Super-pitch
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Super-pitch[idiosyncratic term] 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.
There is, notably, one extension for complex numbers developed by Kneser, which so far seems the best when implemented to the reals.
The term "super-pitch" was proposed by CompactStar.
"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, the term would be superoctave and the equivalence would be enumerated as to 22x, etc. and log2(x), log2(log2(x)), etc. An equal divison of the superoctave (EDSO) 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 or Kneser's extension of super-logarithm.
The super-pitch equivalent of just intonation is intervals of the form logb(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). For example, 8 = 23, 16 = 222, 25 = 52, 27 = 33, 36 = 62, and 81 = 322. 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). Super-vals, super-mappings, and even super-temperaments can be derived by using super-monzos instead of regular monzos. This means that subgroups in super-pitch theory are made of non-perfect powers, like 2.3.5.6.7.10 for example.
Super-pitch division
While there is more than one way to interpolate tetrative numbers, there is one unique function in the complex plane which continuously and differentiably satisfies the recurrence relation f(x+1) = x^f(x), which is arguably also, as a subset, the best way to extend it to the reals. The paper says that "the comparison of other solutions to Kneser's solution force it to be the unique solution"[1]
This is arguably the way to make super-pitch divisions of a given interval. For example, the following table shows 10 super-pitch divisions of the octave [1]:
| 10 equal divisions of the superoctave
per Kneser's solution | |
|---|---|
| Interval | Cents |
| 1.089118 | 147.7924 |
| 1.178977 | 285.0424 |
| 1.270146 | 413.9926 |
| 1.363209 | 536.4081 |
| 1.458782 | 653.7169 |
| 1.557524 | 767.105 |
| 1.660157 | 877.5837 |
| 1.767486 | 986.0384 |
| 1.880419 | 1093.265 |
| 2.00 | 1200 |
Since exponentiation is not commutative and has two inverses - root and logarithm which breed two distinct numbers, likewise tetration has similar inverses which are too sets of disjoint numbers - solution to x^x = 2 is not the same number as solution to slog2(x) = 0.5. A pure interpolative function does not cancel out, for example the interval step 5, 1.458782..., when raised to the power of itself does not yield 2. Likewise, the first step, 1.089118, power-tower-ated 10 times does not yield 2 either.
Individual pages for EDSO
- Main article: EDSO