Logarithmic approximants: Difference between revisions

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">WORK IN PROGRESS

=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

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 reasonably accurate approximation to 5-limit just intonation?

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

* Why does the 3-limit framework produce ~~satisfying~~ scale structures?

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

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

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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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The approximants of superparticular intervals are reciprocals of odd integers:

~~<Figure>^^^~~

[[image:Low-order superparticular intervals.png]]

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

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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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[[math]]

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

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)**=

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=**Quadratic approximants**=

==Definition==

==Definition==

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

[[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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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.

==~~ ~~==

== ==

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

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

===~~ ~~**Theorem**===

===**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.

===**Remarks**===

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.

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.

//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.

===**Proof**===

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which is the geometric mean of their frequency ratios.

===**Examples**===

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,

√2 = 1.41421.

The ratio of the large tone, //T// = __9/8__, to the small tone, //t// = __10/9__, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

//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.

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

###Perfect fourth = __4/3__ = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.

Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

[[math]]

\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))

[[math]]

As a result, if two intervals L and s are tuned in the silver ratio, with //s = L/δ//s, subtracting twice the small interval //s// from the large interval //L// leaves a remainder of size //s/δ//s:

[[math]]

\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):

|| Octave

1200.00

(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

1697.06

(1698.04) || Perfect fifth

702.94

(701.96) || Minor third

291.17

(294.13) || Apotome

120.61

(113.69) || 17-tone comma

49.96

(66.76) ||

where the limma is the Pythagorean semitone.

Thus for example:

###octave = 2×fourth + tone

###fourth = 2×tone + limma

###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:

* Subtracting twice an interval from the interval on its left generates the interval on its right.

* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.

* Adjacent horizontal pairs have ratio δs = √2 + 1.

* Adjacent vertical pairs have ratio √2.

* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.

The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor δs = √2 + 1 produces another interval in the temperament. Any tempered interval //J’// can be split into three parts, two of equal size //J’/////δ//s and the other of size //J’/////δs2//.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver mean produce ratios involving Pell numbers

###√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,

Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

###√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).

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 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).

[[image:Silver temperament graphic.png width="800" height="587"]]

^^^</pre></div>

^^^

= =

= 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~~<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

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 ~~satisfying~~ scale structures?</li></ul>

<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>

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

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

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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>

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

<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

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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The approximants of superparticular intervals are reciprocals of odd integers:

&lt;~~Figure~~&gt;~~^^^~~

<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="toc3"><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="toc4"><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="toc5"><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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--><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).

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="toc6"><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="toc7"><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="toc8"><a name="Quadratic approximants"></a>Quadratic approximants</h1>

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

<h2 id="toc9"><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:

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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="toc10"><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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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"><!-- ws:end:WikiTextHeadingRule:60 --~~> </span~~></h2>

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

<h3 id="toc13"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a><!-- ws:end:WikiTextHeadingRule:64 --~~> </span~~>Theorem</h3>

<h3 id="toc13"><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="toc14"><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.

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.

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="toc15"><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="toc16"><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,

√2 = 1.41421.

The ratio of the large tone, T = 9/8, to the small tone, t = 10/9, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.

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.

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

###Perfect fourth = 4/3 = 497.056 cents

This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).

A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.

Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.

The continued fraction expansion of the silver ratio has a particularly simple form:

<!-- ws:start:WikiTextMathRule:38:

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

\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))</script>

As a result, if two intervals L and s are tuned in the silver ratio, with s = L/δs, subtracting twice the small interval s from the large interval L leaves a remainder of size s/δs:

<!-- ws:start:WikiTextMathRule:39:

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

\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):

<tr>

<td>Octave

1200.00

(1200.00)

</td>

<td>Perfect fourth

497.06

(498.04)

</td>

<td>Tone

205.89

(203.91)

</td>

<td>Limma

85.28

(90.22)

</td>

<td>Pythag comma

35.32

(23.46)

</td>

</tr>

<tr>

<td>Perfect 11th

1697.06

(1698.04)

</td>

<td>Perfect fifth

702.94

(701.96)

</td>

<td>Minor third

291.17

(294.13)

</td>

<td>Apotome

120.61

(113.69)

</td>

<td>17-tone comma

49.96

(66.76)

</td>

</tr>

</table>

where the limma is the Pythagorean semitone.

Thus for example:

###octave = 2×fourth + tone

###fourth = 2×tone + limma

###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:

<ul><li>Subtracting twice an interval from the interval on its left generates the interval on its right.</li><li>An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.</li><li>Adjacent horizontal pairs have ratio δs = √2 + 1.</li><li>Adjacent vertical pairs have ratio √2.</li><li>Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.</li></ul>The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.

