Zetave: Difference between revisions
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{{Mathematical interest | {{Mathematical interest}} | ||
The '''zetave''' | 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 == | == 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 e<sup> | [[Category:Zeta]] | ||
Latest revision as of 13:38, 21 August 2025
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This page presents a topic of primarily mathematical interest.
While it is derived from sound mathematical principles, its applications in terms of utility for actual music may be limited, highly contrived, or as yet unknown. |
The zetave is defined as [math]\displaystyle{ e^{2\pi} }[/math]. Its value is roughly 535.492, or 10877.664 ¢. The zetave is the interval which is equally divided when the zeta function is not scaled so that [math]\displaystyle{ \mathrm{Im}(s) }[/math] corresponds to EDOs, 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 Riemann zeta function correspond to equal divisions of the zetave (EDZ). (i.e. when taking [math]\displaystyle{ \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]\displaystyle{ \tfrac{2\pi}{\ln(2)} }[/math]. More generally, an equal division of an interval x can be expressed as an EDZ via [math]\displaystyle{ \tfrac{2\pi}{\ln(x)} }[/math]. For an equal tuning expressed as an equal division of the natave (e), this reduces to a multiplication by [math]\displaystyle{ 2\pi }[/math]; in other words, the zetave is the result of stacking [math]\displaystyle{ 2\pi }[/math] nataves. The appearance of the zetave in the zeta function's usage in tuning suggests that it has a natural relation to equal-step tunings.
It is extremely well-approximated by 31edo: 281 steps of 31edo is 10877.419 ¢, and [math]\displaystyle{ e^{2\pi} }[/math] is larger than [math]\displaystyle{ 2^{281/31} }[/math] by only 0.245 ¢ (0.0142%, or 1 in 7,066). Another notable approximant is 139edo: 1260 steps of 139edo is 10877.698 ¢, and [math]\displaystyle{ e^{2\pi} }[/math] is smaller than [math]\displaystyle{ 2^{1260/139} }[/math] by only 0.034 ¢. 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 ei𝜏 = 1.