Logarithmic approximants: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...

[[math]]

The function //v(r)// is the order (1,1) Padé approximant of the function //J(r) =//½ ln //r// in the region of //r// = 1, which has the property of matching the function value and its first and second derivatives at this value of //r//. The bimodular approximant function is thus accurate to second order in //r// – 1.

The function //v(r)// is the order (1,1) Padé approximant of the function //J(r) =//½ ln //r// in the region of //r// = 1, which has the property of matching the function value and its first and second derivatives at this value of //r//. The bimodular approximant function is thus accurate to second order in //r// – 1.

As an example, the size of the perfect fifth (in dNp units) is

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\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...

[[math]]

The bimodular approximant for this interval (r = 3/2) is

The bimodular approximant for this interval (//r// = 3/2) is

[[math]]

\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2

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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).

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[file:Bimod Approx 2014-6-8.pdf|this paper]]. See also [[Don Page comma]] (another name for this type of comma).

=**3. Padé approximants of order (1,2)**=

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While //v// is the frequency difference divided by twice the arithmetic frequency mean, //q// is the frequency difference divided by twice the geometric frequency mean:

[[math]]

\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}

\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}

[[math]]

//r// can be retrieved from //q// using

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###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’), //δ//s = √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.

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’), //δ//s = √2 + 1 = 2.4142. On this basis, and by analogy with [[Golden Meantone|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:

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\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:

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 = s/\delta_s

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###octave = 2×fourth + tone

###fourth = 2×tone + limma

###tone = 2×limma + Pythag

###tone = 2×limma + Pythag comma

###perfect 11th (__8/3__) = 2×fifth + minor third

###fifth = 2×(minor third) + apotome

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 horizontal pairs have ratio //δ////s// = √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~~ comparable magnitude – highly desirable properties for the formation of musical scales.

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 are clearly differentiated but of comparable 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//.

In this fractal temperament, multiplying or dividing any interval by the factor //δ////s// = √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

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The accuracy of the silver fifth means that the scheme produces 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 next column containing the 29-tone comma and //minus// the 41-tone comma).

Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

[[image:Continued fraction jigsaw 41edo.png width="800" height="396"]]

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######Figure 2. Continued fraction jigsaw for 41edo

Figure 3 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.

Figure 3 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"]]

######Figure 3. Geometrical representation of silver temperament

==Golden temperaments==

It has been shown in an example above that the ratio of the large tone (//T//= __9/8__) to the small tone (//t// =__10/9__) is closely approximated by

It has been shown in an example above that the ratio of the large tone (//T// = __9/8__) to the small tone (//t// = __10/9__) is closely approximated by

[[math]]

\qquad T/t = \sqrt{5}/2

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where //ϕ// = 1.61803... is the golden ratio.

If a Fibonacci sequence of intervals is formed from the pair of intervals //T// – //t///2 and //t//, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to //ϕ//. The sequence formed in this way is Sequence 1 in the following table.

|| Sequence 1:# || #//t///2 - 3//c//# || #2//c// || #//t///2 //- c// || #//T - t///2 || #//t// || #//T + t///2 || #//M + t///2 || #2//M// ||

|| Sequence 1:# || #//t///2 - 3//c//# || #2//c// || #//t///2 //- c// || #//T - t///2 || #//t// || #//T + t///2# || #//M + t///2# || #2//M// ||

|| Sequence 2:# || #//magic// || #//diesis// || //#chroma#// || #//semitone//# || #//t// || #//mp// || #//f - c// || #//m6p - c//# ||

|| Difference: || #-3//σ///2 || #//σ// || #-//σ///2 || #//σ///2 || #0 || #//σ///2 || #//σ///2 || #//σ// ||

|| Difference: || #-3//σ///2 || #//σ// || #-//σ///2 || #//σ///2 || #0 || #//σ///2 || #//σ///2 || #//σ// ||

|| Seq 1 ratios: || || #1.6120## || #1.6204 || #1.6171 || #1.6184# || #1.6179 || #1.6181 || #1.6180 ||

|| Seq 2 ratios: || || #1.3865 || #1.7212 || #1.5810 || #1.6325 || #1.6125 || #1.6201 || #1.6172 ||

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The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate //ϕ// rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly //ϕ// would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to //ϕ// while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.

