Logarithmic approximants

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WORK IN PROGRESS
=<span style="font-family: "Arial Black",Gadget,sans-serif;">Introduction</span>= 
<span style="font-family: Arial,Helvetica,sans-serif;">The term //logarithmic approximant//  (or //approximant// for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why do certain temperaments such as 12edo provide a reasonably accurate approximation to 5-limit just intonation?</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why are certain commas small, and roughly how small are they?</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why does the 3-limit framework produce aesthetically pleasing scale structures?</span>

The exact size, in cents, of an interval with frequency ratio //r// is
[[math]]
\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}

[[math]]
where for just intervals r is rational and can be written as the ratio of two integers:
[[math]]
\qquad r = n/d

[[math]]
When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as
[[math]]
\qquad J = \tfrac{1}{2} \ln{r}

[[math]]
This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e<span style="vertical-align: super;">2</span> = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.
Comparing the two units of measurement we find
1 dineper = 2400/ln(2) = 3462.468 cents
which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus __3/2__ is the perfect fifth.

Three types of approximants are described here:
* Bimodular approximants (first order rational approximants)
* Padé approximants of order (1,2) (second order rational approximants)
* Quadratic approximants

=**<span style="font-size: 20px;">Bimodular approximants</span>**= 
==<span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Definition</span>== 
The bimodular approximant of an interval with frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r = n/d</span>// is
[[math]]
\qquad v = \frac{r-1}{r+1}

[[math]]
//<span style="font-family: Georgia,serif; font-size: 110%;">v </span>//can thus be expressed as
[[math]]
\qquad v = \frac{n-d}{n+d} \\

[[math]]
<span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum)
<span style="color: #ffffff;">######</span> =½ (frequency difference) / (mean frequency)
<span style="font-family: Georgia,serif; font-size: 110%;">//r// </span>can be retrieved from <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> using the inverse relation
[[math]]
\qquad r = \frac{1+v}{1-v}
[[math]]

<span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Properties</span>
When <span style="font-family: Georgia,serif; font-size: 110%;">//r// </span>is small, <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> 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 <span style="font-family: Georgia,serif; font-size: 110%;">//r//</span> is
[[math]]
\qquad J = \tfrac{1}{2} \ln{r}
[[math]]
the relationship between <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span> can be expressed as
[[math]]
\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...
[[math]]
which shows that <span style="font-family: Georgia,serif; font-size: 110%;">//v// ≈ //J//</span> and provides an indication of the size and sign of the error involved in this approximation.
//<span style="font-family: Georgia;">J</span>// can be expressed in terms of <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span> as
[[math]]
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...
[[math]]
The function <span style="font-family: Georgia,serif; font-size: 110%;">//v(r)//</span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> in the region of //r// = 1, which has the property of matching the function value and its first and second derivatives at this value of //r//. The bimodular approximant function is thus accurate to second order in //r// – 1.

As an example, the size of the perfect fifth (in dNp units) is
[[math]]
\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...
[[math]]
The bimodular approximant for this interval (r = 3/2) is
[[math]]
\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2
[[math]]
and the Taylor series indicates that the error in this value is about
[[math]]
\qquad -\tfrac{1}{3}v^3 = -0.00267...
[[math]]

The approximants of superparticular intervals are reciprocals of odd integers:
[[image:Low-order superparticular intervals.png]]

If <span style="font-family: Georgia,serif; font-size: 110%;">//v//[//J//] </span>denotes the bimodular approximant of an interval <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span> with frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//,
[[math]]
\qquad v[-J] = -v[J] \\
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}
[[math]]
This last result is equivalent to the identity expressing <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 + </span><span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif;">)</span> in terms of <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> and <span style="font-family: Georgia,serif; font-size: 110%;">tanh(//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">).</span>

==**<span style="font-size: 15px;">Bimodular approximants and equal temperaments</span>**== 
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, //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/7) approximate a minor seventh (//r// = 9/5, = 2/7)
Three major thirds (//r// = 5/4, //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/9) or two __7/5__s (//<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/6) or five __8/7__s (//<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// = 1/15) approximate an octave (//r// = 2/1,//<span style="font-family: Georgia,serif; font-size: 110%;"> v</span>// = 1/3)
Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.
Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos alpha.
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos beta.
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos gamma. This temperament has high accuracy because it conforms to the policy noted above.
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.
Tuning the intervals __9/7__, __7/5__ and __5/3__ in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.
Tuning the intervals __11/9__, __9/7__, __3/2__ and __5/3__ in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.
Relationships of this sort can be identified in all equal temperaments.

==<span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Bimodular commas</span>== 
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 <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> (with<span style="font-family: Georgia,serif; font-size: 110%;"> //J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> < <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>) and their approximants <span style="font-family: Georgia,serif; font-size: 110%;">//v//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>, we define the //bimodular residue// as
[[math]]
\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}
[[math]]
and using the Taylor series expansion of <span style="font-family: Georgia,serif; font-size: 110%;">//J//(//v//)</span> we find
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)
[[math]]
The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of <span style="font-family: Georgia,serif; font-size: 110%;">//J//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> with integer coefficients sharing no common factor:
[[math]]
\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)
[[math]]
where
[[math]]
\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}
[[math]]
and (with rare exceptions)
[[math]]
\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}
[[math]]
The bimodular residue is accurately estimated by
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)
[[math]]
and therefore
[[math]]
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m
[[math]]

===Examples=== 
If the source intervals are the perfect fourth (<span style="font-family: Georgia,serif; font-size: 110%;">//f// =</span> __<span style="font-family: Georgia,serif; font-size: 110%;">4/3</span>__//)// and the perfect fifth (<span style="font-family: Georgia,serif; font-size: 110%;">//F// = __3/2__</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">), </span><span style="font-family: Arial,Helvetica,sans-serif;">then</span> <span style="font-family: Georgia,serif; font-size: 110%;">//v//1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">//v//2 = 1/5</span>, and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// 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 (<span style="font-family: Georgia,serif; font-size: 110%;">//f// = __4/3__</span>) and the minor seventh (<span style="font-family: Georgia,serif; font-size: 110%;">//m//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">7 </span><span style="font-family: Georgia,serif; font-size: 110%;">= __9/5__), </span><span style="font-family: Arial,Helvetica,sans-serif;">then </span><span style="font-family: Georgia,serif; font-size: 110%;">//v//</span>1 <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;">//v//2 = 2/7</span>, //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>//r <span style="font-family: Georgia,serif; font-size: 110%;">= 2/7</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// 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
[[math]]

