Zetave

This page presents a topic of primarily mathematical interest. While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited and highly contrived.

The zetave is defined as [math]\displaystyle{ e^{2\pi} }[/math]. Its value is roughly 535.492, or 10877.664 ¢. The zetave is the interval which is equally divided when the zeta function is not scaled so that [math]\displaystyle{ \mathrm{Im}(s) }[/math] corresponds to EDOs, and in that context has first been noticed by Keenan Pepper, referring to it as the "natural interval". In other words, imaginary values on the Riemann zeta function correspond to equal divisions of the zetave (EDZ). (i.e. when taking [math]\displaystyle{ \zeta(\tfrac{1}{2} + it) }[/math], the value t is an equal tuning expressed as an EDZ).

12edo is about 108.7766edz, and any EDO can be converted to an EDZ by multiplying the number by [math]\displaystyle{ \tfrac{2\pi}{\ln(2)} }[/math]. More generally, an equal division of an interval x can be expressed as an EDZ via [math]\displaystyle{ \tfrac{2\pi}{\ln(x)} }[/math]. For an equal tuning expressed as an equal division of the natave (e), this reduces to a multiplication by [math]\displaystyle{ 2\pi }[/math]; in other words, the zetave is the result of stacking [math]\displaystyle{ 2\pi }[/math] nataves. The appearance of the zetave in the zeta function's usage in tuning suggests that it has a natural relation to equal-step tunings.

It is extremely well-approximated by 31edo: 281 steps of 31edo is 10877.419 ¢, and [math]\displaystyle{ e^{2\pi} }[/math] is larger than [math]\displaystyle{ 2^{281/31} }[/math] by only 0.245 ¢ (0.0142%, or 1 in 7,066).

Trivia

• The zetave, as a ratio, can be expressed as an i-th root of 1; this is, in fact, a statement of Euler's identity that e^i𝜏 = 1.