Logarithmic approximants: Difference between revisions

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<h4>Original Wikitext content:</h4>

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

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

=Introduction=

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

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

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* Quadratic approximants

=**Bimodular approximants**=

=**2. Bimodular approximants**=

==Definition==

==Definition==

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

[[math]]

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[[math]]

==Properties==

==Properties==

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

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

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[[math]]

The approximants of superparticular intervals are reciprocals of odd integers:

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

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

######Figure 1. Bimodular approximants for low-order superparticular intervals

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

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

==**Bimodular approximants and equal temperaments**==

==Bimodular approximants and equal temperaments==

While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

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

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

==Bimodular commas==

==Bimodular commas==

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

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

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

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

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

==Definition==

==Definition==

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

[[math]]

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(__3/2__) / (__20/17__) = 2.4949 ≈ (15/74) / (6/74) = 5/2

=**Quadratic approximants**=

=**4. Quadratic approximants**=

==Definition==

==Definition==

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

[[math]]

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

==Properties==

==Properties==

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

[[math]]

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[[math]]

The quadratic approximant //q// of a double interval 2//J// (for example, the ditone) is rational, which suggests using ½ q(r2) as a rational approximant of //J// (where //J// has frequency ratio //r//):

The quadratic approximant //q// of a double interval 2//J// (for example, the ditone) is rational, which suggests using ½ //q//(//r//2) as a rational approximant of //J// (where //J// has frequency ratio //r//):

[[math]]

\qquad \tfrac{1}{2} q(r^2) = \tfrac{1}{4} (r - \frac{1}{r}) = \tfrac{1}{2} \sinh{2J} = J + \tfrac{2}{3}J^3 + \tfrac{2}{15}J^5 + ...

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

== ==

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

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

===**Theorem**===

===Theorem===

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

===**Remarks**===

===Remarks===

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

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

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

===**Proof**===

===Proof===

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

[[math]]

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[[math]]

which is the geometric mean of their frequency ratios.

===**Examples**===

===Examples===

The ratio of the perfect fifth, //F// = __3/2__, to the perfect fourth, //f// = __4/3__, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).

//F/f// = 701.955/498.045 = 1.40942,

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√5/2 = 1.11803.

==**Silver temperament**==

==Silver temperament==

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

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

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

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

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

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

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

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

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

The 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).

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

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="~~395~~"]]

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

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

######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.

[[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

[[math]]

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

[[math]]

It follows that

[[math]]

\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi

[[math]]

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 - 3c//# || #//2////c// || #//t/2 - c// || #//T - t/2// || #//t// || #//T + t/2// || #//M + t/2// || #//2M// ||

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

|| Difference: || #-3σ/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 ||

where f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/327__68.__

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to //ϕ//.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (//σ//), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).

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 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 error in the intervals //s, t, m// and //M// 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).

==Pythagorean triples of quadratic approximants==

If the quadratic approximants //q//1//, q//2 and //q//3 of a set of three intervals //J//1, //J//2 and //J//3 satisfy

[[math]]

\qquad q_1^2 + q_2^2 = q_3^2

[[math]]

they can be said to form a Pythagorean triple.

The following are three examples. In the first and third cases, their counterparts in 12edo, //J//1', //J//2' and //J//3' , are also Pythagorean triples:

|| #//J//1 || #//J//2 || #//J//3 || #//q//1 || #//q//2 || #//q//3 || #//J//1' || #//J//2' || #//J//3'# ||

|| #6/5# || #5/4 || #4/3# || #1/2√30# || #1/4√5 || #1/4√3 || #3 || #4 || #5 ||

|| #4/3 || #12/5# || #5/2# || #1/4√3 || #7/4√15# || #3/2√10# || || || ||

|| #8/5 || #12/5 || #8/3 || #3/4√10 || #7/4√15 || #5/4√6 || #8 || #15 || #17 ||

==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.

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// = //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

[[math]]

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

[[math]]

Estimating //J//2 and //J//1 with their quadratic approximants we then have

[[math]]

\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m

[[math]]

For //gammic//:

###//J//₁= 6/5, //J//₂= 5/4

###//v//₁ = 1/11, //v//₂ = 1/9, //b//m = 1

###//q//₁² = (1/4)(1/30), //q//₂//² =// (1/4)(1/20)

###//gammic// = //b//(//J//₁,//J//₂) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)

and for //semisuper://

###//J//₁= 25/24, //J//₂= 4/3

###//v//₁ = 1/49, //v//₂ = 1/7, //b//m = 1/7

###//q//₁² = (1/4)(1/600), //q//₂//² =// (1/4)(1/12)

###//semisuper// = //b//(//J//₁,//J//₂) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)

Therefore

###//gammic/semisuper// ≈ 10/7

as required.

