Height: Difference between revisions
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=Definition | The '''height''' is a tool to measure the dissonance of [[JI]] [[interval]]s. | ||
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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. 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. | ||
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By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example. | By changing the base of the exponent to a value other than 2, you can set up completely different equivalence relations. Replacing the 2 with a 3 yields tritave-equivalence, for example. | ||
= | == Examples of Height Functions == | ||
=Examples of Height Functions | |||
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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 [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions. | ||
== See also == | |||
* [[Commas by taxicab distance]] | |||
[[Category:Theory]] | [[Category:Theory]] | ||