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

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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== References ==

== External links ==

[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]

[[Category:Theory]]

[[Category:Transcendental]]

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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><ref>[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]</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>
 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" />:
@@ Line 57: / Line 57: @@
 == References ==
+<references />
+== External links ==
+[http://myweb.astate.edu/wpaulsen/tetcalc/tetcalc.html Tetration calculator]
 [[Category:Theory]]
 [[Category:Transcendental]]