To estimate the size of //selenia// we must quantify for the error in this ratio. A more accurate analysis gives

[[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

[[math]]

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

[[math]]

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

[[math]]

Thus a better estimate for //gammic/semisuper// is

[[math]]

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

[[math]]

from which it follows that

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

####### ##≈ 7 //gammic// (1 - //f////semisuper// / //f////gammic//)

Putting in the numbers we find

//###f////gammic// = 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)

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

Therefore

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

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in //q//6//,// which become significant when the //f// values are very similar.)

In summary, the reason //selenia// is small (compared to //gammic// and //semisuper//) is because the quadratic approximants of //gammic// and //semisuper// are in the ratio 10/7. The reason it is //very// small (of order //gammic///1000 rather than //gammic///10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

[[math]]

\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2

[[math]]

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

~~^^^TBC~~

=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>

=~~~~Source~~~~=

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

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Logarithmic approximants</title></head><body>~~WORK IN PROGRESS ~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Logarithmic approximants</title></head><body><h1 id="toc0"><a name="x1. Introduction"></a>1. Introduction</h1>

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

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

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

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Three types of approximants are described here:

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

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

<h1 id="toc1"><a name="x2. Bimodular approximants"></a>2. Bimodular approximants</h1>

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

<h2 id="toc2"><a name="x2. Bimodular approximants-Definition"></a>Definition</h2>

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

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

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

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

<h2 id="toc3"><a name="x2. Bimodular approximants-Properties"></a>Properties</h2>

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

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

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

The approximants of superparticular intervals are reciprocals of odd integers:

The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.

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

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

######Figure 1. Bimodular approximants for low-order superparticular intervals

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

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

<h2 id="toc4"><a name="Bimodular approximants-Bimodular approximants and equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:48 --~~><strong~~>Bimodular approximants and equal temperaments</span~~></strong~~></h2>

<h2 id="toc4"><a name="x2. Bimodular approximants-Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</h2>

While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:

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

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

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

<h2 id="toc5"><a name="x2. Bimodular approximants-Bimodular commas"></a>Bimodular commas</h2>

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

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

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--><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m</script>

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

<h3 id="toc6"><a name="x2. Bimodular approximants-Bimodular commas-Examples"></a>Examples</h3>

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

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

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

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

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

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

<h2 id="toc8"><a name="x3. Padé approximants of order (1,2)-Definition"></a>Definition</h2>

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

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

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

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

<h1 id="toc9"><a name="x4. Quadratic approximants"></a>4. Quadratic approximants</h1>

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

<h2 id="toc10"><a name="x4. Quadratic approximants-Definition"></a>Definition</h2>

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

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

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

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

<h2 id="toc11"><a name="x4. Quadratic approximants-Properties"></a>Properties</h2>

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

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

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\qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6</script>

The quadratic approximant q of a double interval 2J (for example, the ditone) is rational, which suggests using ½ q(r2) as a rational approximant of J (where J has frequency ratio r):

The quadratic approximant q of a double interval 2J (for example, the ditone) is rational, which suggests using ½ q(r2) as a rational approximant of J (where J has frequency ratio r):

<!-- ws:start:WikiTextMathRule:36:

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

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

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

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

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

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

<h2 id="toc13"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a>Relative sizes of intervals between 3 frequencies in arithmetic progression</h2>

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

<h3 id="toc14"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a>Theorem</h3>

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

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

<h3 id="toc15"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a>Remarks</h3>

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

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

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

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

<h3 id="toc16"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a>Proof</h3>

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

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

which is the geometric mean of their frequency ratios.

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

<h3 id="toc17"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"></a>Examples</h3>

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

F/f = 701.955/498.045 = 1.40942,

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√5/2 = 1.11803.

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

<h2 id="toc18"><a name="x4. Quadratic approximants-Silver temperament"></a>Silver temperament</h2>

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

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

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

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

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

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

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

This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).

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

The 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.

<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;" />

######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.

<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

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

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

\qquad T/t = \sqrt{5}/2&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad T/t = \sqrt{5}/2</script>

It follows that

<!-- ws:start:WikiTextMathRule:41:

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

\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi</script>

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.

<tr>

<td>Sequence 1:#

</td>

<td>#t/2 - 3c#

</td>

<td>#2c

</td>

<td>#t/2 - c

</td>

<td>#T - t/2

</td>

<td>#t

</td>

<td>#T + t/2

</td>

<td>#M + t/2

</td>

<td>#2M

</td>

</tr>

<tr>

<td>Sequence 2:#

</td>

<td>#magic

</td>

<td>#diesis

</td>

<td>#chroma#

</td>

<td>#semitone#

</td>

<td>#t

</td>

<td>#mp

</td>

<td>#f - c

</td>

<td>#m6p - c#

</td>

</tr>

<tr>

<td>Difference:

</td>

<td>#-3σ/2

</td>

<td>#σ

</td>

<td>#-σ/2

</td>

<td>#σ/2

</td>

<td>#0

</td>

<td>#σ/2

</td>

<td>#σ/2

</td>

<td>#σ

</td>

</tr>

<tr>

<td>Seq 1 ratios:

</td>

<td>

</td>

<td>#1.6120##

</td>

<td>#1.6204

</td>

<td>#1.6171

</td>

<td>#1.6184#

</td>

<td>#1.6179

</td>

<td>#1.6181

</td>

<td>#1.6180

</td>

</tr>

<tr>

<td>Seq 2 ratios:

</td>

<td>

</td>

<td>#1.3865

</td>

<td>#1.7212

</td>

<td>#1.5810

</td>

<td>#1.6325

</td>

<td>#1.6125

</td>

<td>#1.6201

</td>

<td>#1.6172

</td>

</tr>

</table>

where f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/32768.

The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to ϕ.

Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (σ), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).

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 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 error in the intervals s, t, m and M 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).

<h2 id="toc20"><a name="x4. Quadratic approximants-Pythagorean triples of quadratic approximants"></a>Pythagorean triples of quadratic approximants</h2>

If the quadratic approximants q1, q2 and q3 of a set of three intervals J1, J2 and J3 satisfy

<!-- ws:start:WikiTextMathRule:42:

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

\qquad q_1^2 + q_2^2 = q_3^2&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad q_1^2 + q_2^2 = q_3^2</script>

they can be said to form a Pythagorean triple.

The following are three examples. In the first and third cases, their counterparts in 12edo, J1', J2' and J3' , are also Pythagorean triples:

<tr>

<td>#J1

</td>

<td>#J2

</td>

<td>#J3

</td>

<td>#q1

</td>

<td>#q2

</td>

<td>#q3

</td>

<td>#J1'

</td>

<td>#J2'

</td>

<td>#J3'#

</td>

</tr>

<tr>

<td>#6/5#

</td>

<td>#5/4

</td>

<td>#4/3#

</td>

<td>#1/2√30#

</td>

<td>#1/4√5

</td>

<td>#1/4√3

</td>

<td>#3

</td>

<td>#4

</td>

<td>#5

</td>

</tr>

<tr>

<td>#4/3

</td>

<td>#12/5#

</td>

<td>#5/2#

</td>

<td>#1/4√3

</td>

<td>#7/4√15#

</td>

<td>#3/2√10#

</td>

<td>

</td>

<td>

</td>

<td>

</td>

</tr>

<tr>

<td>#8/5

</td>

<td>#12/5

</td>

<td>#8/3

</td>

<td>#3/4√10

</td>

<td>#7/4√15

</td>

<td>#5/4√6

</td>

<td>#8

</td>

<td>#15

</td>

<td>#17

</td>

</tr>

</table>

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

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

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

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.

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 = 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

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

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

\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m</script>

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

Estimating J2 and J1 with their quadratic approximants we then have

<~~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~~;" />

<!-- ws:start:WikiTextMathRule:44:

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

\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&lt;br/&gt;[[math]]

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

For gammic:

###J₁= 6/5, J₂= 5/4

###v₁ = 1/11, v₂ = 1/9, bm = 1

###q₁² = (1/4)(1/30), q₂² = (1/4)(1/20)

###gammic = b(J₁,J₂) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)

and for semisuper:

###J₁= 25/24, J₂= 4/3

###v₁ = 1/49, v₂ = 1/7, bm = 1/7

###q₁² = (1/4)(1/600), q₂² = (1/4)(1/12)

###semisuper = b(J₁,J₂) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)

Therefore

###gammic/semisuper ≈ 10/7

as required.

To estimate the size of selenia we must quantify for the error in this ratio. A more accurate analysis gives

<!-- ws:start:WikiTextMathRule:45:

[[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]]

--><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>

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

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

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

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

--><script type="math/tex">\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)</script>

Thus a better estimate for gammic/semisuper is

<!-- ws:start:WikiTextMathRule:47:

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

\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 (1 - fsemisuper / fgammic)

Putting in the numbers we find

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

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

###fgammic - fsemisuper = 1/6000

Therefore

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

which is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in q6, which become significant when the f values are very similar.)

In summary, the reason selenia is small (compared to gammic and semisuper) is because the quadratic approximants of gammic and semisuper are in the ratio 10/7. The reason it is very small (of order gammic/1000 rather than gammic/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:

<!-- ws:start:WikiTextMathRule:48:

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

\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2</script>

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

~~^^^TBC ~~

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

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

<h1 id="~~toc20~~"><a name="Source"></a><!-- ws:end:WikiTextHeadingRule:80 --~~>Source~~</h1>

