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: | 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↑↑0 = 1 | ||
a↑↑x = a<sup>a↑↑(x-1)</sup> if x & | a↑↑x = a<sup>a↑↑(x-1)</sup> if x > 0 | ||
For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, a↑↑3 = a<sup>a<sup>a</sup></sup>, a↑↑4 = a<sup>a<sup>a<sup>a</sup></sup></sup>, and son. | For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, a↑↑3 = a<sup>a<sup>a</sup></sup>, a↑↑4 = a<sup>a<sup>a<sup>a</sup></sup></sup>, and son. | ||
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slog<sub>b</sub>(1) = 0 | slog<sub>b</sub>(1) = 0 | ||
slog<sub>b</sub>(x) = slog<sub>b</sub>(log<sub>b</sub>(x)) + 1 | slog<sub>b</sub>(x) = slog<sub>b</sub>(log<sub>b</sub>(x)) + 1 if x > 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: | 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↑↑0 = | a↑↑0 = x + 1 if -1 ≤ x ≤ 0 | ||
a↑↑x = a<sup>a↑↑(x-1)</sup> if x & | a↑↑x = a<sup>a↑↑(x-1)</sup> if x > 0 | ||
For example, a↑↑0.5 = √a, a↑↑1.5 = a<sup>√a</sup>, and a↑↑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 ≤ x ≤ 1 | |||
slog<sub>b</sub>(x) = xslog<sub>b</sub>(log<sub>b</sub>(x)) + 1 if 0 ≤ x ≤ 1 | |||