User:CompactStar/Super-pitch: Difference between revisions

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== "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 or Kneser's extension of super-logarithm.

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).

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 == "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<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 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 or Kneser's extension of super-logarithm.
+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.
 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).