User:CompactStar/Super-pitch: Difference between revisions
m Fredg999 moved page Super-pitch to User:CompactStar/Super-pitch over redirect: OR-3 (idiosyncratic framework and terminology) |
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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. | {{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. | |||
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. | 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. | ||
There is, notably, one extension for complex numbers developed by Kneser, which so far seems the best when implemented to the reals. | 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 [[User:CompactStar|CompactStar]]. | |||
== "Super-pitch equivalents" of different concepts == | == "Super-pitch equivalents" of different concepts == | ||
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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). | 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). | ||
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. | 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 == | == Super-pitch division == | ||
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=== Individual pages for EDSO === | === Individual pages for EDSO === | ||
{{Main|EDSO}} | |||
== References == | == References == | ||