Zetave - Revision history

ArrowHead294 at 21:52, 27 November 2025

2025-11-27T21:52:33Z

← Older revision		Revision as of 21:52, 27 November 2025
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	[[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.		[[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.		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==
			== Approximations ==
	{{interval edo approximation \| interval = 535482/1000\|interval_name = the Zetave}}		{{interval edo approximation \| interval = 535482/1000\|interval_name = the Zetave}}

	== 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>2π''i''</sup> {{=}} 1}}.
	[[Category:Zeta]]		[[Category:Zeta]]

Pailiaq: added approximations

2025-11-27T20:53:28Z

added approximations

← Older revision		Revision as of 20:53, 27 November 2025
Line 6:		Line 6:

	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.		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.
			==Approximations==
			{{interval edo approximation \| interval = 535482/1000\|interval_name = the Zetave}}
	== 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]]

Dummy index: +139edo

2025-08-21T13:38:43Z

+139edo

← Older revision		Revision as of 13:38, 21 August 2025
Line 5:		Line 5:
	[[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.		[[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}}).		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 {{nowrap\|''e''<sup>''i''𝜏</sup> {{=}} 1}}.
	[[Category:Zeta]]		[[Category:Zeta]]

ArrowHead294 at 20:30, 20 April 2025

2025-04-20T20:30:12Z

← Older revision		Revision as of 20:30, 20 April 2025
Line 1:		Line 1:
	{{Mathematical interest}}		{{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 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}}).

	== 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]]

ArrowHead294 at 20:37, 8 April 2025

2025-04-08T20:37:58Z

← Older revision		Revision as of 20:37, 8 April 2025
Line 5:		Line 5:
	[[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\|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.

	It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, 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 is 10877.419{{cent}}, differing from <math>e^{2\pi}</math> by only 0.245{{c}} (0.00225%, or 1 in 44,400).

	== 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]]

ArrowHead294 at 20:37, 8 April 2025

2025-04-08T20:37:42Z

← Older revision		Revision as of 20:37, 8 April 2025
Line 5:		Line 5:
	[[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\|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.

	It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, ~~which is flat of~~ <math>e^{2\pi}</math> by only 0.245{{c}}, ~~which corresponds to~~ 1 in 44,400 ~~or 0.00225%~~.		It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, differing from <math>e^{2\pi}</math> by only 0.245{{c}} (0.00225%, or 1 in 44,400).

	== 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]]

Sintel: bold and emph, remove broken infobox, texify

2025-04-05T19:16:35Z

bold and emph, remove broken infobox, texify

← Older revision		Revision as of 19:16, 5 April 2025
Line 1:		Line 1:
	{{Mathematical interest~~}}{{Infobox interval\|ratio=e^{2\pi}\|cents=10877.6643\|Ratio=e^{\tau}\|Cents=10877.6643\|Name=zetave~~}}		{{Mathematical interest}}

	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 <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).

	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%.		[[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.

			It is extremely well-approximated by [[31edo]]: 281 steps of 31edo is 10877.419{{c}}, which is flat of <math>e^{2\pi}</math> 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]]

ArrowHead294 at 02:22, 27 March 2025

2025-03-27T02:22:00Z

← Older revision		Revision as of 02:22, 27 March 2025
Line 3:		Line 3:
	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}} (1 in 44,400 or 0.00225%).		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]]

ArrowHead294 at 18:34, 26 March 2025

2025-03-26T18:34:22Z

← Older revision		Revision as of 18:34, 26 March 2025
Line 3:		Line 3:
	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}} (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]]

ArrowHead294 at 18:32, 26 March 2025

2025-03-26T18:32:38Z

← Older revision		Revision as of 18:32, 26 March 2025
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	{{Mathematical interest}}{{Infobox interval\|ratio=e^{2\pi}\|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>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~~]].		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}}.

	== 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>i𝜏</sup> = 1.
	[[Category:Zeta]]		[[Category:Zeta]]