User:CompactStar/Super-pitch: Difference between revisions

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(Would have created in mainspace, but this is WIP and the explanation is currently somewhat vague)
'''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 ==
'''Super-pitch''' is a quantity that is equal to the [https://en.wikipedia.org/wiki/Super-logarithm super-logarithm] of frequency, just as pitch is the logarithm of frequency. Super-logarithms are an inverse function of [https://en.wikipedia.org/wiki/Tetration tetration] (iterated exponentiation), defined recursively as:
Tetration is iterated exponentiation, like how exponentiation is iterated multiplication. Tetration can be defined recursively as:
 
a↑↑b = 1 if b = 0
slog<sub>b</sub>(1) = 0
a&uarr;&uarr;b = a<sup>a&uarr;(b-1)</sup> if b > 0
 
For example, a&uarr;&uarr;1 = a, a&uarr;&uarr;2 = a<sup>a</sup>, and a&uarr;&uarr;3 = a<sup>a<sup>a</sup></sup>.
slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1
 
For example, slog<sub>b</sub>(b) = 1, slog<sub>b</sub>(b<sup>b</sup>) = 2 and slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3. Super-logarithms are only defined for inputs that are a power tower of the base (1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc.). However, there are various generalizations of the super-logarithm to other inputs, the most common of which is the linear approximation, defined as:
 
slog<sub>b</sub>(x) = x - 1 if 0 &le; x &le; 1
 
slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1

Revision as of 02:40, 15 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.

Tetration and super-logarithm

Tetration is iterated exponentiation, like how exponentiation is iterated multiplication. Tetration can be defined recursively as: a↑↑b = 1 if b = 0 a↑↑b = aa↑(b-1) if b > 0 For example, a↑↑1 = a, a↑↑2 = aa, and a↑↑3 = aaa.

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