Tempering the Sequence 2 ratios to //ϕ// while tuning the octave pure and tempering out the syntonic comma yields [[Golden Meantone|golden meantone]] temperament.

Tempering the Sequence 1 ratios to //ϕ// yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals //s, t//, //M// and //m//=__6/5__ are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at //ϕ// in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

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==A small 34edo comma==

As [[Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (‘//selenia//’) is remarkably small at just 0.004764 cents.~~~~

As [[Gene Ward Smith]] has noted, the 5-limit comma |-433 -137 280> (‘//selenia//’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

~~~~The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable [[Comma-based lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the //gammic// comma |-29 -11 20> (4.769 cents) and the //semisuper// comma |23 6 -14> (3.338 cents). In particular,

###//selenia// = 7 //gammic// – 10 //semisuper//

So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.

//Gammic// and //semisuper// are both ~~__bimodular commas__~~:

//Gammic// and //semisuper// are both bimodular commas:

###//gammic// = //b//(__6/5__,__5/4__)

###//semisuper// = //b//(__25/24__,__4/3__)

Using a result given in the section on bimodular commas, the size of ~~the bimodular comma~~ //b//(//J//1,//J//2) can be estimated using

Using a result given in the section on bimodular commas, the size of //b//(//J//1,//J//2) can be estimated using

[[math]]

\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m

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[[math]]

\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m

[[math]]

So to improve our estimates of //b//(//J//1,//J//2) we should multiply them by

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\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}

[[math]]

from which it follows that

###//selenia// = 7 //gammic// - 10 //semisuper//

######## ≈ 7 //gammic// (//f////gammic// - //f////semisuper~~//)~~///fgammic//

######## ≈ 7 //gammic// (//f////gammic - fsemisuper)/fgammic//

Putting in the numbers:

//###f//~~//gammic// =~~ 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120

//###fgammic //= 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120

###//f///~~/semisuper//~~ = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)

###//fsemisuper //= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)

###//fgammic - fsemisuper //= 1/6000

###//fgammic - fsemisuper //= 1/6000

Therefore

###//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561 cents

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and (//q//(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.

=Source=

=Source=

This article is based on 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>

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

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

The function v(r) is the order (1,1) Padé approximant of the function J(r) =½ ln r in the region of r = 1, which has the property of matching the function value and its first and second derivatives at this value of r. The bimodular approximant function is thus accurate to second order in r – 1.

The function v(r) is the order (1,1) Padé approximant of the function J(r) =½ ln r in the region of r = 1, which has the property of matching the function value and its first and second derivatives at this value of r. The bimodular approximant function is thus accurate to second order in r – 1.

As an example, the size of the perfect fifth (in dNp units) is

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\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...</script>

The bimodular approximant for this interval (r = 3/2) is

The bimodular approximant for this interval (r = 3/2) is

<!-- ws:start:WikiTextMathRule:10:

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

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--><script type="math/tex">\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f</script>

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

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to <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>. See also <a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a> (another name for this type of comma).

<h1 id="toc7"><a name="x3. Padé approximants of order (1,2)"></a>3. Padé approximants of order (1,2)</h1>

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<!-- ws:start:WikiTextMathRule:28:

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

\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;br/&gt;[[math]]

\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}</script>

--><script type="math/tex">\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}</script>

r can be retrieved from q using

<!-- ws:start:WikiTextMathRule:29:

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###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.

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 <a class="wiki_link" href="/Golden%20Meantone">golden meantone</a> 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:

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\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:

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;

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###octave = 2×fourth + tone

###fourth = 2×tone + limma

###tone = 2×limma + Pythag

###tone = 2×limma + Pythag comma

###perfect 11th (8/3) = 2×fifth + minor third

###fifth = 2×(minor third) + apotome

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~~ comparable magnitude – highly desirable properties for the formation of musical scales.

<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 are clearly differentiated but of comparable 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.

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

Line 1,043:

Line 1,045:

The accuracy of the silver fifth means that the scheme produces 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 next column containing the 29-tone comma and minus the 41-tone comma).

Figure 2 is a continued fraction jigsaw showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

Figure 2 is a continued fraction jigsaw showing the sizes of the octave (o), fourth (f), tone (T), limma (sp), Pythagorean comma (p) and 29-tone comma (p29) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

<img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" />

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######Figure 2. Continued fraction jigsaw for 41edo

Figure 3 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.

Figure 3 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;" />

######Figure 3. Geometrical representation of silver temperament

<h2 id="toc19"><a name="x4. Quadratic approximants-Golden temperaments"></a>Golden temperaments</h2>

It has been shown in an example above that the ratio of the large tone (T= 9/8) to the small tone (t =10/9) is closely approximated by

It has been shown in an example above that the ratio of the large tone (T = 9/8) to the small tone (t = 10/9) is closely approximated by

<!-- ws:start:WikiTextMathRule:40:

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

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<td>#t

</td>

<td>#T + t/2

<td>#T + t/2#

</td>

<td>#M + t/2

<td>#M + t/2#

</td>

<td>#2M

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<td>#-σ/2

</td>

<td>#σ/2

<td>#σ/2

</td>

<td>#0

<td>#0

</td>

<td>#σ/2

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The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate ϕ rather less accurately.

A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly ϕ would be ‘golden temperaments’.

Tempering the Sequence 2 ratios to ϕ while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.

Tempering the Sequence 2 ratios to ϕ while tuning the octave pure and tempering out the syntonic comma yields <a class="wiki_link" href="/Golden%20Meantone">golden meantone</a> temperament.

Tempering the Sequence 1 ratios to ϕ yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals s, t, M and m=6/5 are all ±0.02106 cents.

Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at ϕ in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

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<h2 id="toc21"><a name="x4. Quadratic approximants-A small 34edo comma"></a>A small 34edo comma</h2>

As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the 5-limit comma |-433 -137 280&gt; (‘selenia’) is remarkably small at just 0.004764 cents.~~ ~~

As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the 5-limit comma |-433 -137 280&gt; (‘selenia’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using quadratic approximants.

