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:MartinGough|MartinGough]] and made on <tt>2015-02-~~19 13~~:07:32 UTC</tt>.

: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 14:08:56 UTC</tt>.

: The original revision id was <tt>~~541464324~~</tt>.

: The original revision id was <tt>541592976</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.

<h4>Original Wikitext content:</h4>

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

=Introduction=

=Introduction=

The term //logarithmic approximant//[[xenharmonic/Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]] (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as

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

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

* Why is the ~~ratio of the perfect fifth to the perfect fourth close to √2~~?

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

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

Line 41:

=**Bimodular approximants**=

==**Definition**==

==Definition==

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

[[math]]

Line 59:

[[math]]

==**Properties ~~of bimodular approximants~~**==

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

Line 99:

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)

Line 114:

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

Line 146:

===Examples===

If the source intervals are the perfect fourth (//f = ~~__4~~/~~3__~~)// and the perfect fifth (//F = __3/2__//), //v//1 = 1/7, //v//2 = 1/5, and //b// is the Pythagorean comma:

If the source intervals are the perfect fourth (//f// = __4/3__//)// and the perfect fifth (//F// = __3/2__), then //v//1 = 1/7, //v//2 = 1/5, and //b// is the Pythagorean comma:

[[math]]

\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f

[[math]]

If the source intervals are the perfect fourth (//f// //= __4/3__//) and the minor seventh (//m7 = __9/5__/~~/),~~ //v//1 = 1/7, //v//2 = 2/7, //b//r = 2/7 and //b// is the syntonic comma:

If the source intervals are the perfect fourth (//f// = __4/3__) and the minor seventh (//m//7 = __9/5__), then //v//1 = 1/7, //v//2 = 2/7, //b//r = 2/7 and //b// is the syntonic comma:

[[math]]

\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f

Line 158:

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

==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]]

Line 171:

Line 172:

\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))

[[math]]

The first convergent of this continued fraction is //v//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is ~~3//v///(~~3-//v~~//~~2).

The first convergent of this continued fraction is //v//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

[[math]]

\qquad y = \frac{3v}{3-v^2}

[[math]]

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

|| //Interval// || //(1,2) Padé approximant// ||

|| //Interval J//########### || //(1,2) Padé approximant y//# ||

|| Perfect twelfth __3/1__ || 6/11 ||

|| Perfect twelfth = __3/1__ || 6/11 ||

|| Octave 2/1 || 9/26 ||

|| Octave = __2/1__ || 9/26 ||

|| Major sixth 5/3 || 12 15/ ||

|| Major sixth = __5/3__ || 12/47 ||

|| Perfect fifth 3/2 || 74~~/47~~ ||

|| Perfect fifth = __3/2__ || 15/74 ||

|| Perfect fourth 4/3 || 21/146 ||

|| Perfect fourth = __4/3__ || 21/146 ||

|| Major third 5/4 || 27/~~242~~ ||

|| Major third = __5/4__ || 27/242 ||

The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(__3/1__) / (__6/5__) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(__3/1__) / (__7/4__) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = __49/48__)

(__2/1__) / (__7/6__) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9> comma)

(__2/1__) / (__27/25__) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(__5/3__) / (__49/45__) = 5.9986 ≈ (12/47) / (2/47) = 6

(__5/3__) / (__25/22__) = 3.9960 ≈ (12/47) / (3/47) = 4

(__5/3__) / (__26/21__) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(__5/3__) / (__27/20__) = 1.7022 ≈ (12/47) / (7/47) = 12/7

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

=**Quadratic approximants**=

==Definition==

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

[[math]]

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

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

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

[[math]]

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

[[math]]

\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...

[[math]]

it is apparent that //q// is about twice as accurate as //v//, with an error of opposite sign.

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

[[math]]

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

[[math]]

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

[[math]]

\qquad \sqrt{r} = q + \sqrt{1+q^2}

[[math]]

The following are the quadratic approximants of some simple 5-limit intervals:

|| //Interval// //J//##################### || //Quadratic approximant// //q// ## ||

|| Perfect twelfth = __3/1__ || 1/√3 ||

|| Octave = __2/1__ || 1/2√2 ||

|| Minor seventh = __9/5__ || 2/3√5 ||

|| Pythagorean minor seventh = __16/9__ || 7/24 ||

|| Major sixth = __5/3__ || 1/√15 ||

|| Minor sixth = __8/5__ || 3/4√10 ||

|| Perfect fifth = __3/2__ || 1/2√6 ||

|| Perfect fourth = __4/3__ || 1/4√3 ||

|| Major third = __5/4__ || 1/4√5 ||

|| Minor third = __6/5__ || 1/2√30 ||

|| Pythagorean minor third = __32/27__ || 5/24√6 ||

|| Large tone = __9/8__ || 1/12√2 ||

|| Small tone = __10/9__ || 1/6√10 ||

|| Diatonic semitone = __16/15__ || 1/8√15 ||

|| Chroma = __25/24__ || 1/20√6 ||

|| Syntonic comma = __81/80__ || 1/72√5 ||

Expressed in terms of the bimodular approximant,// v = j/g//,

[[math]]

\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}

[[math]]

Quadratic approximants of just intervals thus have the form //q = j/√k//, where //j// and //k// are integers and //j//2// + k = g//2 is a perfect square.