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

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 18:21:56 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-21 11:17:47 UTC</tt>.<br>
-: The original revision id was <tt>541616972</tt>.<br>
+: The original revision id was <tt>541645916</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 <h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">WORK IN PROGRESS
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;1. Introduction&lt;/span&gt;=
-=&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant// (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;
@@ Line 40: / Line 39: @@
 * Quadratic approximants
-=**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**=
+=**&lt;span style="font-size: 20px;"&gt;2. Bimodular approximants&lt;/span&gt;**=
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;==
 The bimodular approximant of an interval with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;// is
 [[math]]
@@ Line 59: / Line 58: @@
 [[math]]
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Properties&lt;/span&gt;==
 When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r//&lt;/span&gt; is
@@ Line 89: / Line 88: @@
 [[math]]
-The approximants of superparticular intervals are reciprocals of odd integers:
+The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.
 [[image:Low-order superparticular intervals.png]]
+&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 1. Bimodular approximants for low-order superparticular intervals
 If &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//[//J//] &lt;/span&gt;denotes the bimodular approximant of an interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt; with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//,
@@ Line 99: / Line 99: @@
 This last result is equivalent to the identity expressing &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1 + &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;)&lt;/span&gt; in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;).&lt;/span&gt;
-==**&lt;span style="font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;**==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;==
 While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:
 Two perfect fourths (//r// = 4/3, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)
@@ Line 114: / Line 114: @@
 Relationships of this sort can be identified in all equal temperaments.
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Bimodular commas&lt;/span&gt;==
 As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the //bimodular residue// as
@@ Line 157: / Line 157: @@
 Further examples of bimodular commas are provided in Reference 1. See also [[Don Page comma]] (another name for this type of comma).
-=**&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;**=
+=**&lt;span style="font-size: 21.33px;"&gt;3. Padé approximants of order (1,2)&lt;/span&gt;**=
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;==
 In the section on bimodular approximants it was shown than an interval of logarithmic size //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// (measured in dineper units) is related to its bimodular approximant by
 [[math]]
@@ Line 196: / Line 196: @@
 (__3/2__) / (__20/17__) = 2.4949 ≈ (15/74) / (6/74) = 5/2
-=**&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;**=
+=**&lt;span style="font-size: 21.33px;"&gt;4. Quadratic approximants&lt;/span&gt;**=
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;==
 The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = //n/////d//&lt;/span&gt; is
 [[math]]
@@ Line 245: / Line 245: @@
 The presence of a square root in the denominator of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// (except where //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.
-==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Properties&lt;/span&gt;==
 If //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[//J//]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//[//J//n]&lt;/span&gt; , then
 [[math]]
@@ Line 279: / Line 279: @@
 [[math]]
-The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of a double interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2//J//&lt;/span&gt; (for example, the ditone) is rational, which suggests using &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;½ q(r&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; as a rational approximant of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// (where //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// has frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//):
+The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of a double interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2//J//&lt;/span&gt; (for example, the ditone) is rational, which suggests using &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;½ //q//(//r//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; as a rational approximant of //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// (where //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// has frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;//):
 [[math]]
 \qquad \tfrac{1}{2} q(r^2) = \tfrac{1}{4} (r - \frac{1}{r}) = \tfrac{1}{2} \sinh{2J} = J + \tfrac{2}{3}J^3 + \tfrac{2}{15}J^5 + ...
@@ Line 286: / Line 286: @@
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
 == ==
-==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
+==&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
-===**Theorem**===
+===&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Theorem&lt;/span&gt;===
 If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.
-===**Remarks**===
+===&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Remarks&lt;/span&gt;===
 If the harmonics have indices //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n – m, n&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n + m&lt;/span&gt;//, the two intervals have reduced frequency ratios //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n/(n – m)&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(n + m)/n&lt;/span&gt;//. It can be assumed that //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// have no common factor.
 //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// is the epimoricity of the intervals. When //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;// = 1 the intervals are adjacent superparticular intervals.
 The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.
-===**Proof**===
+===&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Proof&lt;/span&gt;===
 The ratio of the intervals as estimated from their quadratic approximants is
 [[math]]
@@ Line 299: / Line 299: @@
 [[math]]
 which is the geometric mean of their frequency ratios.
-===**Examples**===
+===&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Examples&lt;/span&gt;===
 The ratio of the perfect fifth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//F// = __3/2__&lt;/span&gt;, to the perfect fourth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//f// = __4/3__&lt;/span&gt;, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//F/f// = 701.955/498.045 = 1.40942,&lt;/span&gt;
@@ Line 307: / Line 307: @@
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√5/2 = 1.11803.&lt;/span&gt;
-==**Silver temperament**==
+==&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Silver temperament&lt;/span&gt;==
-As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.
+As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.
 It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fifth = __3/2__ = 702.944 cents
@@ Line 364: / Line 364: @@
 Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,
-This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).
+This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).
-The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.
+The 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).
-The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.
+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.
-[[image:Continued fraction jigsaw 41edo.png width="800" height="395"]]
+[[image:Continued fraction jigsaw 41edo.png width="800" height="396"]]
-The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).
+&lt;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.
 [[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
+[[math]]
+\qquad T/t = \sqrt{5}/2
+[[math]]
+It follows that
+[[math]]
+\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi
+[[math]]
+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 - 3c//&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;//2M// &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;-3σ/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: "Palatino Linotype","Book Antiqua",Palatino,serif;"&gt;&lt;span style="color: #ffffff;"&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: "Lucida Sans Unicode","Lucida Grande",sans-serif; font-size: 80%;"&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; ||
+where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/327__68.__&lt;/span&gt;
+The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;//.
+Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;σ&lt;/span&gt;//), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).
+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 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 error in the intervals //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t, m&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&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).
+==&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%; vertical-align: sub;"&gt;Pythagorean triples of quadratic approximants&lt;/span&gt;==
+If the quadratic approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//1//, q//2&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//3&lt;/span&gt; of a set of three intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//1, //J//2&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;//3 satisfy
+[[math]]
+\qquad q_1^2 + q_2^2 = q_3^2
+[[math]]
+they can be said to form a Pythagorean triple.
+The following are three examples. In the first and third cases, their counterparts in 12edo, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//1', //J//2'&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//3'&lt;/span&gt; , are also Pythagorean triples:
+|| &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; line-height: 0px; overflow: hidden;"&gt;#&lt;/span&gt;//J//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//3&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//1&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//2&lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//q//3 &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//1' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//2' &lt;/span&gt; || &lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;//J//3'&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; ||
+|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;6/5&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/2√30&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5 ||
+|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/2&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/2√10&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt; ||   ||   ||   ||
+|| &lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/3 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/4√10 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4√6 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;15 || &lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;17 ||
+==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;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: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//Gammic// and //semisuper// are both __bimodular commas__:&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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;
+[[math]]
+\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m
+[[math]]
+&lt;span style="color: #333333;"&gt;Estimating &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//J//2&lt;/span&gt;&lt;span style="color: #333333;"&gt; and &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;//J//1&lt;/span&gt;&lt;span style="color: #333333;"&gt; with their quadratic approximants we then have&lt;/span&gt;
+[[math]]
+\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m
+[[math]]
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For //gammic//:&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//J//₁= 6/5, //J//₂= 5/4&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//v//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//q//&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂//² =// &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;//gammic// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and for //semisuper://&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;//J//₁= 25/24, //J//₂= 4/3&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//v//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;//&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;//&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//q//&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/600),&lt;/span&gt; //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₂//² =// &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/12)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;//semisuper// = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//b//(//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,//J//&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Therefore&lt;/span&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;//gammic/semisuper// ≈ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;10/7&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;as required.&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;To estimate the size of //selenia// we must quantify for the error in this ratio. A more accurate analysis gives&lt;/span&gt;
+[[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
+[[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;
+[[math]]
+\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)
+[[math]]
+&lt;span style="color: #333333;"&gt;Thus a better estimate for //gammic/semisuper// is&lt;/span&gt;
+[[math]]
+\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;"&gt;####### ##&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 //gammic// (1 - //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;/ //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; font-size: 110%;"&gt;)&lt;/span&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers we find&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; 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-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: #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 (1/6000) (120/119) //gammic// = //gammic///850 = 0.00561&lt;/span&gt;&lt;span style="color: #333333;"&gt; cents&lt;/span&gt;
+&lt;span style="color: #333333;"&gt;which &lt;/span&gt;is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;"&gt;6&lt;/span&gt;//,// which become significant when the //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;// values are very similar.)
+In summary, the reason //selenia// is small (compared to //gammic// and //semisuper//) is because the quadratic approximants of //gammic// and //semisuper// are in the ratio 10/7. The reason it is //very// small (of order //gammic///1000 rather than //gammic///10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:
+[[math]]
+\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2
+[[math]]
+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.
-^^^TBC
+=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>
-=&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;=
-&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;This article summarises original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.&lt;/span&gt;</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;WORK IN PROGRESS&lt;br /&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:49:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x1. Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:49 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;1. Introduction&lt;/span&gt;&lt;/h1&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
   &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:&lt;/span&gt;&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments such as 12edo provide a good approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce aesthetically pleasing scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
@@ Line 413: / Line 513: @@
 Three types of approximants are described here:&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;Bimodular approximants (first order rational approximants)&lt;/li&gt;&lt;li&gt;Padé approximants of order (1,2) (second order rational approximants)&lt;/li&gt;&lt;li&gt;Quadratic approximants&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:51:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x2. Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:51 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;2. Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:53:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x2. Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:53 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The bimodular approximant of an interval with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;&lt;/em&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:3:
@@ Line 437: / Line 537: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad r = \frac{1+v}{1-v}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:5 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:55:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x2. Bimodular approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:55 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
   When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.&lt;br /&gt;
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
