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, slog_b(1) = 0, slog_b(b) = 1, slog_b(b^b) = 2, slog_b(b^{b^b}) = 3, and so on. This definition only allows for inputs of the form 1, b, b^b, b^{b^b}, 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 2^{2^x}, 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 intervals of the form log_b(x) for positive integers b and x. This includes all of just intonation, since all just intervals 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. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS A007916 (non-perfect powers, or the equivalent of primes). For example, 8 = 2³, 16 = 2^2², 25 = 5², 27 = 3³, 36 = 6², and 81 = 3^2². From this, it is straightforward to define the super-pitch equivalent of monzos, or "super-monzos" (just substitute prime factorization for this power tower representation).