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

=**<span style="font-size: 21.33px;">Padé approximants of order (1,2)</span>**= 
==Definition== 
In the section on bimodular approximants it was shown than an interval of logarithmic size //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// (measured in dineper units) is related to its bimodular approximant by
[[math]]
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...
[[math]]
where
[[math]]
\qquad v = \frac{r-1}{r+1}
[[math]]
and //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>// is the interval’s frequency ratio.
Another way to express this relationship is with a continued fraction:
[[math]]
\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 //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is
[[math]]
<span style="font-family: Georgia,serif; font-size: 110%;">\qquad y = \frac{3v}{3-v^2</span>}
[[math]]
Values of this rational approximant for some simple 5-limit intervals are shown in the table below.
|| //Interval <span style="font-family: Georgia,serif; font-size: 110%;">J</span>//<span style="color: #ffffff;">###########</span> || //(1,2) Padé approximant <span style="font-family: Georgia,serif; font-size: 110%;">y</span>//<span style="color: #ffffff;">#</span> ||
|| Perfect twelfth = __3/1__ || 6/11 ||
|| Octave = __2/1__ || 9/26 ||
|| Major sixth = __5/3__ || 12/47 ||
|| Perfect fifth = __3/2__ || 15/74 ||
|| Perfect fourth = __4/3__ || 21/146 ||
|| 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

=**<span style="font-size: 21.33px;">Quadratic approximants</span>**= 
==<span style="font-family: "Arial Black",Gadget,sans-serif;">Definition</span>== 
The quadratic approximant //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// of an interval //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;">//r// = //n/////d//</span> 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 //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// is about twice as accurate as //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//, with an error of opposite sign.
While //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>// is the frequency difference divided by twice the arithmetic frequency mean, //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// 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]]
//<span style="font-family: Georgia,serif; font-size: 110%;">r</span>// can be retrieved from //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// using
[[math]]
\qquad \sqrt{r} = q + \sqrt{1+q^2}
[[math]]

The following are the quadratic approximants of some simple 5-limit intervals:
|| //Interval// //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>//<span style="color: #ffffff;">##################### </span> || //Quadratic approximant// <span style="font-family: Georgia,serif; font-size: 110%;">//q//</span><span style="color: #ffffff; font-family: Georgia,serif; font-size: 110%;"> ##</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;">Perfect twelfth = __3/1__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/√3</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Octave = __2/1__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√2</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Minor seventh = __9/5__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 2/3√5</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor seventh = __16/9__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 7/24</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Major sixth = __5/3__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/√15</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Minor sixth = __8/5__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 3/4√10</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fifth = __3/2__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√6</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fourth = __4/3__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√3</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Major third = __5/4__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√5</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Minor third = __6/5__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√30</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor third = __32/27__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 5/24√6</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Large tone = __9/8__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/12√2</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Small tone = __10/9__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/6√10</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Diatonic semitone = __16/15__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/8√15</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Chroma = __25/24__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/20√6</span> ||
|| <span style="font-family: Arial,Helvetica,sans-serif;"> Syntonic comma = __81/80__</span> || <span style="font-family: Arial,Helvetica,sans-serif;"> 1/72√5</span> ||

Expressed in terms of the bimodular approximant,//<span style="font-family: Georgia,serif; font-size: 110%;"> v = j/g</span>//,
[[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 //<span style="font-family: Georgia,serif; font-size: 110%;">q = j/√k</span>//, where //<span style="font-family: Georgia,serif; font-size: 110%;">j</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">k</span>// are integers and //<span style="font-family: Georgia,serif; font-size: 110%;">j</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>//<span style="font-family: Georgia,serif; font-size: 110%;"> + k = g</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> is a perfect square.
The presence of a square root in the denominator of //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// (except where //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// 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.

==<span style="font-family: "Arial Black",Gadget,sans-serif;">Properties</span>== 
If //<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//<span style="font-family: Georgia,serif; font-size: 110%;">[//J//]</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//q//[//J//]</span> denote, respectively, the bimodular and quadratic approximants of an interval //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// with frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//, and //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>//<span style="font-family: Georgia,serif; font-size: 80%;">n</span> denotes <span style="font-family: Georgia,serif; font-size: 110%;">//q//[//J//n]</span> , 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 <span style="font-family: Georgia,serif; font-size: 110%;">//J//2 = F =__3/2__ </span><span style="font-family: Arial,Helvetica,sans-serif;">and</span> <span style="font-family: Georgia,serif; font-size: 110%;">//J//1 = //f// = __4/3__</span> 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 //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>// of a double interval <span style="font-family: Georgia,serif; font-size: 110%;">2//J//</span> (for example, the ditone) is rational, which suggests using <span style="font-family: Georgia,serif; font-size: 110%;">½ q(r</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> as a rational approximant of //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// (where //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>// has frequency ratio //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//):
[[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.
== == 
==<span style="font-family: "Arial Black",Gadget,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span>== 
===**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 //<span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">n + m</span>//, the two intervals have reduced frequency ratios //<span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span>//. It can be assumed that //<span style="font-family: Georgia,serif; font-size: 110%;">n</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// have no common factor.
//<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// is the epimoricity of the intervals. When //<span style="font-family: Georgia,serif; font-size: 110%;">m</span>// = 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**=== 
The ratio of the perfect fifth, <span style="font-family: Georgia,serif; font-size: 110%;">//F// = __3/2__</span>, to the perfect fourth, <span style="font-family: Georgia,serif; font-size: 110%;">//f// = __4/3__</span>, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).
<span style="font-family: Georgia,serif; font-size: 110%;">//F/f// = 701.955/498.045 = 1.40942,</span>
<span style="font-family: Georgia,serif; font-size: 110%;">√2 = 1.41421.</span>
The ratio of the large tone, <span style="font-family: Georgia,serif; font-size: 110%;">//T// = __9/8__</span>, to the small tone, <span style="font-family: Georgia,serif; font-size: 110%;">//t// = __10/9__</span>, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.
<span style="font-family: Georgia,serif; font-size: 110%;">//T/////t// = 203.910/182.404 = 1.11790,</span>
<span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span>
===**Silver temperament**=== 
As the first example above shows, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.
It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to
<span style="color: #ffffff;">###</span>Perfect fifth = __3/2__ = 702.944 cents
<span style="color: #ffffff;">###</span>Perfect fourth = __4/3__ = 497.056 cents
This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).
A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.
Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.
The continued fraction expansion of the silver ratio has a particularly simple form:
[[math]]
\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))
[[math]]
As a result, if two intervals L and s are tuned in the silver ratio, with <span style="font-family: Georgia,serif; font-size: 110%;">//s = L/δ//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>, subtracting twice the small interval //<span style="font-family: Georgia,serif; font-size: 110%;">s</span>// from the large interval //<span style="font-family: Georgia,serif; font-size: 110%;">L</span>// leaves a remainder of size <span style="font-family: Georgia,serif; font-size: 110%;">//s/δ//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>:
[[math]]
\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}
[[math]]
and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows (the names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):
|| Octave
1200.00
(1200.00) || Perfect fourth 
497.06
(498.04) || Tone
205.89
(203.91) || Limma 
85.28
(90.22) || Pythag comma 
35.32
(23.46) ||
|| Perfect 11th
1697.06
(1698.04) || Perfect fifth
702.94
(701.96) || Minor third
291.17
(294.13) || Apotome
120.61
(113.69) || 17-tone comma
49.96
(66.76) ||
where the limma is the Pythagorean semitone.
Thus for example:
<span style="color: #ffffff;">###</span>octave = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>fourth + tone
<span style="color: #ffffff;">###</span>fourth = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>tone + limma
<span style="color: #ffffff;">###</span>tone = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>limma + Pythag
When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.
The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:
* Subtracting twice an interval from the interval on its left generates the interval on its right.
* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
* Adjacent horizontal pairs have ratio <span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1.</span>
* Adjacent vertical pairs have ratio <span style="font-family: Georgia,serif; font-size: 110%;">√2</span>.
* Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.
The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.
In this fractal temperament, multiplying or dividing any interval by the factor <span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>// can be split into three parts, two of equal size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size //<span style="font-family: Georgia,serif; font-size: 110%;">J’</span>//<span style="font-family: Georgia,serif; font-size: 110%;">/</span>//<span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span>//.
A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.
Successive convergents of the silver mean produce ratios involving Pell numbers
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,
Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,
This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).
The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.

