Zetave: Difference between revisions
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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 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}} | 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}}, which corresponds to 1 in 44,400 or 0.00225%. | ||
== Trivia == | == 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>''i''𝜏</sup> {{=}} 1}}. | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||