Logarithmic approximants

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=**<span style="font-size: 20px;">1. Introduction</span>**= 
<span style="font-family: Arial,Helvetica,sans-serif;">A //logarithmic approximant// (or //approximant// for short) is an algebraic approximation to the logarithm function. By approximating interval sizes, logarithmic approximants can shed light on questions such as:</span>
* <span style="font-family: Arial,Helvetica,sans-serif;">Why do certain temperaments such as 12edo provide a good 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 <span style="font-family: Georgia,serif; font-size: 110%;">e</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 7.38906...</span> 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. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then __r__ = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).

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;">2. Bimodular approximants</span>**= 
==<span style="font-family: Arial,Helvetica,sans-serif;">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,Helvetica,sans-serif;">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) [[http://en.wikipedia.org/wiki/Pad%C3%A9_approximant|Padé approximant]] of the function <span style="font-family: Georgia,serif; font-size: 110%;"> //J(r) =//½ ln //r// </span> in the region of <span style="font-family: Georgia,serif; font-size: 110%;">//r// = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of //<span style="font-family: Georgia,serif; font-size: 110%;">r</span>//. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;">//r// – 1</span>.

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 (<span style="font-family: Georgia,serif; font-size: 110%;">//r// = 3/2</span>) 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, as shown in Figure 1.
[[image:Low-order superparticular intervals.png]]
<span style="color: #ffffff;">######</span>Figure 1. Bimodular approximants for low-order superparticular intervals

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-family: Arial,Helvetica,sans-serif; 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 [[Bohlen-Pierce|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 [[88cET|88 cent equal temperament]].
Relationships of this sort can be identified in all equal temperaments.

==<span style="font-family: Arial,Helvetica,sans-serif;">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(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]]

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

=**<span style="font-size: 21.33px;">3. Padé approximants of order (1,2)</span>**= 
==<span style="font-family: Arial,Helvetica,sans-serif;">Definition</span>== 
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]]
\qquad y = \frac{v}{1-v^2/3}
[[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;">4. Quadratic approximants</span>**= 
==<span style="font-family: Arial,Helvetica,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}) \\
\qquad = \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 q = \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,Helvetica,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]]
where //large tone// = __9/8__.
However, 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//</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2 </span><span style="font-family: Georgia,serif; font-size: 110%;">= //F// =__3/2__</span> <span style="font-family: Arial,Helvetica,sans-serif;">and</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; font-size: 110%;">= //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,Helvetica,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span>== 
===<span style="font-family: Arial,Helvetica,sans-serif;">Theorem</span>=== 
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.
===<span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span>=== 
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 [[Superpartient|degree of 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.
===<span style="font-family: Arial,Helvetica,sans-serif;">Proof</span>=== 
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.
===<span style="font-family: Arial,Helvetica,sans-serif;">Examples</span>=== 
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>

==<span style="font-family: Arial,Helvetica,sans-serif;">Argent temperament</span>== 
As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.
It can be shown that the error in a pair of intervals tuned in the ratio of their approximants is minimised if the sum of the intervals is normalised – in this case to a pure octave. If this is done while maintaining the √2 ratio the perfect fifth and fourth are tempered to
<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 [[http://en.wikipedia.org/wiki/Silver_ratio|silver ratio]] (sometimes called the silver mean), //<span style="font-family: Georgia,serif; font-size: 110%;">δ</span>//<span style="vertical-align: sub;">s </span>= <span style="font-family: Georgia,serif; font-size: 110%;">√2 + 1 = 2.4142</span>. On this basis, and by analogy with [[Golden Meantone|golden meantone]] temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead.
Argent 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 //<span style="font-family: Georgia,serif; font-size: 110%;">L</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">s</span>// 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 = s/\delta_s
[[math]]
(since 1///<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 = //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</span>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone __<span style="font-family: Arial,Helvetica,sans-serif;">256/243</span>__) followed by tempered and just sizes in cents:
|| Octave
1200.00
(1200.00) || Perfect fourth<span style="color: #ffffff;">##</span>
497.06
(498.04) || Tone
205.89
(203.91) || Limma
85.28
(90.22) || Pythag comma
35.32
(23.46) ||
|| Perfect 11th<span style="color: #ffffff;">##</span>
1697.06
(1698.04) || Perfect fifth
702.94
(701.96) || Minor third<span style="color: #ffffff;">##</span>
291.17
(294.13) || Apotome<span style="color: #ffffff;">##</span>
120.61
(113.69) || 17-tone comma<span style="color: #ffffff;">##</span>
49.96
(66.76) ||
Thus for example:
<span style="color: #ffffff;">###</span>octave = 2×fourth + tone
<span style="color: #ffffff;">###</span>fourth = 2×tone + limma
<span style="color: #ffffff;">###</span>tone = 2×limma + Pythag comma
<span style="color: #ffffff;">###</span>perfect 11th (__8/3__) = 2×fifth + minor third
<span style="color: #ffffff;">###</span>fifth = 2×(minor third) + apotome
When picturing these relationships it makes most musical sense to place the small interval between the two larger ones, as in the ‘continued fraction jigsaw’ below.
The following relationships hold in the table, the first two being valid for the pure intervals as well as their tempered counterparts:
* Subtracting twice an interval from the interval on its left generates the interval on its right.
* An interval in the second row is the sum of the interval immediately above and the interval diagonally above and to the right.
* Adjacent horizontal pairs have ratio //<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 are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.
In this fractal temperament, multiplying or dividing any interval by the factor //<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 ratio produce ratios involving [[http://en.wikipedia.org/wiki/Pell_number|Pell numbers]].
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,
Other approximations to the silver ratio 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, 41edo, 70edo etc.).
The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and //minus// the 41-tone comma).

Figure 2 is a //continued fraction jigsaw// showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.

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

<span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo

Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, m<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean minor third, s<span style="font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean limma, X<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.
[[image:Silver temperament graphic.png width="800" height="587"]]
<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent temperament

By the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem|Gelfond-Schneider theorem ]] the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;">//r// = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2//a//+//b//</span>, where//<span style="font-family: Georgia,serif; font-size: 110%;"> a</span>// and //<span style="font-family: Georgia,serif; font-size: 110%;">b</span>// are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;">//a// = 0</span>). The frequency ratio of the tempered perfect eleventh (<span style="font-family: Georgia,serif; font-size: 110%;">__8/3__ = __2.6666...__</span>) is the [[http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant|Gelfond-Schneider constant ]]or Hilbert number, <span style="font-family: Georgia,serif; font-size: 110%;">2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 2.665144</span>...