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


The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).
[[image:Silver temperament graphic.png width="800" height="587"]]


^^^
= = 
=<span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;">Source</span>= 
<span style="font-family: Arial,Helvetica,sans-serif;">This article summarises original research by Martin Gough. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.</span>

Original HTML content:

<html><head><title>Logarithmic approximants</title></head><body>WORK IN PROGRESS<br />
<!-- ws:start:WikiTextHeadingRule:40:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:40 --><span style="font-family: "Arial Black",Gadget,sans-serif;">Introduction</span></h1>
 <span style="font-family: Arial,Helvetica,sans-serif;">The term <em>logarithmic approximant</em>  (or <em>approximant</em> for short) denotes an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as</span><br />
<ul><li><span style="font-family: Arial,Helvetica,sans-serif;">Why do certain temperaments such as 12edo provide a reasonably accurate approximation to 5-limit just intonation?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why are certain commas small, and roughly how small are they?</span></li><li><span style="font-family: Arial,Helvetica,sans-serif;">Why does the 3-limit framework produce aesthetically pleasing scale structures?</span></li></ul><br />
The exact size, in cents, of an interval with frequency ratio <em>r</em> is<br />
<!-- ws:start:WikiTextMathRule:0:
[[math]]&lt;br/&gt;
\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J_c = 1200 \log_2{r} = 1200 \ln{r}/\ln{2}
</script><!-- ws:end:WikiTextMathRule:0 --><br />
where for just intervals r is rational and can be written as the ratio of two integers:<br />
<!-- ws:start:WikiTextMathRule:1:
[[math]]&lt;br/&gt;
\qquad r = n/d&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad r = n/d
</script><!-- ws:end:WikiTextMathRule:1 --><br />
When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as<br />
<!-- ws:start:WikiTextMathRule:2:
[[math]]&lt;br/&gt;
\qquad J = \tfrac{1}{2} \ln{r}&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{r}
</script><!-- ws:end:WikiTextMathRule:2 --><br />
This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e<span style="vertical-align: super;">2</span> = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.<br />
Comparing the two units of measurement we find<br />
1 dineper = 2400/ln(2) = 3462.468 cents<br />
which is about 1.4 semitones short of three octaves.<br />
<br />
The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus <u>3/2</u> is the perfect fifth.<br />
<br />
Three types of approximants are described here:<br />
<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><br />
<!-- ws:start:WikiTextHeadingRule:42:&lt;h1&gt; --><h1 id="toc1"><a name="Bimodular approximants"></a><!-- ws:end:WikiTextHeadingRule:42 --><strong><span style="font-size: 20px;">Bimodular approximants</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:44:&lt;h2&gt; --><h2 id="toc2"><a name="Bimodular approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:44 --><span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Definition</span></h2>
 The bimodular approximant of an interval with frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r = n/d</span></em> is<br />
<!-- ws:start:WikiTextMathRule:3:
[[math]]&lt;br/&gt;
\qquad v = \frac{r-1}{r+1}&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v = \frac{r-1}{r+1}
</script><!-- ws:end:WikiTextMathRule:3 --><br />
<em><span style="font-family: Georgia,serif; font-size: 110%;">v </span></em>can thus be expressed as<br />
<!-- ws:start:WikiTextMathRule:4:
[[math]]&lt;br/&gt;
\qquad v = \frac{n-d}{n+d} \\&lt;br /&gt;
&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v = \frac{n-d}{n+d} \\
</script><!-- ws:end:WikiTextMathRule:4 --><br />
<span style="color: #ffffff;">######</span> = (frequency difference) / (frequency sum)<br />
<span style="color: #ffffff;">######</span> =½ (frequency difference) / (mean frequency)<br />
<span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> </span>can be retrieved from <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> using the inverse relation<br />
<!-- ws:start:WikiTextMathRule:5:
[[math]]&lt;br/&gt;
\qquad r = \frac{1+v}{1-v}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad r = \frac{1+v}{1-v}</script><!-- ws:end:WikiTextMathRule:5 --><br />
<br />
<span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Properties</span><br />
When <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> </span>is small, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> 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.<br />
Noting that the exact size (in dineper units) of the interval with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em></span> is<br />
<!-- ws:start:WikiTextMathRule:6:
[[math]]&lt;br/&gt;
\qquad J = \tfrac{1}{2} \ln{r}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{r}</script><!-- ws:end:WikiTextMathRule:6 --><br />
the relationship between <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span> can be expressed as<br />
<!-- ws:start:WikiTextMathRule:7:
[[math]]&lt;br/&gt;
\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v = \frac{r-1}{r+1} = \frac{e^{2J}-1}{e^{2J}+1} = \tanh{J} = J - \tfrac{1}{3}J^3 + \tfrac{2}{15}J^5 - ...</script><!-- ws:end:WikiTextMathRule:7 --><br />
which shows that <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em> ≈ <em>J</em></span> and provides an indication of the size and sign of the error involved in this approximation.<br />
<em><span style="font-family: Georgia;">J</span></em> can be expressed in terms of <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span> as<br />
<!-- ws:start:WikiTextMathRule:8:
[[math]]&lt;br/&gt;
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:8 --><br />
The function <span style="font-family: Georgia,serif; font-size: 110%;"><em>v(r)</em></span> is the order (1,1) Padé approximant of the function <span style="font-family: Georgia,serif; font-size: 110%;"> <em>J(r) =</em>½ ln <em>r</em> </span> in the region of <em>r</em> = 1, which has the property of matching the function value and its first and second derivatives at this value of <em>r</em>. The bimodular approximant function is thus accurate to second order in <em>r</em> – 1.<br />
<br />
As an example, the size of the perfect fifth (in dNp units) is<br />
<!-- ws:start:WikiTextMathRule:9:
[[math]]&lt;br/&gt;
\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J = \tfrac{1}{2} \ln{3/2} = 0.20273...</script><!-- ws:end:WikiTextMathRule:9 --><br />
The bimodular approximant for this interval (r = 3/2) is<br />
<!-- ws:start:WikiTextMathRule:10:
[[math]]&lt;br/&gt;
\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2</script><!-- ws:end:WikiTextMathRule:10 --><br />
and the Taylor series indicates that the error in this value is about<br />
<!-- ws:start:WikiTextMathRule:11:
[[math]]&lt;br/&gt;
\qquad -\tfrac{1}{3}v^3 = -0.00267...&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad -\tfrac{1}{3}v^3 = -0.00267...</script><!-- ws:end:WikiTextMathRule:11 --><br />
<br />
The approximants of superparticular intervals are reciprocals of odd integers:<br />
<!-- ws:start:WikiTextLocalImageRule:282:&lt;img src=&quot;/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Low-order%20superparticular%20intervals.png/541610692/Low-order%20superparticular%20intervals.png" alt="Low-order superparticular intervals.png" title="Low-order superparticular intervals.png" /><!-- ws:end:WikiTextLocalImageRule:282 --><br />
<br />
If <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>[<em>J</em>] </span>denotes the bimodular approximant of an interval <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span> with frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>,<br />
<!-- ws:start:WikiTextMathRule:12:
[[math]]&lt;br/&gt;
\qquad v[-J] = -v[J] \\&lt;br /&gt;
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v[-J] = -v[J] \\
\qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}</script><!-- ws:end:WikiTextMathRule:12 --><br />
This last result is equivalent to the identity expressing <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 + </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif;">)</span> in terms of <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> and <span style="font-family: Georgia,serif; font-size: 110%;">tanh(<em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">).</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc3"><a name="Bimodular approximants-Bimodular approximants and equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:46 --><strong><span style="font-size: 15px;">Bimodular approximants and equal temperaments</span></strong></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:<br />
Two perfect fourths (<em>r</em> = 4/3, <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/7) approximate a minor seventh (<em>r</em> = 9/5, = 2/7)<br />
