User:CompactStar/Super-pitch

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 = a^a↑↑(x-1) if x > 0

For example, a↑↑1 = a, a↑↑2 = a^a, a↑↑3 = a^aa, a↑↑4 = a^aaa, and son.

The super-logarithm is an inverse function of tetration, defined as:

slog_b(1) = 0

slog_b(x) = slog_b(log_b(x)) + 1 if x > 1

So slog_b(b) = 1, slog_b(b^b) = 2, slog_b(b^bb) = 3, slog_b(b^bbb) = 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 = x + 1 if -1 ≤ x ≤ 0

a↑↑x = a^a↑↑(x-1) if x > 0

For example, a↑↑0.5 = √a, a↑↑1.5 = a^√a, and a↑↑2.5 = a^a√a. The corresponding extension of the super-logarithm is:

slog_b(x) = x - 1 if 0 ≤ x ≤ 1

slog_b(x) = slog_b(log_b(x)) + 1 if 0 ≤ x ≤ 1