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''1 and ''q''2 are considered equivalent if ~~the following is true:~~

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''1 and ''q''2 are considered equivalent if they differ only by factors of 2.

We can also consider other equivalences. For example, we can assume tritave equivalence by ignoring factors of 3.

~~<math>\displaystyle 2^{-v_2 \left( {q_1} \right)} q_1~~ = ~~2^{-v_2 \left( {q_2} \right)} q_2</math>~~

== 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.

~~where v''p''(''q'') is the~~ [[~~Wikipedia: P-adic valuation|''p''-adic valuation~~]] ~~of ''q''.~~

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.

~~Or equivalently~~, ~~if ''n'' has any integer solutions:~~

~~<math>\displaystyle q_1 = 2^n q_2</math>~~

If the ~~above~~ condition is ~~met, we may then establish the following equivalence relation:~~

~~<math>\displaystyle q_1 \equiv q_2</math>~~

~~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 ==

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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 [[Wikipedia: Grassmannian|Grassmann varieties]], complexity measures of regular temperaments are also height functions.

See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities]] for an extensive discussion of heights and semi-heights used in regular temperament theory.

== History ==

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~~[[Category:Height| ]] ~~

[[Category:Math]]

[[Category:Interval complexity measures]]

@@ 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:
+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 they differ only by factors of 2.
+We can also consider other equivalences. For example, we can assume tritave equivalence by ignoring factors of 3.
-<math>\displaystyle 2^{-v_2 \left( {q_1} \right)} q_1 = 2^{-v_2 \left( {q_2} \right)} q_2</math>
+== 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.
-where v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic valuation|''p''-adic valuation]] of ''q''.
+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.
-Or equivalently, if ''n'' has any integer solutions:
-<math>\displaystyle q_1 = 2^n q_2</math>
-If the above condition is met, we may then establish the following equivalence relation:
-<math>\displaystyle q_1 \equiv q_2</math>
-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 ==
@@ Line 89: / Line 81: @@
 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 [[Wikipedia: Grassmannian|Grassmann varieties]], complexity measures of regular temperaments are also height functions.
+See [[Dave Keenan & Douglas Blumeyer's guide to RTT/Alternative complexities]] for an extensive discussion of heights and semi-heights used in regular temperament theory.
 == History ==
@@ Line 101: / Line 95: @@
 <references/>
-[[Category:Height| ]] <!-- main article -->
 [[Category:Math]]
+[[Category:Interval complexity measures]]