Three major thirds (<em>r</em> = 5/4, <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/9) or two <u>7/5</u>s (<em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/6) or five <u>8/7</u>s (<em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> = 1/15) approximate an octave (<em>r</em> = 2/1,<em><span style="font-family: Georgia,serif; font-size: 110%;"> v</span></em> = 1/3)<br />
Bimodular approximants (abbreviated to ‘approximants’ here) also provide simple explanations for the properties of certain equal temperaments.<br />
Tuning the perfect fourth and perfect fifth in the ratio of their approximants (1/7 : 1/5 = 5 : 7) and adjusting their sum to a pure octave yields 12edo (considered as a 3-limit temperament). This is an example of the high accuracy typically obtainable from a tempering policy which takes two intervals which are similar in size and not too large, tunes them in their approximant ratio, and normalises their sum to a pure interval.<br />
Aspects of 12edo considered as a 5-limit temperament can be explained by noting that it tunes the major third, major sixth and octave in the ratio of their approximants (1/9 : 1/4 : 1/3 = 4 : 9 : 12). The accuracy here is lower because the octave is of a size where the approximant has a significant error, and tuning the octave pure assigns the entire error to the smaller intervals.<br />
Tuning the major third and perfect fifth in the in the ratio of their approximants (1/9 : 1/5) and tuning the fifth pure yields Carlos alpha.<br />
Tuning the minor third and perfect fifth in the in the ratio of their approximants (1/11 : 1/5) and tuning the fifth pure yields Carlos beta.<br />
Tuning the minor third and major third in the ratio of their approximants (1/11 : 1/9) and adjusting their sum to a perfect fifth yields Carlos gamma. This temperament has high accuracy because it conforms to the policy noted above.<br />
Tuning the octave pure while preserving the ratios specified above yields, respectively, 31edo, 19edo and 34edo.<br />
Tuning the intervals <u>9/7</u>, <u>7/5</u> and <u>5/3</u> in the ratio of their approximants (1/8 : 1/6 : 1/4 = 3 : 4 : 6) and adjusting their sum to a perfect twelfth yields the equally tempered Bohlen-Pierce scale.<br />
Tuning the intervals <u>11/9</u>, <u>9/7</u>, <u>3/2</u> and <u>5/3</u> in the ratio of their approximants (1/10 : 1/8 : 1/5 : 1/4 = 4 : 5 : 8 : 10) and adjusting their sum to a major tenth yields 88 cent equal temperament.<br />
Relationships of this sort can be identified in all equal temperaments.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:48:&lt;h2&gt; --><h2 id="toc4"><a name="Bimodular approximants-Bimodular commas"></a><!-- ws:end:WikiTextHeadingRule:48 --><span style="font-family: "Arial Black",Gadget,sans-serif; font-size: 15px;">Bimodular commas</span></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.<br />
Given two intervals <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> (with<span style="font-family: Georgia,serif; font-size: 110%;"> <em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> &lt; <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>) and their approximants <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span>, we define the <em>bimodular residue</em> as<br />
<!-- ws:start:WikiTextMathRule:13:
[[math]]&lt;br/&gt;
\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b_r(J_1,J_2) = \frac{J_2}{v_2} - \frac{J_1}{v_1}</script><!-- ws:end:WikiTextMathRule:13 --><br />
and using the Taylor series expansion of <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>(<em>v</em>)</span> we find<br />
<!-- ws:start:WikiTextMathRule:14:
[[math]]&lt;br/&gt;
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (v_2^2 – v_1^2) = \tfrac{1}{3} (v_2 + v_1)(v_2 – v_1)</script><!-- ws:end:WikiTextMathRule:14 --><br />
The bimodular comma is obtained from the bimodular residue by means of a rational multiplier which ensures that the result (in line with the usual convention applied to commas) is a linear combination of <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2</span> with integer coefficients sharing no common factor:<br />
<!-- ws:start:WikiTextMathRule:15:
[[math]]&lt;br/&gt;
\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b(J_1,J_2) ≈ b_m(J_1,J_2) b_r(J_1,J_2)</script><!-- ws:end:WikiTextMathRule:15 --><br />
where<br />
<!-- ws:start:WikiTextMathRule:16:
[[math]]&lt;br/&gt;
\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v_1 = \frac{j_1}{g_1}, v_2 = \frac{j_2}{g_2}</script><!-- ws:end:WikiTextMathRule:16 --><br />
and (with rare exceptions)<br />
<!-- ws:start:WikiTextMathRule:17:
[[math]]&lt;br/&gt;
\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b_m(J_1,J_2) ≈ \frac{LCM(j_1,j_2)}{GCD(g_1,g_2)}</script><!-- ws:end:WikiTextMathRule:17 --><br />
The bimodular residue is accurately estimated by<br />
<!-- ws:start:WikiTextMathRule:18:
[[math]]&lt;br/&gt;
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1)</script><!-- ws:end:WikiTextMathRule:18 --><br />
and therefore<br />
<!-- ws:start:WikiTextMathRule:19:
[[math]]&lt;br/&gt;
\qquad b_r(J_1,J_2) ≈ \tfrac{1}{3} (J_1+J_2)(J_2-J_1) b_m&lt;br/&gt;[[math]]
 --><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><!-- ws:end:WikiTextMathRule:19 --><br />
<br />
<!-- ws:start:WikiTextHeadingRule:50:&lt;h3&gt; --><h3 id="toc5"><a name="Bimodular approximants-Bimodular commas-Examples"></a><!-- ws:end:WikiTextHeadingRule:50 -->Examples</h3>
 If the source intervals are the perfect fourth (<span style="font-family: Georgia,serif; font-size: 110%;"><em>f</em> =</span> <u><span style="font-family: Georgia,serif; font-size: 110%;">4/3</span></u><em>)</em> and the perfect fifth (<span style="font-family: Georgia,serif; font-size: 110%;"><em>F</em> = <u>3/2</u></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">), </span><span style="font-family: Arial,Helvetica,sans-serif;">then</span> <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>1 = 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>2 = 1/5</span>, and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> is the Pythagorean comma:<br />
<!-- 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><!-- ws:end:WikiTextMathRule:20 --><br />
If the source intervals are the perfect fourth (<span style="font-family: Georgia,serif; font-size: 110%;"><em>f</em> = <u>4/3</u></span>) and the minor seventh (<span style="font-family: Georgia,serif; font-size: 110%;"><em>m</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">7 </span><span style="font-family: Georgia,serif; font-size: 110%;">= <u>9/5</u>), </span><span style="font-family: Arial,Helvetica,sans-serif;">then </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em></span>1 <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7</span>, <span style="font-family: Georgia,serif; font-size: 110%;"><em>v</em>2 = 2/7</span>, <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em>r <span style="font-family: Georgia,serif; font-size: 110%;">= 2/7</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> is the syntonic comma:<br />
<!-- ws:start:WikiTextMathRule:21:
[[math]]&lt;br/&gt;
\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b(m_7,f) = b_r(m_7,f) = \tfrac{2}{7} \left( \frac{m_7}{\tfrac{2}{7}} - \frac{f}{\tfrac{1}{7}} \right) = m_7 – 2f</script><!-- ws:end:WikiTextMathRule:21 --><br />
<br />
Further examples of bimodular commas are provided in Reference 1. See also <a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a> (another name for this type of comma).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc6"><a name="Padé approximants of order (1,2)"></a><!-- ws:end:WikiTextHeadingRule:52 --><strong><span style="font-size: 21.33px;">Padé approximants of order (1,2)</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc7"><a name="Padé approximants of order (1,2)-Definition"></a><!-- ws:end:WikiTextHeadingRule:54 -->Definition</h2>
 In the section on bimodular approximants it was shown than an interval of logarithmic size <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> (measured in dineper units) is related to its bimodular approximant by<br />
<!-- ws:start:WikiTextMathRule:22:
[[math]]&lt;br/&gt;
\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad J = \tanh^{-1}{v} = v + \tfrac{1}{3}v^3 + \tfrac{1}{5}v^5 - ...</script><!-- ws:end:WikiTextMathRule:22 --><br />
where<br />
<!-- ws:start:WikiTextMathRule:23:
[[math]]&lt;br/&gt;
\qquad v = \frac{r-1}{r+1}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad v = \frac{r-1}{r+1}</script><!-- ws:end:WikiTextMathRule:23 --><br />
and <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em> is the interval’s frequency ratio.<br />
Another way to express this relationship is with a continued fraction:<br />
<!-- ws:start:WikiTextMathRule:24:
[[math]]&lt;br/&gt;
\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><!-- ws:end:WikiTextMathRule:24 --><br />
The first convergent of this continued fraction is <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em>, the bimodular approximant. The second convergent, and the Padé approximant of order (1,2), is<br />
<!-- 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"><span style="font-family: Georgia,serif; font-size: 110%;">\qquad y = \frac{3v}{3-v^2</span>}</script><!-- ws:end:WikiTextMathRule:25 --><br />
Values of this rational approximant for some simple 5-limit intervals are shown in the table below.<br />


