User:Sintel/Zeta working page

Derivation

Our goal is to derive a function that quantifies how effectively an equal division of the octave (EDO) approximates just intonation (JI). We will use the variable x to denote an EDO where x=12 represents 12edo. Importantly, we allow x to take continuous values, enabling it to represent any equal-step tuning where the step size is 1200/x cents. For example, x=8.202 corresponds to 13 equal divisions of the tritave (3/1).

To assess how well an equal temperament approximates the harmonic series, we need a function that measures the accuracy of approximation for each overtone.

For any overtone n, the quantity [math]\displaystyle{ x \log_2(n) }[/math] represents how many steps in our equal temperament are needed to approximate that overtone. When this value is close to an integer, the approximation is good; when it deviates significantly from an integer, the approximation is poor.

To quantify this proximity to integers, we can use the cosine function [math]\displaystyle{ f(x) = \cos(2 \pi x) }[/math], which peaks at integer values. This provides a simple metric where values close to 1 indicate good approximations and values close to -1 indicate poor ones.

Let's begin by considering only the first few harmonics and look at [math]\displaystyle{ f(x \log_2 2) }[/math], [math]\displaystyle{ f(x \log_2 3) }[/math], [math]\displaystyle{ f(x \log_2 4) }[/math], [math]\displaystyle{ f(x \log_2 5) }[/math].

The sum of these values gives us an overall quality measure:

[math]\displaystyle{ \sum_{n=1}^5 f(x \log_2 n) }[/math]

Todo: add a figure here

Ideally, we would like to extend this function to sum over the entire harmonic series:

[math]\displaystyle{ \sum_{n=1}^{\infty} f(x \log_2 n) }[/math]

However, this infinite sum does not converge. To address this issue, we can introduce a weighting factor that diminishes the contribution of higher harmonics:

[math]\displaystyle{ \begin{aligned} F(x) &= \sum_{n=1}^{\infty} \frac{1}{n^\sigma} f(x \log_2 n) \\ &= \sum_{n=1}^{\infty} \frac{1}{n^\sigma} \cos(2 \pi x \log_2 n) \end{aligned} }[/math]

This series converges provided that [math]\displaystyle{ \sigma \gt 1 }[/math], with the parameter sigma controlling how quickly the influence of higher harmonics declines.

We can reformulate this expression using Euler's formula [math]\displaystyle{ e^{ix} = \cos x + i \sin x }[/math], and replace the cosine with the real part of a complex exponential:

[math]\displaystyle{ \begin{aligned} F(x) &= \sum_{n=1}^{\infty} \frac{1}{n^\sigma} \mathrm{Re} \left( e^{-2 \pi i x \log_2 n} \right) \\ &= \mathrm{Re} \, \sum_{n=1}^{\infty} \frac{1}{n^\sigma} e^{-2 \pi i x \log_2 n} \end{aligned} }[/math]

Using properties of logarithms, we know that [math]\displaystyle{ e^{x \log_2 n} = n^{\frac{x}{\ln 2}} }[/math], which gives us:

[math]\displaystyle{ \begin{aligned} F(x) &= \mathrm{Re} \, \sum_{n=1}^{\infty} \frac{1}{n^\sigma} n^{-\frac{2 \pi}{\ln 2} i x} \\ \end{aligned} }[/math]

To clean this up a bit we will make a substitution and set [math]\displaystyle{ t = \tfrac{2 \pi}{\ln 2} x }[/math].

[math]\displaystyle{ \begin{aligned} F(x) &= \mathrm{Re} \, \sum_{n=1}^{\infty} \frac{1}{n^\sigma} n^{-i t} \\ &= \mathrm{Re} \, \sum_{n=1}^{\infty} n^{-(\sigma + it)} \end{aligned} }[/math]

This final expression is precisely the definition of the Riemann zeta function [math]\displaystyle{ \zeta(s) }[/math] evaluated at the complex value [math]\displaystyle{ s = \sigma + i t = \sigma + \tfrac{2 \pi i}{\ln(2)} x }[/math]. Thus, our measure of tuning quality can be expressed directly in terms of this fundamental mathematical function:

[math]\displaystyle{ F(x) = \mathrm{Re} \, \zeta \left( \sigma + \frac{2 \pi i}{\ln(2)} x \right) }[/math]