Zetave: Difference between revisions

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.

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 ==

* 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]]

@@ 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.
-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 ==
 * 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]]