<table class="wiki_table">
    <tr>
        <td><em>Interval <span style="font-family: Georgia,serif; font-size: 110%;">J</span></em><span style="color: #ffffff;">###########</span><br />
</td>
        <td><em>(1,2) Padé approximant <span style="font-family: Georgia,serif; font-size: 110%;">y</span></em><span style="color: #ffffff;">#</span><br />
</td>
    </tr>
    <tr>
        <td>Perfect twelfth = <u>3/1</u><br />
</td>
        <td>6/11<br />
</td>
    </tr>
    <tr>
        <td>Octave = <u>2/1</u><br />
</td>
        <td>9/26<br />
</td>
    </tr>
    <tr>
        <td>Major sixth = <u>5/3</u><br />
</td>
        <td>12/47<br />
</td>
    </tr>
    <tr>
        <td>Perfect fifth = <u>3/2</u><br />
</td>
        <td>15/74<br />
</td>
    </tr>
    <tr>
        <td>Perfect fourth = <u>4/3</u><br />
</td>
        <td>21/146<br />
</td>
    </tr>
    <tr>
        <td>Major third = <u>5/4</u><br />
</td>
        <td>27/242<br />
</td>
    </tr>
</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:<br />
<br />
(<u>3/1</u>) / (<u>6/5</u>) = 6.0257 ≈ (6/11) / (1/11) = 6 (kleisma)<br />
(<u>3/1</u>) / (<u>7/4</u>) = 1.9632 ≈ (6/11) / (3/11) = 2 (septimal diesis = <u>49/48</u>)<br />
(<u>2/1</u>) / (<u>7/6</u>) = 4.4966 ≈ (9/26) / (1/13) = 9/2 (|-11 -9 0 9&gt; comma)<br />
(<u>2/1</u>) / (<u>27/25</u>) = 9.0065 ≈ (9/26) / (1/26) = 9 (ennealimma)<br />
(<u>5/3</u>) / (<u>49/45</u>) = 5.9986 ≈ (12/47) / (2/47) = 6<br />
(<u>5/3</u>) / (<u>25/22</u>) = 3.9960 ≈ (12/47) / (3/47) = 4<br />
(<u>5/3</u>) / (<u>26/21</u>) = 2.3918 ≈ (12/47) / (5/47) = 12/5<br />
(<u>5/3</u>) / (<u>27/20</u>) = 1.7022 ≈ (12/47) / (7/47) = 12/7<br />
(<u>3/2</u>) / (<u>20/17</u>) = 2.4949 ≈ (15/74) / (6/74) = 5/2<br />
<br />
<!-- ws:start:WikiTextHeadingRule:56:&lt;h1&gt; --><h1 id="toc8"><a name="Quadratic approximants"></a><!-- ws:end:WikiTextHeadingRule:56 --><strong><span style="font-size: 21.33px;">Quadratic approximants</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:58:&lt;h2&gt; --><h2 id="toc9"><a name="Quadratic approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:58 --><span style="font-family: "Arial Black",Gadget,sans-serif;">Definition</span></h2>
 The quadratic approximant <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> of an interval <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> with frequency ratio <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = <em>n</em><em>/d</em></span> is<br />
<!-- 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><!-- ws:end:WikiTextMathRule:26 --><br />
If this is compared with the expression for the bimodular approximant,<br />
<!-- 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><!-- ws:end:WikiTextMathRule:27 --><br />
it is apparent that <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> is about twice as accurate as <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em>, with an error of opposite sign.<br />
While <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em> is the frequency difference divided by twice the arithmetic frequency mean, <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> is the frequency difference divided by twice the geometric frequency mean:<br />
<!-- 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><!-- ws:end:WikiTextMathRule:28 --><br />
<em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em> can be retrieved from <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> using<br />
<!-- 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><!-- ws:end:WikiTextMathRule:29 --><br />
<br />
The following are the quadratic approximants of some simple 5-limit intervals:<br />