==Golden temperaments== 
It has been shown in an example above that the ratio of the large tone (//T// <span style="font-family: Georgia,serif; font-size: 110%;">= __9/8__</span>) to the small tone (<span style="font-family: Georgia,serif; font-size: 110%;">//t// = __10/9__</span>) is closely approximated by
[[math]]
\qquad T/t = \sqrt{5}/2
[[math]]
It follows that
[[math]]
\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi
[[math]]
where <span style="font-family: Georgia,serif; font-size: 110%;">//ϕ// = 1.61803</span>... is the golden ratio.
If a Fibonacci sequence of intervals is formed from the pair of intervals <span style="font-family: Georgia,serif; font-size: 110%;">//T// – //t///2</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">t</span>//, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>//. The sequence formed in this way is Sequence 1 in the following table.
|| Sequence 1:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff;">#</span>//t///2 - 3//c//<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t///2 //- c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T - t///2 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//T + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//M + t///2<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2//M// </span> ||
|| Sequence 2:<span style="color: #ffffff;">#</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//magic//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//diesis//</span> || //<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>chroma<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span>// || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//semitone//<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//t//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//mp//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//f - c//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//m6p - c//</span><span style="color: #ffffff;">#</span> ||
|| Difference: || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-3//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ//</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="color: #ffffff; font-family: Georgia,serif;">#</span>0 || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ///2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>//σ// </span> ||
|| Seq 1 ratios: ||   || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6120</span><span style="color: #ffffff; font-family: Georgia,serif;">##</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6204</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6171</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6184<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6179</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6181</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6180</span> ||
|| Seq 2 ratios: ||   || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.3865</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.7212</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.5810</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6325</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6125</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6201 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6172 </span> ||
where <span style="font-family: Georgia,serif; font-size: 110%;">//f// = __4/3__, //T// = __9/8__, //t// = __10/9__, //M// = __5/4__, //magic// = __3125/3072__, //diesis// = __128/125__, //chroma// = __25/24__, //semitone// = __16/15__, //mp// = __32/27__, //c// = //syntonic comma// = __81/80__, //m6p// = __128/81__, //σ// = //schisma// = __32805/32768.__</span>
The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>//.
Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (//<span style="font-family: Georgia,serif; font-size: 110%;">σ</span>//), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).
The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>// rather less accurately.
A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>// would be ‘golden temperaments’.
Tempering the Sequence 2 ratios to //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>// while tuning the octave pure and tempering out the syntonic comma yields [[Golden Meantone|golden meantone]] temperament.
Tempering the Sequence 1 ratios to //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>// yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals //<span style="font-family: Georgia,serif; font-size: 110%;">s, t</span>//, //<span style="font-family: Georgia,serif; font-size: 110%;">M</span>// and <span style="font-family: Georgia,serif; font-size: 110%;">//m//=__6/5__</span> are all ±0.02106 cents.
Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at //<span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span>// in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).

==<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%; vertical-align: sub;">Pythagorean triples of quadratic approximants</span>== 
If the quadratic approximants <span style="font-family: Georgia,serif; font-size: 110%;">//q//1//, q//2</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//q//3</span> of a set of three intervals <span style="font-family: Georgia,serif; font-size: 110%;">//J//1, //J//2</span> and //<span style="font-family: Georgia,serif; font-size: 110%;">J</span>//3 satisfy
[[math]]
\qquad q_1^2 + q_2^2 = q_3^2
[[math]]
they can be said to form a [[http://en.wikipedia.org/wiki/Pythagorean_triple|Pythagorean triple]].
The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;">//J//1', //J//2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;">//J//3'</span>, are also Pythagorean triples:
|| <span style="font-family: Georgia,serif;"><span style="color: #ffffff; line-height: 0px; overflow: hidden;">#</span>//J//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//1</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//2</span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//q//3 </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//1' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//2' </span> || <span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span>//J//3'<span style="color: #ffffff; font-family: Georgia;">#</span></span> ||
|| <span style="color: #ffffff; font-family: Georgia;">#</span>__6/5__<span style="color: #ffffff;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>__5/4__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__4/3__<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/2√30<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√5 || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3 || <span style="color: #ffffff; font-family: Georgia;">#</span>3 || <span style="color: #ffffff; font-family: Georgia;">#</span>4 || <span style="color: #ffffff; font-family: Georgia;">#</span>5 ||
|| <span style="color: #ffffff; font-family: Georgia;">#</span>__4/3__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__12/5__<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>__5/2__<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3 || <span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15<span style="color: #ffffff; font-family: Georgia;">#</span> || <span style="color: #ffffff; font-family: Georgia;">#</span>3/2√10<span style="color: #ffffff; font-family: Georgia;">#</span> ||   ||   ||   ||
|| <span style="color: #ffffff; font-family: Georgia;">#</span>__8/5__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__12/5__ || <span style="color: #ffffff; font-family: Georgia;">#</span>__8/3__ || <span style="color: #ffffff; font-family: Georgia;">#</span>3/4√10 || <span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15 || <span style="color: #ffffff; font-family: Georgia;">#</span>5/4√6 || <span style="color: #ffffff; font-family: Georgia;">#</span>8 || <span style="color: #ffffff; font-family: Georgia;">#</span>15 || <span style="color: #ffffff; font-family: Georgia;">#</span>17 ||

