Height: Difference between revisions

Line 29:

{| class="wikitable"

! Name

! Type

! H(n/d)

! H(q)

! H(q) simplified by equivalence relation

|-

| | <~~u>Name:</u~~>

| [[Benedetti height]] (or [[Tenney height]])

| ~~| Type:~~

| Height

| | H(n/d):

| <math>n d</math>

| | H(q):

| <math>2^{\large{\|q\|_{T1}}}</math>

| | H(q~~) simplified by equivalence relation:~~

| <math>\|q\|_{T1}</math>

|-

| | [[~~Benedetti_height|Benedetti~~ height]]

| [[Wilson height]]

| Height

~~(or [[Tenney_Height|Tenney Height]])~~

| <math>\text{sopf}(n d)</math>

| | Height

| <math>2^{\large{\text{sopf}(n d)}}</math>

| | <math>n d</math>

| <math>\text{sopf}(q)</math>

| | <math>2^{\large{\~~|q\|_~~{T1}}}</math>

| | <math>\~~|q\|_~~{T1}</math>

|-

| ~~| Wilson Height~~

| Weil height

| | Height

| Height

| | <math>\~~text~~{~~sopf}(~~n d)</math>

| <math>\max \left( {n , d} \right)</math>

| | <math>2^{\large{\~~text~~{~~sopf~~}(~~n d~~)}}</math>

| <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math>

| | <math>\~~text~~{~~sopf~~}(q)</math>

| <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math>

|-

| ~~| Weil Height~~

| Arithmetic height

| | Height

| Height

| | <math>~~\max \left( {~~n , d~~} \right)~~</math>

| <math>n + d</math>

| | <math>2^{\large{\frac{1}{2}(\|q\|_{T1} ~~+ \mid \log_2(\mid q \mid)\mid)~~}}</math>

| <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>

| | <math>\|q\|_{T1} + \~~mid~~ \log_2(~~\mid~~ q \~~mid~~)~~\mid~~</math>

| <math>\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</math>

|-

| ~~| Arithmetic Height~~

| Harmonic height

| | Height

| Semi-Height

| | <math>n + d</math>

| <math>\dfrac {n d} {n + d}</math>

| | <math>\dfrac {\left( {q + 1} \right)~~} {\sqrt{q}~~} 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>

|-

~~| | Harmonic Height~~

| [[Kees height]]

~~| | Semi-Height~~

| Semi-Height

~~| | <math>\dfrac {n d} {n + d}</math>~~

| <math>\max \left( {2^{-v_2 \left( {n} \right)} n, 2^{-v_2 \left( {d} \right)} d} \right)</math>

~~| | <math>\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</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>\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_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>

|-

| | [[~~Kees_Height|~~Kees ~~Height~~]]

| | Semi-Height

| | <math>\max \left( {2^{-v_2 \left( {n} \right)} n ,

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^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math>

|}

Line 79:

Line 75:

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."

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>

~~<math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</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 [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.

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

@@ Line 29: / Line 29: @@
 {| 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>n d</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]]
+| [[Wilson height]]
+| Height
-(or [[Tenney_Height|Tenney Height]])
+| <math>\text{sopf}(n d)</math>
-| | Height
+| <math>2^{\large{\text{sopf}(n d)}}</math>
-| | <math>n d</math>
+| <math>\text{sopf}(q)</math>
-| | <math>2^{\large{\|q\|_{T1}}}</math>
-| | <math>\|q\|_{T1}</math>
 |-
-| | Wilson Height
+| Weil height
-| | Height
+| Height
-| | <math>\text{sopf}(n d)</math>
+| <math>\max \left( {n , d} \right)</math>
-| | <math>2^{\large{\text{sopf}(n d)}}</math>
+| <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math>
-| | <math>\text{sopf}(q)</math>
+| <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math>
 |-
-| | Weil Height
+| Arithmetic height
-| | Height
+| Height
-| | <math>\max \left( {n , d} \right)</math>
+| <math>n + d</math>
-| | <math>2^{\large{\frac{1}{2}(\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid)}}</math>
+| <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</math>
-| | <math>\|q\|_{T1} + \mid \log_2(\mid q \mid)\mid</math>
+| <math>\|q\|_{T1} + 2 \log_2 \left( {q + 1} \right) - \log_2 \left( {q} \right)</math>
 |-
-| | Arithmetic Height
+| Harmonic height
-| | Height
+| Semi-Height
-| | <math>n + d</math>
+| <math>\dfrac {n d} {n + d}</math>
-| | <math>\dfrac {\left( {q + 1} \right)} {\sqrt{q}} 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>
 |-
-| | Harmonic Height
+| [[Kees height]]
-| | Semi-Height
+| Semi-Height
-| | <math>\dfrac {n d} {n + d}</math>
+| <math>\max \left( {2^{-v_2 \left( {n} \right)} n, 2^{-v_2 \left( {d} \right)} d} \right)</math>
-| | <math>\dfrac {\sqrt{q}} {\left( {q + 1} \right)} 2^{\large{\frac{1}{2} {\|q\|_{T1}}}}</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>\|q\|_{T1} - 2 \log_2 \left( {q + 1} \right) + \log_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>
-|-
-| | [[Kees_Height|Kees Height]]
-| | Semi-Height
-| | <math>\max \left( {2^{-v_2 \left( {n} \right)} n ,
-^{-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^{-v_2 \left( {q} \right)} q}\|_{T1} + | \log_2 \left( {q} \right) - v_2 \left( {q} \right) |</math>
 |}
@@ Line 79: / Line 75: @@
 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."
 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>
-<math>n = 2^{\large{\frac{1}{2}(\|q\|_{T1} + \log_2(q))}}</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 [http://en.wikipedia.org/wiki/Grassmannian Grassmann varieties], complexity measures of regular temperaments are also height functions.
-<math>d = 2^{\large{\frac{1}{2}(\|q\|_{T1} - \log_2(q))}}</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|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:Theory]]