In this fractal temperament, multiplying or dividing any interval by the factor δs = √2 + 1 produces another interval in the temperament. Any tempered interval J’ can be split into three parts, two of equal size J’/δs and the other of size J’/δs2.

A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.

Successive convergents of the silver mean produce ratios involving Pell numbers

###√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,

Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers

###√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).

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 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;" />

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

^^^

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

<h1 id="toc19"><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 14:08:56 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 17:40:46 UTC</tt>.<br>
-: The original revision id was <tt>541592976</tt>.<br>
+: The original revision id was <tt>541614438</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//[[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;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 reasonably accurate 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 satisfying scale structures?&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;
 The exact size, in cents, of an interval with frequency ratio //r// is
@@ 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; 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 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; font-size: 15px;"&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 90: / Line 90: @@
 The approximants of superparticular intervals are reciprocals of odd integers:
-&lt;Figure&gt;^^^
+[[image:Low-order superparticular intervals.png]]
 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;//,
@@ 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; 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 &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 155: / Line 155: @@
 [[math]]
-Further examples of bimodular commas are provided in Reference 1^^^. See also __Don Page comma^^^__ (another name for this type of comma).
+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;**=
@@ 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]]
@@ Line 244: / Line 244: @@
 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 283: / Line 283: @@
 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.
-==&lt;span style="font-family: Times New Roman;"&gt; &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;==
+==&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: Times New Roman;"&gt; &lt;/span&gt;**Theorem**===
+===**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.
 ===**Remarks**===
-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.
+If the harmonics have indices //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n – m, n&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n + m&lt;/span&gt;//, the two intervals have reduced frequency ratios //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n/(n – m)&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(n + m)/n&lt;/span&gt;//. It can be assumed that //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// have no common factor.
-m is the epimoricity of the intervals. When m = 1 the intervals are adjacent superparticular intervals.
+//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// is the epimoricity of the intervals. When //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// = 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.
 ===**Proof**===
@@ Line 298: / Line 298: @@
 which is the geometric mean of their frequency ratios.
 ===**Examples**===
+The ratio of the perfect fifth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//F// = __3/2__&lt;/span&gt;, to the perfect fourth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//f// = __4/3__&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;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//F/f// = 701.955/498.045 = 1.40942,&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 = 1.41421.&lt;/span&gt;
+The ratio of the large tone, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//T// = __9/8__&lt;/span&gt;, to the small tone, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//t// = __10/9__&lt;/span&gt;, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.
+&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.
+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
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fourth = __4/3__ = 497.056 cents
+This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).
+A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.
+Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.
+The continued fraction expansion of the silver ratio has a particularly simple form:
+[[math]]
+\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))
+[[math]]
+As a result, if two intervals L and s are tuned in the silver ratio, with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//s = L/δ//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;, subtracting twice the small interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s&lt;/span&gt;// from the large interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;L&lt;/span&gt;// leaves a remainder of size &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//s/δ//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;:
+[[math]]
+\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):
+|| Octave
+.00
+(1200.00) || Perfect fourth
+.06
+(498.04) || Tone
+.89
+(203.91) || Limma
+.28
+(90.22) || Pythag comma
+.32
+(23.46) ||
+|| Perfect 11th
+.06
+(1698.04) || Perfect fifth
+.94
+(701.96) || Minor third
+.17
+(294.13) || Apotome
+.61
+(113.69) || 17-tone comma
+.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;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;tone = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;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:
+* Subtracting twice an interval from the interval on its left generates the interval on its right.
+* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
+* Adjacent horizontal pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1.&lt;/span&gt;
+* Adjacent vertical pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2&lt;/span&gt;.
+* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.
+The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.
+In this fractal temperament, multiplying or dividing any interval by the factor &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1&lt;/span&gt; produces another interval in the temperament. Any tempered interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;// can be split into three parts, two of equal size //&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%;"&gt;/&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt; and the other of size //&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%;"&gt;/&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;//.
+A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.
+Successive convergents of the silver mean produce ratios involving Pell numbers
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,
+Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,
+This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).
+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 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).
+[[image:Silver temperament graphic.png width="800" height="587"]]
-^^^</pre></div>
+^^^
+= =
+=&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,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:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&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;&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;  (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 satisfying 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 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;
 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 337: / Line 411: @@
 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;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&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: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:42:&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:42 --&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; font-size: 15px;"&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:
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   --&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;span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;"&gt;Properties&lt;/span&gt;&lt;br /&gt;
 When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.&lt;br /&gt;
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
@@ Line 398: / Line 472: @@
 &lt;br /&gt;
 The approximants of superparticular intervals are reciprocals of odd integers:&lt;br /&gt;
