Logarithmic approximants: Difference between revisions

Line 1:

<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:~~genewardsmith~~|~~genewardsmith~~]] and made on <tt>2015-02-~~26 13~~:03:00 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 05:58:43 UTC</tt>.

: The original revision id was <tt>~~542277300~~</tt>.

: The original revision id was <tt>563714785</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Line 378:

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

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

Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

[[math]]

\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.

[[math]]

This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.

Another way to express the first of these relationships is

[[math]]

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},

[[math]]

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (//r// = 2√2//a//+//b//, where// a// and //b// are integers) are transcendental, with the exception of octave multiples (//a// = 0). The frequency ratio of the tempered perfect eleventh (__8/3__ = __2.6666...__) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, 2√2 = 2.665144...

Line 482:

Line 494:

Thanks to [[Gene Ward Smith]] for the Gelfond-Schneider result.</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><h1 id="toc0"><a name="x1. Introduction"></a>1. Introduction</h1>

<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>

A logarithmic approximant (or approximant for short) is 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>

Line 515:

Line 527:

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

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

<h2 id="toc2"><a name="x2. 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:

Line 539:

Line 551:

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

<h2 id="toc3"><a name="x2. 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

Line 576:

Line 588:

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

Line 588:

Line 600:

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="x2. Bimodular approximants-Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</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)

Line 603:

Line 615:

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

<h2 id="toc5"><a name="x2. 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

Line 641:

Line 653:

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

<h3 id="toc6"><a name="x2. 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:

Line 655:

Line 667:

For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a>. See also <a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a> (another name for this type of comma).

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

<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="x3. 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:

Line 738:

Line 750:

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

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

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

<h2 id="toc10"><a name="x4. 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:

Line 885:

Line 897:

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="x4. 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:

Line 940:

Line 952:

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="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>

<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="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a>Theorem</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="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a>Remarks</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 <a class="wiki_link" href="/Superpartient">degree of epimoricity</a> 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="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a>Proof</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

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

Line 955:

Line 967:

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

Line 963:

Line 975:

√5/2 = 1.11803.

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

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

Line 1,051:

Line 1,063:

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 argent 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;" />

<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 argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, mp#= Pythagorean minor third, sp#= Pythagorean limma, Xp#= Pythagorean apotome, p = Pythagorean comma.

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

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

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

Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:

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

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

\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.</script>

This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.

Another way to express the first of these relationships is

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

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

&lt;br /&gt;

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&lt;br/&gt;[[math]]

--><script type="math/tex">

\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},</script>

which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).

By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (r = 2√2a+b, where a and b are integers) are transcendental, with the exception of octave multiples (a = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow">Gelfond-Schneider constant </a>or Hilbert number, 2√2 = 2.665144...

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

<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:

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

[[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>

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

It follows that

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

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

[[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>

--><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.

Line 1,188:

Line 1,215:

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>

<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:

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

[[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>

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

they can be said to form a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow">Pythagorean triple</a>.

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

Line 1,282:

Line 1,309:

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

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

As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the 5-limit comma |-433 -137 280&gt; (‘selenia’) is remarkably small at just 0.004764 cents. 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 <a class="wiki_link" href="/Gammic%20node">//gammic// comma </a>|-29 -11 20&gt; (4.769 cents) and the semisuper comma (AKA <a class="wiki_link" href="/vishnuzma">vishnuzma</a>) |23 6 -14&gt; (3.338 cents). In particular,

Line 1,291:

Line 1,318:

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

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

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

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

[[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>

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

Estimating J2 and J1 with their quadratic approximants we then have

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

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

[[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>

--><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

Line 1,315:

Line 1,342:

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

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

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

[[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}{5} (q_1^2 + q_2^2) ) b_m&lt;br/&gt;[[math]]

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

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

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

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

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

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

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

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

--><script type="math/tex">\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)</script>

--><script type="math/tex">\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)</script>

Thus a better estimate for gammic/semisuper is

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

<!-- ws:start:WikiTextMathRule:49:

[[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>

--><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

Line 1,342:

Line 1,369:

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:

<!-- ws:start:WikiTextMathRule:50:

[[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>

--><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.

