×tone~~ + limma

###fourth = 2×tone + limma

###tone = ~~2×limma~~ + Pythag

###tone = 2×limma + Pythag

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

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The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.

The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament. ~~The octave is represented by squares partly visible at the upper and lower edges of the diagram~~.

The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.

[[image:Continued fraction jigsaw 41edo.png width="800" height="395"]]

The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).

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^^^

^^^TBC

= =

= Source=

=Source=

This article summarises original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.</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

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

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

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

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

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

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

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

<!-- ws:start:WikiTextMathRule:0:

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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

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

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

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

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

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

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

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

Properties

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

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

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

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

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The approximants of superparticular intervals are reciprocals of odd integers:

<img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" />

<img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" />

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

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

<h2 id="~~toc3~~"><a name="Bimodular approximants-Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</h2>

<h2 id="toc4"><a name="Bimodular approximants-Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</h2>

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

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

<h2 id="~~toc4~~"><a name="Bimodular approximants-Bimodular commas"></a>Bimodular commas</h2>

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

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

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

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--><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>

<h3 id="~~toc5~~"><a name="Bimodular approximants-Bimodular commas-Examples"></a>Examples</h3>

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

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

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

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Further examples of bimodular commas are provided in Reference 1. See also <a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a> (another name for this type of comma).

<h1 id="~~toc6~~"><a name="Padé approximants of order (1,2)"></a>Padé approximants of order (1,2)</h1>

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

<h2 id="~~toc7~~"><a name="Padé approximants of order (1,2)-Definition"></a>Definition</h2>

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

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

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

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(3/2) / (20/17) = 2.4949 ≈ (15/74) / (6/74) = 5/2

<h1 id="~~toc8~~"><a name="Quadratic approximants"></a>Quadratic approximants</h1>

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

<h2 id="~~toc9~~"><a name="Quadratic approximants-Definition"></a>Definition</h2>

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

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

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

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

\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\&lt;br /&gt;

\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\&lt;br /&gt;

\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\&lt;br /&gt;

\qquad = \sinh{J} \\&lt;br /&gt;

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

--><script type="math/tex">\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\

\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\

\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\

\qquad = \sinh{J} \\

\qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...</script>

If this is compared with the expression for the bimodular approximant,

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

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

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

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

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

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<!-- ws:start:WikiTextMathRule:32:

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

\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}&lt;br/&gt;[[math]]

\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}</script>

--><script type="math/tex">\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}</script>

Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.

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\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6</script>

~~but~~ this can also be derived from bimodular approximants. Using

where octave = 2/1, large tone = 9/8.

However, this can also be derived from bimodular approximants. Using

<!-- ws:start:WikiTextMathRule:34:

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

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

--><script type="math/tex">\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}</script>

with J2 = F =3/2 and J1 = f = 4/3 this gives

with J2 = F =3/2 and J1 = f = 4/3 this gives

<!-- ws:start:WikiTextMathRule:35:

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

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However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.

The most interesting approximate interval ratios derivable from quadratic approximants are irrational.

<h2 id="~~toc11~~"> </h2>

<h2 id="toc12"> </h2>

<h2 id="~~toc12~~"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a>Relative sizes of intervals between 3 frequencies in arithmetic progression</h2>

<h2 id="toc13"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a>Relative sizes of intervals between 3 frequencies in arithmetic progression</h2>

<h3 id="~~toc13~~"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a>Theorem</h3>

<h3 id="toc14"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a>Theorem</h3>

If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.

<h3 id="~~toc14~~"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a>Remarks</h3>

<h3 id="toc15"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a>Remarks</h3>

If the harmonics have indices n – m, n and n + m, the two intervals have reduced frequency ratios n/(n – m) and (n + m)/n. It can be assumed that n and m have no common factor.

m is the epimoricity of the intervals. When m = 1 the intervals are adjacent superparticular intervals.

The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.

<h3 id="~~toc15~~"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a>Proof</h3>

<h3 id="toc16"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a>Proof</h3>

The ratio of the intervals as estimated from their quadratic approximants is

<!-- ws:start:WikiTextMathRule:37:

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--><script type="math/tex">\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}</script>

which is the geometric mean of their frequency ratios.

