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 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.

The '''zetave''' is defined as <math>e^{2\pi}</math>. Its value is roughly 535.492, or 10877.664{{c}}. The zetave is the interval which is equally divided when the [[zeta]] function is ''not'' scaled so that <math>\mathrm{Im}(s)</math> 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(\tfrac{1}{2} + it)</math>, the value ''t'' is an equal tuning expressed as an EDZ).

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}}.

[[12edo]] is about 108.7766edz, and any EDO can be converted to an EDZ by multiplying the number by <math>\tfrac{2\pi}{\ln(2)}</math>. More generally, an equal division of an interval ''x'' can be expressed as an EDZ via <math>\tfrac{2\pi}{\ln(x)}</math>. For an equal tuning expressed as an [[EDN|equal division of the natave]] (''e''), this reduces to a multiplication by <math>2\pi</math>; in other words, the zetave is the result of stacking <math>2\pi</math> [[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{{cent}}, and <math>e^{2\pi}</math> is larger than <math>2^{281/31}</math> by only 0.245{{c}} (0.0142%, or {{nowrap|1 in 7,066}}). Another notable approximant is [[139edo]]: 1260 steps of 139edo is 10877.698{{c}}, and <math>e^{2\pi}</math> is smaller than <math>2^{1260/139}</math> by only 0.034{{c}}. In other words, it is 1260edz, a highly composite EDZ.

== 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}}.

[[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}}
+{{Mathematical interest}}
-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.
+The '''zetave''' is defined as <math>e^{2\pi}</math>. Its value is roughly 535.492, or 10877.664{{c}}. The zetave is the interval which is equally divided when the [[zeta]] function is ''not'' scaled so that <math>\mathrm{Im}(s)</math> 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(\tfrac{1}{2} + it)</math>, the value ''t'' is an equal tuning expressed as an EDZ).
-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}}.
+[[12edo]] is about 108.7766edz, and any EDO can be converted to an EDZ by multiplying the number by <math>\tfrac{2\pi}{\ln(2)}</math>. More generally, an equal division of an interval ''x'' can be expressed as an EDZ via <math>\tfrac{2\pi}{\ln(x)}</math>. For an equal tuning expressed as an [[EDN|equal division of the natave]] (''e''), this reduces to a multiplication by <math>2\pi</math>; in other words, the zetave is the result of stacking <math>2\pi</math> [[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{{cent}}, and <math>e^{2\pi}</math> is larger than <math>2^{281/31}</math> by only 0.245{{c}} (0.0142%, or {{nowrap|1 in 7,066}}). Another notable approximant is [[139edo]]: 1260 steps of 139edo is 10877.698{{c}}, and <math>e^{2\pi}</math> is smaller than <math>2^{1260/139}</math> by only 0.034{{c}}. In other words, it is 1260edz, a highly composite EDZ.
 == 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}}.
 [[Category:Zeta]]