<h1 id="toc22"><a name="Sources and acknowledgements"></a>Sources and acknowledgements</h1>

<h1 id="toc22"><a name="Sources and acknowledgements"></a>Sources and acknowledgements</h1>

This article is based on original research by <a class="wiki_link" href="/Martin%20Gough">Martin Gough</a>. 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.

The tuning referred to here as argent temperament was described by <a class="wiki_link" href="/graham%20breed">Graham Breed</a> and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.

Thanks to <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> for the Gelfond-Schneider result.</body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-02-26 13:03:00 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-10-24 05:58:43 UTC</tt>.<br>
-: The original revision id was <tt>542277300</tt>.<br>
+: The original revision id was <tt>563714785</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 378: / Line 378: @@
 [[image:Silver temperament graphic.png width="800" height="587"]]
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament
+Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:
+[[math]]
+\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.
+[[math]]
+This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.
+Another way to express the first of these relationships is
+[[math]]
+\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},
+[[math]]
+which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).
 By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2//a//+//b//&lt;/span&gt;, where//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;// and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//a// = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;__8/3__ = __2.6666...__&lt;/span&gt;) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&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; = 2.665144&lt;/span&gt;...
@@ Line 482: / Line 494: @@
 Thanks to [[Gene Ward Smith]] for the Gelfond-Schneider result.</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;&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;strong&gt;&lt;span style="font-size: 20px;"&gt;1. Introduction&lt;/span&gt;&lt;/strong&gt;&lt;/h1&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:51:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x1. Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:51 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;1. Introduction&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
   &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;A &lt;em&gt;logarithmic approximant&lt;/em&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) is 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 515: / Line 527: @@
 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: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:53:&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:53 --&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: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;
+  &lt;!-- ws:start:WikiTextHeadingRule:55:&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:55 --&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 539: / Line 551: @@
   --&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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:57:&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:57 --&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 576: / Line 588: @@
 &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: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;!-- ws:start:WikiTextLocalImageRule:483:&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:483 --&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;
@@ Line 588: / Line 600: @@
 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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:59:&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:59 --&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 603: / Line 615: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&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;
+&lt;!-- ws:start:WikiTextHeadingRule:61:&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:61 --&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 641: / Line 653: @@
   --&gt;&lt;script type="math/tex"&gt;\qquad b(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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:63:&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:63 --&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 655: / Line 667: @@
 For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to &lt;a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');"&gt;this paper&lt;/a&gt;. See also &lt;a class="wiki_link" href="/Don%20Page%20comma"&gt;Don Page comma&lt;/a&gt; (another name for this type of comma).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:63:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="x3. Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:63 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;3. Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:65:&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:65 --&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: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;
+  &lt;!-- ws:start:WikiTextHeadingRule:67:&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:67 --&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:
@@ Line 738: / Line 750: @@
 (&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: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:69:&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:69 --&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: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;
+  &lt;!-- ws:start:WikiTextHeadingRule:71:&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:71 --&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 885: / Line 897: @@