<h3 id="~~toc16~~"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"></a>Examples</h3>

<h3 id="toc17"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"></a>Examples</h3>

The ratio of the perfect fifth, F = 3/2, to the perfect fourth, f = 4/3, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

F/f = 701.955/498.045 = 1.40942,

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T/t = 203.910/182.404 = 1.11790,

√5/2 = 1.11803.

<h3 id="~~toc17~~"><a name="Quadratic approximants~~-Relative sizes of intervals between 3 frequencies in arithmetic progression~~-Silver temperament"></a>Silver temperament</h3>

As the first example above ~~shows~~, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.

<h2 id="toc18"><a name="Quadratic approximants-Silver temperament"></a>Silver temperament</h2>

As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.

It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to

###Perfect fifth = 3/2 = 702.944 cents

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\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}</script>

and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows ~~(the~~ names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):

and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone) followed by tempered and just sizes in cents:

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(1200.00)

</td>

<td>Perfect fourth

<td>Perfect fourth##

497.06

(498.04)

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(203.91)

</td>

<td>Limma

<td>Limma

85.28

(90.22)

</td>

<td>Pythag comma

<td>Pythag comma

35.32

(23.46)

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</tr>

<tr>

<td>Perfect 11th

<td>Perfect 11th##

1697.06

(1698.04)

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(701.96)

</td>

<td>Minor third

<td>Minor third##

291.17

(294.13)

</td>

<td>Apotome

<td>Apotome##

120.61

(113.69)

</td>

<td>17-tone comma

<td>17-tone comma##

49.96

(66.76)

Line 922:

Line 928:

</table>

~~where the limma is the Pythagorean semitone. ~~

Thus for example:

###octave = ~~2×fourth~~ + tone

###octave = 2×fourth + tone

###fourth = ~~2×tone~~ + limma

###fourth = 2×tone + limma

###tone = ~~2×limma~~ + Pythag

###tone = 2×limma + Pythag

When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.

The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:

Line 939:

Line 944:

The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.

The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament~~. The octave is represented by squares partly visible at the upper and lower edges of the diagram~~.

The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.

<img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x395/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 395px; width: 800px;" />

The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).