@@ Line 473: / Line 573: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad -\tfrac{1}{3}v^3 = -0.00267...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:11 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-The approximants of superparticular intervals are reciprocals of odd integers:&lt;br /&gt;
+The approximants of superparticular intervals are reciprocals of odd integers, as shown in Figure 1.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:284:&amp;lt;img src=&amp;quot;/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:284 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:481:&amp;lt;img src=&amp;quot;/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:481 --&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 1. Bimodular approximants for low-order superparticular intervals&lt;br /&gt;
 &lt;br /&gt;
 If &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;] &lt;/span&gt;denotes the bimodular approximant of an interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt; with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;,&lt;br /&gt;
@@ Line 485: / Line 586: @@
 This last result is equivalent to the identity expressing &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1 + &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;)&lt;/span&gt; in terms of &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;tanh(&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;).&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Bimodular approximants-Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:57:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="x2. Bimodular approximants-Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:57 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/h2&gt;
   While bimodular approximants have historically been used as a means of estimating the sizes of very small intervals, they remain reasonably accurate as the interval size is increased to an octave or more. And being easily computable, they provide a quick means of comparing the relative sizes of intervals. For example:&lt;br /&gt;
 Two perfect fourths (&lt;em&gt;r&lt;/em&gt; = 4/3, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; = 1/7) approximate a minor seventh (&lt;em&gt;r&lt;/em&gt; = 9/5, = 2/7)&lt;br /&gt;
@@ Line 500: / Line 601: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:50:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:59:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x2. Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:59 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
   As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.&lt;br /&gt;
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; &lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &amp;lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the &lt;em&gt;bimodular residue&lt;/em&gt; as&lt;br /&gt;
@@ Line 538: / Line 639: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:19 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Bimodular approximants-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Examples&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:61:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x2. Bimodular approximants-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:61 --&gt;Examples&lt;/h3&gt;
   If the source intervals are the perfect fourth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;f&lt;/em&gt; =&lt;/span&gt; &lt;u&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;4/3&lt;/span&gt;&lt;/u&gt;&lt;em&gt;)&lt;/em&gt; and the perfect fifth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F&lt;/em&gt; = &lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;), &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;then&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;2 = 1/5&lt;/span&gt;, and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; is the Pythagorean comma:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:20:
@@ Line 552: / Line 653: @@
 Further examples of bimodular commas are provided in Reference 1. See also &lt;a class="wiki_link" href="/Don%20Page%20comma"&gt;Don Page comma&lt;/a&gt; (another name for this type of comma).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule: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;
-  &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="x3. Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   In the section on bimodular approximants it was shown than an interval of logarithmic size &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; (measured in dineper units) is related to its bimodular approximant by&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:22:
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 (&lt;u&gt;3/2&lt;/u&gt;) / (&lt;u&gt;20/17&lt;/u&gt;) = 2.4949 ≈ (15/74) / (6/74) = 5/2&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:67:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="x4. Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:67 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;4. Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:69:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="x4. Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:69 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = &lt;em&gt;n&lt;/em&gt;&lt;em&gt;/d&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:26:
@@ Line 782: / Line 883: @@
 The presence of a square root in the denominator of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; (except where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:71:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="x4. Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:71 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
   If &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;]&lt;/span&gt; denote, respectively, the bimodular and quadratic approximants of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;, and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 80%;"&gt;n&lt;/span&gt; denotes &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;[&lt;em&gt;J&lt;/em&gt;n]&lt;/span&gt; , then&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:31:
@@ Line 830: / Line 931: @@
 \qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:35 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of a double interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;em&gt;J&lt;/em&gt;&lt;/span&gt; (for example, the ditone) is rational, which suggests using &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;½ q(r&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; as a rational approximant of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; (where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; has frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;):&lt;br /&gt;
+The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of a double interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&lt;em&gt;J&lt;/em&gt;&lt;/span&gt; (for example, the ditone) is rational, which suggests using &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;½ &lt;em&gt;q&lt;/em&gt;(&lt;em&gt;r&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt; as a rational approximant of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; (where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; has frequency ratio &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt;):&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:36:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 837: / Line 938: @@
 However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.&lt;br /&gt;
 The most interesting approximate interval ratios derivable from quadratic approximants are irrational.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt; &lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:73:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;!-- ws:end:WikiTextHeadingRule:73 --&gt; &lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&gt;&lt;span style="font-family: "Arial Black",Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/h3&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:77:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:77 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Theorem&lt;/span&gt;&lt;/h3&gt;
   If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;&lt;strong&gt;Remarks&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:79:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:79 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Remarks&lt;/span&gt;&lt;/h3&gt;
   If the harmonics have indices &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n – m, n&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n + m&lt;/span&gt;&lt;/em&gt;, the two intervals have reduced frequency ratios &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n/(n – m)&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(n + m)/n&lt;/span&gt;&lt;/em&gt;. It can be assumed that &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;n&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; have no common factor.&lt;br /&gt;
 &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; is the epimoricity of the intervals. When &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;m&lt;/span&gt;&lt;/em&gt; = 1 the intervals are adjacent superparticular intervals.&lt;br /&gt;
 The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:72:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:72 --&gt;&lt;strong&gt;Proof&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:81:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:81 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Proof&lt;/span&gt;&lt;/h3&gt;
   The ratio of the intervals as estimated from their quadratic approximants is&lt;br /&gt;
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   --&gt;&lt;script type="math/tex"&gt;\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:37 --&gt;&lt;br /&gt;
 which is the geometric mean of their frequency ratios.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:74:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:74 --&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:83:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:83 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Examples&lt;/span&gt;&lt;/h3&gt;
   The ratio of the perfect fifth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F&lt;/em&gt; = &lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt;, to the perfect fourth, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt;, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).&lt;br /&gt;
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;F/f&lt;/em&gt; = 701.955/498.045 = 1.40942,&lt;/span&gt;&lt;br /&gt;
@@ Line 860: / Line 961: @@
 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;√5/2 = 1.11803.&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:76:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Quadratic approximants-Silver temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:76 --&gt;&lt;strong&gt;Silver temperament&lt;/strong&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="x4. Quadratic approximants-Silver temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Silver temperament&lt;/span&gt;&lt;/h2&gt;
-  As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.&lt;br /&gt;
+  As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.&lt;br /&gt;
 It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;Perfect fifth = &lt;u&gt;3/2&lt;/u&gt; = 702.944 cents&lt;br /&gt;
@@ Line 941: / Line 1,042: @@
 Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,&lt;br /&gt;
-This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).&lt;br /&gt;
+This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, 41edo, 70edo etc.).&lt;br /&gt;
-The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.&lt;br /&gt;
+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;
+&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;
+&lt;br /&gt;
+&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;
+&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;
+&lt;!-- ws:start:WikiTextMathRule:40:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad T/t = \sqrt{5}/2&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad T/t = \sqrt{5}/2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:40 --&gt;&lt;br /&gt;
+It follows that&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:41:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:41 --&gt;&lt;br /&gt;
+where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;ϕ&lt;/em&gt; = 1.61803&lt;/span&gt;... is the golden ratio.&lt;br /&gt;
+If a Fibonacci sequence of intervals is formed from the pair of intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;T&lt;/em&gt; – &lt;em&gt;t&lt;/em&gt;/2&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;t&lt;/span&gt;&lt;/em&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;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt;. The sequence formed in this way is Sequence 1 in the following table.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Sequence 1:&lt;span style="color: #ffffff;"&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;"&gt;#&lt;/span&gt;&lt;em&gt;t/2 - 3c&lt;/em&gt;&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;2&lt;/em&gt;&lt;em&gt;c&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/2 - c&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/2&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&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/2&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;M + t/2&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;2M&lt;/em&gt; &lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Sequence 2:&lt;span style="color: #ffffff;"&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;magic&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;diesis&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;chroma&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;semitone&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;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;mp&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;f - c&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;m6p - c&lt;/em&gt;&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Difference:&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;-3σ/2&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;σ&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;-σ/2&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: "Palatino Linotype","Book Antiqua",Palatino,serif;"&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;σ/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;&lt;span style="font-family: "Lucida Sans Unicode","Lucida Grande",sans-serif; font-size: 80%;"&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;σ/2&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;σ/2&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;σ &lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Seq 1 ratios:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&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;1.6120&lt;/span&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&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;1.6204&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;1.6171&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;1.6184&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;1.6179&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;1.6181&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;1.6180&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;Seq 2 ratios:&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&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;1.3865&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;1.7212&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;1.5810&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;1.6325&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;1.6125&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;1.6201 &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;1.6172 &lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+where &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f = 4/3, T = 9/8, t = 10/9, M = 5/4, magic = 3125/3072, diesis = 128/125, semitone = 16/15, mp = 32/27, c = syntonic comma = 81/80, m6p = 128/81, σ = schisma = 32805/327&lt;u&gt;68.&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;ϕ&lt;/span&gt;&lt;/em&gt;.&lt;br /&gt;
+Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;σ&lt;/span&gt;&lt;/em&gt;), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).&lt;br /&gt;
+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 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 error in the intervals &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;s, t, m&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;M&lt;/span&gt;&lt;/em&gt; 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;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:89:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc20"&gt;&lt;a name="x4. Quadratic approximants-Pythagorean triples of quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:89 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%; vertical-align: sub;"&gt;Pythagorean triples of quadratic approximants&lt;/span&gt;&lt;/h2&gt;
+ If the quadratic approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;1&lt;em&gt;, q&lt;/em&gt;2&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;3&lt;/span&gt; of a set of three intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1, &lt;em&gt;J&lt;/em&gt;2&lt;/span&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt;3 satisfy&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:42:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad q_1^2 + q_2^2 = q_3^2&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad q_1^2 + q_2^2 = q_3^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:42 --&gt;&lt;br /&gt;
+they can be said to form a Pythagorean triple.&lt;br /&gt;
