Height: Difference between revisions

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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>~~

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

~~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~~.

== Height versus norm ==

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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 v''p''(''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 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>[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 L1 norm on monzos, but where each prime is weighted by ''p'' rather than log (''p'').

Some useful identities:

@@ Line 18: / Line 18: @@
 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>
-where v<sub>''p''</sub>(''q'') is the [[Wikipedia: P-adic valuation|''p''-adic valuation]] of ''q''.
-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.
 == Height versus norm ==
@@ Line 86: / Line 73: @@
 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: