User:CompactStar/Super-pitch: Difference between revisions

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== 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>.

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><ref>[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]</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" />:

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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.

== References ==

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 == 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>.
+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><ref>[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]</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" />:
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 |}
 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.
 == References ==