User:CompactStar/Super-pitch: Difference between revisions

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'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.
'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] (inverse [https://en.wikipedia.org/wiki/Tetration tetration]) of frequency, just as pitch is the logarithm of frequency.
== Tetration and super-logarithm ==
Tetration (most commonly represented with ↑↑) is an operator that is iterated exponentiation, like how exponentiation is iterated multiplication. Tetration can be defined recursively as:
a↑↑0 = 1
a&uarr;&uarr;x = a<sup>a&uarr;&uarr;(x-1)</sup> if x &gt; 0
For example, a&uarr;&uarr;1 = a, a&uarr;&uarr;2 = a<sup>a</sup>, a&uarr;&uarr;3 = a<sup>a<sup>a</sup></sup>, a&uarr;&uarr;4 = a<sup>a<sup>a<sup>a</sup></sup></sup>, and son.
The super-logarithm is an inverse function of tetration, defined as:
slog<sub>b</sub>(1) = 0
slog<sub>b</sub>(x) = slog<sub>b</sub>(log<sub>b</sub>(x)) + 1  if x &gt; 1
So slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, slog<sub>b</sub>(b<sup>b<sup>b<sup>b</sup></sup></sup>) = 4, and so on. Tetration is only defined for integer inputs, while super-logarithm is only defined for integer outputs. However, there are various extensions of tetration, the most common of which is the linear approximation:
a&uarr;&uarr;0 = x + 1 if -1 &le; x &le; 0
a&uarr;&uarr;x = a<sup>a&uarr;&uarr;(x-1)</sup> if x &gt; 0
For example, a&uarr;&uarr;0.5 = √a, a&uarr;&uarr;1.5 = a<sup>√a</sup>, and a&uarr;&uarr;2.5 = a<sup>a<sup>√a</sup></sup>. The corresponding extension of the super-logarithm is:
slog<sub>b</sub>(x) = x - 1 if 0 &le; x &le; 1
slog<sub>b</sub>(x) = slog<sub>b</sub>(log<sub>b</sub>(x)) + 1  if 0 &le; x &le; 1

Revision as of 02:45, 20 June 2023

Super-pitch is a quantity that is equal to the super-logarithm (inverse tetration) of frequency, just as pitch is the logarithm of frequency.

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