+The following are three examples. In the first and third cases, their counterparts in 12edo, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1', &lt;em&gt;J&lt;/em&gt;2'&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;3'&lt;/span&gt; , are also Pythagorean triples:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; line-height: 0px; overflow: hidden;"&gt;#&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;1&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;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2&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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3&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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;1&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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;2&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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;3 &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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;1' &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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;2' &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 style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;3'&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;6/5&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/2√30&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;4/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/2&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;1/4√3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/2√10&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;12/5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;3/4√10&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;7/4√15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;5/4√6&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;15&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif;"&gt;&lt;span style="font-family: Georgia;"&gt;#&lt;/span&gt;&lt;/span&gt;17&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
 &lt;br /&gt;
-The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the octave (o), fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament.&lt;br /&gt;
+&lt;!-- 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;!-- ws:start:WikiTextLocalImageRule:285:&amp;lt;img src=&amp;quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x395/Continued%20fraction%20jigsaw%2041edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 395px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x395/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 395px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:285 --&gt;&lt;br /&gt;
+ &lt;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;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: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&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;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&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;!-- ws:start:WikiTextMathRule:43:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:43 --&gt;&lt;br /&gt;
 &lt;br /&gt;
-The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;Estimating &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;2&lt;/span&gt;&lt;span style="color: #333333;"&gt; and &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1&lt;/span&gt;&lt;span style="color: #333333;"&gt; with their quadratic approximants we then have&lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:286:&amp;lt;img src=&amp;quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 587px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:286 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:44:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:44 --&gt;&lt;br /&gt;
 &lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;For &lt;em&gt;gammic&lt;/em&gt;:&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 6/5, &lt;em&gt;J&lt;/em&gt;₂= 5/4&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/11, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/9, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/30),&lt;/span&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;₂&lt;em&gt;² =&lt;/em&gt; &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;(1/4)(1/20)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;em&gt;gammic&lt;/em&gt; = &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;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)&lt;/span&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and for &lt;em&gt;semisuper:&lt;/em&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;/span&gt;&lt;em&gt;J&lt;/em&gt;₁= 25/24, &lt;em&gt;J&lt;/em&gt;₂= 4/3&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/49, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;v&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7, &lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif;"&gt;b&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 60%;"&gt;m&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; = 1/7&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Georgia,serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₁² = &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/600),&lt;/span&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;₂&lt;em&gt;² =&lt;/em&gt; &lt;/span&gt;&lt;span style="font-family: Georgia,serif;"&gt;(1/4)(1/12)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &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;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₁&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;,&lt;em&gt;J&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;"&gt;₂&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Therefore&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #ffffff;"&gt;###&lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;&lt;em&gt;gammic/semisuper&lt;/em&gt; ≈ &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;10/7&lt;/span&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;as required.&lt;/span&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;To estimate the size of &lt;em&gt;selenia&lt;/em&gt; we must quantify for the error in this ratio. A more accurate analysis gives&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:45:
+[[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]]
+ --&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;
+&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:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad f = 1 – \tfrac{2}{15} (q_1^2 + q_2^2)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:46 --&gt;&lt;br /&gt;
+&lt;span style="color: #333333;"&gt;Thus a better estimate for &lt;em&gt;gammic/semisuper&lt;/em&gt; is&lt;/span&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:47:
+[[math]]&amp;lt;br/&amp;gt;
+\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;"&gt;####### ##&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;≈ 7 &lt;em&gt;gammic&lt;/em&gt; (1 - &lt;em&gt;f&lt;/em&gt;&lt;/span&gt;&lt;em&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;/ &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;gammic&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;)&lt;/span&gt;&lt;br /&gt;
+&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Putting in the numbers we find&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;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-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: #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 (1/6000) (120/119) &lt;em&gt;gammic&lt;/em&gt; = &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;
+&lt;span style="color: #333333;"&gt;which &lt;/span&gt;is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;"&gt;6&lt;/span&gt;&lt;em&gt;,&lt;/em&gt; which become significant when the &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;f&lt;/span&gt;&lt;/em&gt; values are very similar.)&lt;br /&gt;
+In summary, the reason &lt;em&gt;selenia&lt;/em&gt; is small (compared to &lt;em&gt;gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt;) is because the quadratic approximants of &lt;em&gt;gammic&lt;/em&gt; and &lt;em&gt;semisuper&lt;/em&gt; are in the ratio 10/7. The reason it is &lt;em&gt;very&lt;/em&gt; small (of order &lt;em&gt;gammic&lt;/em&gt;/1000 rather than &lt;em&gt;gammic&lt;/em&gt;/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:48:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:48 --&gt;&lt;br /&gt;
+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;
-^^^TBC&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;Source&lt;/h1&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:78:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc19"&gt;&lt;!-- ws:end:WikiTextHeadingRule:78 --&gt; &lt;/h1&gt;
+  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>
- &lt;!-- ws:start:WikiTextHeadingRule:80:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc20"&gt;&lt;a name="Source"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:80 --&gt;&lt;span style="font-family: Arial Black;"&gt;Source&lt;/span&gt;&lt;/h1&gt;
-  &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;This article summarises original research by Martin Gough. See &lt;a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');"&gt;this paper&lt;/a&gt; for a fuller account of bimodular approximants.&lt;/span&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>