 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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:73:&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:73 --&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 940: / Line 952: @@
 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: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:75:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;!-- ws:end:WikiTextHeadingRule:75 --&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,Helvetica,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:77:&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:77 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&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;
+  &lt;!-- ws:start:WikiTextHeadingRule:79:&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:79 --&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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:81:&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:81 --&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 &lt;a class="wiki_link" href="/Superpartient"&gt;degree of epimoricity&lt;/a&gt; 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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:83:&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:83 --&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;
 &lt;!-- ws:start:WikiTextMathRule:37:
@@ Line 955: / Line 967: @@
   --&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: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;
+&lt;!-- ws:start:WikiTextHeadingRule:85:&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:85 --&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 963: / Line 975: @@
 &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:85:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="x4. Quadratic approximants-Argent temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:85 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Argent temperament&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:87:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="x4. Quadratic approximants-Argent temperament"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:87 --&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Argent 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, 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;
@@ Line 1,051: / Line 1,063: @@
 Figure 2 is a &lt;em&gt;continued fraction jigsaw&lt;/em&gt; showing the sizes of the octave (o), fourth (f), tone (T), limma (s&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;), Pythagorean comma (p) and 29-tone comma (p&lt;span style="font-size: 60%;"&gt;29&lt;/span&gt;) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextLocalImageRule:482:&amp;lt;img src=&amp;quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 396px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:482 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextLocalImageRule:484:&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:484 --&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 argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, m&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= Pythagorean minor third, s&lt;span style="font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= Pythagorean limma, X&lt;span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;"&gt;p&lt;/span&gt;&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt;= Pythagorean 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;!-- ws:start:WikiTextLocalImageRule:485:&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:485 --&gt;&lt;br /&gt;
 &lt;span style="color: #ffffff;"&gt;######&lt;/span&gt;Figure 3. Geometrical representation of argent temperament&lt;br /&gt;
+&lt;br /&gt;
+Argent temperament tunes the augmented fourth (tritone) and diminished fifth (double minor third) in the ratio 3/2√2, which is also the ratio of the quadratic approximants of 10/7 and 7/5:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:40:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \frac{q[10/7]}{q[7/5]}= \frac{3 / \sqrt{70}} {2 / \sqrt{35}} = \tfrac{3}{2\sqrt{2}}.&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:40 --&gt;&lt;br /&gt;
+This means that in Argent temperament the augmented fourth is very close to 10/7 and the diminished fifth is very close to 7/5. The discrepancy in each case is just 0.175 cents.&lt;br /&gt;
+Another way to express the first of these relationships is&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:41:
+[[math]]&amp;lt;br/&amp;gt;
+ &amp;lt;br /&amp;gt;
+\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;
+\qquad 3 (\tfrac{1}{\sqrt{6}} – \tfrac{2}{\sqrt{3}}) ≈ \tfrac{3}{\sqrt{70}},&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:41 --&gt;&lt;br /&gt;
+which after squaring both sides leads to √2 ≈ 99/70, a well-known approximation which can be confirmed by noting that 99/70 = √(2 + 1/4900).&lt;br /&gt;
 &lt;br /&gt;
 By the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow"&gt;Gelfond-Schneider theorem &lt;/a&gt; the frequency ratios of all argent intervals (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = 2&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;√2&lt;em&gt;a&lt;/em&gt;+&lt;em&gt;b&lt;/em&gt;&lt;/span&gt;, where&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; a&lt;/span&gt;&lt;/em&gt; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt; are integers) are transcendental, with the exception of octave multiples (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;a&lt;/em&gt; = 0&lt;/span&gt;). The frequency ratio of the tempered perfect eleventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;u&gt;8/3&lt;/u&gt; = &lt;u&gt;2.6666...&lt;/u&gt;&lt;/span&gt;) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow"&gt;Gelfond-Schneider constant &lt;/a&gt;or Hilbert number, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;2&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; = 2.665144&lt;/span&gt;...&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;
