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. | ||
The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slog<sub>b</sub>(1) = 0, 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></sup>) = 3. This definition only allows for inputs of the form 1, b, b<sup>b</sup>, b<sup>b<sup>b</sup></sup>, etc., although there are various contionus extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions. | |||
== Super-pitch equivalents of different concepts == | == Super-pitch equivalents of different concepts == | ||
If super-pitch is used instead of pitch, equivalence works differently. For example, in a pitch-based system, the frequency x would be octave-equivalent to 2*x, 2*2*x, etc. and x/2, x/2/2, etc. But in a super-pitch based system, it would be octave-equivalent to 2<sup>2<sup>x</sup></sup>, etc. and log<sub>2</sub>(x), log<sub>2</sub>(log<sub>2</sub>(x)), etc. An "equal super-pitch divisions of the octave" is identical to an [[EDO]] within the range [[1/1]]-[[2/1]] if using the linear approximation of super-logarithm, but it is distinct if using the quadratic approximation of super-logarithm. | If super-pitch is used instead of pitch, equivalence works differently. For example, in a pitch-based system, the frequency x would be octave-equivalent to 2*x, 2*2*x, etc. and x/2, x/2/2, etc. But in a super-pitch based system, it would be octave-equivalent to 2<sup>2<sup>x</sup></sup>, etc. and log<sub>2</sub>(x), log<sub>2</sub>(log<sub>2</sub>(x)), etc. An "equal super-pitch divisions of the octave" is identical to an [[EDO]] within the range [[1/1]]-[[2/1]] if using the linear approximation of super-logarithm, but it is distinct if using the quadratic approximation of super-logarithm. | ||
Revision as of 02:59, 20 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.
The super-logarithm is traditionally defined the number of times a logarithm must be iterated to get to 1. For example, slogb(1) = 0, slogb(b) = 1, slogb(bb) = 2, and slogb(bbb) = 3. This definition only allows for inputs of the form 1, b, bb, bbb, etc., although there are various contionus extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions.
Super-pitch equivalents of different concepts
If super-pitch is used instead of pitch, equivalence works differently. For example, in a pitch-based system, the frequency x would be octave-equivalent to 2*x, 2*2*x, etc. and x/2, x/2/2, etc. But in a super-pitch based system, it would be octave-equivalent to 22x, etc. and log2(x), log2(log2(x)), etc. An "equal super-pitch divisions of the octave" is identical to an EDO within the range 1/1-2/1 if using the linear approximation of super-logarithm, but it is distinct if using the quadratic approximation of super-logarithm.