Height: Difference between revisions

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

~~__TOC__~~

== 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.

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<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''1 and ''q''2 are considered equivalent if the following is true:

~~<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''.

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.

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

== 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.

~~<math>\displaystyle q_1 = 2^n q_2</math>~~

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.

~~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 ==

{| class="wikitable"

! Name

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| Height

| <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>

|-

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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.

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>.

== See also ==

* [[Commas by taxicab distance]]

* [[Harmonic entropy]]

== References ==

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~~[[Category:Theory]]~~

~~[[Category:Height| ]] ~~

[[Category:Math]]

[[Category:~~Measure~~]]

[[Category:Interval complexity measures]]

@@ Line 1: / Line 1: @@
 The '''height''' is a mathematical tool to measure the [[complexity]] of [[JI]] [[interval]]s.
-__TOC__
 == 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.
@@ Line 19: / Line 16: @@
 <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:
-<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''.
+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.
-Or equivalently, if ''n'' has any integer solutions:
+== 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.
-<math>\displaystyle q_1 = 2^n q_2</math>
+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.
-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 ==
 {| class="wikitable"
 ! Name
@@ Line 55: / Line 43: @@
 | Height
 | <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>
 |-
@@ Line 93: / Line 81: @@
 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>.
 == See also ==
 * [[Commas by taxicab distance]]
+* [[Harmonic entropy]]
 == References ==
@@ Line 106: / Line 95: @@
 <references/>
-[[Category:Theory]]
-[[Category:Height| ]] <!-- main article -->
 [[Category:Math]]
-[[Category:Measure]]
+[[Category:Interval complexity measures]]