Height: Difference between revisions

Line 46:

| [[Wilson 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}(n d)}}</math>

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

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

|-

| Weil height

Line 77:

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.

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 <math>\text{sopfr}(nd)</math> is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repeition"] 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]] "Division of the Tetrachord."<ref>See http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf, page 55</ref>

Some useful identities:

@@ Line 46: / Line 46: @@
 | [[Wilson 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}(n d)}}</math>
-| <math>\text{sopf}(q)</math>
+| <math>\text{sopfr}(q)</math>
 |-
 | Weil height
@@ Line 77: / Line 77: @@
 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.
-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 <math>\text{sopfr}(nd)</math> is the [https://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors with repeition"] 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]] "Division of the Tetrachord."<ref>See http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf, page 55</ref>
 Some useful identities: