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<math>\displaystyle H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>
<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 ==
== Examples of height functions ==
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Where ||''q''||<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''.
Where ||''q''||<sub>T1</sub> is the [[Generalized Tenney norms and Tp interval space #The Tenney Norm (T1 norm)|tenney norm]] of ''q'' in monzo form, and v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of ''q''.


The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> 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>[http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf ''Division of the Tetrachord''], page 55. John Chalmers. </ref>.
The function sopfr (''nd'') is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repetition"] of ''n''·''d''. Equivalently, this is the L<sub>1</sub> norm on monzos, but where each prime is weighted by ''p'' rather than log (''p'').  


Some useful identities:
Some useful identities:
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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.
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 ==
== History ==
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<references/>
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[[Category:Height| ]] <!-- main article -->
[[Category:Math]]
[[Category:Math]]
[[Category:Interval complexity measures]]
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