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==

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]]

\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\

\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\

\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\

\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\

\qquad q[-J] = -q[J] \\

\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\

\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\

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

\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\

[[math]]

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

[[math]]

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

[[math]]

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

For example

[[math]]

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

[[math]]

but this can also be derived from bimodular approximants. Using

[[math]]

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

[[math]]

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

[[math]]

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

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

[[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//):

[[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 + ...

[[math]]

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

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

== ==

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

=== **Theorem**===

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

===**Remarks**===

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

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**===

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

[[math]]

\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}

[[math]]

which is the geometric mean of their frequency ratios.

===**Examples**===

Line 185:

Line 304:

<h4>Original HTML content:</h4>

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

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

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

The term logarithmic approximant<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1">{1}</a> (or approximant for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as

<ul><li>Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?</li><li>Why are certain commas small, and roughly how small are they?</li><li>Why is the ~~ratio of the perfect fifth to the perfect fourth close to √2~~?</li></ul>

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

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

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

Line 218:

Line 337:

Three types of approximants are described here:

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

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

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

<h2 id="toc2"><a name="Bimodular approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:29 --~~><strong~~>Definition</span~~></strong~~></h2>

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

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

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

Line 242:

Line 361:

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

<~~!-- ws:start:WikiTextHeadingRule:31:&lt;h2&gt; --><h2 id~~="~~toc3"><a name="Bimodular approximants-Properties of bimodular approximants"></a><!~~-~~- ws~~:~~end:WikiTextHeadingRule:31~~ -~~-><strong&gt~~;~~Properties ~~of bimodular approximants~~</~~strong></h2>

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

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

Line 290:

Line 409:

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

<h1 id="~~toc4~~"><a name="Bimodular approximants and equal temperaments"></a>Bimodular approximants and equal temperaments</h1>

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

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

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

Line 305:

Line 424:

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

<h2 id="~~toc5~~"><a name="Bimodular approximants ~~and equal temperaments~~-Bimodular commas"></a>Bimodular commas</h2>

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

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

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

Line 343:

Line 462:

--><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 ~~and equal temperaments~~-Bimodular commas-Examples"></a>Examples</h3>

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

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

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:

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

\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f</script>

If the source intervals are the perfect fourth (f = 4/3) and the minor seventh (m7 = 9/5), v1 = 1/7, v2 = 2/7, br = 2/7 and b is the syntonic comma:

If the source intervals are the perfect fourth (f = 4/3) and the minor seventh (m7 = 9/5), then v1 = 1/7, v2 = 2/7, br = 2/7 and b is the syntonic comma:

<!-- ws:start:WikiTextMathRule:21:

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

Line 357:

Line 476:

Further examples of bimodular commas are provided in Reference 1^^^. See also Don Page comma^^^ (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="toc6"><a name="Padé approximants of order (1,2)"></a>Padé approximants of order (1,2)</h1>

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

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 374:

Line 494:

\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))</script>

The first convergent of this continued fraction is v, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is 3<~~em&~~gt;v~~/(3-~~~~v</em~~></~~span~~>2).

The first convergent of this continued fraction is v, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is

<!-- ws:start:WikiTextMathRule:25:

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

&lt;span style=&quot;font-family: Georgia,serif; font-size: 110%;&quot;&gt;\qquad y = \frac{3v}{3-v^2&lt;/span&gt;}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad y = \frac{3v}{3-v^2}</script>

Values of this rational approximant for some simple 5-limit intervals are shown in the table below.

Line 380:

Line 504:

<tr>

<td>Interval

<td>Interval J###########

</td>

<td>(1,2) Padé approximant

<td>(1,2) Padé approximant y#

</td>

</tr>

<tr>

<td>Perfect twelfth 3/1

<td>Perfect twelfth = 3/1

</td>

<td>6/11

Line 392:

Line 516:

</tr>

<tr>

<td>Octave 2/1

<td>Octave = 2/1

</td>

<td>9/26

Line 398:

Line 522:

</tr>

<tr>

<td>Major sixth 5/3

<td>Major sixth = 5/3

</td>

<td>12 15/

<td>12/47

</td>

</tr>

<tr>

<td>Perfect fifth 3/2

<td>Perfect fifth = 3/2

</td>

<td>74~~/47~~

<td>15/74

</td>

</tr>

<tr>

<td>Perfect fourth 4/3

<td>Perfect fourth = 4/3

</td>

<td>21/146

Line 416:

Line 540:

</tr>

<tr>

<td>Major third 5/4

<td>Major third = 5/4

</td>

<td>27/242

Line 423:

Line 547:

</table>

The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:

(3/1) / (6/5) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)

(3/1) / (7/4) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = 49/48)

(2/1) / (7/6) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9&gt; comma)

(2/1) / (27/25) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)

(5/3) / (49/45) = 5.9986 ≈ (12/47) / (2/47) = 6

(5/3) / (25/22) = 3.9960 ≈ (12/47) / (3/47) = 4

(5/3) / (26/21) = 2.3918 ≈ (12/47) / (5/47) = 12/5

(5/3) / (27/20) = 1.7022 ≈ (12/47) / (7/47) = 12/7

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

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

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

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

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

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

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

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

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

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

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

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

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

<!-- ws:start:WikiTextMathRule:27:

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

\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</script>

it is apparent that q is about twice as accurate as v, with an error of opposite sign.

