Height: Difference between revisions

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== Definition ==

A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.

A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. (See [[Wikipedia:Height function]].) For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.

A height function H(q) on the positive rationals q should fulfill the following criteria:

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|}

Where ||q||T1 is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and vp(x) is the [~~http~~://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x.

Where ||q||T1 is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and vp(x) is the [https://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x.

The function <math>\text{sopfr}(nd)</math> is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repeition"] of n*d. Equivalently, this is the L1 norm on monzos, but where each prime is weighted by "p" rather than "log(p)". This is called "Wilson's Complexity" in [[John Chalmers]]'s "Divisions of the Tetrachord."<ref>See http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf, page 55</ref>

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* <math>n d = 2^{\|q\|_{T1}}</math>

Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [~~http~~://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.

Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [https://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.

== History ==

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 == Definition ==
-A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.
+A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. (See [[Wikipedia:Height function]].)  For example we can assign each element of the positive rational numbers a height, and hence a complexity. While there is no consensus on the restrictions of a height, we will attempt to create a definition for positive rational numbers which is practical for musical purposes.
 A height function H(q) on the positive rationals q should fulfill the following criteria:
@@ Line 75: / Line 75: @@
 |}
-Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x.
+Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [https://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x.
 The function <math>\text{sopfr}(nd)</math> is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repeition"] of n*d. Equivalently, this is the L1 norm on monzos, but where each prime is weighted by "p" rather than "log(p)". This is called "Wilson's Complexity" in [[John Chalmers]]'s "Divisions of the Tetrachord."<ref>See http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf, page 55</ref>
@@ Line 84: / Line 84: @@
 * <math>n d = 2^{\|q\|_{T1}}</math>
-Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.
+Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[abstract regular temperament]]s can be identified with rational points on [https://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.
 == History ==