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.
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, slogb(bbb) = 3, and so on. This definition only allows for inputs of the form 1, b, bb, bbb, etc., although there are various continuous 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.
The super-pitch equivalent of just intonation is frequencies of the form logb(x) for positive integers b and x. This includes all of just intonation, since all just ratios can be described as logarithms (e.g. 3/2 = log4(8)), in addition to some irrational numbers such as log2(3).
It is possible to construct super-pitch equivalents of most concepts in regular temperament theory.