+&lt;!-- ws:start:WikiTextHeadingRule:89:&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:89 --&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:
+&lt;!-- ws:start:WikiTextMathRule:42:
 [[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;
+  --&gt;&lt;script type="math/tex"&gt;\qquad T/t = \sqrt{5}/2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:42 --&gt;&lt;br /&gt;
 It follows that&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMathRule:41:
+&lt;!-- ws:start:WikiTextMathRule:43:
 [[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;
+  --&gt;&lt;script type="math/tex"&gt;\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:43 --&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;
@@ Line 1,188: / Line 1,215: @@
 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;
+&lt;!-- ws:start:WikiTextHeadingRule:91:&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:91 --&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:
+&lt;!-- ws:start:WikiTextMathRule:44:
 [[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;
+  --&gt;&lt;script type="math/tex"&gt;\qquad q_1^2 + q_2^2 = q_3^2&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:44 --&gt;&lt;br /&gt;
 they can be said to form a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow"&gt;Pythagorean triple&lt;/a&gt;.&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;
@@ Line 1,282: / Line 1,309: @@
 &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:WikiTextHeadingRule:93:&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:93 --&gt;A small 34edo comma&lt;/h2&gt;
   &lt;span style="color: #333333;"&gt;As &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; has noted, the &lt;/span&gt;5-limit comma &lt;span style="color: #333333;"&gt;|-433 -137 280&amp;gt; (‘&lt;em&gt;selenia&lt;/em&gt;’) is remarkably small at just 0.004764 cents. 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;a class="wiki_link" href="/Gammic%20node"&gt;//gammic// comma &lt;/a&gt;|-29 -11 20&amp;gt; (4.769 cents) and the &lt;em&gt;semisuper&lt;/em&gt; comma (AKA &lt;em&gt;&lt;a class="wiki_link" href="/vishnuzma"&gt;vishnuzma&lt;/a&gt;&lt;/em&gt;) |23 6 -14&amp;gt; (3.338 cents). In particular,&lt;br /&gt;
@@ Line 1,291: / Line 1,318: @@
 &lt;span style="color: #333333;"&gt;&lt;span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&gt;###&lt;/span&gt;&lt;em&gt;semisuper&lt;/em&gt; = &lt;/span&gt;&lt;span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;b&lt;/em&gt;(&lt;u&gt;25/24&lt;/u&gt;,&lt;u&gt;4/3&lt;/u&gt;)&lt;/span&gt;&lt;br /&gt;
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Using a result given in the section on bimodular commas, the size of &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:
+&lt;!-- ws:start:WikiTextMathRule:45:
 [[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;
+  --&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:45 --&gt;&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:WikiTextMathRule:44:
+&lt;!-- ws:start:WikiTextMathRule:46:
 [[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;
+  --&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:46 --&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="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;"&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;
@@ Line 1,315: / Line 1,342: @@
 &lt;br /&gt;
 &lt;span style="color: #333333;"&gt;To estimate the size of &lt;em&gt;selenia&lt;/em&gt; we must quantify the error in this ratio. A more accurate analysis gives&lt;/span&gt;&lt;br /&gt;
-&lt;!-- ws:start:WikiTextMathRule:45:
+&lt;!-- ws:start:WikiTextMathRule:47:
 [[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}{5} (q_1^2 + q_2^2) ) b_m&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
-\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:45 --&gt;&lt;br /&gt;
+\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:47 --&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:
+&lt;!-- ws:start:WikiTextMathRule:48:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&amp;lt;br/&amp;gt;[[math]]
-  --&gt;&lt;script type="math/tex"&gt;\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:46 --&gt;&lt;br /&gt;
+  --&gt;&lt;script type="math/tex"&gt;\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:48 --&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:
+&lt;!-- ws:start:WikiTextMathRule:49:
 [[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;
+  --&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:49 --&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;
@@ Line 1,342: / Line 1,369: @@
 &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:
+&lt;!-- ws:start:WikiTextMathRule:50:
 [[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;
+  --&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:50 --&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;
-&lt;!-- ws:start:WikiTextHeadingRule:93:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Sources and acknowledgements"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:93 --&gt;Sources and acknowledgements&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:95:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc22"&gt;&lt;a name="Sources and acknowledgements"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:95 --&gt;Sources and acknowledgements&lt;/h1&gt;
   This article is based on original research by &lt;a class="wiki_link" href="/Martin%20Gough"&gt;Martin Gough&lt;/a&gt;. 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;br /&gt;
 The tuning referred to here as argent temperament was described by &lt;a class="wiki_link" href="/graham%20breed"&gt;Graham Breed&lt;/a&gt; and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.&lt;br /&gt;
 Thanks to &lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&gt;Gene Ward Smith&lt;/a&gt; for the Gelfond-Schneider result.&lt;/body&gt;&lt;/html&gt;</pre></div>