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

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

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

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

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

r can be retrieved from q using

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

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

\qquad \sqrt{r} = q + \sqrt{1+q^2}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \sqrt{r} = q + \sqrt{1+q^2}</script>

The following are the quadratic approximants of some simple 5-limit intervals:

<tr>

<td>Interval J#####################

</td>

<td>Quadratic approximant q ##

</td>

</tr>

<tr>

<td>Perfect twelfth = 3/1

</td>

<td> 1/√3

</td>

</tr>

<tr>

<td> Octave = 2/1

</td>

<td> 1/2√2

</td>

</tr>

<tr>

<td> Minor seventh = 9/5

</td>

<td> 2/3√5

</td>

</tr>

<tr>

<td> Pythagorean minor seventh = 16/9

</td>

<td> 7/24

</td>

</tr>

<tr>

<td> Major sixth = 5/3

</td>

<td> 1/√15

</td>

</tr>

<tr>

<td> Minor sixth = 8/5

</td>

<td> 3/4√10

</td>

</tr>

<tr>

<td> Perfect fifth = 3/2

</td>

<td> 1/2√6

</td>

</tr>

<tr>

<td> Perfect fourth = 4/3

</td>

<td> 1/4√3

</td>

</tr>

<tr>

<td> Major third = 5/4

</td>

<td> 1/4√5

</td>

</tr>

<tr>

<td> Minor third = 6/5

</td>

<td> 1/2√30

</td>

</tr>

<tr>

<td> Pythagorean minor third = 32/27

</td>

<td> 5/24√6

</td>

</tr>

<tr>

<td> Large tone = 9/8

</td>

<td> 1/12√2

</td>

</tr>

<tr>

<td> Small tone = 10/9

</td>

<td> 1/6√10

</td>

</tr>

<tr>

<td> Diatonic semitone = 16/15

</td>

<td> 1/8√15

</td>

</tr>

<tr>

<td> Chroma = 25/24

</td>

<td> 1/20√6

</td>

</tr>

<tr>

<td> Syntonic comma = 81/80

</td>

<td> 1/72√5

</td>

</tr>

</table>

Expressed in terms of the bimodular approximant, v = j/g,

<!-- ws:start:WikiTextMathRule:30:

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

\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}</script>

Quadratic approximants of just intervals thus have the form q = j/√k, where j and k are integers and j2 + k = g2 is a perfect square.

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

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

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:

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

\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\&lt;br /&gt;

\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\&lt;br /&gt;

\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\&lt;br /&gt;

\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\&lt;br /&gt;

\qquad q[-J] = -q[J] \\&lt;br /&gt;

\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\&lt;br /&gt;

\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\&lt;br /&gt;

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

\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\

\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\

\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\

\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\

\qquad q[-J] = -q[J] \\

\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\

\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\

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

\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\</script>

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

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

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

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

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

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

For example

<!-- ws:start:WikiTextMathRule:33:

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

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

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

but this can also be derived from bimodular approximants. Using

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

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

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

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

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

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

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

\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\&lt;br /&gt;

\qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6&lt;br/&gt;[[math]]

--><script type="math/tex">\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\

\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):

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

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

\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 + ...&lt;br/&gt;[[math]]

--><script type="math/tex">\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 + ...</script>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}&lt;br/&gt;[[math]]

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

which is the geometric mean of their frequency ratios.