<table class="wiki_table">
    <tr>
        <td><em>Interval</em> <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em><span style="color: #ffffff;">##################### </span><br />
</td>
        <td><em>Quadratic approximant</em> <span style="font-family: Georgia,serif; font-size: 110%;"><em>q</em></span><span style="color: #ffffff; font-family: Georgia,serif; font-size: 110%;"> ##</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;">Perfect twelfth = <u>3/1</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/√3</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Octave = <u>2/1</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√2</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Minor seventh = <u>9/5</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 2/3√5</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor seventh = <u>16/9</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 7/24</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Major sixth = <u>5/3</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/√15</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Minor sixth = <u>8/5</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 3/4√10</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fifth = <u>3/2</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√6</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Perfect fourth = <u>4/3</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√3</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Major third = <u>5/4</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/4√5</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Minor third = <u>6/5</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/2√30</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Pythagorean minor third = <u>32/27</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 5/24√6</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Large tone = <u>9/8</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/12√2</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Small tone = <u>10/9</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/6√10</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Diatonic semitone = <u>16/15</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/8√15</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Chroma = <u>25/24</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/20√6</span><br />
</td>
    </tr>
    <tr>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> Syntonic comma = <u>81/80</u></span><br />
</td>
        <td><span style="font-family: Arial,Helvetica,sans-serif;"> 1/72√5</span><br />
</td>
    </tr>
</table>

