Wilson norm: Difference between revisions

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The '''Wilson norm''', otherwise known as '''Wilson height''', is a measure of [[complexity]] for [[just interval]]s, similar to the [[Tenney norm]], but has some very beneficial properties that make it an excellent metric to look at.

If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq'').

If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq''). This is called ''Wilson's complexity'' in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>

Note that we have {{nowrap| sopfr(''pq'') {{=}} sopfr(''p'') + sopfr(''q'') }}, similar to {{w|logarithm}} – as a result, this function is sometimes even referred to as the "integer logarithm". So, equivalently, we can define the Wilson norm of a rational number ''p''/''q'' as the Wilson norm of ''p'', plus the Wilson norm of ''q''.

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* 98/97: 113 (97-limit)

* 83/82: 126 (83-limit)

== External links ==

* [https://github.com/FloraCanou/temperament_evaluator/wiki/Parametric-Tenney%E2%80%93Wilson-norm Github | ''Parametric Tenney–Wilson norm'' · FloraCanou/temperament_evaluator Wiki] – a unified, parametrized Tenney–Wilson norm used by [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator].

== References ==

[[Category:Interval complexity measures]]

[[Category:Consonance and dissonance]]

@@ Line 2: / Line 2: @@
 The '''Wilson norm''', otherwise known as '''Wilson height''', is a measure of [[complexity]] for [[just interval]]s, similar to the [[Tenney norm]], but has some very beneficial properties that make it an excellent metric to look at.
-If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq'').
+If ''p''/''q'' is a positive rational number reduced to its lowest terms, then the Wilson norm is the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the number ''pq'', counting multiplicity. This function is often written sopfr(''pq''). This is called ''Wilson's complexity'' in [[John Chalmers]]'s ''Divisions of the Tetrachord''<ref>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>
 Note that we have {{nowrap| sopfr(''pq'') {{=}} sopfr(''p'') + sopfr(''q'') }}, similar to {{w|logarithm}} – as a result, this function is sometimes even referred to as the "integer logarithm". So, equivalently, we can define the Wilson norm of a rational number ''p''/''q'' as the Wilson norm of ''p'', plus the Wilson norm of ''q''.
@@ Line 174: / Line 174: @@
 * 98/97: 113 (97-limit)
 * 83/82: 126 (83-limit)
+== External links ==
+* [https://github.com/FloraCanou/temperament_evaluator/wiki/Parametric-Tenney%E2%80%93Wilson-norm Github | ''Parametric Tenney–Wilson norm'' · FloraCanou/temperament_evaluator Wiki] – a unified, parametrized Tenney–Wilson norm used by [[Flora Canou]]'s [https://github.com/FloraCanou/temperament_evaluator Temperament Evaluator].
+== References ==
 [[Category:Interval complexity measures]]
 [[Category:Consonance and dissonance]]