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

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

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-19 13:07:32 UTC</tt>.<br>
+: This revision was by author [[User:MartinGough|MartinGough]] and made on <tt>2015-02-20 14:08:56 UTC</tt>.<br>
-: The original revision id was <tt>541464324</tt>.<br>
+: The original revision id was <tt>541592976</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
-=Introduction=
+=&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;=
 &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term //logarithmic approximant//[[xenharmonic/Mike's Lecture on Vector Spaces and Dual Spaces#ref1|{1}]] (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?&lt;/span&gt;
 * &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;
-* &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why is the ratio of the perfect fifth to the perfect fourth close to √2?&lt;/span&gt;
+* &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce satisfying scale structures?&lt;/span&gt;
 The exact size, in cents, of an interval with frequency ratio //r// is
@@ Line 41: / Line 41: @@
 =**&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;**=
-==**&lt;span style="font-size: 15px;"&gt;Definition&lt;/span&gt;**==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Definition&lt;/span&gt;==
 The bimodular approximant of an interval with frequency ratio //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r = n/d&lt;/span&gt;// is
 [[math]]
@@ Line 59: / Line 59: @@
 [[math]]
-==**&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;**==
+&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Properties&lt;/span&gt;
 When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r//&lt;/span&gt; is
@@ Line 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: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;**=
+==**&lt;span style="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-size: 15px;"&gt;Bimodular commas&lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;==
 As a consequence of the near-rational interval relationships implied by approximants, any pair of source intervals can be used to define a comma.
 Given two intervals &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt; (with&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; //J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; &lt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;) and their approximants &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;1&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;2&lt;/span&gt;, we define the //bimodular residue// as
@@ Line 146: / Line 146: @@
 ===Examples===
-If the source intervals are the perfect fourth (//f = __4/3__)// and the perfect fifth (//F = __3/2__//), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 1/5&lt;/span&gt;, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the Pythagorean comma:
+If the source intervals are the perfect fourth (&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%;"&gt;4/3&lt;/span&gt;__//)// and the perfect fifth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//F// = __3/2__&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;//v//1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 1/5&lt;/span&gt;, and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the Pythagorean comma:
 [[math]]
 \qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f
 [[math]]
-If the source intervals are the perfect fourth (//f// //= __4/3__//) and the minor seventh (//m&lt;span style="vertical-align: sub;"&gt;7 = __9/5__&lt;/span&gt;//), &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//1 = 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 2/7&lt;/span&gt;, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the syntonic comma:
+If the source intervals are the perfect fourth (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//f// = __4/3__&lt;/span&gt;) and the minor seventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//m//&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;7 &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= __9/5__), &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;//v//&lt;/span&gt;1 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//v//2 = 2/7&lt;/span&gt;, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;//r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&lt;/span&gt; and //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;// is the syntonic comma:
 [[math]]
 \qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f
@@ Line 158: / Line 158: @@
 =**&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;**=
+==Definition==
 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 171: / Line 172: @@
 \qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))
 [[math]]
-The first convergent of this continued fraction is //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; 3//v///(3-//v//&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;).
+The first convergent of this continued fraction is //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
+[[math]]
+&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;\qquad y = \frac{3v}{3-v^2&lt;/span&gt;}
+[[math]]
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.
-|| //Interval// || //(1,2) Padé approximant// ||
+|| //Interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;//&lt;span style="color: #ffffff;"&gt;###########&lt;/span&gt; || //(1,2) Padé approximant &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;y&lt;/span&gt;//&lt;span style="color: #ffffff;"&gt;#&lt;/span&gt; ||
-|| Perfect twelfth __3/1__ || 6/11 ||
+|| Perfect twelfth = __3/1__ || 6/11 ||
-|| Octave 2/1 || 9/26 ||
+|| Octave = __2/1__ || 9/26 ||
-|| Major sixth 5/3 || 12 15/ ||
+|| Major sixth = __5/3__ || 12/47 ||
-|| Perfect fifth 3/2 || 74/47 ||
+|| Perfect fifth = __3/2__ || 15/74 ||
-|| Perfect fourth 4/3 || 21/146 ||
+|| Perfect fourth = __4/3__ || 21/146 ||
-|| Major third 5/4 || 27/242 ||
+|| Major third = __5/4__ || 27/242 ||
+The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:
+(__3/1__) / (__6/5__) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)
+(__3/1__) / (__7/4__) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = __49/48__)
+(__2/1__) / (__7/6__) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9&gt; comma)
+(__2/1__) / (__27/25__) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)
+(__5/3__) / (__49/45__) = 5.9986 ≈ (12/47) / (2/47) = 6
+(__5/3__) / (__25/22__) = 3.9960 ≈ (12/47) / (3/47) = 4
+(__5/3__) / (__26/21__) = 2.3918 ≈ (12/47) / (5/47) = 12/5
+(__5/3__) / (__27/20__) = 1.7022 ≈ (12/47) / (7/47) = 12/7
+(__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-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;==
+The quadratic approximant //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// of an interval //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;// with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//r// = //n/////d//&lt;/span&gt; is
+[[math]]
+\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\
+\qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...
+[[math]]
+If this is compared with the expression for the bimodular approximant,
+[[math]]
+\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...
+[[math]]
+it is apparent that //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// is about twice as accurate as //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;//, with an error of opposite sign.
+While //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;// is the frequency difference divided by twice the arithmetic frequency mean, //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// is the frequency difference divided by twice the geometric frequency mean:
+[[math]]
+\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}
+[[math]]
+//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;// can be retrieved from //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;// using
+[[math]]
+\qquad \sqrt{r} = q + \sqrt{1+q^2}
+[[math]]
+The following are the quadratic approximants of some simple 5-limit intervals:
+|| //Interval// //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;//&lt;span style="color: #ffffff;"&gt;##################### &lt;/span&gt; || //Quadratic approximant// &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//q//&lt;/span&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif; font-size: 110%;"&gt; ##&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Perfect twelfth = __3/1__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/√3&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Octave = __2/1__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√2&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor seventh = __9/5__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 2/3√5&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Pythagorean minor seventh = __16/9__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 7/24&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Major sixth = __5/3__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/√15&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor sixth = __8/5__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 3/4√10&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Perfect fifth = __3/2__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√6&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Perfect fourth = __4/3__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/4√3&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Major third = __5/4__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/4√5&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor third = __6/5__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√30&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Pythagorean minor third = __32/27__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 5/24√6&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Large tone = __9/8__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/12√2&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Small tone = __10/9__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/6√10&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Diatonic semitone = __16/15__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/8√15&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Chroma = __25/24__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/20√6&lt;/span&gt; ||
+|| &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Syntonic comma = __81/80__&lt;/span&gt; || &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/72√5&lt;/span&gt; ||
+Expressed in terms of the bimodular approximant,//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; v = j/g&lt;/span&gt;//,
+[[math]]
+\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}
+[[math]]
+Quadratic approximants of just intervals thus have the form //&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q = j/√k&lt;/span&gt;//, where //&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;k&lt;/span&gt;// are integers 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: super;"&gt;2&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; + k = g&lt;/span&gt;//&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; is a perfect square.
+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;==
+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]]
+\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\
+\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\
+\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\
+\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\
+\qquad q[-J] = -q[J] \\
+\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\
+\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\
+\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1} \\
+\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\
+[[math]]
+The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity
+[[math]]
+\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}
+[[math]]
+Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.
+For example
+[[math]]
+\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6
+[[math]]
+but this can also be derived from bimodular approximants. Using
+[[math]]
+\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}
+[[math]]
+with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//2 = F =__3/2__ &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;//J//1 = //f// = __4/3__&lt;/span&gt; this gives
+[[math]]
+\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\
+\qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6
+[[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;//):
+[[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 + ...
+[[math]]
+However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.
+The most interesting approximate interval ratios derivable from quadratic approximants are irrational.
+==&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;==
+==&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;==
+===&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;**Theorem**===
+If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.
+===**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**===
+The ratio of the intervals as estimated from their quadratic approximants is
+[[math]]
+\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}
+[[math]]
+which is the geometric mean of their frequency ratios.
+===**Examples**===
@@ Line 185: / Line 304: @@
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Logarithmic approximants&lt;/title&gt;&lt;/head&gt;&lt;body&gt;WORK IN PROGRESS&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:25:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:25 --&gt;Introduction&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Introduction&lt;/span&gt;&lt;/h1&gt;
   &lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;The term &lt;em&gt;logarithmic approximant&lt;/em&gt;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Mike%27s%20Lecture%20on%20Vector%20Spaces%20and%20Dual%20Spaces#ref1"&gt;{1}&lt;/a&gt; (or &lt;em&gt;approximant&lt;/em&gt; for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as&lt;/span&gt;&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why is the ratio of the perfect fifth to the perfect fourth close to √2?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why do certain temperaments (such as 12edo) provide a reasonably accurate approximation to 5-limit just intonation?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why are certain commas small, and roughly how small are they?&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Why does the 3-limit framework produce satisfying scale structures?&lt;/span&gt;&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 The exact size, in cents, of an interval with frequency ratio &lt;em&gt;r&lt;/em&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:0:
@@ Line 218: / Line 337: @@
 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:27:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:27 --&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:40:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:29:&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:29 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Definition&lt;/span&gt;&lt;/strong&gt;&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:42:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Bimodular approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:42 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