<img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" />

<img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" />

^^^

^^^TBC

<h1 id="~~toc18~~"> </h1>

<h1 id="toc19"> </h1>

<h1 id="~~toc19~~"><a name="Source"></a> </span>Source</h1>

<h1 id="toc20"><a name="Source"></a>Source</h1>

This article summarises original research by Martin Gough. See <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a> for a fuller account of bimodular approximants.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 17:40:46 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:05:26 UTC</tt>.<br>
-: The original revision id was <tt>541614438</tt>.<br>
+: The original revision id was <tt>541616090</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">WORK IN PROGRESS
-=&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
+=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant//  (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant// (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;
-* &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a reasonably accurate approximation to 5-limit just intonation?&lt;/span&gt;
+* &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;
@@ Line 28: / Line 28: @@
 [[math]]
-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.
+This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;e&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 7.38906...&lt;/span&gt; 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
@@ Line 41: / Line 41: @@
 =**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**=
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 The bimodular approximant of an interval with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;// is
 [[math]]
@@ Line 59: / Line 59: @@
 [[math]]
-&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Properties&lt;/span&gt;
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
 When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r//&lt;/span&gt; is
@@ Line 114: / Line 114: @@
 Relationships of this sort can be identified in all equal temperaments.
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;==
 As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the //bimodular residue// as
@@ Line 158: / Line 158: @@
 =**&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;**=
-==Definition==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 In the section on bimodular approximants it was shown than an interval of logarithmic size //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// (measured in dineper units) is related to its bimodular approximant by
 [[math]]
@@ Line 197: / Line 197: @@
 =**&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;**=
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
 The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = //n/////d//&lt;/span&gt; is
 [[math]]
 \qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
-\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\
+\qquad = \sinh{J} \\
 \qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...
 [[math]]
@@ Line 244: / Line 245: @@
 The presence of a square root in the denominator of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// (except where //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
 If //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[//J//]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//n]&lt;/span&gt; , then
 [[math]]
@@ Line 259: / Line 260: @@
 The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity
 [[math]]
-\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}
+\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}
 [[math]]
@@ Line 267: / Line 268: @@
 \qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
 [[math]]
-but this can also be derived from bimodular approximants. Using
+where //octave// = __2/1__, //large tone// = __9/8__.
+However, this can also be derived from bimodular approximants. Using
 [[math]]
 \qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}
 [[math]]
-with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//2 = F =__3/2__ &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//1 = //f// = __4/3__&lt;/span&gt; this gives
+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;2 &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= //F// =__3/2__&lt;/span&gt; &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&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; font-size: 110%;"&gt;= //f// = __4/3__&lt;/span&gt; this gives
 [[math]]
 \qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\
@@ Line 284: / Line 286: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
 == ==
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
 ===**Theorem**===
 If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.
@@ Line 304: / Line 306: @@
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//T/////t// = 203.910/182.404 = 1.11790,&lt;/span&gt;
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√5/2 = 1.11803.&lt;/span&gt;
-===**Silver temperament**===
-As the first example above shows, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.
+==**Silver temperament**==
+As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.
 It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fifth = __3/2__ = 702.944 cents
@@ Line 320: / Line 323: @@
 \qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}
 [[math]]
-and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows (the names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):
+and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone) followed by tempered and just sizes in cents:
 || Octave
 .00
-(1200.00) || Perfect fourth
+(1200.00) || Perfect fourth&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;
 .06
 (498.04) || Tone
 .89
 (203.91) || Limma
 .28
 (90.22) || Pythag comma
 .32
 (23.46) ||
-|| Perfect 11th
+|| Perfect 11th&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;
 .06
 (1698.04) || Perfect fifth
 .94
-(701.96) || Minor third
+(701.96) || Minor third&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;
 .17
-(294.13) || Apotome
+(294.13) || Apotome&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;
 .61
-(113.69) || 17-tone comma
+(113.69) || 17-tone comma&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;
 .96
 (66.76) ||
-where the limma is the Pythagorean semitone.
 Thus for example:
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;octave = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;fourth + tone
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;octave = 2×fourth + tone
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fourth = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;tone + limma
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fourth = 2×tone + limma
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;limma + Pythag
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag
 When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.
 The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:
@@ Line 365: / Line 367: @@
 The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.
-The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament. The octave is represented by squares partly visible at the upper and lower edges of the diagram.
+The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.
+[[image:Continued fraction jigsaw 41edo.png width="800" height="395"]]
 The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).
@@ Line 372: / Line 374: @@
-^^^
+^^^TBC
 = =
