User:CompactStar/Super-pitch: Difference between revisions
Jump to navigation
Jump to search
CompactStar (talk | contribs) No edit summary |
CompactStar (talk | contribs) No edit summary |
||
| Line 9: | Line 9: | ||
For example, 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. Super-logarithms are only defined for inputs that are a power tower of the base (1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc.). However, there are various generalizations of the super-logarithm to other inputs, the most common of which is the linear approximation, defined as: | For example, 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. Super-logarithms are only defined for inputs that are a power tower of the base (1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, 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<sub>b</sub>(x) = x - 1 if 0 ≤ x ≤ | slog<sub>b</sub>(x) = x - 1 if 0 ≤ x ≤ 1 | ||
slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1 | slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1 | ||
Revision as of 04:12, 14 June 2023
(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:
slogb(1) = 0
slogb(bx) = slogb(x) + 1
For example, slogb(b) = 1, slogb(bb) = 2 and slogb(bbb) = 3. Super-logarithms are only defined for inputs that are a power tower of the base (1, b, bb, bbb, etc.). However, there are various generalizations of the super-logarithm to other inputs, the most common of which is the linear approximation, defined as:
slogb(x) = x - 1 if 0 ≤ x ≤ 1
slogb(bx) = slogb(x) + 1