Wilson norm: Difference between revisions
sopf -> sopfr (sum of prime factors with *repetition*) |
|||
| Line 1: | Line 1: | ||
The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at. | The '''Wilson height''' is a different way of weighting rational numbers than the [[Tenney height]], but has some very beneficial properties that make it an excellent metric to look at. | ||
If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of the number p*q, counting multiplicity. This function is often written <math>\text{ | If p/q is a positive rational number reduced to its lowest terms, then the Wilson height is the [http://mathworld.wolfram.com/SumofPrimeFactors.html "sum of prime factors"] of the number p*q, counting multiplicity. This function is often written <math>\text{sopfr}(pq)</math>. | ||
Note that we have <math>\text{ | Note that we have <math>\text{sopfr}(pq) = \text{sopfr}(p) + \text{sopfr}(q)</math>, similar to the logarithm -- as a result, this function is sometimes even referred to as the "integer logarithm." So, equivalently, we can define the Wilson height of a rational number p/q as the Wilson height of p, plus the Wilson height of q. | ||
== Example == | == Example == | ||
The sum of prime factors function is fairly simple: for some number, simply list all the prime factors (multiple times if they appear more than once), and add them together. For instance, for the number "81", we have 81=3*3*3*3, so the Wilson height is 3+3+3+3 = 12. Likewise, for the number "80", we have 80 = 2*2*2*2*5, so the Wilson height is 2+2+2+2+5 = 13. The sum of both is 25, which is the Wilson height of 81/80 - also obtainable by saying that 81*80 = 6480 = 2*2*2*2*3*3*3*3*5, for which the | The sum of prime factors function is fairly simple: for some number, simply list all the prime factors (multiple times if they appear more than once), and add them together. For instance, for the number "81", we have 81=3*3*3*3, so the Wilson height is 3+3+3+3 = 12. Likewise, for the number "80", we have 80 = 2*2*2*2*5, so the Wilson height is 2+2+2+2+5 = 13. The sum of both is 25, which is the Wilson height of 81/80 - also obtainable by saying that 81*80 = 6480 = 2*2*2*2*3*3*3*3*5, for which the sopfr is 2+2+2+2+3+3+3+3+5 = 25. | ||
This measure can similarly be extended to JI chords, so that the Wilson height of a:b:c is equal to the sum of the | This measure can similarly be extended to JI chords, so that the Wilson height of a:b:c is equal to the sum of the sopfr's of a, b, and c. This definition can be scaled by a constant depending only on the size of the chord, so as to make it easier to compare chords of different cardinalities. | ||
== Uses== | == Uses== | ||
| Line 27: | Line 27: | ||
4. When used on JI intervals, such as 15/8, this measures how well the interval can fit into simple JI chords with simple subsets. For instance, 15/8 fits into 8:10:15, 8:12:15, 8:10:12:15, each of which has simple subsets such as 2:3, 4:5, 4:5:6, etc. It has a Wilson height of 14. In comparison, 13/6 does not have quite as many simple-subset triads and tetrads that it can fit into, and has a Wilson height of 18 (which is not that much worse). | 4. When used on JI intervals, such as 15/8, this measures how well the interval can fit into simple JI chords with simple subsets. For instance, 15/8 fits into 8:10:15, 8:12:15, 8:10:12:15, each of which has simple subsets such as 2:3, 4:5, 4:5:6, etc. It has a Wilson height of 14. In comparison, 13/6 does not have quite as many simple-subset triads and tetrads that it can fit into, and has a Wilson height of 18 (which is not that much worse). | ||
The common theme in #1, #3, and #4, on a mathematical level, is that the | The common theme in #1, #3, and #4, on a mathematical level, is that the sopfr function measures in some sense "how composite" a number, ratio, or chord is. This property is what makes it useful in indirectly measuring the subsets of the chord. Note that #2 is also a useful property that seems unrelated to this. | ||
== L1 Norm on Monzos == | == L1 Norm on Monzos == | ||
| Line 33: | Line 33: | ||
The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by | The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by | ||
<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{ | <math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math> | ||
which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. Both of these norms can be extended to the exterior algebra, so that we can use it as a measure of the complexity of a temperament. | which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. Both of these norms can be extended to the exterior algebra, so that we can use it as a measure of the complexity of a temperament. | ||