Ie/me

Introduction

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:

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

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

$J_c=1200{\log }_{2}r=1200\ln r/\ln 2$

where for just intervals r is rational and can be written as the ratio of two integers:

$r=n/d$

When manipulating approximants it is convenient to work with a different logarithmic base, in which the interval is defined as

$J=\frac{1}{2}\ln r$

This is equivalent to replacing the cent with a unit of interval measurement having a frequency ratio e2 = 7.38906... This unit interval can conveniently be termed the dineper (dNp), being twice the size of the natural unit for logarithmic measurement, the Neper.

Comparing the two units of measurement we find

1 dineper = 2400/ln(2) = 3462.468 cents

which is about 1.4 semitones short of three octaves.

The logarithmic size of an interval with a given frequency ratio can be conveniently notated as that ratio underlined. Thus 3/2 is the perfect fifth. 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

Definition

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

$v=\frac{r-1}{r+1}$

v can thus be expressed as

解析失败 (语法错误): {\displaystyle \qquad v = \frac{n-d}{n+d} \\ }

###### = (frequency difference) / (frequency sum)

###### =½ (frequency difference) / (mean frequency)

r can be retrieved from v using the inverse relation

$r=\frac{1+v}{1-v}$

Properties

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

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

$J=\frac{1}{2}\ln r$

the relationship between v and J can be expressed as

$v=\frac{r-1}{r+1}=\frac{e^{2J}-1}{e^{2J}+1}=\tanh J=J-\frac{1}{3}{J}^3+\frac{2}{15}{J}^5-...$

which shows that v ≈ J and provides an indication of the size and sign of the error involved in this approximation.

J can be expressed in terms of v as

$J={\tanh }^{-1}v=v+\frac{1}{3}{v}^3+\frac{1}{5}{v}^5-...$

The function v(r) is the order (1,1) Padé approximant of the function J(r) =½ ln r in the region of r = 1, which has the property of matching the function value and its first and second derivatives at this value of r. The bimodular approximant function is thus accurate to second order in r – 1.

As an example, the size of the perfect fifth (in dNp units) is

$J=\frac{1}{2}\ln 3/2=0.20273...$

The bimodular approximant for this interval (r = 3/2) is

解析失败 (语法错误): {\displaystyle \qquad v = (3/2 – 1)/(3/2 + 1) = (3 – 2)/(3 + 2) = 1/5 = 0.2}

and the Taylor series indicates that the error in this value is about

$-\frac{1}{3}{v}^3=-0.00267...$

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

Low-order superparticular intervals.png

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

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

解析失败 (语法错误): {\displaystyle \qquad v[-J] = -v[J] \\ \qquad v[J_1 +J_2] = \frac{v_1+v_2}{1+v_1 v_2}}

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