~~~~The minute size of this comma can be explained using quadratic approximants.

It can be shown, using a suitable <a class="wiki_link" href="/Comma-based%20lattices">comma-based lattice</a>, that every comma tempered out by 34edo can be expressed as an integer linear combination of the gammic comma |-29 -11 20&gt; (4.769 cents) and the semisuper comma |23 6 -14&gt; (3.338 cents). In particular,

###selenia = 7 gammic – 10 semisuper

So to prove that selenia is small we must show that gammic/semisuper ≈ 10/7.

Gammic and semisuper are both ~~~~bimodular commas~~~~:

Gammic and semisuper are both bimodular commas:

###gammic = b(6/5,5/4)

###semisuper = b(25/24,4/3)

Using a result given in the section on bimodular commas, the size of ~~the bimodular comma~~ b(J1,J2) can be estimated using

Using a result given in the section on bimodular commas, the size of b(J1,J2) can be estimated using

<!-- ws:start:WikiTextMathRule:43:

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

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

\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\&lt;br /&gt;

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m&lt;br/&gt;[[math]]

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m</script>

\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m</script>

So to improve our estimates of b(J1,J2) we should multiply them by

<!-- ws:start:WikiTextMathRule:46:

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\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}</script>

~~ ~~

from which it follows that

###selenia = 7 gammic - 10 semisuper

######## ≈ 7 gammic (fgammic</span~~></em~~> - ~~~~f~~~~<em~~>semisuper~~)~~/</span~~><em>fgammic

######## ≈ 7 gammic (fgammic - fsemisuper)/fgammic

Putting in the numbers:

###f~~gammic~~<~~/em~~> = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120

###fgammic = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120

###f~~semisuper~~<~~/em~~> = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)

###fsemisuper = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)

###fgammic - fsemisuper = 1/6000

###fgammic - fsemisuper = 1/6000

Therefore

###selenia ≈ 7 gammic (1/6000) (120/119) = gammic/850 = 0.00561 cents

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and (q(25/24))2 , being small in comparison to the other terms, compromises this equality only slightly.

