Zetave: Difference between revisions

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

The '''zetave''', e^~~tau~~ or ~~~535~~.~~49/1~~ 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). 12edo is about 108.~~776643404edz~~. The appearance of the zetave in the zeta function's usage in tuning suggests that it has ~~some sort of~~ natural relation to [[~~Equal~~-step tuning|~~equal~~-~~step tunings~~]].

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]]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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-{{Infobox interval|ratio=e^{\tau}|cents=10877.6643|Ratio=e^{2\pi}|Cents=10877.6643|Name=zetave}}
+{{Mathematical interest}}
-The '''zetave''', e^tau or ~535.49/1 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). 12edo is about 108.776643404edz. The appearance of the zetave in the zeta function's usage in tuning suggests that it has some sort of natural relation to [[Equal-step tuning|equal-step tunings]].
+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]]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]]