-=&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;=
+=&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;=
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;This article summarises original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.&lt;/span&gt;</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;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
-  &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt;  (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as&lt;/span&gt;&lt;br /&gt;
+  &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a 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 does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 The exact size, in cents, of an interval with frequency ratio &lt;em&gt;r&lt;/em&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
@@ Line 402: / Line 404: @@
   --&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 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;
+This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;e&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 7.38906...&lt;/span&gt; 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;
@@ Line 412: / Line 414: @@
 &lt;ul&gt;&lt;li&gt;Bimodular approximants (first order rational approximants)&lt;/li&gt;&lt;li&gt;Padé approximants of order (1,2) (second order rational approximants)&lt;/li&gt;&lt;li&gt;Quadratic approximants&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The bimodular approximant of an interval with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;&lt;/em&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
@@ Line 435: / Line 437: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad r = \frac{1+v}{1-v}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:5 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Properties&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
-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 472: / Line 474: @@
 &lt;br /&gt;
 The approximants of superparticular intervals are reciprocals of odd integers:&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:282:&amp;lt;img src=&amp;quot;/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:282 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:284:&amp;lt;img src=&amp;quot;/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:284 --&gt;&lt;br /&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;
@@ Line 483: / Line 485: @@
 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;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Bimodular approximants-Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h2&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, &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;
@@ Line 498: / Line 500: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
   As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.&lt;br /&gt;
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &amp;lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the &lt;em&gt;bimodular residue&lt;/em&gt; as&lt;br /&gt;
@@ Line 536: / Line 538: @@
   --&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;
-&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;Examples&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Bimodular approximants-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Examples&lt;/h3&gt;
   If the source intervals are the perfect fourth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;f&lt;/em&gt; =&lt;/span&gt; &lt;u&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;4/3&lt;/span&gt;&lt;/u&gt;&lt;em&gt;)&lt;/em&gt; and the perfect fifth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F&lt;/em&gt; = &lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;), &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;then&lt;/span&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;
 &lt;!-- ws:start:WikiTextMathRule:20:
@@ Line 550: / Line 552: @@
 Further examples of bimodular commas are provided in Reference 1. See also &lt;a class="wiki_link" href="/Don%20Page%20comma"&gt;Don Page comma&lt;/a&gt; (another name for this type of comma).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;Definition&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   In the section on bimodular approximants it was shown than an interval of logarithmic size &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; (measured in dineper units) is related to its bimodular approximant by&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:22:
@@ Line 633: / Line 635: @@
 (&lt;u&gt;3/2&lt;/u&gt;) / (&lt;u&gt;20/17&lt;/u&gt;) = 2.4949 ≈ (15/74) / (6/74) = 5/2&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = &lt;em&gt;n&lt;/em&gt;&lt;em&gt;/d&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:26:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\&amp;lt;br /&amp;gt;
-\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\&amp;lt;br /&amp;gt;
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\&amp;lt;br /&amp;gt;
+\qquad = \sinh{J} \\&amp;lt;br /&amp;gt;
 \qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
-\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) \\
+\qquad = \sinh{J} \\
 \qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:26 --&gt;&lt;br /&gt;
 If this is compared with the expression for the bimodular approximant,&lt;br /&gt;
@@ Line 778: / Line 782: @@
 The presence of a square root in the denominator of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; (except where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
   If &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;, and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;n]&lt;/span&gt; , then&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:31:
@@ Line 803: / Line 807: @@
 &lt;!-- ws:start:WikiTextMathRule:32:
 [[math]]&amp;lt;br/&amp;gt;
-\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}&amp;lt;br/&amp;gt;[[math]]
+\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:32 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad \sinh{(J_2 + J_1)} \sinh{(J_2 - J_1)} = \sinh^2{J_2} - \sinh^2{J_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:32 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.&lt;br /&gt;
@@ Line 812: / Line 816: @@
 \qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:33 --&gt;&lt;br /&gt;
-but this can also be derived from bimodular approximants. Using&lt;br /&gt;
+where &lt;em&gt;octave&lt;/em&gt; = &lt;u&gt;2/1&lt;/u&gt;, &lt;em&gt;large tone&lt;/em&gt; = &lt;u&gt;9/8&lt;/u&gt;.&lt;br /&gt;
+However, this can also be derived from bimodular approximants. Using&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:34:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:34 --&gt;&lt;br /&gt;
-with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;2 = F =&lt;u&gt;3/2&lt;/u&gt; &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1 = &lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt; this gives&lt;br /&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;2 &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= &lt;em&gt;F&lt;/em&gt; =&lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt; &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&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; font-size: 110%;"&gt;= &lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt; this gives&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:35:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 832: / Line 837: @@
