Wilson height

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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 "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{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

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 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

There are several benefits of using the Wilson height, and the same measure can be arrived at for different reasons.

One particularly important property is that it behaves somewhat like a combined measure of the Tenney height of the ratio, and the smallest prime-limit that the ratio fits into. A good way to see this is an example: 81/80 has a Wilson height of 25, whereas the neighboring ratio 80/79 has a Wilson height of 92, and 82/81 has a Wilson height of 55. Note that 81/80 fits into the 5-limit, whereas you need to go to the 79-limit before getting 80/79, and the 41-limit before getting 82/81, and that this seems reflected in the Wilson height of the ratios. In comparison, the Tenney heights of all three are virtually equal: log2(79*80) = 12.63, log2(80*81) = 12.66, and log2(81*82) = 12.70.

There are several reasons why such a metric may be desirable:

1. When looking for good vanishing commas for subgroup temperaments, it is very important to look at not only the comma's complexity and associated error, but also whether the comma defines temperaments on simple subgroups. 81/80 and 80/79 are virtually equivalent in complexity and error, but the former fits into the 5-limit, and hence defines a 5-limit temperament, a 2.9.5 subgroup temperament, as well as a 7-limit rank-3 temperament, and so on. In comparison, 80/79 has the prime 79 in the denominator, so you will not see it define any temperaments on relatively simple subgroups at all.

2. As an entirely separate reason, when doing tuning optimizations, sometimes the 1/log(p) Tenney-weighting can "roll off" very slowly. For instance, with Tenney weighting, all primes between 25 and 125 have only ~1/3 to ~1/2 as much weighting on prime 5 - and there are 20 of them. Trying to balance all this can cause tuning optimization routines to place more mistuning on primes 2, 3, 5, etc, for the benefit of minimizing the weighted average (or max) error of this huge, heavy-weighted "tail" of primes. Wilson height rectifies this by having the weighting roll off much more quickly.

3. When used on JI chords, this measures how well the chord breaks into simple subsets. For example, 7:9:11:14:17 is slightly lower in the harmonic series than 8:10:12:15:18, but the second (JI major 9 chord) has lots of simple subsets of 2:3, 4:5:6, etc. This metric quantifies this property; the latter scores much better (36) than the former (54). This can also be used on JI scales, treated simply as huge chords; the Wilson height of the entire scale will quantify in some sense how simple the chords of the scale are.

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 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

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{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.

We can likewise keep the [math]p[/math] and [math]1/p[/math] weighting, but change things so that we have a weighted math>\ell_2</math> norm instead of a weighted [math]\ell_1[/math]. We can call this the Wilson-Euclidean norm, and likewise use it to create metrics similar to the Tenney-Euclidean_metrics, including a Wilson-weighted version of the Cangwu_badness.


Superparticular Ratios

Below is a list of superparticular ratios from 2/1 to 100/99, ranked by this "Wilson height," as well as a note about the smallest prime limit each ratio fits into.

You can see that as the Wilson height increases, you get ratios that are generally get higher in Tenney height, but where these "simple-prime-limit" ratios are given an increase in the ranking. For instance, note how 100/99 (11-limit) is ahead of 24/23 (23-limit), for instance.

2/1: 2 (2-limit)
3/2: 5 (3-limit)
4/3: 7 (3-limit)
5/4: 9 (5-limit)
6/5: 10 (5-limit)
7/6: 12 (7-limit)
9/8: 12 (3-limit)
8/7: 13 (7-limit)
10/9: 13 (5-limit)
16/15: 16 (5-limit)
15/14: 17 (7-limit)
11/10: 18 (11-limit)
12/11: 18 (11-limit)
21/20: 19 (7-limit)
25/24: 19 (5-limit)
13/12: 20 (13-limit)
28/27: 20 (7-limit)
14/13: 22 (13-limit)
36/35: 22 (7-limit)
22/21: 23 (11-limit)
27/26: 24 (13-limit)
33/32: 24 (11-limit)
17/16: 25 (17-limit)
18/17: 25 (17-limit)
26/25: 25 (13-limit)
49/48: 25 (7-limit)
64/63: 25 (7-limit)
81/80: 25 (5-limit)
45/44: 26 (11-limit)
50/49: 26 (7-limit)
19/18: 27 (19-limit)
40/39: 27 (13-limit)
55/54: 27 (11-limit)
20/19: 28 (19-limit)
56/55: 29 (11-limit)
65/64: 30 (13-limit)
35/34: 31 (17-limit)
100/99: 31 (11-limit)
24/23: 32 (23-limit)
51/50: 32 (17-limit)
34/33: 33 (17-limit)
91/90: 33 (13-limit)
99/98: 33 (11-limit)
66/65: 34 (13-limit)
57/56: 35 (19-limit)
23/22: 36 (23-limit)
46/45: 36 (23-limit)
76/75: 36 (19-limit)
78/77: 36 (13-limit)
85/84: 36 (17-limit)
39/38: 37 (19-limit)
52/51: 37 (17-limit)
96/95: 37 (19-limit)
30/29: 39 (29-limit)
29/28: 40 (29-limit)
70/69: 40 (23-limit)
31/30: 41 (31-limit)
32/31: 41 (31-limit)
77/76: 41 (19-limit)
63/62: 46 (31-limit)
37/36: 47 (37-limit)
69/68: 47 (23-limit)
92/91: 47 (23-limit)
88/87: 49 (29-limit)
41/40: 52 (41-limit)
75/74: 52 (37-limit)
42/41: 53 (41-limit)
58/57: 53 (29-limit)
43/42: 55 (43-limit)
82/81: 55 (41-limit)
38/37: 58 (37-limit)
44/43: 58 (43-limit)
48/47: 58 (47-limit)
93/92: 61 (31-limit)
54/53: 64 (53-limit)
86/85: 67 (43-limit)
53/52: 70 (53-limit)
60/59: 71 (59-limit)
47/46: 72 (47-limit)
61/60: 73 (61-limit)
95/94: 73 (47-limit)
87/86: 77 (43-limit)
67/66: 83 (67-limit)
72/71: 83 (71-limit)
94/93: 83 (47-limit)
71/70: 85 (71-limit)
73/72: 85 (73-limit)
68/67: 88 (67-limit)
59/58: 90 (59-limit)
80/79: 92 (79-limit)
62/61: 94 (61-limit)
79/78: 97 (79-limit)
84/83: 97 (83-limit)
90/89: 102 (89-limit)
89/88: 106 (89-limit)
97/96: 110 (97-limit)
74/73: 112 (73-limit)
98/97: 113 (97-limit)
83/82: 126 (83-limit)