<h1 id="toc22"><a name="Source"></a>Source</h1>

<h1 id="toc22"><a name="Source"></a>Source</h1>

This article is based on 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>2015-02-21 12:57:21 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-21 16:01:05 UTC</tt>.<br>
-: The original revision id was <tt>541651356</tt>.<br>
+: The original revision id was <tt>541661880</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>
@@ Line 73: / Line 73: @@
 \qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...
 [[math]]
-The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v(r)//&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J(r) =//½ ln //r// &lt;/span&gt; in the region of //r// = 1, which has the property of matching the function value and its first and second derivatives at this value of //r//. The bimodular approximant function is thus accurate to second order in //r// – 1.
+The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v(r)//&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J(r) =//½ ln //r// &lt;/span&gt; in the region of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 1&lt;/span&gt;, which has the property of matching the function value and its first and second derivatives at this value of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//. The bimodular approximant function is thus accurate to second order in &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// – 1&lt;/span&gt;.
 As an example, the size of the perfect fifth (in dNp units) is
@@ Line 79: / Line 79: @@
 \qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...
 [[math]]
-The bimodular approximant for this interval (r = 3/2) is
+The bimodular approximant for this interval (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 3/2&lt;/span&gt;) is
 [[math]]
 \qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2
@@ 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).
+For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to [[file:Bimod Approx 2014-6-8.pdf|this paper]]. See also [[Don Page comma]] (another name for this type of comma).
 =**&lt;span style="font-size: 21.33px;"&gt;3. Padé approximants of order (1,2)&lt;/span&gt;**=
@@ Line 212: / Line 212: @@
 While //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// is the frequency difference divided by twice the arithmetic frequency mean, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// is the frequency difference divided by twice the geometric frequency mean:
 [[math]]
-\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
+\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
 [[math]]
 //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;// can be retrieved from //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// using
@@ Line 313: / Line 313: @@
 &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’), //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;//&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. 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.
+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’), //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;//&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. On this basis, and by analogy with [[Golden Meantone|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:
@@ Line 319: / Line 319: @@
 \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;:
+As a result, if two intervals //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;L&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s&lt;/span&gt;// 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 = s/\delta_s
@@ Line 349: / Line 349: @@
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;octave = 2×fourth + tone
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fourth = 2×tone + limma
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag comma
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;perfect 11th (__8/3__) = 2×fifth + minor third
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fifth = 2×(minor third) + apotome
 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 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 comparable magnitude – highly desirable properties for the formation of musical scales.
+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 are clearly differentiated but of comparable 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;//.
+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
@@ Line 367: / Line 369: @@
 The accuracy of the silver fifth means that the scheme produces 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 next column containing the 29-tone comma and //minus// the 41-tone comma).
-Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.
+Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.
 [[image:Continued fraction jigsaw 41edo.png width="800" height="396"]]
@@ Line 373: / Line 375: @@
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 2. Continued fraction jigsaw for 41edo
-Figure 3 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, m&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= pythag minor third, s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= limma, X&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= apotome, p = Pythagorean comma.
+Figure 3 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, m&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= pythag minor third, s&lt;span style="font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= limma, X&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= apotome, p = Pythagorean comma.
 [[image:Silver temperament graphic.png width="800" height="587"]]
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of silver temperament
 ==Golden temperaments==
-It has been shown in an example above that the ratio of the large tone (//T//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= __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;) is closely approximated by
+It has been shown in an example above that the ratio of the large tone (//T// &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= __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;) is closely approximated by
 [[math]]
 \qquad T/t = \sqrt{5}/2