==A small 34edo comma== 
<span style="color: #333333;">As [[Gene Ward Smith]] has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280> (‘//selenia//’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants.
It can be shown, using a suitable [[Comma-based lattices|comma-based lattice]], that every comma tempered out by 34edo can be expressed as an integer linear combination of the [[Gammic node|//gammic// comma ]]|-29 -11 20> (4.769 cents) and the //semisuper// comma (AKA //[[vishnuzma]]//) |23 6 -14> (3.338 cents). In particular,
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//selenia// = 7 //gammic// – 10 //semisuper//</span>
<span style="color: #333333;">So to prove that //selenia// is small we must show that //gammic/////semisuper// ≈ 10/7.</span>
<span style="color: #333333;">//Gammic// and //semisuper// are both bimodular commas:</span>
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//gammic// = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">//b//(__6/5__,__5/4__)</span>
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//semisuper// = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">//b//(__25/24__,__4/3__)</span>
<span style="font-family: Arial,Helvetica,sans-serif;">Using a result given in the section on bimodular commas, the size of </span><span style="font-family: Georgia,serif; font-size: 110%;">//b//(//J//1,//J//2)</span><span style="color: #333333;"> can be estimated using</span>
[[math]]
\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m
[[math]]
<span style="color: #333333;">Estimating </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">//J//2</span><span style="color: #333333;"> and </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;">//J//1</span><span style="color: #333333;"> with their quadratic approximants we then have</span>
[[math]]
\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m
[[math]]
<span style="font-family: Arial,Helvetica,sans-serif;">For //gammic//:</span>
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//J//₁= 6/5, //J//₂= 5/4</span>
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//v//</span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/11, </span>//<span style="font-family: Georgia,serif; font-size: 110%;">v</span>//₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/9, </span>//<span style="font-family: Georgia,serif; font-size: 110%;">b</span>//<span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1</span>
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//q//</span><span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₁² = </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/30),</span> //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>//<span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₂//² =// </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/20)</span>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span>//gammic// = </span><span style="font-family: Georgia,serif; font-size: 110%;">//b//(//J//</span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,//J//</span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)</span>
<span style="font-family: Arial,Helvetica,sans-serif;">For //semisuper://</span>
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//J//₁= 25/24, //J//₂= 4/3</span>
<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//v//</span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/49, </span>//<span style="font-family: Georgia,serif;">v</span>//₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7, </span>//<span style="font-family: Georgia,serif;">b</span>//<span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1/7</span>
<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//q//</span><span style="font-family: Arial,Helvetica,sans-serif;">₁² = </span><span style="font-family: Georgia,serif;">(1/4)(1/600),</span> //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>//<span style="font-family: Arial,Helvetica,sans-serif;">₂//² =// </span><span style="font-family: Georgia,serif;">(1/4)(1/12)</span>
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//semisuper// = </span><span style="font-family: Georgia,serif; font-size: 110%;">//b//(//J//</span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,//J//</span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)</span>
<span style="font-family: Arial,Helvetica,sans-serif;">Therefore</span>
<span style="color: #ffffff;">###</span><span style="font-family: Arial,Helvetica,sans-serif;">//gammic/semisuper// ≈ </span><span style="font-family: Georgia,serif; font-size: 110%;">10/7</span>
<span style="color: #333333;">as required.</span>

<span style="color: #333333;">To estimate the size of //selenia// we must quantify the error in this ratio. A more accurate analysis gives</span>
[[math]]
\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m
[[math]]
<span style="color: #333333;">So to improve our estimates of </span><span style="font-family: Georgia,serif; font-size: 110%;">//b//(//J//1,//J//2)</span> <span style="color: #333333;">we should multiply them by</span>
[[math]]
\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)
[[math]]
<span style="color: #333333;">Thus a better estimate for //gammic/semisuper// is</span>
[[math]]
\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}
[[math]]
<span style="font-family: Arial,Helvetica,sans-serif;">from which it follows that</span>
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">//selenia// = 7 //gammic// - 10 //semisuper//</span>
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">######## </span><span style="font-family: Georgia,serif; font-size: 110%;">≈ 7 //gammic//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"> (//f//</span></span>//<span style="font-family: Georgia,serif; font-size: 70%; vertical-align: sub;">gammic</span><span style="font-family: Georgia,serif; font-size: 110%;"> - f</span><span style="font-size: 70%; vertical-align: sub;">semisuper</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">)</span><span style="font-family: Georgia,serif; font-size: 110%;">/f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span>//
<span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span>
//<span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span>//<span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span>
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)</span>
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span>//<span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span>//<span style="font-family: Georgia,serif; font-size: 110%;">= 1/6000</span>
<span style="color: #333333;">Therefore</span>
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">//selenia// ≈ 7 //gammic// (1/6000) (120/119) = //gammic///850 = 0.00561</span><span style="color: #333333;"> cents</span>
<span style="color: #333333;">which </span>is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in //<span style="font-family: Georgia,serif; font-size: 110%;">q</span>//<span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;">6</span>//,// which become significant when the //<span style="font-family: Georgia,serif; font-size: 110%;">f</span>// values are very similar.)
In summary, the reason //selenia// is small (compared to //gammic// and //semisuper//) is because the quadratic approximants of //gammic// and //semisuper// are in the ratio 10/7. The reason it is //very// small (of order //gammic///1000 rather than //gammic///10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:
[[math]]
\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2
[[math]]
and <span style="font-family: Georgia,serif; font-size: 110%;">(//q//(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly.

=Sources and acknowledgements= 
This article is based on original research by [[Martin Gough]]. See [[file:Bimod Approx 2014-6-8.pdf|this paper]] for a fuller account of bimodular approximants.
The tuning referred to here as argent temperament was described by [[graham breed|Graham Breed]] and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.
Thanks to [[Gene Ward Smith]] for the Gelfond-Schneider result.

Original HTML content:

