Zetave: Difference between revisions
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{{Mathematical interest}}{{Infobox interval|ratio=e^{\ | {{Mathematical interest}}{{Infobox interval|ratio=e^{2\pi}|cents=10877.6643|Ratio=e^{\tau}|Cents=10877.6643|Name=zetave}} | ||
The '''zetave''' is defined as e<sup> | The '''zetave''' is defined as e<sup>2π</sup>, 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 [[EDO]]s, 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 [[The Riemann zeta function and tuning|Riemann zeta function]] correspond to equal divisions of the zetave (EDZ) (i.e. when taking <math>\zeta(0.5 + zi)</math>, the value ''z'' 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 <math>\frac{2π}{ln(2)}</math> (and in general, an equal division of an interval ''x'' can be expressed as an EDZ via <math>\frac{2π}{ln(x)}</math>. For an equal tuning expressed as an [[EDN|equal division of the natave]] (e), this reduces to a multiplication by 2π; in other words, the zetave is the result of stacking 2π [[Natave|nataves]]. The appearance of the zetave in the zeta function's usage in tuning suggests that it has a natural relation to [[Equal-step tuning|equal-step tunings]]. | ||
== Trivia == | == Trivia == | ||
Revision as of 13:11, 26 March 2025
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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 |
The zetave is defined as e2π, 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 [math]\displaystyle{ \zeta(0.5 + zi) }[/math], the value z 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 [math]\displaystyle{ \frac{2π}{ln(2)} }[/math] (and in general, an equal division of an interval x can be expressed as an EDZ via [math]\displaystyle{ \frac{2π}{ln(x)} }[/math]. 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.
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 ei𝜏 = 1.