Height: Difference between revisions

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<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>

Exponentiation and logarithm are such functions commonly used for converting a height between ~~the~~ arithmetic and logarithmic ~~scale~~.

Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.

A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:

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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.

== Height versus norm ==

Height functions are applied to ratios, whereas norms are measurements on interval lattices [[wikipedia: embedding|embedded]] in [[wikipedia: Normed vector space|normed vector spaces]]. Some height functions are essentially norms, and they are numerically equal. For example, the [[Tenney height]] is also the Tenney norm.

However, not all height functions are norms, and not all norms are height functions. The [[Benedetti height]] is not a norm, since it does not satisfy the condition of absolute homogeneity. The [[taxicab distance]] is not a height, since there can be infinitely many intervals below a given bound.

== Examples of height functions ==

@@ Line 16: / Line 16: @@
 <math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>
-Exponentiation and logarithm are such functions commonly used for converting a height between the arithmetic and logarithmic scale.
+Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.
 A '''semi-height''' is a function which does not obey criterion #3 above, so that there is a rational number ''q'' ≠ 1 such that H(''q'') = H(1), resulting in an equivalence relation on its elements, under which #1 is modified to a finite number of equivalence classes. An example would be [[octave equivalence]], where two ratios ''q''<sub>1</sub> and ''q''<sub>2</sub> are considered equivalent if the following is true:
@@ Line 33: / Line 33: @@
 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.
+== Height versus norm ==
+Height functions are applied to ratios, whereas norms are measurements on interval lattices [[wikipedia: embedding|embedded]] in [[wikipedia: Normed vector space|normed vector spaces]]. Some height functions are essentially norms, and they are numerically equal. For example, the [[Tenney height]] is also the Tenney norm.
+However, not all height functions are norms, and not all norms are height functions. The [[Benedetti height]] is not a norm, since it does not satisfy the condition of absolute homogeneity. The [[taxicab distance]] is not a height, since there can be infinitely many intervals below a given bound.
 == Examples of height functions ==