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equivalence section: no need to talk about p-adic valuations here
 
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=Definition:=
The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.
A '''height''' is a function on members of an algebraically defined object which maps elements to real numbers, yielding a type of complexity measurement. 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:
== Definition ==
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.


<ol><li>Given any constant C, there are finitely many elements q such that H(q) ≤ C.</li><li>H(q) is bounded below by H(1), so that H(q) ≥ H(1) for all q.</li><li>H(q) = H(1) iff q = 1.</li><li>H(q) = H(1/q)</li><li>H(q^n) ≥ H(q) for any non-negative integer n</li></ol>
A height function H(''q'') on the positive rationals ''q'' should fulfill the following criteria:


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:
# 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''<sup>''n''</sup>) ≥ H(''q'') for any non-negative integer ''n''.  


<math>H \left( {q} \right) \equiv F \left( {H} \left( {q} \right) \right)</math>
If we have a function F 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:


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


<math>2^{-v_2 \left( {p} \right)} p = 2^{-v_2 \left( {q} \right)} q</math>
Exponentiation and logarithm are such functions commonly used for converting a height between arithmetic and logarithmic scales.


Or equivalently, if n has any integer solutions:
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>p = 2^n q</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.


If the above condition is met, we may then establish the following equivalence relation:
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.  
 
<math>p \equiv q</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 ==
{| class="wikitable"
{| class="wikitable"
! Name
! Type
! H(''n''/''d'')
! H(''q'')
! H(''q'') simplified by equivalence relation
|-
|-
| | <u>Name:</u>
| [[Benedetti height]] <br> (or [[Tenney height]])
| | <u>Type:</u>
| Height
| | <u>H(n/d):</u>
| <math>nd</math>
| | <u>H(q):</u>
| <math>2^{\large{\|q\|_{T1}}}</math>
| | <u>H(q) simplified by equivalence relation:</u>
| <math>\|q\|_{T1}</math>
|-
| | [[Benedetti_height|Benedetti height]]
 
(or [[Tenney_Height|Tenney Height]])
| | Height
| | <math>n d</math>
| | <math>2^{\large{\|q\|_{T1}}}</math>
| | <math>\|q\|_{T1}</math>
|-
|-
| | Wilson Height
| [[Wilson height]]
| | Height
| Height
| | <math>\text{sopf}(n d)</math>
| <math>\text{sopfr}(n d)</math>
| | <math>2^{\large{\text{sopf}(n d)}}</math>
| <math>2^{\large{\text{sopfr}(q)}}</math>
| | <math>\text{sopf}(q)</math>
| <math>\text{sopfr}(q)</math>
|-
|-
| | Weil Height
| [[Weil height]]
| | Height
| Height
| | <math>\max \left( {n , d} \right)</math>
| <math>\max \left( {n , d} \right)</math>
| | <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math>
| <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math>
| | <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math>
| <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math>
|-
|-
| | Arithmetic Height
| Arithmetic height
| | Height
| Height
| | <math>n + d</math>
| <math>n + d</math>
| | <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>
| <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>
| | <math>\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</math>
| <math>\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</math>
|-
|-
| | Harmonic Height
| Harmonic semi-height
| | Semi-Height
| Semi-Height
| | <math>\dfrac {n d} {n + d}</math>
| <math>\dfrac {n d} {n + d}</math>
| | <math>\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>
| <math>\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>
| | <math>\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)</math>
| <math>\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_2 \left( {q} \right)</math>
|-
|-
| | [[Kees_Height|Kees Height]]
| [[Kees semi-height]]
| | Semi-Height
| Semi-Height
| | <math>\max \left( {2^{-v_2 \left( {n} \right)} n ,
| <math>\max \left( {2^{-v_2 \left( {n} \right)} n, 2^{-v_2 \left( {d} \right)} d} \right)</math>
2^{-v_2 \left( {d} \right)} d} \right)</math>
| <math>2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(q) - v_2(q) \mid \right)\right)}}</math>
| | <math>2^{\large{\left(\frac{1}{2}\left(\|2^{-v_2 \left( {q} \right)} q\|_{T1} + \mid \log_2(q) - v_2(q) \mid \right)\right)}}</math>
| <math>\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math>
| | <math>\|{2^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math>
|}
|}


Where ||q||<span style="font-size: 80%; vertical-align: sub;">T1</span> is the [[Generalized_Tenney_Norms_and_Tp_Interval_Space#The Tenney Norm (T1 norm)|tenney norm]] of q in monzo form, and v<span style="vertical-align: sub;">p</span>(x) is the [http://en.wikipedia.org/wiki/P-adic_order p-adic valuation] of x.
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 <math>\text{sopf}(nd)</math> is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] 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 Complexity" in John Chalmers "Division of the Tetrachord."
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>.


Some useful identities:
* <math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</math>
* <math>d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</math>
* <math>n d = 2^{\|q\|_{T1}}</math>


Some useful identities:
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 ==
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>.


<math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</math>
== See also ==
* [[Commas by taxicab distance]]
* [[Harmonic entropy]]


<math>d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</math>
== References ==


<math>n d = 2^{\|q\|_{T1}}</math>
<references/>


Height functions can also be put on the points of [http://planetmath.org/encyclopedia/QuasiProjectiveVariety.html projective varieties]. Since [[Abstract_regular_temperament|abstract regular temperaments]] can be identified with rational points on [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.
[[Category:Math]]
[[Category:height]]
[[Category:Interval complexity measures]]
[[Category:math]]
[[Category:measure]]
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