@@ Line 388: / Line 390: @@
 where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//ϕ// = 1.61803&lt;/span&gt;... is the golden ratio.
 If a Fibonacci sequence of intervals is formed from the pair of intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//T// – //t///2&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;t&lt;/span&gt;//, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;//. The sequence formed in this way is Sequence 1 in the following table.
-|| Sequence 1:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//t///2 - 3//c//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t///2 //- c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T - t///2 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T + t///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//M + t///2 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//M// &lt;/span&gt; ||
+|| Sequence 1:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//t///2 - 3//c//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t///2 //- c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T - t///2 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//T + t///2&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//M + t///2&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2//M// &lt;/span&gt; ||
 || Sequence 2:&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//magic//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//diesis//&lt;/span&gt; || //&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;chroma&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;// || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//semitone//&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//t//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//mp//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//f - c//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//m6p - c//&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; ||
-|| Difference: || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-//σ///2&lt;/span&gt; || &lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;//σ///2 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;0 || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ// &lt;/span&gt; ||
+|| Difference: || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-3//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ//&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;span style="font-family: "Palatino Linotype","Book Antiqua",Palatino,serif; font-size: 90%;"&gt;0&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ///2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;//σ// &lt;/span&gt; ||
 || Seq 1 ratios: ||   || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6120&lt;/span&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;##&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6204&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6171&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6184&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6179&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6181&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6180&lt;/span&gt; ||
 || Seq 2 ratios: ||   || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.3865&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.7212&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.5810&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6325&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6125&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6201 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;1.6172 &lt;/span&gt; ||
@@ Line 398: / Line 400: @@
 The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// rather less accurately.
 A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// would be ‘golden temperaments’.
-Tempering the Sequence 2 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.
+Tempering the Sequence 2 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// while tuning the octave pure and tempering out the syntonic comma yields [[Golden Meantone|golden meantone]] temperament.
 Tempering the Sequence 1 ratios to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t&lt;/span&gt;//, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;// and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//m//=__6/5__&lt;/span&gt; are all ±0.02106 cents.
 Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;// in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).
@@ Line 415: / Line 417: @@
 ==A small 34edo comma==
-&lt;span style="color: #333333;"&gt;As [[Gene Ward Smith]] has noted, the &lt;/span&gt;5-limit comma &lt;span style="color: #333333;"&gt;|-433 -137 280&gt; (‘//selenia//’) is remarkably small at just 0.004764 cents.&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;As [[Gene Ward Smith]] has noted, the &lt;/span&gt;5-limit comma &lt;span style="color: #333333;"&gt;|-433 -137 280&gt; (‘//selenia//’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.
-&lt;span style="color: #333333;"&gt;The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.
 It can be shown, using a suitable [[Comma-based lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the //gammic// comma |-29 -11 20&gt; (4.769 cents) and the //semisuper// comma |23 6 -14&gt; (3.338 cents). In particular,
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//selenia// = 7 //gammic// – 10 //semisuper//&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.&lt;/span&gt;
-&lt;span style="color: #333333;"&gt;//Gammic// and //semisuper// are both __bimodular commas__:&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;//Gammic// and //semisuper// are both bimodular commas:&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__6/5__,__5/4__)&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//b//(__25/24__,__4/3__)&lt;/span&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of the bimodular comma&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //b//(//J//1,//J//2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//1,//J//2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;
 [[math]]
 \qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m
@@ Line 448: / Line 449: @@
 [[math]]