<br />
Expressed in terms of the bimodular approximant,<em><span style="font-family: Georgia,serif; font-size: 110%;"> v = j/g</span></em>,<br />
<!-- 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><!-- ws:end:WikiTextMathRule:30 --><br />
Quadratic approximants of just intervals thus have the form <em><span style="font-family: Georgia,serif; font-size: 110%;">q = j/√k</span></em>, where <em><span style="font-family: Georgia,serif; font-size: 110%;">j</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">k</span></em> are integers and <em><span style="font-family: Georgia,serif; font-size: 110%;">j</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><em><span style="font-family: Georgia,serif; font-size: 110%;"> + k = g</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> is a perfect square.<br />
The presence of a square root in the denominator of <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> (except where <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> 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.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc10"><a name="Quadratic approximants-Properties"></a><!-- ws:end:WikiTextHeadingRule:60 --><span style="font-family: "Arial Black",Gadget,sans-serif;">Properties</span></h2>
 If <em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em><span style="font-family: Georgia,serif; font-size: 110%;">[<em>J</em>]</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>q</em>[<em>J</em>]</span> denote, respectively, the bimodular and quadratic approximants of an interval <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> with frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>, and <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em><span style="font-family: Georgia,serif; font-size: 80%;">n</span> denotes <span style="font-family: Georgia,serif; font-size: 110%;"><em>q</em>[<em>J</em>n]</span> , then<br />
<!-- 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><!-- ws:end:WikiTextMathRule:31 --><br />
The last two expressions are rational for just intervals, and the last result is equivalent to the hyperbolic trigonometric identity<br />
<!-- 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><!-- ws:end:WikiTextMathRule:32 --><br />
<br />
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.<br />
For example<br />
<!-- 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><!-- ws:end:WikiTextMathRule:33 --><br />
but this can also be derived from bimodular approximants. Using<br />
<!-- 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><!-- ws:end:WikiTextMathRule:34 --><br />
with <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>2 = F =<u>3/2</u> </span><span style="font-family: Arial,Helvetica,sans-serif;">and</span> <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>1 = <em>f</em> = <u>4/3</u></span> this gives<br />
<!-- 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><!-- ws:end:WikiTextMathRule:35 --><br />
<br />
The quadratic approximant <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em> of a double interval <span style="font-family: Georgia,serif; font-size: 110%;">2<em>J</em></span> (for example, the ditone) is rational, which suggests using <span style="font-family: Georgia,serif; font-size: 110%;">½ q(r</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;">)</span> as a rational approximant of <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> (where <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em> has frequency ratio <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>):<br />
<!-- 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><!-- ws:end:WikiTextMathRule:36 --><br />
However, this approximant is both less accurate and more complex than the corresponding bimodular approximant, and consequently of limited value.<br />
The most interesting approximate interval ratios derivable from quadratic approximants are irrational.<br />
<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc11"><!-- ws:end:WikiTextHeadingRule:62 --> </h2>
 <!-- ws:start:WikiTextHeadingRule:64:&lt;h2&gt; --><h2 id="toc12"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a><!-- ws:end:WikiTextHeadingRule:64 --><span style="font-family: "Arial Black",Gadget,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span></h2>
 <!-- ws:start:WikiTextHeadingRule:66:&lt;h3&gt; --><h3 id="toc13"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a><!-- ws:end:WikiTextHeadingRule:66 --><strong>Theorem</strong></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.<br />
<!-- ws:start:WikiTextHeadingRule:68:&lt;h3&gt; --><h3 id="toc14"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a><!-- ws:end:WikiTextHeadingRule:68 --><strong>Remarks</strong></h3>
 If the harmonics have indices <em><span style="font-family: Georgia,serif; font-size: 110%;">n – m, n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">n + m</span></em>, the two intervals have reduced frequency ratios <em><span style="font-family: Georgia,serif; font-size: 110%;">n/(n – m)</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">(n + m)/n</span></em>. It can be assumed that <em><span style="font-family: Georgia,serif; font-size: 110%;">n</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> have no common factor.<br />
<em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> is the epimoricity of the intervals. When <em><span style="font-family: Georgia,serif; font-size: 110%;">m</span></em> = 1 the intervals are adjacent superparticular intervals.<br />
The geometric mean of the frequency ratios is the frequency ratio corresponding to the arithmetic mean of the intervals.<br />
<!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc15"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a><!-- ws:end:WikiTextHeadingRule:70 --><strong>Proof</strong></h3>
 The ratio of the intervals as estimated from their quadratic approximants is<br />
<!-- 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><!-- ws:end:WikiTextMathRule:37 --><br />
which is the geometric mean of their frequency ratios.<br />
<!-- ws:start:WikiTextHeadingRule:72:&lt;h3&gt; --><h3 id="toc16"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"></a><!-- ws:end:WikiTextHeadingRule:72 --><strong>Examples</strong></h3>
 The ratio of the perfect fifth, <span style="font-family: Georgia,serif; font-size: 110%;"><em>F</em> = <u>3/2</u></span>, to the perfect fourth, <span style="font-family: Georgia,serif; font-size: 110%;"><em>f</em> = <u>4/3</u></span>, as estimated by their quadratic approximants (1/2√6 and 1/4√3) is √2, which is the frequency ratio of the arithmetic mean of these intervals (the half-octave).<br />
<span style="font-family: Georgia,serif; font-size: 110%;"><em>F/f</em> = 701.955/498.045 = 1.40942,</span><br />
<span style="font-family: Georgia,serif; font-size: 110%;">√2 = 1.41421.</span><br />
The ratio of the large tone, <span style="font-family: Georgia,serif; font-size: 110%;"><em>T</em> = <u>9/8</u></span>, to the small tone, <span style="font-family: Georgia,serif; font-size: 110%;"><em>t</em> = <u>10/9</u></span>, as estimated by their quadratic approximants (1/12√2 and 1/6√10) is √5/2, which is the frequency ratio of the mean tone.<br />
<span style="font-family: Georgia,serif; font-size: 110%;"><em>T</em><em>/t</em> = 203.910/182.404 = 1.11790,</span><br />
<span style="font-family: Georgia,serif; font-size: 110%;">√5/2 = 1.11803.</span><br />
<!-- ws:start:WikiTextHeadingRule:74:&lt;h3&gt; --><h3 id="toc17"><a name="Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Silver temperament"></a><!-- ws:end:WikiTextHeadingRule:74 --><strong>Silver temperament</strong></h3>
 As the first example above shows, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, namely 1.4000.<br />
It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to<br />
<span style="color: #ffffff;">###</span>Perfect fifth = <u>3/2</u> = 702.944 cents<br />
<span style="color: #ffffff;">###</span>Perfect fourth = <u>4/3</u> = 497.056 cents<br />
This fifth is wide by 0.989 cents, and the fourth narrow by the same amount. These errors are of about half the magnitude, and of opposite sign, as their counterparts in 12edo (where these intervals are tuned in the ratio of their bimodular approximants).<br />
A 3-limit temperament constructed on this tuning sets the octave and the perfect fourth (and many other intervals) in the ‘silver ratio’ (sometimes called the ‘silver mean’), δS = √2 + 1 = 2.4142. On this basis, and by analogy with ‘golden meantone’ temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘silver temperament’ is proposed instead.<br />
Silver temperament has interesting fractal properties which help to explain why 3-limit tuning forms aesthetically pleasing scales.<br />
The continued fraction expansion of the silver ratio has a particularly simple form:<br />
<!-- ws:start:WikiTextMathRule:38:
[[math]]&lt;br/&gt;
\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad \delta_s = √2 + 1 = 2 + 1/(2 + 1/(2 + 1/(2 + ...)))</script><!-- ws:end:WikiTextMathRule:38 --><br />
As a result, if two intervals L and s are tuned in the silver ratio, with <span style="font-family: Georgia,serif; font-size: 110%;"><em>s = L/δ</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>, subtracting twice the small interval <em><span style="font-family: Georgia,serif; font-size: 110%;">s</span></em> from the large interval <em><span style="font-family: Georgia,serif; font-size: 110%;">L</span></em> leaves a remainder of size <span style="font-family: Georgia,serif; font-size: 110%;"><em>s/δ</em></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span>:<br />
<!-- ws:start:WikiTextMathRule:39:
[[math]]&lt;br/&gt;
\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = \frac{s}{\delta_s}</script><!-- ws:end:WikiTextMathRule:39 --><br />
and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows (the names are assigned according to Pythagorean conventions, followed by tempered and just sizes in cents):<br />