   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 242: / Line 361: @@
   --&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:31:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Properties of bimodular approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:31 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Properties of bimodular approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h2&gt;
+&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Properties&lt;/span&gt;&lt;br /&gt;
- When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.&lt;br /&gt;
+When &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; &lt;/span&gt;is small, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;&lt;/span&gt; provides an approximate relative measure of the logarithmic size of the interval. This approximation was exploited by Joseph Sauveur in 1701 and later by Euler and others.&lt;br /&gt;
 Noting that the exact size (in dineper units) of the interval with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:6:
@@ Line 290: / Line 409: @@
 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:33:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:33 --&gt;&lt;strong&gt;&lt;span style="font-size: 20px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:44:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Bimodular approximants-Bimodular approximants and equal temperaments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:44 --&gt;&lt;strong&gt;&lt;span style="font-size: 15px;"&gt;Bimodular approximants and equal temperaments&lt;/span&gt;&lt;/strong&gt;&lt;/h2&gt;
   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 305: / Line 424: @@
 Relationships of this sort can be identified in all equal temperaments.&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:35:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Bimodular approximants and equal temperaments-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:35 --&gt;&lt;span style="font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:46:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Bimodular approximants-Bimodular commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:46 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif; font-size: 15px;"&gt;Bimodular commas&lt;/span&gt;&lt;/h2&gt;
   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 343: / Line 462: @@
   --&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:37:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Bimodular approximants and equal temperaments-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:37 --&gt;Examples&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:48:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Bimodular approximants-Bimodular commas-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:48 --&gt;Examples&lt;/h3&gt;
-  If the source intervals are the perfect fourth (&lt;em&gt;f = &lt;u&gt;4/3&lt;/u&gt;)&lt;/em&gt; and the perfect fifth (&lt;em&gt;F = &lt;u&gt;3/2&lt;/u&gt;&lt;/em&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;
+  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:
 [[math]]&amp;lt;br/&amp;gt;
 \qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad b(F,f) = b_r(F,f) = \frac{F}{\tfrac{1}{5}} - \frac{f}{\tfrac{1}{7}} = 5F – 7f&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:20 --&gt;&lt;br /&gt;
-If the source intervals are the perfect fourth (&lt;em&gt;f&lt;/em&gt; &lt;em&gt;= &lt;u&gt;4/3&lt;/u&gt;&lt;/em&gt;) and the minor seventh (&lt;em&gt;m&lt;span style="vertical-align: sub;"&gt;7 = &lt;u&gt;9/5&lt;/u&gt;&lt;/span&gt;&lt;/em&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 = 2/7&lt;/span&gt;, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&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 syntonic comma:&lt;br /&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;u&gt;4/3&lt;/u&gt;&lt;/span&gt;) and the minor seventh (&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;m&lt;/em&gt;&lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"&gt;7 &lt;/span&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= &lt;u&gt;9/5&lt;/u&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;&lt;/span&gt;1 &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 1/7&lt;/span&gt;, &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;v&lt;/em&gt;2 = 2/7&lt;/span&gt;, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;b&lt;/span&gt;&lt;/em&gt;r &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;= 2/7&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 syntonic comma:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextMathRule:21:
 [[math]]&amp;lt;br/&amp;gt;
@@ Line 357: / Line 476: @@
 Further examples of bimodular commas are provided in Reference 1^^^. See also &lt;u&gt;Don Page comma^^^&lt;/u&gt; (another name for this type of comma).&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:39:&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:39 --&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:50:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Padé approximants of order (1,2)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:50 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Padé approximants of order (1,2)&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:52:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Padé approximants of order (1,2)-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:52 --&gt;Definition&lt;/h2&gt;
   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 374: / Line 494: @@
 \qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&amp;lt;br/&amp;gt;[[math]]
   --&gt;&lt;script type="math/tex"&gt;\qquad J = \tanh^(-1){v} = v / (1-v^2/(3 – 4v^2/(5 – 9v^2/(7 - ...)))&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:24 --&gt;&lt;br /&gt;
-The first convergent of this continued fraction is &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; 3&lt;em&gt;v&lt;/em&gt;/(3-&lt;em&gt;v&lt;/em&gt;&lt;/span&gt;&lt;span style="vertical-align: super;"&gt;2&lt;/span&gt;).&lt;br /&gt;
+The first convergent of this continued fraction is &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:25:
+[[math]]&amp;lt;br/&amp;gt;
+&amp;lt;span style=&amp;quot;font-family: Georgia,serif; font-size: 110%;&amp;quot;&amp;gt;\qquad y = \frac{3v}{3-v^2&amp;lt;/span&amp;gt;}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;\qquad y = \frac{3v}{3-v^2&lt;/span&gt;}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:25 --&gt;&lt;br /&gt;
 Values of this rational approximant for some simple 5-limit intervals are shown in the table below.&lt;br /&gt;
@@ Line 380: / Line 504: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;em&gt;Interval&lt;/em&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;em&gt;Interval &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt;&lt;span style="color: #ffffff;"&gt;###########&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;em&gt;(1,2) Padé approximant&lt;/em&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;em&gt;(1,2) Padé approximant &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;y&lt;/span&gt;&lt;/em&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;Perfect twelfth &lt;u&gt;3/1&lt;/u&gt;&lt;br /&gt;
+         &lt;td&gt;Perfect twelfth = &lt;u&gt;3/1&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;6/11&lt;br /&gt;
@@ Line 392: / Line 516: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Octave 2/1&lt;br /&gt;
+         &lt;td&gt;Octave = &lt;u&gt;2/1&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;9/26&lt;br /&gt;
@@ Line 398: / Line 522: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Major sixth 5/3&lt;br /&gt;