 \qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
-\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m
+\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m
 [[math]]
 &lt;span style="color: #333333;"&gt;So to improve our estimates of &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//1,//J//2)&lt;/span&gt; &lt;span style="color: #333333;"&gt;we should multiply them by&lt;/span&gt;
@@ Line 458: / Line 459: @@
 \qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}
 [[math]]
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;from which it follows that&lt;/span&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//selenia// = 7 //gammic// - 10 //semisuper//&lt;/span&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 //gammic//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (//f//&lt;/span&gt;&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - //f//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//semisuper//)&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;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;//
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 //gammic//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (//f//&lt;/span&gt;&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper&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: Georgia,serif; font-size: 110%;"&gt;/f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt;//
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers:&lt;/span&gt;
-//&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//gammic// = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;
+//&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic &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%; vertical-align: sub;"&gt;1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;//semisuper// = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper &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: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/6000&lt;/span&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt; &lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper &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;1/6000&lt;/span&gt;
 &lt;span style="color: #333333;"&gt;Therefore&lt;/span&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;
@@ Line 475: / Line 475: @@
 and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(//q//(25/24))&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; , being small in comparison to the other terms, compromises this equality only slightly.
-=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Source&lt;/span&gt;=
+=&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Source&lt;/span&gt;=
 This article is based on 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>
@@ Line 553: / Line 553: @@
 \qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:8 --&gt;&lt;br /&gt;
-The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v(r)&lt;/em&gt;&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J(r) =&lt;/em&gt;½ ln &lt;em&gt;r&lt;/em&gt; &lt;/span&gt; in the region of &lt;em&gt;r&lt;/em&gt; = 1, which has the property of matching the function value and its first and second derivatives at this value of &lt;em&gt;r&lt;/em&gt;. The bimodular approximant function is thus accurate to second order in &lt;em&gt;r&lt;/em&gt; – 1.&lt;br /&gt;
+The function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v(r)&lt;/em&gt;&lt;/span&gt; is the order (1,1) Padé approximant of the function &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J(r) =&lt;/em&gt;½ ln &lt;em&gt;r&lt;/em&gt; &lt;/span&gt; in the region of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 1&lt;/span&gt;, which has the property of matching the function value and its first and second derivatives at this value of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;. The bimodular approximant function is thus accurate to second order in &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; – 1&lt;/span&gt;.&lt;br /&gt;
 &lt;br /&gt;
 As an example, the size of the perfect fifth (in dNp units) is&lt;br /&gt;
@@ Line 560: / Line 560: @@
 \qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:9 --&gt;&lt;br /&gt;
-The bimodular approximant for this interval (r = 3/2) is&lt;br /&gt;
+The bimodular approximant for this interval (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 3/2&lt;/span&gt;) is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:10:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 649: / Line 649: @@
   --&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;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;
+For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to &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;. 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:63:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="x3. Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;3. Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
@@ Line 756: / Line 756: @@
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 [[math]]&amp;lt;br/&amp;gt;
-\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&amp;lt;br/&amp;gt;[[math]]
+\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:28 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:28 --&gt;&lt;br /&gt;
 &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt; can be retrieved from &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; using&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:29:
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 &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’), &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;/em&gt;&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. 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;
+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’), &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;δ&lt;/span&gt;&lt;/em&gt;&lt;span style="vertical-align: sub;"&gt;s &lt;/span&gt;= &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√2 + 1 = 2.4142&lt;/span&gt;. On this basis, and by analogy with &lt;a class="wiki_link" href="/Golden%20Meantone"&gt;golden meantone&lt;/a&gt; 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;