<table class="wiki_table">
    <tr>
        <td>Octave<br />
1200.00<br />
(1200.00)<br />
</td>
        <td>Perfect fourth <br />
497.06<br />
(498.04)<br />
</td>
        <td>Tone<br />
205.89<br />
(203.91)<br />
</td>
        <td>Limma <br />
85.28<br />
(90.22)<br />
</td>
        <td>Pythag comma <br />
35.32<br />
(23.46)<br />
</td>
    </tr>
    <tr>
        <td>Perfect 11th<br />
1697.06<br />
(1698.04)<br />
</td>
        <td>Perfect fifth<br />
702.94<br />
(701.96)<br />
</td>
        <td>Minor third<br />
291.17<br />
(294.13)<br />
</td>
        <td>Apotome<br />
120.61<br />
(113.69)<br />
</td>
        <td>17-tone comma<br />
49.96<br />
(66.76)<br />
</td>
    </tr>
</table>

where the limma is the Pythagorean semitone.<br />
Thus for example:<br />
<span style="color: #ffffff;">###</span>octave = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>fourth + tone<br />
<span style="color: #ffffff;">###</span>fourth = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>tone + limma<br />
<span style="color: #ffffff;">###</span>tone = 2<span style="font-family: "Calibri","sans-serif"; font-size: 14.66px;">×</span>limma + Pythag<br />
When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.<br />
The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:<br />
<ul><li>Subtracting twice an interval from the interval on its left generates the interval on its right.</li><li>An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.</li><li>Adjacent horizontal pairs have ratio <span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1.</span></li><li>Adjacent vertical pairs have ratio <span style="font-family: Georgia,serif; font-size: 110%;">√2</span>.</li><li>Extending the table to a third row yields consisting of the intervals in the first row multiplied by 2, and so on.</li></ul>The regularity of this scheme, combined with the fact that the ratios between closely related intervals are of order 2, means that its intervals form orderly sequences in which successive terms have similar magnitude – highly desirable properties for the formation of musical scales.<br />
In this fractal temperament, multiplying or dividing any interval by the factor <span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s </span><span style="font-family: Georgia,serif; font-size: 110%;">= √2 + 1</span> produces another interval in the temperament. Any tempered interval <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em> can be split into three parts, two of equal size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span> and the other of size <em><span style="font-family: Georgia,serif; font-size: 110%;">J’</span></em><span style="font-family: Georgia,serif; font-size: 110%;">/</span><em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">s</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span></em>.<br />
A similar principle applies to multiplication and division by the factor √2, except that intervals in the top row of the table cannot be divided by √2 to yield another interval in the temperament. These properties means that the temperament would support compositional techniques based on novel types of intervallic augmentation and diminution.<br />
Successive convergents of the silver mean produce ratios involving Pell numbers<br />
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,<br />
Other approximations to the silver mean are provided by ratios of consecutive half Pell-Lucas numbers, which are formed by adding consecutive Pell numbers<br />
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 3, 7/3, 17/7, 41/17, 99/41…,<br />
This accounts for the frequent occurrence of Pell numbers and half Pell-Lucas numbers representing Pythagorean intervals in equal temperaments (5edo, 7edo, 12edo, 17edo, 29edo, etc.).<br />
The accuracy of the silver fifth means that the scheme produces very workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur.<br />
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The diagram below is a ‘continued fraction jigsaw’ showing the sizes of the fourth (f), tone (T), limma (sp) and Pythagorean comma (p) as tempered by 41edo - an approximation to silver temperament. The octave is represented by squares partly visible at the upper and lower edges of the diagram.<br />
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The next diagram is a geometrical representation of silver temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = tone, mp = pythag minor third, sp = limma, Xp = apotome, p = Pythagorean comma).<br />
<!-- ws:start:WikiTextLocalImageRule:283:&lt;img src=&quot;/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 587px; width: 800px;&quot; /&gt; --><img src="/file/view/Silver%20temperament%20graphic.png/541613984/800x587/Silver%20temperament%20graphic.png" alt="Silver temperament graphic.png" title="Silver temperament graphic.png" style="height: 587px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:283 --><br />
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^^^<br />
<!-- ws:start:WikiTextHeadingRule:76:&lt;h1&gt; --><h1 id="toc18"><!-- ws:end:WikiTextHeadingRule:76 --> </h1>
 <!-- ws:start:WikiTextHeadingRule:78:&lt;h1&gt; --><h1 id="toc19"><a name="Source"></a><!-- ws:end:WikiTextHeadingRule:78 --><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;"> </span><span style="font-family: Arial Black;">Source</span></h1>
 <span style="font-family: Arial,Helvetica,sans-serif;">This article summarises original research by Martin Gough. See <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a> for a fuller account of bimodular approximants.</span></body></html>