<html><head><title>Logarithmic approximants</title></head><body><!-- ws:start:WikiTextHeadingRule:49:&lt;h1&gt; --><h1 id="toc0"><a name="x1. Introduction"></a><!-- ws:end:WikiTextHeadingRule:49 --><strong><span style="font-size: 20px;">1. Introduction</span></strong></h1>
 <span style="font-family: Arial,Helvetica,sans-serif;">A <em>logarithmic approximant</em> (or <em>approximant</em> for short) is 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 good 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 <span style="font-family: Georgia,serif; font-size: 110%;">e</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 7.38906...</span> 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. This can also be expressed by an explicit function: if bim(r) = (r-1)/(r+1), then <u>r</u> = bim(r). The inverse function can be written mib(v) = (1+v)/(1-v).<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:51:&lt;h1&gt; --><h1 id="toc1"><a name="x2. Bimodular approximants"></a><!-- ws:end:WikiTextHeadingRule:51 --><strong><span style="font-size: 20px;">2. Bimodular approximants</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:53:&lt;h2&gt; --><h2 id="toc2"><a name="x2. Bimodular approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:53 --><span style="font-family: Arial,Helvetica,sans-serif;">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 />
<!-- ws:start:WikiTextHeadingRule:55:&lt;h2&gt; --><h2 id="toc3"><a name="x2. Bimodular approximants-Properties"></a><!-- ws:end:WikiTextHeadingRule:55 --><span style="font-family: Arial,Helvetica,sans-serif;">Properties</span></h2>
 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) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pad%C3%A9_approximant" rel="nofollow">Padé approximant</a> 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 <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 1</span>, which has the property of matching the function value and its first and second derivatives at this value of <em><span style="font-family: Georgia,serif; font-size: 110%;">r</span></em>. The bimodular approximant function is thus accurate to second order in <span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> – 1</span>.<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 (<span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 3/2</span>) 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, as shown in Figure 1.<br />
<!-- ws:start:WikiTextLocalImageRule:481:&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:481 --><br />
<span style="color: #ffffff;">######</span>Figure 1. Bimodular approximants for low-order superparticular intervals<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:57:&lt;h2&gt; --><h2 id="toc4"><a name="x2. Bimodular approximants-Bimodular approximants and equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:57 --><span style="font-family: Arial,Helvetica,sans-serif; font-size: 15px;">Bimodular approximants and equal temperaments</span></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 <a class="wiki_link" href="/Carlos%20Alpha">Carlos Alpha</a>.<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 <a class="wiki_link" href="/Carlos%20Beta">Carlos Beta</a>.<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 <a class="wiki_link" href="/Carlos%20Gamma">Carlos Gamma</a> . 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 <a class="wiki_link" href="/Bohlen-Pierce">equally tempered Bohlen-Pierce scale</a>.<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 <a class="wiki_link" href="/88cET">88 cent equal temperament</a>.<br />
Relationships of this sort can be identified in all equal temperaments.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:59:&lt;h2&gt; --><h2 id="toc5"><a name="x2. Bimodular approximants-Bimodular commas"></a><!-- ws:end:WikiTextHeadingRule:59 --><span style="font-family: Arial,Helvetica,sans-serif;">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(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(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:61:&lt;h3&gt; --><h3 id="toc6"><a name="x2. Bimodular approximants-Bimodular commas-Examples"></a><!-- ws:end:WikiTextHeadingRule:61 -->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 />
For further examples of bimodular commas, and a fuller account of bimodular approximant theory, refer to <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a>. See also <a class="wiki_link" href="/Don%20Page%20comma">Don Page comma</a> (another name for this type of comma).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:63:&lt;h1&gt; --><h1 id="toc7"><a name="x3. Padé approximants of order (1,2)"></a><!-- ws:end:WikiTextHeadingRule:63 --><strong><span style="font-size: 21.33px;">3. Padé approximants of order (1,2)</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:65:&lt;h2&gt; --><h2 id="toc8"><a name="x3. Padé approximants of order (1,2)-Definition"></a><!-- ws:end:WikiTextHeadingRule:65 --><span style="font-family: Arial,Helvetica,sans-serif;">Definition</span></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;
\qquad y = \frac{v}{1-v^2/3}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad y = \frac{v}{1-v^2/3}</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:67:&lt;h1&gt; --><h1 id="toc9"><a name="x4. Quadratic approximants"></a><!-- ws:end:WikiTextHeadingRule:67 --><strong><span style="font-size: 21.33px;">4. Quadratic approximants</span></strong></h1>
 <!-- ws:start:WikiTextHeadingRule:69:&lt;h2&gt; --><h2 id="toc10"><a name="x4. Quadratic approximants-Definition"></a><!-- ws:end:WikiTextHeadingRule:69 --><span style="font-family: Arial,Helvetica,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}) \\&lt;br /&gt;
\qquad = \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}) \\
\qquad = \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 q = \frac{r-1}{2\sqrt{r}} = \frac{n-d}{2\sqrt{nd}}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad q = \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:71:&lt;h2&gt; --><h2 id="toc11"><a name="x4. Quadratic approximants-Properties"></a><!-- ws:end:WikiTextHeadingRule:71 --><span style="font-family: Arial,Helvetica,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 />
where <em>large tone</em> = <u>9/8</u>.<br />
However, 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></span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">2 </span><span style="font-family: Georgia,serif; font-size: 110%;">= <em>F</em> =<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></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>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 <em>q</em>(<em>r</em></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:73:&lt;h2&gt; --><h2 id="toc12"><!-- ws:end:WikiTextHeadingRule:73 --> </h2>
 <!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc13"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression"></a><!-- ws:end:WikiTextHeadingRule:75 --><span style="font-family: Arial,Helvetica,sans-serif;">Relative sizes of intervals between 3 frequencies in arithmetic progression</span></h2>
 <!-- ws:start:WikiTextHeadingRule:77:&lt;h3&gt; --><h3 id="toc14"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Theorem"></a><!-- ws:end:WikiTextHeadingRule:77 --><span style="font-family: Arial,Helvetica,sans-serif;">Theorem</span></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:79:&lt;h3&gt; --><h3 id="toc15"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Remarks"></a><!-- ws:end:WikiTextHeadingRule:79 --><span style="font-family: Arial,Helvetica,sans-serif;">Remarks</span></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 <a class="wiki_link" href="/Superpartient">degree of epimoricity</a> 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:81:&lt;h3&gt; --><h3 id="toc16"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Proof"></a><!-- ws:end:WikiTextHeadingRule:81 --><span style="font-family: Arial,Helvetica,sans-serif;">Proof</span></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:83:&lt;h3&gt; --><h3 id="toc17"><a name="x4. Quadratic approximants-Relative sizes of intervals between 3 frequencies in arithmetic progression-Examples"></a><!-- ws:end:WikiTextHeadingRule:83 --><span style="font-family: Arial,Helvetica,sans-serif;">Examples</span></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 />
<br />
<!-- ws:start:WikiTextHeadingRule:85:&lt;h2&gt; --><h2 id="toc18"><a name="x4. Quadratic approximants-Argent temperament"></a><!-- ws:end:WikiTextHeadingRule:85 --><span style="font-family: Arial,Helvetica,sans-serif;">Argent temperament</span></h2>
 As shown in the first example above, the estimate of the ratio of the perfect fifth to the perfect fourth derived from quadratic approximants is √2 = 1.4142. This is a little larger than the exact ratio, 1.4094, which in turn is larger than the ratio of the intervals as tuned in 12edo, 1.4000.<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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Silver_ratio" rel="nofollow">silver ratio</a> (sometimes called the silver mean), <em><span style="font-family: Georgia,serif; font-size: 110%;">δ</span></em><span style="vertical-align: sub;">s </span>= <span style="font-family: Georgia,serif; font-size: 110%;">√2 + 1 = 2.4142</span>. On this basis, and by analogy with <a class="wiki_link" href="/Golden%20Meantone">golden meantone</a> temperament (in which the ratios of certain pairs of intervals are matched to the golden ratio) the temperament might be named ‘silver meantone’. However, the term meantone is inappropriate here since the temperament has a slightly enlarged fifth and makes no claim to accuracy in the 5-limit. So the name ‘argent temperament' is proposed instead.<br />
Argent 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 <em><span style="font-family: Georgia,serif; font-size: 110%;">L</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">s</span></em> 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 = s/\delta_s&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad L – 2s = (\delta_s – 2)s = s/\delta_s</script><!-- ws:end:WikiTextMathRule:39 --><br />
(since 1<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><span style="font-family: Georgia,serif; font-size: 110%;">= √2 - 1 = <em>δ</em></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</span>) and consequently this process can be continued indefinitely to generate sequences of decreasing intervals as follows. The names are assigned according to Pythagorean conventions (the limma being the Pythagorean semitone <u><span style="font-family: Arial,Helvetica,sans-serif;">256/243</span></u>) 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<span style="color: #ffffff;">##</span><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<span style="color: #ffffff;">##</span><br />
1697.06<br />
(1698.04)<br />
</td>
        <td>Perfect fifth<br />
702.94<br />
(701.96)<br />
</td>
        <td>Minor third<span style="color: #ffffff;">##</span><br />
291.17<br />
(294.13)<br />
</td>
        <td>Apotome<span style="color: #ffffff;">##</span><br />
120.61<br />
(113.69)<br />
</td>
        <td>17-tone comma<span style="color: #ffffff;">##</span><br />
49.96<br />
(66.76)<br />
</td>
    </tr>
</table>

