Zetave: Difference between revisions

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

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

The '''zetave''' is defined as e𝜏, where e is the exponential constant ~~and 𝜏 is the circle constant 2π~~. 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~~. 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 zeta(0.5+zi), the value z is an equal tuning expressed as an EDZ). 12edo is about 108.~~776643404edz~~, and an EDO can be converted to an EDZ by multiplying the number by <~~sup>𝜏/<sub~~>ln(2)</~~sub~~> (and in general, an equal division of an interval ''x'' can be expressed as an EDZ via <~~sup>𝜏/<sub~~>ln(x)</~~sub~~>. For an equal tuning expressed as an [[EDN|equal division of the natave]] (e), this reduces to a multiplication by 𝜏; in other words, the zetave is the result of stacking 𝜏 [[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 <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 ==

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-{{Mathematical interest}}{{Infobox interval|ratio=e^{\tau}|cents=10877.6643|Ratio=e^{\tau}|Cents=10877.6643|Name=zetave}}
+{{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>𝜏</sup>, where e is the exponential constant and 𝜏 is the circle constant 2π. 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. 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 zeta(0.5+zi), the value z is an equal tuning expressed as an EDZ). 12edo is about 108.776643404edz, and an EDO can be converted to an EDZ by multiplying the number by <sup>𝜏</sup>/<sub>ln(2)</sub> (and in general, an equal division of an interval ''x'' can be expressed as an EDZ via <sup>𝜏</sup>/<sub>ln(x)</sub>. For an equal tuning expressed as an [[EDN|equal division of the natave]] (e), this reduces to a multiplication by 𝜏; in other words, the zetave is the result of stacking 𝜏 [[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 <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 ==