User:CompactStar/Super-pitch: Difference between revisions
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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 ≥ 1 | ||
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>(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 | ||
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, and so on. Tetration | 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<sup>x</sup> if 0 < x < 1 | |||
a↑↑x = a<sup>a↑↑(x-1)</sup> if x ≥ 1 | |||
Revision as of 03:05, 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 (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↑↑x = aa↑↑(x-1) if x ≥ 1
For example, a↑↑1 = a, a↑↑2 = aa, a↑↑3 = aaa, a↑↑4 = aaaa, and son.
The super-logarithm is an inverse function of tetrartion, defined as:
slogb(1) = 0
slogb(x) = slogb(logb(x)) + 1
So slogb(b) = 1, slogb(bb) = 2, slogb(bbb) = 3, slogb(bbbb) = 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 = ax if 0 < x < 1
a↑↑x = aa↑↑(x-1) if x ≥ 1