Thus for example:<br />
<span style="color: #ffffff;">###</span>octave = 2×fourth + tone<br />
<span style="color: #ffffff;">###</span>fourth = 2×tone + limma<br />
<span style="color: #ffffff;">###</span>tone = 2×limma + Pythag comma<br />
<span style="color: #ffffff;">###</span>perfect 11th (<u>8/3</u>) = 2×fifth + minor third<br />
<span style="color: #ffffff;">###</span>fifth = 2×(minor third) + apotome<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 <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;"><em>s</em> </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 are clearly differentiated but of comparable magnitude – highly desirable properties for the formation of musical scales.<br />
In this fractal temperament, multiplying or dividing any interval by the factor <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;"><em>s</em> </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 ratio produce ratios involving <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pell_number" rel="nofollow">Pell numbers</a>.<br />
<span style="color: #ffffff;">###</span>√2 + 1 ≈ 2, 5/2, 12/5, 29/12, 70/29…,<br />
Other approximations to the silver ratio 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, 41edo, 70edo etc.).<br />
The accuracy of the argent fifth means that the scheme produces workable approximations to the true sizes of the 3-limit intervals featured in the table. However, if the table is extended one further step to the right, errors of sign begin to occur (the next column containing the 29-tone comma and <em>minus</em> the 41-tone comma).<br />
<br />
Figure 2 is a <em>continued fraction jigsaw</em> showing the sizes of the octave (o), fourth (f), tone (T), limma (s<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span>), Pythagorean comma (p) and 29-tone comma (p<span style="font-size: 60%;">29</span>) as tempered by 41edo - an approximation to argent temperament. The same diagram with different labelling can also represent 5edo, 7edo, 12edo, 17edo, 29edo, etc.<br />
<br />
<!-- ws:start:WikiTextLocalImageRule:482:&lt;img src=&quot;/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 396px; width: 800px;&quot; /&gt; --><img src="/file/view/Continued%20fraction%20jigsaw%2041edo.png/541636098/800x396/Continued%20fraction%20jigsaw%2041edo.png" alt="Continued fraction jigsaw 41edo.png" title="Continued fraction jigsaw 41edo.png" style="height: 396px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:482 --><br />
<br />
<span style="color: #ffffff;">######</span>Figure 2. Continued fraction jigsaw for 41edo<br />
<br />
Figure 3 is a geometrical representation of argent temperament in which the size of an interval is proportional to the length of the corresponding line (o = octave, F = fifth, f = fourth, T = large tone, m<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean minor third, s<span style="font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean limma, X<span style="font-family: Arial,Helvetica,sans-serif; font-size: 80%; vertical-align: super;">p</span><span style="color: #ffffff;">#</span>= Pythagorean apotome, p = Pythagorean comma.<br />
<!-- ws:start:WikiTextLocalImageRule:483:&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:483 --><br />
<span style="color: #ffffff;">######</span>Figure 3. Geometrical representation of argent temperament<br />
<br />
By the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem" rel="nofollow">Gelfond-Schneider theorem </a> the frequency ratios of all argent intervals (<span style="font-family: Georgia,serif; font-size: 110%;"><em>r</em> = 2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2<em>a</em>+<em>b</em></span>, where<em><span style="font-family: Georgia,serif; font-size: 110%;"> a</span></em> and <em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em> are integers) are transcendental, with the exception of octave multiples (<span style="font-family: Georgia,serif; font-size: 110%;"><em>a</em> = 0</span>). The frequency ratio of the tempered perfect eleventh (<span style="font-family: Georgia,serif; font-size: 110%;"><u>8/3</u> = <u>2.6666...</u></span>) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant" rel="nofollow">Gelfond-Schneider constant </a>or Hilbert number, <span style="font-family: Georgia,serif; font-size: 110%;">2</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">√2</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 2.665144</span>...<br />
<br />
<!-- ws:start:WikiTextHeadingRule:87:&lt;h2&gt; --><h2 id="toc19"><a name="x4. Quadratic approximants-Golden temperaments"></a><!-- ws:end:WikiTextHeadingRule:87 -->Golden temperaments</h2>
 It has been shown in an example above that the ratio of the large tone (<em>T</em> <span style="font-family: Georgia,serif; font-size: 110%;">= <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>) is closely approximated by<br />
<!-- ws:start:WikiTextMathRule:40:
[[math]]&lt;br/&gt;
\qquad T/t = \sqrt{5}/2&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad T/t = \sqrt{5}/2</script><!-- ws:end:WikiTextMathRule:40 --><br />
It follows that<br />
<!-- ws:start:WikiTextMathRule:41:
[[math]]&lt;br/&gt;
\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad (T + t/2)/t = (\sqrt{5}+1)/2 = \phi</script><!-- ws:end:WikiTextMathRule:41 --><br />
where <span style="font-family: Georgia,serif; font-size: 110%;"><em>ϕ</em> = 1.61803</span>... is the golden ratio.<br />
If a Fibonacci sequence of intervals is formed from the pair of intervals <span style="font-family: Georgia,serif; font-size: 110%;"><em>T</em> – <em>t</em>/2</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">t</span></em>, and extended in both directions, it can thus be expected that the ratios between successive intervals in this sequence will also be close to <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em>. The sequence formed in this way is Sequence 1 in the following table.<br />


