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The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.
The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.
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== Definition ==
== 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.


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:


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


<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>
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:


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>


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


Or equivalently, if n has any integer solutions:
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 = 2^n q</math>
 
If the above condition is met, we may then establish the following equivalence relation:
 
<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
! Name
! Type
! Type
! H(n/d)
! H(''n''/''d'')
! H(q)
! H(''q'')
! H(q) simplified by equivalence relation
! H(''q'') simplified by equivalence relation
|-
|-
| [[Benedetti height]] <br> (or [[Tenney height]])
| [[Benedetti height]] <br> (or [[Tenney height]])
| Height
| Height
| <math>n d</math>
| <math>nd</math>
| <math>2^{\large{\|q\|_{T1}}}</math>
| <math>2^{\large{\|q\|_{T1}}}</math>
| <math>\|q\|_{T1}</math>
| <math>\|q\|_{T1}</math>
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| Height
| Height
| <math>\text{sopfr}(n d)</math>
| <math>\text{sopfr}(n d)</math>
| <math>2^{\large{\text{sopfr}(n d)}}</math>
| <math>2^{\large{\text{sopfr}(q)}}</math>
| <math>\text{sopfr}(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>
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|}
|}


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 [https://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{sopfr}(nd)</math> 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>See http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf, page 55</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''). 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:
Some useful identities:
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* <math>n d = 2^{\|q\|_{T1}}</math>
* <math>n d = 2^{\|q\|_{T1}}</math>


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 [https://en.wikipedia.org/wiki/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 ==
 
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>.
The concept of height was introduced to xenharmonics by [[Gene Ward Smith]] in 2001<ref>https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31418#31488</ref>; it comes from the mathematical field of number theory (for more information, see [[Wikipedia:Height_function|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, 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>.


== See also ==
== See also ==
* [[Commas by taxicab distance]]
* [[Commas by taxicab distance]]
* [[Harmonic entropy]]


== References ==
== References ==
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<references/>
<references/>


[[Category:Theory]]
[[Category:Height| ]] <!-- main article -->
[[Category:Math]]
[[Category:Math]]
[[Category:Measure]]
[[Category:Interval complexity measures]]
Retrieved from "https://en.xen.wiki/w/Height"