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^𝜏, where e is the exponential constant and 𝜏 is the circle constant 2π. 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. In other words, imaginary values on the Riemann zeta function correspond to equal divisions of the zetave (EDZ) (i.e. when taking zeta(0.5+zi), the value z is an equal tuning expressed as an EDZ). 12edo is about 108.776643404edz, and an EDO can be converted to an EDZ by multiplying the number by ^𝜏/_ln(2) (and in general, an equal division of an interval x can be expressed as an EDZ via ^𝜏/_ln(x). For an equal tuning expressed as an equal division of the natave (e), this reduces to a multiplication by 𝜏; in other words, the zetave is the result of stacking 𝜏 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.

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.