<table class="wiki_table">
    <tr>
        <td>Sequence 1:<span style="color: #ffffff;">#</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff;">#</span><em>t</em>/2 - 3<em>c</em><span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2<em>c</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>t</em>/2 <em>- c</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>T - t</em>/2 </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>t</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>T + t</em>/2<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>M + t</em>/2<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>2<em>M</em> </span><br />
</td>
    </tr>
    <tr>
        <td>Sequence 2:<span style="color: #ffffff;">#</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>magic</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>diesis</em></span><br />
</td>
        <td><em><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>chroma<span style="color: #ffffff; font-family: Georgia,serif;">#</span></span></em><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>semitone</em><span style="color: #ffffff; font-family: Georgia,serif;">#</span></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>t</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>mp</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>f - c</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>m6p - c</em></span><span style="color: #ffffff;">#</span><br />
</td>
    </tr>
    <tr>
        <td>Difference:<br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-3<em>σ</em>/2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em></span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>-<em>σ</em>/2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia,serif;">#</span>0<br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em>/2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span><em>σ</em> </span><br />
</td>
    </tr>
    <tr>
        <td>Seq 1 ratios:<br />
</td>
        <td><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6120</span><span style="color: #ffffff; font-family: Georgia,serif;">##</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6204</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6171</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6184<span style="color: #ffffff; font-family: Georgia,serif;">#</span> </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6179</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6181</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6180</span><br />
</td>
    </tr>
    <tr>
        <td>Seq 2 ratios:<br />
</td>
        <td><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.3865</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.7212</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.5810</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6325</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6125</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6201 </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia,serif;">#</span>1.6172 </span><br />
</td>
    </tr>
</table>

where <span style="font-family: Georgia,serif; font-size: 110%;"><em>f</em> = <u>4/3</u>, <em>T</em> = <u>9/8</u>, <em>t</em> = <u>10/9</u>, <em>M</em> = <u>5/4</u>, <em>magic</em> = <u>3125/3072</u>, <em>diesis</em> = <u>128/125</u>, <em>chroma</em> = <u>25/24</u>, <em>semitone</em> = <u>16/15</u>, <em>mp</em> = <u>32/27</u>, <em>c</em> = <em>syntonic comma</em> = <u>81/80</u>, <em>m6p</em> = <u>128/81</u>, <em>σ</em> = <em>schisma</em> = <u>32805/32768.</u></span><br />
The ratios between successive intervals in Sequence 1 are shown in the row labelled ‘Seq 1 ratios’, and are indeed close to <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em>.<br />
Sequence 2 is another Fibonacci sequence of intervals which differ from those in Sequence 1 by small amounts of the order of one schisma (<em><span style="font-family: Georgia,serif; font-size: 110%;">σ</span></em>), as indicated by the row marked ‘Difference’ (which is itself a Fibonacci sequence).<br />
The ratios of consecutive pairs of intervals in Sequence 2 are shown in the row labelled ‘Seq 2 ratios’. They approximate <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em> rather less accurately.<br />
A suitable name for 5-limit tunings in which the intervals in either Sequence 1 or Sequence 2, or both, are tempered to exactly <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em> would be ‘golden temperaments’.<br />
Tempering the Sequence 2 ratios to <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em> while tuning the octave pure and tempering out the syntonic comma yields <a class="wiki_link" href="/Golden%20Meantone">golden meantone</a> temperament.<br />
Tempering the Sequence 1 ratios to <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em> yields a range of temperaments which can be made extremely accurate by, for example, tuning the octave and fifth (and therefore all Pythagorean intervals) pure. In this temperament the errors in the intervals <em><span style="font-family: Georgia,serif; font-size: 110%;">s, t</span></em>, <em><span style="font-family: Georgia,serif; font-size: 110%;">M</span></em> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>m</em>=<u>6/5</u></span> are all ±0.02106 cents.<br />
Tempering out the schisma tunes Sequences 1 and 2 identically so that the ratios between consecutive intervals can be fixed at <em><span style="font-family: Georgia,serif; font-size: 110%;">ϕ</span></em> in both sequences. Normalised to a pure octave, the resulting temperament, ‘golden schismatic’, has a fifth of 701.791061 cents (error -0.163 cents) and a major third of 385.671509 cents (error -0.642 cents).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:89:&lt;h2&gt; --><h2 id="toc20"><a name="x4. Quadratic approximants-Pythagorean triples of quadratic approximants"></a><!-- ws:end:WikiTextHeadingRule:89 --><span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%; vertical-align: sub;">Pythagorean triples of quadratic approximants</span></h2>
 If the quadratic approximants <span style="font-family: Georgia,serif; font-size: 110%;"><em>q</em>1<em>, q</em>2</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>q</em>3</span> of a set of three intervals <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>1, <em>J</em>2</span> and <em><span style="font-family: Georgia,serif; font-size: 110%;">J</span></em>3 satisfy<br />
<!-- ws:start:WikiTextMathRule:42:
[[math]]&lt;br/&gt;
\qquad q_1^2 + q_2^2 = q_3^2&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad q_1^2 + q_2^2 = q_3^2</script><!-- ws:end:WikiTextMathRule:42 --><br />
they can be said to form a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Pythagorean_triple" rel="nofollow">Pythagorean triple</a>.<br />
The following are three examples. In the first and third cases, their counterparts in 12edo, <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>1', <em>J</em>2'</span> and <span style="font-family: Georgia,serif; font-size: 110%;"><em>J</em>3'</span>, are also Pythagorean triples:<br />


