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 and super-logarithm == | ||
Tetration (most commonly represented with ↑↑) 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↑↑ | a↑↑0 = 1 | ||
a↑↑ | a↑↑x = a<sup>a↑↑(x-1)</sup> if x > 0 | ||
For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, and a↑↑ | For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, a↑↑3 = a<sup>a<sup>a</sup></sup>, and a↑↑4 = a<sup>a<sup>a<sup>a</sup></sup></sup>. | ||
The super-logarithm is an inverse function of tetrartion, defined as: | |||
slog<sub>b</sub>(1) = 0 | |||
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, and slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3. | |||
Revision as of 02:44, 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 (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 > 0
For example, a↑↑1 = a, a↑↑2 = aa, a↑↑3 = aaa, and a↑↑4 = aaaa.
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, and slogb(bbb) = 3.