@@ Line 972: / Line 972: @@
 \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;
+As a result, if two intervals &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;L&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s&lt;/span&gt;&lt;/em&gt; 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;
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@@ Line 1,030: / Line 1,030: @@
 &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×tone + limma&lt;br /&gt;
-&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;tone = 2×limma + Pythag comma&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;perfect 11th (&lt;u&gt;8/3&lt;/u&gt;) = 2×fifth + minor third&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;fifth = 2×(minor third) + apotome&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 comparable magnitude – highly desirable properties for the formation of musical scales.&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;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;&lt;em&gt;s&lt;/em&gt; &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 are clearly differentiated but of comparable 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;
+In this fractal temperament, multiplying or dividing any interval by the factor &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;&lt;em&gt;s&lt;/em&gt; &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;
@@ Line 1,043: / Line 1,045: @@
 The accuracy of the silver fifth means that the scheme produces 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 next column containing the 29-tone comma and &lt;em&gt;minus&lt;/em&gt; the 41-tone comma).&lt;br /&gt;
 &lt;br /&gt;
-Figure 2 is a &lt;em&gt;continued fraction jigsaw&lt;/em&gt; showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.&lt;br /&gt;
+Figure 2 is a &lt;em&gt;continued fraction jigsaw&lt;/em&gt; showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to silver temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:482:&amp;lt;img src=&amp;quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 396px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:482 --&gt;&lt;br /&gt;
@@ Line 1,049: / Line 1,051: @@
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 2. Continued fraction jigsaw for 41edo&lt;br /&gt;
 &lt;br /&gt;
-Figure 3 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, m&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= pythag minor third, s&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= limma, X&lt;span style="vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= apotome, p = Pythagorean comma.&lt;br /&gt;
+Figure 3 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, m&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= pythag minor third, s&lt;span style="font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= limma, X&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= apotome, p = Pythagorean comma.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:483:&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:483 --&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of silver temperament&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc19"&gt;&lt;a name="x4. Quadratic approximants-Golden temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;Golden temperaments&lt;/h2&gt;
-  It has been shown in an example above that the ratio of the large tone (&lt;em&gt;T&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&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;) is closely approximated by&lt;br /&gt;
+  It has been shown in an example above that the ratio of the large tone (&lt;em&gt;T&lt;/em&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&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;) is closely approximated by&lt;br /&gt;
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@@ Line 1,082: / Line 1,084: @@
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;t&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T + t&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;T + t&lt;/em&gt;/2&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;M + t&lt;/em&gt;/2 &lt;/span&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;M + t&lt;/em&gt;/2&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt; &lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;2&lt;em&gt;M&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
@@ Line 1,118: / Line 1,120: @@
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;-&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;br /&gt;
+         &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;0&lt;br /&gt;
+         &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;span style="font-family: "Palatino Linotype","Book Antiqua",Palatino,serif; font-size: 90%;"&gt;0&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;#&lt;/span&gt;&lt;em&gt;σ&lt;/em&gt;/2&lt;/span&gt;&lt;br /&gt;
@@ Line 1,176: / Line 1,178: @@
 The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; rather less accurately.&lt;br /&gt;
 A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; would be ‘golden temperaments’.&lt;br /&gt;
-Tempering the Sequence 2 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; while tuning the octave pure and tempering out the syntonic comma yields golden meantone temperament.&lt;br /&gt;
+Tempering the Sequence 2 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; while tuning the octave pure and tempering out the syntonic comma yields &lt;a class="wiki_link" href="/Golden%20Meantone"&gt;golden meantone&lt;/a&gt; temperament.&lt;br /&gt;
 Tempering the Sequence 1 ratios to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t&lt;/span&gt;&lt;/em&gt;, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;&lt;/em&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;m&lt;/em&gt;=&lt;u&gt;6/5&lt;/u&gt;&lt;/span&gt; are all ±0.02106 cents.&lt;br /&gt;
 Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt; in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).&lt;br /&gt;
@@ Line 1,275: / Line 1,277: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:91:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc21"&gt;&lt;a name="x4. Quadratic approximants-A small 34edo comma"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:91 --&gt;A small 34edo comma&lt;/h2&gt;
