Zetave: Difference between revisions

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The '''zetave''' is defined as <math>e^{2\pi}</math>. Its value is roughly 535.49, or 10877.66{{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).

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).

[[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~~|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]]s.

[[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}}, ~~differing from~~ <math>e^{2\pi}</math> by only 0.245{{c}} (0.~~00225~~%, or 1 in 44,~~400~~).

It is extremely well-approximated by [[31edo]]: 281 steps of 31edo (<math>2^{281/31}</math>) is 10877.419{{cent}}, and falls short of <math>e^{2\pi}</math> by only 0.245{{c}} (0.0142%, or {{nowrap|1 in 7,067}}). Another notable approximant is [[139edo]]: 1260 steps of 139edo (<math>2^{1260/139}</math>) is 10877.698{{c}}, and exceeds <math>e^{2\pi}</math> by only 0.034{{c}} (0.00194%, or {{nowrap|1 in 51,676}}); in other words, it is 1260edz, a highly composite EDZ.

== Approximations ==

{{interval edo approximation | interval = 535482/1000|interval_name = the Zetave}}

== 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 {{nowrap|''e''<sup>2π''i''</sup> {{=}} 1}}.

[[Category:Zeta]]

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 {{Mathematical interest}}
-The '''zetave''' is defined as <math>e^{2\pi}</math>. Its value is roughly 535.49, or 10877.66{{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).
+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).
-[[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|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]]s.
+[[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}}, differing from <math>e^{2\pi}</math> by only 0.245{{c}} (0.00225%, or 1 in 44,400).
+It is extremely well-approximated by [[31edo]]: 281 steps of 31edo (<math>2^{281/31}</math>) is 10877.419{{cent}}, and falls short of <math>e^{2\pi}</math> by only 0.245{{c}} (0.0142%, or {{nowrap|1 in 7,067}}). Another notable approximant is [[139edo]]: 1260 steps of 139edo (<math>2^{1260/139}</math>) is 10877.698{{c}}, and exceeds <math>e^{2\pi}</math> by only 0.034{{c}} (0.00194%, or {{nowrap|1 in 51,676}}); in other words, it is 1260edz, a highly composite EDZ.
+== Approximations ==
+{{interval edo approximation | interval = 535482/1000|interval_name = the Zetave}}
 == 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 {{nowrap|''e''<sup>2π''i''</sup> {{=}} 1}}.
 [[Category:Zeta]]