+         &lt;td&gt;Major sixth = &lt;u&gt;5/3&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;12 15/&lt;br /&gt;
+         &lt;td&gt;12/47&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Perfect fifth 3/2&lt;br /&gt;
+         &lt;td&gt;Perfect fifth = &lt;u&gt;3/2&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;74/47&lt;br /&gt;
+         &lt;td&gt;15/74&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Perfect fourth 4/3&lt;br /&gt;
+         &lt;td&gt;Perfect fourth = &lt;u&gt;4/3&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;21/146&lt;br /&gt;
@@ Line 416: / Line 540: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;Major third 5/4&lt;br /&gt;
+         &lt;td&gt;Major third = &lt;u&gt;5/4&lt;/u&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;27/242&lt;br /&gt;
@@ Line 423: / Line 547: @@
 &lt;/table&gt;
+The denominators of these fractions rapidly get large, so this type of approximant has limited usefulness. However, when combined with bimodular approximants it has occasional value in explaining apparent numerical coincidences and the smallness of the associated commas. For example:&lt;br /&gt;
+&lt;br /&gt;
+(&lt;u&gt;3/1&lt;/u&gt;) / (&lt;u&gt;6/5&lt;/u&gt;) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)&lt;br /&gt;
+(&lt;u&gt;3/1&lt;/u&gt;) / (&lt;u&gt;7/4&lt;/u&gt;) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = &lt;u&gt;49/48&lt;/u&gt;)&lt;br /&gt;
+(&lt;u&gt;2/1&lt;/u&gt;) / (&lt;u&gt;7/6&lt;/u&gt;) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9&amp;gt; comma)&lt;br /&gt;
+(&lt;u&gt;2/1&lt;/u&gt;) / (&lt;u&gt;27/25&lt;/u&gt;) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)&lt;br /&gt;
+(&lt;u&gt;5/3&lt;/u&gt;) / (&lt;u&gt;49/45&lt;/u&gt;) = 5.9986 ≈ (12/47) / (2/47) = 6&lt;br /&gt;
+(&lt;u&gt;5/3&lt;/u&gt;) / (&lt;u&gt;25/22&lt;/u&gt;) = 3.9960 ≈ (12/47) / (3/47) = 4&lt;br /&gt;
+(&lt;u&gt;5/3&lt;/u&gt;) / (&lt;u&gt;26/21&lt;/u&gt;) = 2.3918 ≈ (12/47) / (5/47) = 12/5&lt;br /&gt;
+(&lt;u&gt;5/3&lt;/u&gt;) / (&lt;u&gt;27/20&lt;/u&gt;) = 1.7022 ≈ (12/47) / (7/47) = 12/7&lt;br /&gt;
+(&lt;u&gt;3/2&lt;/u&gt;) / (&lt;u&gt;20/17&lt;/u&gt;) = 2.4949 ≈ (15/74) / (6/74) = 5/2&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:54:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Quadratic approximants"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:54 --&gt;&lt;strong&gt;&lt;span style="font-size: 21.33px;"&gt;Quadratic approximants&lt;/span&gt;&lt;/strong&gt;&lt;/h1&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:56:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Quadratic approximants-Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:56 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Definition&lt;/span&gt;&lt;/h2&gt;
+ The quadratic approximant &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; of an interval &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; with frequency ratio &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;r&lt;/em&gt; = &lt;em&gt;n&lt;/em&gt;&lt;em&gt;/d&lt;/em&gt;&lt;/span&gt; is&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:26:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\&amp;lt;br /&amp;gt;
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\&amp;lt;br /&amp;gt;
+\qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad q(r) = \tfrac{1}{2} (r^{1/2} – r^{-1/2}) \\
+\qquad = \tfrac{1}{2} (e^J - e^{-J}) = \sinh{J} \\
+\qquad = J + \tfrac{1}{3!} J^3 + \tfrac{1}{5!} J^5 + ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:26 --&gt;&lt;br /&gt;
+If this is compared with the expression for the bimodular approximant,&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:27:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad v = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:27 --&gt;&lt;br /&gt;
+it is apparent that &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; is about twice as accurate as &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt;, with an error of opposite sign.&lt;br /&gt;
+While &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;v&lt;/span&gt;&lt;/em&gt; is the frequency difference divided by twice the arithmetic frequency mean, &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; is the frequency difference divided by twice the geometric frequency mean:&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:28:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad v = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:28 --&gt;&lt;br /&gt;
+&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;r&lt;/span&gt;&lt;/em&gt; can be retrieved from &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; using&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:29:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \sqrt{r} = q + \sqrt{1+q^2}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \sqrt{r} = q + \sqrt{1+q^2}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:29 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+The following are the quadratic approximants of some simple 5-limit intervals:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;em&gt;Interval&lt;/em&gt; &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt;&lt;span style="color: #ffffff;"&gt;##################### &lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;em&gt;Quadratic approximant&lt;/em&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;q&lt;/em&gt;&lt;/span&gt;&lt;span style="color: #ffffff; font-family: Georgia,serif; font-size: 110%;"&gt; ##&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;Perfect twelfth = &lt;u&gt;3/1&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/√3&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Octave = &lt;u&gt;2/1&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√2&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor seventh = &lt;u&gt;9/5&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 2/3√5&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Pythagorean minor seventh = &lt;u&gt;16/9&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 7/24&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Major sixth = &lt;u&gt;5/3&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/√15&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor sixth = &lt;u&gt;8/5&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 3/4√10&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Perfect fifth = &lt;u&gt;3/2&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√6&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Perfect fourth = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/4√3&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Major third = &lt;u&gt;5/4&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/4√5&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Minor third = &lt;u&gt;6/5&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/2√30&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Pythagorean minor third = &lt;u&gt;32/27&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 5/24√6&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Large tone = &lt;u&gt;9/8&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/12√2&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Small tone = &lt;u&gt;10/9&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/6√10&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Diatonic semitone = &lt;u&gt;16/15&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/8√15&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Chroma = &lt;u&gt;25/24&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/20√6&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; Syntonic comma = &lt;u&gt;81/80&lt;/u&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt; 1/72√5&lt;/span&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;br /&gt;
