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, highly contrived, or as yet unknown.

Interval information

Expression	[math]\displaystyle{ e^{\tau} }[/math]
Size in cents	10877.66¢
Name	zetave

The zetave is defined as e^2π, where e is the exponential constant. In terms of a ratio, it is roughly ~535.49/1. The zetave is the interval which is equally divided when the zeta function is not scaled so that Im(s) 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 ζ(1⁄2 + it), the value t is an equal tuning expressed as an EDZ). 12edo is about 108.7766edz, and in general an EDO can be converted to an EDZ by multiplying the number by ⁠2π/ln(2)⁠ (and in general, an equal division of an interval x can be expressed as an EDZ via ⁠2π/ln(x)⁠. For an equal tuning expressed as an equal division of the natave (e), this reduces to a multiplication by 2π; in other words, the zetave is the result of stacking 2π 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 ¢, which is flat of e^2π by only 0.245 ¢ (1 in 44,400 or 0.00225%).

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.