-  &lt;span style="color: #333333;"&gt;As &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; has noted, the &lt;/span&gt;5-limit comma &lt;span style="color: #333333;"&gt;|-433 -137 280&amp;gt; (‘&lt;em&gt;selenia&lt;/em&gt;’) is remarkably small at just 0.004764 cents.&lt;/span&gt;&lt;br /&gt;
+  &lt;span style="color: #333333;"&gt;As &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; has noted, the &lt;/span&gt;5-limit comma &lt;span style="color: #333333;"&gt;|-433 -137 280&amp;gt; (‘&lt;em&gt;selenia&lt;/em&gt;’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;The minute size of this comma can be explained using qu&lt;/span&gt;adratic approximants.&lt;br /&gt;
 It can be shown, using a suitable &lt;a class="wiki_link" href="/Comma-based%20lattices"&gt;comma-based lattice&lt;/a&gt;, that every comma tempered out by 34edo can be expressed as an integer linear combination of the &lt;em&gt;gammic&lt;/em&gt; comma |-29 -11 20&amp;gt; (4.769 cents) and the &lt;em&gt;semisuper&lt;/em&gt; comma |23 6 -14&amp;gt; (3.338 cents). In particular,&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;selenia&lt;/em&gt; = 7 &lt;em&gt;gammic&lt;/em&gt; – 10 &lt;em&gt;semisuper&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;So to prove that &lt;em&gt;selenia&lt;/em&gt; is small we must show that &lt;em&gt;gammic&lt;/em&gt;&lt;em&gt;/semisuper&lt;/em&gt; ≈ 10/7.&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #333333;"&gt;&lt;em&gt;Gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are both &lt;u&gt;bimodular commas&lt;/u&gt;:&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;&lt;em&gt;Gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are both bimodular commas:&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;6/5&lt;/u&gt;,&lt;u&gt;5/4&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;25/24&lt;/u&gt;,&lt;u&gt;4/3&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of the bimodular comma&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;1,&lt;em&gt;J&lt;/em&gt;2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;1,&lt;em&gt;J&lt;/em&gt;2)&lt;/span&gt;&lt;span style="color: #333333;"&gt; can be estimated using&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:43:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 1,311: / Line 1,312: @@
 [[math]]&amp;lt;br/&amp;gt;
 \qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\&amp;lt;br /&amp;gt;
-\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m&amp;lt;br/&amp;gt;[[math]]
+\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
-\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{15} (q_1^2 + q_2^2) ) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:45 --&gt;&lt;br /&gt;
+\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:45 --&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;So to improve our estimates of &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;em&gt;J&lt;/em&gt;1,&lt;em&gt;J&lt;/em&gt;2)&lt;/span&gt; &lt;span style="color: #333333;"&gt;we should multiply them by&lt;/span&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:46:
@@ Line 1,324: / Line 1,325: @@
 \qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:47 --&gt;&lt;br /&gt;
-&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;from which it follows that&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;selenia&lt;/em&gt; = 7 &lt;em&gt;gammic&lt;/em&gt; - 10 &lt;em&gt;semisuper&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 &lt;em&gt;gammic&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (&lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - &lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;semisuper&lt;/em&gt;)&lt;/span&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;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;/em&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;######## &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 &lt;em&gt;gammic&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt; (&lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; - f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper&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: Georgia,serif; font-size: 110%;"&gt;/f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt;&lt;/em&gt;&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers:&lt;/span&gt;&lt;br /&gt;
-&lt;em&gt;&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;gammic&lt;/em&gt; = 1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;&lt;br /&gt;
+&lt;em&gt;&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic &lt;/span&gt;&lt;/em&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%; vertical-align: sub;"&gt;1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;&lt;em&gt;semisuper&lt;/em&gt; = 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;=&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)&lt;/span&gt;&lt;br /&gt;
-&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;gammic &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/6000&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;gammic&lt;/span&gt; &lt;span style="font-family: Georgia,serif;"&gt;- &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;span style="font-size: 70%; vertical-align: sub;"&gt;semisuper &lt;/span&gt;&lt;/em&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;1/6000&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #333333;"&gt;Therefore&lt;/span&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;selenia&lt;/em&gt; ≈ 7 &lt;em&gt;gammic&lt;/em&gt; (1/6000) (120/119) = &lt;em&gt;gammic&lt;/em&gt;/850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;&lt;br /&gt;
@@ Line 1,342: / Line 1,342: @@
 and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(&lt;em&gt;q&lt;/em&gt;(25/24))&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; , being small in comparison to the other terms, compromises this equality only slightly.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
   This article is based on 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;/body&gt;&lt;/html&gt;</pre></div>