 However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.&lt;br /&gt;
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt; &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt; &lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/h3&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/h3&gt;
   If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;&lt;strong&gt;Remarks&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;&lt;strong&gt;Remarks&lt;/strong&gt;&lt;/h3&gt;
   If the harmonics have indices &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n – m, n&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n + m&lt;/span&gt;&lt;/em&gt;, the two intervals have reduced frequency ratios &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n/(n – m)&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(n + m)/n&lt;/span&gt;&lt;/em&gt;. It can be assumed that &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; have no common factor.&lt;br /&gt;
 &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; is the epimoricity of the intervals. When &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; = 1 the intervals are adjacent superparticular intervals.&lt;br /&gt;
 The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;&lt;strong&gt;Proof&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;&lt;strong&gt;Proof&lt;/strong&gt;&lt;/h3&gt;
   The ratio of the intervals as estimated from their quadratic approximants is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:37:
@@ Line 847: / Line 852: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:37 --&gt;&lt;br /&gt;
 which is the geometric mean of their frequency ratios.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/h3&gt;
   The ratio of the perfect fifth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F&lt;/em&gt; = &lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt;, to the perfect fourth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt;, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).&lt;br /&gt;
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F/f&lt;/em&gt; = 701.955/498.045 = 1.40942,&lt;/span&gt;&lt;br /&gt;
@@ Line 854: / Line 859: @@
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;T&lt;/em&gt;&lt;em&gt;/t&lt;/em&gt; = 203.910/182.404 = 1.11790,&lt;/span&gt;&lt;br /&gt;
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√5/2 = 1.11803.&lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Silver temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;&lt;strong&gt;Silver temperament&lt;/strong&gt;&lt;/h3&gt;
+&lt;br /&gt;
-  As the first example above shows, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Quadratic approximants-Silver temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;&lt;strong&gt;Silver temperament&lt;/strong&gt;&lt;/h2&gt;
+  As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.&lt;br /&gt;
 It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fifth = &lt;u&gt;3/2&lt;/u&gt; = 702.944 cents&lt;br /&gt;
@@ Line 872: / Line 878: @@
 \qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:39 --&gt;&lt;br /&gt;
-and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows (the names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):&lt;br /&gt;
+and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone) followed by tempered and just sizes in cents:&lt;br /&gt;
@@ Line 881: / Line 887: @@
 (1200.00)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Perfect fourth &lt;br /&gt;
+         &lt;td&gt;Perfect fourth&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;&lt;br /&gt;
 .06&lt;br /&gt;
 (498.04)&lt;br /&gt;
@@ Line 889: / Line 895: @@
 (203.91)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Limma &lt;br /&gt;
+         &lt;td&gt;Limma&lt;br /&gt;
 .28&lt;br /&gt;
 (90.22)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Pythag comma &lt;br /&gt;
+         &lt;td&gt;Pythag comma&lt;br /&gt;
 .32&lt;br /&gt;
 (23.46)&lt;br /&gt;
@@ Line 899: / Line 905: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Perfect 11th&lt;br /&gt;
+         &lt;td&gt;Perfect 11th&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;&lt;br /&gt;
 .06&lt;br /&gt;
 (1698.04)&lt;br /&gt;
@@ Line 907: / Line 913: @@
 (701.96)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Minor third&lt;br /&gt;
+         &lt;td&gt;Minor third&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;&lt;br /&gt;
 .17&lt;br /&gt;
 (294.13)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Apotome&lt;br /&gt;
+         &lt;td&gt;Apotome&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;&lt;br /&gt;
 .61&lt;br /&gt;
 (113.69)&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;17-tone comma&lt;br /&gt;
+         &lt;td&gt;17-tone comma&lt;span style="color: #ffffff;"&gt;##&lt;/span&gt;&lt;br /&gt;
 .96&lt;br /&gt;
 (66.76)&lt;br /&gt;
@@ Line 922: / Line 928: @@
 &lt;/table&gt;
-where the limma is the Pythagorean semitone.&lt;br /&gt;
 Thus for example:&lt;br /&gt;
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;octave = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;fourth + tone&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;octave = 2×fourth + tone&lt;br /&gt;
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fourth = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;tone + limma&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fourth = 2×tone + limma&lt;br /&gt;
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;limma + Pythag&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag&lt;br /&gt;
 When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.&lt;br /&gt;
 The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:&lt;br /&gt;
@@ Line 939: / Line 944: @@
 The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.&lt;br /&gt;
 &lt;br /&gt;
-The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament. The octave is represented by squares partly visible at the upper and lower edges of the diagram.&lt;br /&gt;
+The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.&lt;br /&gt;
-&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:285:&amp;lt;img src=&amp;quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x395/Continued%20fraction%20jigsaw%2041edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 395px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x395/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 395px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:285 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:283:&amp;lt;img src=&amp;quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 587px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:283 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:286:&amp;lt;img src=&amp;quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 587px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:286 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
-^^^&lt;br /&gt;
+^^^TBC&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc18"&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt; &lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt; &lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt; &lt;/span&gt;&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
   &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;This article summarises original research by Martin Gough. See &lt;a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');"&gt;this paper&lt;/a&gt; for a fuller account of bimodular approximants.&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>