<table class="wiki_table">
    <tr>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; line-height: 0px; overflow: hidden;">#</span><em>J</em>1</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>J</em>2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>J</em>3</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>q</em>1</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>q</em>2</span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>q</em>3 </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>J</em>1' </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>J</em>2' </span><br />
</td>
        <td><span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Georgia;">#</span><em>J</em>3'<span style="color: #ffffff; font-family: Georgia;">#</span></span><br />
</td>
    </tr>
    <tr>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>6/5</u><span style="color: #ffffff;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>5/4</u><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>4/3</u><span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>1/2√30<span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>1/4√5<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>3<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>4<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>5<br />
</td>
    </tr>
    <tr>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>4/3</u><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>12/5</u><span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>5/2</u><span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>1/4√3<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15<span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>3/2√10<span style="color: #ffffff; font-family: Georgia;">#</span><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>8/5</u><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>12/5</u><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span><u>8/3</u><br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>3/4√10<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>7/4√15<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>5/4√6<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>8<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>15<br />
</td>
        <td><span style="color: #ffffff; font-family: Georgia;">#</span>17<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:91:&lt;h2&gt; --><h2 id="toc21"><a name="x4. Quadratic approximants-A small 34edo comma"></a><!-- ws:end:WikiTextHeadingRule:91 -->A small 34edo comma</h2>
 <span style="color: #333333;">As <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> has noted, the </span>5-limit comma <span style="color: #333333;">|-433 -137 280&gt; (‘<em>selenia</em>’) is remarkably small at just 0.004764 cents. The minute size of this comma can be explained using qu</span>adratic approximants.<br />
It can be shown, using a suitable <a class="wiki_link" href="/Comma-based%20lattices">comma-based lattice</a>, that every comma tempered out by 34edo can be expressed as an integer linear combination of the <a class="wiki_link" href="/Gammic%20node">//gammic// comma </a>|-29 -11 20&gt; (4.769 cents) and the <em>semisuper</em> comma (AKA <em><a class="wiki_link" href="/vishnuzma">vishnuzma</a></em>) |23 6 -14&gt; (3.338 cents). In particular,<br />
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>selenia</em> = 7 <em>gammic</em> – 10 <em>semisuper</em></span><br />
<span style="color: #333333;">So to prove that <em>selenia</em> is small we must show that <em>gammic</em><em>/semisuper</em> ≈ 10/7.</span><br />
<span style="color: #333333;"><em>Gammic</em> and <em>semisuper</em> are both bimodular commas:</span><br />
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>gammic</em> = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<u>6/5</u>,<u>5/4</u>)</span><br />
<span style="color: #333333;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>semisuper</em> = </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<u>25/24</u>,<u>4/3</u>)</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;">Using a result given in the section on bimodular commas, the size of </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<em>J</em>1,<em>J</em>2)</span><span style="color: #333333;"> can be estimated using</span><br />
<!-- ws:start:WikiTextMathRule:43:
[[math]]&lt;br/&gt;
\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b(J_1,J_2) ≈ \frac{1}{3} (J_2^2 – J_1^2) b_m</script><!-- ws:end:WikiTextMathRule:43 --><br />
<span style="color: #333333;">Estimating </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"><em>J</em>2</span><span style="color: #333333;"> and </span><span style="color: #333333; font-family: Georgia,serif; font-size: 110%;"><em>J</em>1</span><span style="color: #333333;"> with their quadratic approximants we then have</span><br />
<!-- ws:start:WikiTextMathRule:44:
[[math]]&lt;br/&gt;
\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b(J_1,J_2) ≈ \frac{1}{3} (q_2^2 – q_1^2) b_m</script><!-- ws:end:WikiTextMathRule:44 --><br />
<span style="font-family: Arial,Helvetica,sans-serif;">For <em>gammic</em>:</span><br />
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>J</em>₁= 6/5, <em>J</em>₂= 5/4</span><br />
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>v</em></span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/11, </span><em><span style="font-family: Georgia,serif; font-size: 110%;">v</span></em>₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/9, </span><em><span style="font-family: Georgia,serif; font-size: 110%;">b</span></em><span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1</span><br />
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>q</em></span><span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₁² = </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/30),</span> <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em><span style="font-family: Arial,Helvetica,sans-serif; font-size: 110%;">₂<em>² =</em> </span><span style="font-family: Georgia,serif; font-size: 110%;">(1/4)(1/20)</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">###</span><em>gammic</em> = </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<em>J</em></span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,<em>J</em></span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/30 – 1/20) = (1/12) (1/60)</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;">For <em>semisuper:</em></span><br />
<span style="font-family: Georgia,serif; font-size: 110%;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em>J</em>₁= 25/24, <em>J</em>₂= 4/3</span><br />
<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em>v</em></span>₁ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/49, </span><em><span style="font-family: Georgia,serif;">v</span></em>₂ <span style="font-family: Georgia,serif; font-size: 110%;">= 1/7, </span><em><span style="font-family: Georgia,serif;">b</span></em><span style="font-family: Georgia,serif; font-size: 60%;">m</span><span style="font-family: Georgia,serif; font-size: 110%;"> = 1/7</span><br />