-&amp;lt;Figure&amp;gt;^^^&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;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 409: / Line 483: @@
 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:44:&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:44 --&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: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;
   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 424: / Line 498: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:46:&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:46 --&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: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;
   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 462: / Line 536: @@
   --&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:48:&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:48 --&gt;Examples&lt;/h3&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;
   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;
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   --&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;br /&gt;
+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:50:&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:50 --&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: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:52:&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:52 --&gt;Definition&lt;/h2&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;
   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;
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 (&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:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&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: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:56:&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:56 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&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;
   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;
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 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:58:&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:58 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&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-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;
   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;
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 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:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;/h2&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:62:&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:62 --&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: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:64:&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:64 --&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/h3&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;
   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:66:&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:66 --&gt;&lt;strong&gt;Remarks&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-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;&lt;strong&gt;Remarks&lt;/strong&gt;&lt;/h3&gt;
-  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.&lt;br /&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;
-m is the epimoricity of the intervals. When m = 1 the intervals are adjacent superparticular intervals.&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:68:&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:68 --&gt;&lt;strong&gt;Proof&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-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&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;
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   --&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:70:&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:70 --&gt;&lt;strong&gt;Examples&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-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/h3&gt;
-  &lt;br /&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;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 = 1.41421.&lt;/span&gt;&lt;br /&gt;
+The ratio of the large tone, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;T&lt;/em&gt; = &lt;u&gt;9/8&lt;/u&gt;&lt;/span&gt;, to the small tone, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;t&lt;/em&gt; = &lt;u&gt;10/9&lt;/u&gt;&lt;/span&gt;, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.&lt;br /&gt;
+&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;
+ 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;
+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;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fourth = &lt;u&gt;4/3&lt;/u&gt; = 497.056 cents&lt;br /&gt;
+This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).&lt;br /&gt;
+A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.&lt;br /&gt;
+Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.&lt;br /&gt;
+The continued fraction expansion of the silver ratio has a particularly simple form:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:38:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:38 --&gt;&lt;br /&gt;
+As a result, if two intervals L and s are tuned in the silver ratio, with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;s = L/δ&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;, subtracting twice the small interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s&lt;/span&gt;&lt;/em&gt; from the large interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;L&lt;/span&gt;&lt;/em&gt; leaves a remainder of size &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;s/δ&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:39:
+[[math]]&amp;lt;br/&amp;gt;
+\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;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Octave&lt;br /&gt;
+.00&lt;br /&gt;
+(1200.00)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Perfect fourth &lt;br /&gt;
+.06&lt;br /&gt;
+(498.04)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Tone&lt;br /&gt;
+.89&lt;br /&gt;
+(203.91)&lt;br /&gt;
+&lt;/td&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;
+.32&lt;br /&gt;
+(23.46)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Perfect 11th&lt;br /&gt;
+.06&lt;br /&gt;
+(1698.04)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Perfect fifth&lt;br /&gt;
+.94&lt;br /&gt;
+(701.96)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Minor third&lt;br /&gt;
+.17&lt;br /&gt;
+(294.13)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;Apotome&lt;br /&gt;
+.61&lt;br /&gt;
+(113.69)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;17-tone comma&lt;br /&gt;
+.96&lt;br /&gt;
+(66.76)&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&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;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;tone = 2&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;×&lt;/span&gt;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;
+&lt;ul&gt;&lt;li&gt;Subtracting twice an interval from the interval on its left generates the interval on its right.&lt;/li&gt;&lt;li&gt;An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.&lt;/li&gt;&lt;li&gt;Adjacent horizontal pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1.&lt;/span&gt;&lt;/li&gt;&lt;li&gt;Adjacent vertical pairs have ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2&lt;/span&gt;.&lt;/li&gt;&lt;li&gt;Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.&lt;/li&gt;&lt;/ul&gt;The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.&lt;br /&gt;
+In this fractal temperament, multiplying or dividing any interval by the factor &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= √2 + 1&lt;/span&gt; produces another interval in the temperament. Any tempered interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J’&lt;/span&gt;&lt;/em&gt; can be split into three parts, two of equal size &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%;"&gt;/&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt; and the other of size &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%;"&gt;/&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;s&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;/em&gt;.&lt;br /&gt;
+A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.&lt;br /&gt;
+Successive convergents of the silver mean produce ratios involving Pell numbers&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,&lt;br /&gt;
+Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,&lt;br /&gt;
+This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).&lt;br /&gt;
+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;
+&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;br /&gt;
 &lt;br /&gt;
-^^^&lt;/body&gt;&lt;/html&gt;</pre></div>
+^^^&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;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;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>