+Expressed in terms of the bimodular approximant,&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; v = j/g&lt;/span&gt;&lt;/em&gt;,&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:30:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad q = \frac{v}{\sqrt{1-v^2}} = \frac{j}{\sqrt{g^2-j^2}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:30 --&gt;&lt;br /&gt;
+Quadratic approximants of just intervals thus have the form &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q = j/√k&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; and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;k&lt;/span&gt;&lt;/em&gt; are integers and &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;j&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt;&lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt; + k = g&lt;/span&gt;&lt;/em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;"&gt;2&lt;/span&gt; is a perfect square.&lt;br /&gt;
+The presence of a square root in the denominator of &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;q&lt;/span&gt;&lt;/em&gt; (except where &lt;em&gt;&lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;J&lt;/span&gt;&lt;/em&gt; is a double interval) means that quadratic approximants do not, on the whole, imply approximate rational ratios between just intervals or commas of the conventional type. Their interest stems from the fact that ratios involving integer square roots are expressible as repeating continued fractions.&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:58:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Quadratic approximants-Properties"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:58 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Properties&lt;/span&gt;&lt;/h2&gt;
+ 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:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\&amp;lt;br /&amp;gt;
+\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\&amp;lt;br /&amp;gt;
+\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\&amp;lt;br /&amp;gt;
+\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\&amp;lt;br /&amp;gt;
+\qquad q[-J] = -q[J] \\&amp;lt;br /&amp;gt;
+\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\&amp;lt;br /&amp;gt;
+\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\&amp;lt;br /&amp;gt;
+\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1} \\&amp;lt;br /&amp;gt;
+\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad v = \tanh{J}, q = \sinh{J}, \frac{q}{v} = \cosh{J} \\
+\qquad \sqrt{r} = e^J = q(\frac{1}{v} + 1) \\
+\qquad \frac{1}{\sqrt{r}} = e^{-J} = q(\frac{1}{v} - 1) \\
+\qquad \frac{1}{q^2} = \frac{1}{v^2} – 1 \\
+\qquad q[-J] = -q[J] \\
+\qquad q[J_2 + J_1] = q_1 q_2 (\frac{1}{v_2} + \frac{1}{v_1}) \\
+\qquad q[J_2 - J_1] = q_1 q_2 (\frac{1}{v_2} - \frac{1}{v_1}) \\
+\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1} \\
+\qquad q[J_2 + J_1] q[J_2 - J_1] = q_2^2 - q_1^2 \\&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:31 --&gt;&lt;br /&gt;
+The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:32:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \sinh{J_2 + J_1} \sinh{J_2 - J_1} = \sinh^2{J_2} - \sinh^2{J_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:32 --&gt;&lt;br /&gt;
+&lt;br /&gt;
+Where two quadratic approximants have the same square root in the denominator their ratio is rational. This seems to suggest a new source of approximate rational interval ratios, and therefore a new source of commas, but in this situation the approximants always represent the sum and difference of a pair of just intervals, and their ratio can be derived by an alternative route using the bimodular approximants of those intervals.&lt;br /&gt;
+For example&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:33:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \frac{octave}{large \, tone} ≈ \frac{1}{2√2} / \frac{1}{12√2} = 6&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:33 --&gt;&lt;br /&gt;
+but this can also be derived from bimodular approximants. Using&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:34:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \frac {q[J_2 + J_1]}{q[J_2 - J_1]} = \frac{v_2+v_1}{v_2-v_1}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:34 --&gt;&lt;br /&gt;
+with &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;2 = F =&lt;u&gt;3/2&lt;/u&gt; &lt;/span&gt;&lt;span style="font-family: Arial,Helvetica,sans-serif;"&gt;and&lt;/span&gt; &lt;span style="font-family: Georgia,serif; font-size: 110%;"&gt;&lt;em&gt;J&lt;/em&gt;1 = &lt;em&gt;f&lt;/em&gt; = &lt;u&gt;4/3&lt;/u&gt;&lt;/span&gt; this gives&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:35:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\&amp;lt;br /&amp;gt;
+\qquad = \frac{v[F] + v[f]}{v[F] - v[f]} = \frac{1/5 + 1/7}{1/5 - 1/7} = 6&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \frac{octave}{large \, tone} ≈ \frac{q[F+f]}{q[F-f]} \\
+\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;
+&lt;!-- ws:start:WikiTextMathRule:36:
+[[math]]&amp;lt;br/&amp;gt;
+\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 + ...&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\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 + ...&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:36 --&gt;&lt;br /&gt;
+However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.&lt;br /&gt;
+The most interesting approximate interval ratios derivable from quadratic approximants are irrational.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:60:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;!-- ws:end:WikiTextHeadingRule:60 --&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;/h2&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;&lt;span style="font-family: 'Arial Black',Gadget,sans-serif;"&gt;Relative sizes of intervals between 3 frequencies in arithmetic progression&lt;/span&gt;&lt;/h2&gt;
+ &lt;!-- ws:start:WikiTextHeadingRule:64:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:64 --&gt;&lt;span style="font-family: Times New Roman;"&gt; &lt;/span&gt;&lt;strong&gt;Theorem&lt;/strong&gt;&lt;/h3&gt;
+ If three harmonics of a fundamental frequency form an arithmetic progression, then the ratio of the logarithmic sizes of the intervals formed between the lower and upper pairs of harmonics is close to the geometric mean of these intervals’ frequency ratios.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:66:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:66 --&gt;&lt;strong&gt;Remarks&lt;/strong&gt;&lt;/h3&gt;
+ If the harmonics have indices n – m, n and n + m, the two intervals have reduced frequency ratios n/(n – m) and (n + m)/n. It can be assumed that n and m have no common factor.&lt;br /&gt;
+m is the epimoricity of the intervals. When m = 1 the intervals are adjacent superparticular intervals.&lt;br /&gt;
+The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:68:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:68 --&gt;&lt;strong&gt;Proof&lt;/strong&gt;&lt;/h3&gt;
+ The ratio of the intervals as estimated from their quadratic approximants is&lt;br /&gt;
+&lt;!-- ws:start:WikiTextMathRule:37:
+[[math]]&amp;lt;br/&amp;gt;
+\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}&amp;lt;br/&amp;gt;[[math]]
+ --&gt;&lt;script type="math/tex"&gt;\qquad \tfrac{m}{2\sqrt{n(n-m)}} / \tfrac{m}{2\sqrt{(n+m)n}} = \sqrt{\frac{n+m}{n-m}}&lt;/script&gt;&lt;!-- ws:end:WikiTextMathRule:37 --&gt;&lt;br /&gt;
+which is the geometric mean of their frequency ratios.&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:70:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:70 --&gt;&lt;strong&gt;Examples&lt;/strong&gt;&lt;/h3&gt;
+ &lt;br /&gt;
 &lt;br /&gt;
 &lt;br /&gt;
 ^^^&lt;/body&gt;&lt;/html&gt;</pre></div>