User:Lériendil/Nataves, bimodular approximants, and Don Page theory

Bimodular approximants

We start with the following series for the natural logarithm, from Euler.

ln[(1+x)/(1-x)] = 2x + 2(x^3)/3 + 2(x^5)/5 + ..., for -1 < x < 1.

Now consider that the argument of the logarithm function is a fraction, a/b. In this case, x = (a-b)/(a+b). In the specific case that a-b is 1 or 2, x is a unit fraction. This creates a series of approximations to the natural logarithms of odd-particulars.

ln(3/1) = 2/2 + 2/(3*2^3) + 2/(5*2^5) + ...;

ln(2/1) = 2/3 + 2/(3*3^3) + 2/(5*3^5) + ...;

ln(5/3) = 2/4 + 2/(3*4^3) + 2/(5*4^5) + ...;

ln(3/2) = 2/5 + 2/(3*5^3) + 2/(5*5^5) + ...;

etc.

As the fraction grows smaller, the terms beyond the first become negligible compared to the first. Therefore, we can define the bimodular approximant to the logarithm of a fraction a/b to be 2x = 2(a-b)/(a+b), the first term in Euler's series.

The important insight then is that, for some integer c, c * ln[(c+1)/(c-1)] is approximated to c * 2/c = 2 = ln(e^2); at the same time, note that this is equal to ln[((c+1)/(c-1))^c]. Removing the logarithm sign from both expressions, we find that ((c+1)/(c-1))^c, the odd-particular (c+1)/(c-1) stacked c times, approximates e^2, the acoustic double natave, and as the bimodular approximant increases in accuracy as the fraction under the logarithm gets smaller, that this approximation holds better and better as c gets larger.

The usefulness of composite divisions of sharp nataves

Expressed otherwise, consider an approximation of e^2, in an EDO or other tempered system. Say that this approximation is equi-divisible into c parts, for some integer c. Then, by the above, it turns out that there is a good approximation of the fraction (c+1)/(c-1), particularly when this happens to be a (logarithmically) small fraction. By extension, if e^2 is divided into a very composite number of parts, there are many different integers c that it can be divided into.

An example occurs in 131edo, which has an extremely accurate approximation of e at 189 steps; therefore e^2 is at 378 steps, 378 being 2 * 3^3 * 7. Dividing e^2 into 7, 9, 14, 18, 21, 27, and 63 equal parts give rise to approximations of 4/3, 5/4, 15/13, 19/17, 11/10, 14/13, and 32/31 that are increasingly accurate as the fraction gets smaller; all of them are patent in 131edo. This is of more interest than merely in the accuracy of the approximations, however; the convenient logarithmic ratios between the approximations of these fractions lead to surprisingly nice regular temperament and DR properties, as explained later on in the section on Don Page commas. Therefore, there is a strong case for seeking out compositely divided nataves.

For added accuracy on non-tiny intervals, it is best to turn the natave sharp. The next-leading terms in the approximation to the natural logarithm are all positive, and therefore when the logarithm is multiplied by the integer c, the number 2 has a positive correction factor; therefore they stack to a number larger than e^2; the "double natave" they stack to is sharp.

Natural intonation

tbd