Height: Difference between revisions

A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement (see [[Wikipedia: Height function]]). 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 function H(''q'') on the positive rationals ''q'' should fulfill the following criteria:

# Given any constant ''C'', there are finitely many elements ''q'' such that H(''q'') ≤ ''C''.

# H(''q'') is bounded below by H(1), so that H(''q'') ≥ H(1) for all q.

# H(''q'') = H(1) iff ''q'' = 1.

# H(''q'') = H(1/''q'')

# H(''q''^''n'') ≥ H(''q'') for any non-negative integer ''n''.

If we have a function F(''x'') which is strictly increasing on the positive reals, then F(H(''q'')) will rank elements in the same order as H(''q''). We can therefore establish the following equivalence relation:

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

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 ''p'' and ''q'' are considered equivalent if the following is true:

<math>\displaystyle 2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q</math>

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

<math>\displaystyle p = 2^n q</math>

<math>\displaystyle p \equiv q</math>

! H(''n''/''d'')

! H(''q'')

! H(''q'') simplified by equivalence relation

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''(''x'') is the [[Wikipedia: P-adic order|''p''-adic valuation]] of x.

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>.

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.

The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488 Yahoo! Tuning Group | ''Super Particular Stepsize'']</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia: Height function]]). It is not to be confused with the musical notion of [[Wikipedia: Pitch (music) #Theories of pitch perception|''pitch height'' (as opposed to ''pitch chroma'')]]<ref>Though it has also been used to refer to the size of an interval in cents. On page 23 of [https://www.plainsound.org/pdfs/JC&ToH.pdf ''John Cage and the Theor of Harmony''], Tenney writes: "The one-dimensional continuum of pitch-height (i.e. 'pitch' as ordinarily defined)", and graphs it ''as opposed to'' his concept of "harmonic distance", which was ironically the first measurement named by Gene Ward Smith as a "height": "Tenney height".</ref>.