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 is iterated exponentiation, like how exponentiation is iterated multiplication. Tetration can be defined recursively as: | Tetration (most commonly represented with ↑↑) is iterated exponentiation, like how exponentiation is iterated multiplication. Tetration can be defined recursively as: | ||
a↑↑b = 1 if b = 0 | a↑↑b = 1 if b = 0 | ||
a↑↑b = a<sup>a↑(b-1)</sup> if b > 0 | a↑↑b = a<sup>a↑(b-1)</sup> if b > 0 | ||
For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, and a↑↑3 = a<sup>a<sup>a</sup></sup>. | For example, a↑↑1 = a, a↑↑2 = a<sup>a</sup>, and a↑↑3 = a<sup>a<sup>a</sup></sup>. | ||
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 (most commonly represented with ↑↑) 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.