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m Fredg999 moved page Super-pitch to User:CompactStar/Super-pitch over redirect: OR-3 (idiosyncratic framework and terminology)
 
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(Would have created in mainspace, but this is WIP and the explanation is currently somewhat vague)
{{Mathematical interest}}
'''Super-pitch'''{{idiosyncratic}} 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] of frequency, just as pitch is the logarithm of frequency. Super-logarithms are an inverse function of [https://en.wikipedia.org/wiki/Tetration tetration] (iterated exponentiation), defined recursively as:
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, slog<sub>b</sub>(b<sup>b<sup>b</sup></sup>) = 3, and so on. 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 continuous extensions of it for other outputs (most commonly the linear and quadratic approximations, as mentioned on the Wikipedia article) which have differing definitions. 


slog<sub>b</sub>(1) = 0
There is, notably, one extension for complex numbers developed by Kneser, which so far seems the best when implemented to the reals.


slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1
The term "super-pitch" was proposed by [[User:CompactStar|CompactStar]].


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:
== "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 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 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.


slog<sub>b</sub>(x) = x - 1 if 0 &le; x &le; l
The super-pitch equivalent of [[just intonation]] is intervals of the form log<sub>b</sub>(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]] = log<sub>4</sub>(8)), in addition to some irrational numbers such as log<sub>2</sub>(3).


slog<sub>b</sub>(b<sup>x</sup>) = slog<sub>b</sub>(x) + 1
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 [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 2<sup>3</sup>, 16 = 2<sup>2<sup>2</sup></sup>, 25 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 36 = 6<sup>2</sup>, and 81 = 3<sup>2<sup>2</sup></sup>. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s 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"<ref name=":0">http://myweb.astate.edu/wpaulsen/tetration2.pdf</ref>
 
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 <ref name=":0" />:
{| class="wikitable"
! colspan="2" |10 equal divisions of the superoctave
<small>per Kneser's solution</small>
|-
!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|EDSO}}
 
== References ==
<references />
== External links ==
[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]
[[Category:Theory]]
[[Category:Transcendental]]
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