Zetave: Difference between revisions

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{{Mathematical interest}}{{Infobox interval|ratio=e^{2\pi}|cents=10877.6643|Ratio=e^{\tau}|Cents=10877.6643|Name=zetave}}

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 [[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~~]].

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 [[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 {{nowrap|ζ({{frac|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 {{sfrac|2π|ln(2)}} (and in general, an equal division of an interval ''x'' can be expressed as an EDZ via {{sfrac|2π|ln(''x'')}}. 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]]s. The appearance of the zetave in the zeta function's usage in tuning suggests that it has a natural relation to [[equal-step tuning]]s.

It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, which is flat of {{nowrap|''e''2π}} by only 0.245{{c}}.

== 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 {{nowrap|''e''''i''𝜏 {{=}} 1}}.

* 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.

[[Category:Zeta]]

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 {{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>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]].
+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 {{nowrap|ζ({{frac|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 {{sfrac|2π|ln(2)}} (and in general, an equal division of an interval ''x'' can be expressed as an EDZ via {{sfrac|2π|ln(''x'')}}. 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]]s. The appearance of the zetave in the zeta function's usage in tuning suggests that it has a natural relation to [[equal-step tuning]]s.
+It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, which is flat of {{nowrap|''e''<sup>2π</sup>}} by only 0.245{{c}}.
 == 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 {{nowrap|''e''<sup>''i''𝜏</sup> {{=}} 1}}.
-* 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<sup>i𝜏</sup> = 1.
 [[Category:Zeta]]