User:CompactStar/Super-pitch

(Would have created in mainspace, but this is WIP and the explanation is currently somewhat vague)

Super-pitch is a quantity that is equal to the super-logarithm of frequency, just as pitch is the logarithm of frequency. Super-logarithms are an inverse function of tetration (iterated exponentiation), defined recursively as:

slog_b(1) = 0

slog_b(b^x) = slog_b(x) + 1

For example, slog_b(b) = 1, slog_b(b^b) = 2 and slog_b(b^bb) = 3. Super-logarithms are only defined for inputs that are a power tower of the base (1, b, b^b, b^bb, etc.). However, there are various generalizations of the super-logarithm to other inputs, the most common of which is the linear approximation, defined as:

slog_b(x) = x - 1 if 0 ≤ x ≤ l

slog_b(b^x) = slog_b(x) + 1