<span style="font-family: Georgia,serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em>q</em></span><span style="font-family: Arial,Helvetica,sans-serif;">₁² = </span><span style="font-family: Georgia,serif;">(1/4)(1/600),</span> <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em><span style="font-family: Arial,Helvetica,sans-serif;">₂<em>² =</em> </span><span style="font-family: Georgia,serif;">(1/4)(1/12)</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em>semisuper</em> = </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<em>J</em></span>₁<span style="font-family: Georgia,serif; font-size: 110%;">,<em>J</em></span>₂<span style="font-family: Georgia,serif; font-size: 110%;">) ≈ (1/12) (1/12 – 1/600)(1/7) = (1/12) (7/600)</span><br />
<span style="font-family: Arial,Helvetica,sans-serif;">Therefore</span><br />
<span style="color: #ffffff;">###</span><span style="font-family: Arial,Helvetica,sans-serif;"><em>gammic/semisuper</em> ≈ </span><span style="font-family: Georgia,serif; font-size: 110%;">10/7</span><br />
<span style="color: #333333;">as required.</span><br />
<br />
<span style="color: #333333;">To estimate the size of <em>selenia</em> we must quantify the error in this ratio. A more accurate analysis gives</span><br />
<!-- ws:start:WikiTextMathRule:45:
[[math]]&lt;br/&gt;
\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\&lt;br /&gt;
\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad b(J_1,J_2) ≈ \left( \tfrac{1}{3} (q_2^2 – q_1^2) – \tfrac{2}{15} (q_2^4 – q_1^4) \right) b_m \\
\qquad = \tfrac{1}{3} (q_2^2 – q_1^2)(1 – \tfrac{2}{5} (q_1^2 + q_2^2) ) b_m</script><!-- ws:end:WikiTextMathRule:45 --><br />
<span style="color: #333333;">So to improve our estimates of </span><span style="font-family: Georgia,serif; font-size: 110%;"><em>b</em>(<em>J</em>1,<em>J</em>2)</span> <span style="color: #333333;">we should multiply them by</span><br />
<!-- ws:start:WikiTextMathRule:46:
[[math]]&lt;br/&gt;
\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad f = 1 – \tfrac{2}{5} (q_1^2 + q_2^2)</script><!-- ws:end:WikiTextMathRule:46 --><br />
<span style="color: #333333;">Thus a better estimate for <em>gammic/semisuper</em> is</span><br />
<!-- ws:start:WikiTextMathRule:47:
[[math]]&lt;br/&gt;
\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad \frac{gammic}{semisuper} ≈ \frac{10 f_{gamma}} {7 f_{semisuper}}</script><!-- ws:end:WikiTextMathRule:47 --><br />
<span style="font-family: Arial,Helvetica,sans-serif;">from which it follows that</span><br />
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;"><em>selenia</em> = 7 <em>gammic</em> - 10 <em>semisuper</em></span><br />
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif; font-size: 110%;">######## </span><span style="font-family: Georgia,serif; font-size: 110%;">≈ 7 <em>gammic</em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;"> (<em>f</em></span></span><em><span style="font-family: Georgia,serif; font-size: 70%; vertical-align: sub;">gammic</span><span style="font-family: Georgia,serif; font-size: 110%;"> - f</span><span style="font-size: 70%; vertical-align: sub;">semisuper</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">)</span><span style="font-family: Georgia,serif; font-size: 110%;">/f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span></em><br />
<span style="font-family: Arial,Helvetica,sans-serif;">Putting in the numbers:</span><br />
<em><span style="color: #ffffff;">###</span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic </span></em><span style="font-family: Georgia,serif; font-size: 110%;">=</span> <span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">1 – (2/5) (1/4) (1/30 + 1/20) = 1 – 1/120</span><br />
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: sub;">= 1 – (2/5)(1/4) (1/600 + 1/12) = 1 – (1/120) (51/50)</span><br />
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><em><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">gammic</span> <span style="font-family: Georgia,serif;">- </span><span style="font-family: Georgia,serif; font-size: 110%;">f</span><span style="font-size: 70%; vertical-align: sub;">semisuper </span></em><span style="font-family: Georgia,serif; font-size: 110%;">= 1/6000</span><br />
<span style="color: #333333;">Therefore</span><br />
<span style="color: #ffffff; font-family: Arial,Helvetica,sans-serif;">###</span><span style="font-family: Georgia,serif; font-size: 110%;"><em>selenia</em> ≈ 7 <em>gammic</em> (1/6000) (120/119) = <em>gammic</em>/850 = 0.00561</span><span style="color: #333333;"> cents</span><br />
<span style="color: #333333;">which </span>is within 20% of the accurate value, 0.00476 cents. (The discrepancy is due to the influence of terms in <em><span style="font-family: Georgia,serif; font-size: 110%;">q</span></em><span style="font-family: Georgia,serif; font-size: 80%; vertical-align: super;">6</span><em>,</em> which become significant when the <em><span style="font-family: Georgia,serif; font-size: 110%;">f</span></em> values are very similar.)<br />
In summary, the reason <em>selenia</em> is small (compared to <em>gammic</em> and <em>semisuper</em>) is because the quadratic approximants of <em>gammic</em> and <em>semisuper</em> are in the ratio 10/7. The reason it is <em>very</em> small (of order <em>gammic</em>/1000 rather than <em>gammic</em>/10) is because the fractional errors in those approximants are almost the same. That in turn is because the squares of the source intervals of these bimodular commas have nearly the same sum. Note that the quadratic approximants of three of these intervals form a Pythagorean triple:<br />
<!-- ws:start:WikiTextMathRule:48:
[[math]]&lt;br/&gt;
\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2&lt;br/&gt;[[math]]
 --><script type="math/tex">\qquad \left( q(\tfrac{6}{5}) \right)^2 + \left( q(\tfrac{5}{4}) \right)^2 = \left( q(\tfrac{4}{3}) \right)^2</script><!-- ws:end:WikiTextMathRule:48 --><br />
and <span style="font-family: Georgia,serif; font-size: 110%;">(<em>q</em>(25/24))</span><span style="font-family: Georgia,serif; font-size: 110%; vertical-align: super;">2</span> , being small in comparison to the other terms, compromises this equality only slightly.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:93:&lt;h1&gt; --><h1 id="toc22"><a name="Sources and acknowledgements"></a><!-- ws:end:WikiTextHeadingRule:93 -->Sources and acknowledgements</h1>
 This article is based on original research by <a class="wiki_link" href="/Martin%20Gough">Martin Gough</a>. See <a href="/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf" onclick="ws.common.trackFileLink('/file/view/Bimod%20Approx%202014-6-8.pdf/541604262/Bimod%20Approx%202014-6-8.pdf');">this paper</a> for a fuller account of bimodular approximants.<br />
The tuning referred to here as argent temperament was described by <a class="wiki_link" href="/graham%20breed">Graham Breed</a> and Paul Hahn in posts (#12599, #12670) to the Yahoo tuning list on 10 and 12 August 2000.<br />
Thanks to <a class="wiki_link" href="/Gene%20Ward%20Smith">Gene Ward Smith</a> for the Gelfond-Schneider result.</body></html>