Wilson height
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 with repetition" 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.
One important theorem is that the Wilson-optimal tuning happens to also be the Benedetti optimal tuning for subgroups with a pairwise coprime basis (e.g. prime limits and some others); see also BOP Tuning.
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
- 80/79 has a Wilson height of 92
- 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 is 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:
- 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.
- 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, and is also related to the BOP tuning.
- As yet another reason, when used on JI chords, this metric provides an indirect measure of 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. This metric isn't always perfect as it will even treat individual intervals as better if they fit into a lower prime-limit, but it's still something of a useful heuristic.
- The interpretation of Wilson height for individual JI intervals is less direct, but can be thought of as a composite measure of the interval's psychoacoustic complexity and its prime limit. One way to think of it is that it measures how well the interval "could potentially 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.
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.
Wilson Height and Tenney Height: A Psychoacoustic Comparison
Note that the Wilson height doesn't really correspond directly to the psychoacoustic concordance, perhaps thought of as "crunchiness," of a simple JI interval or chord in the same way that the Tenney height does.
For instance, you will note that 7/4 has a Wilson height of 11 and 9/4 has a Wilson height of 10. This is because the 9/4 score is improved because it fits into a smaller prime-limit than 7/4 does (namely the 3-limit). Thus, if one is only interested in quantifying this kind of psychoacoustic measure, the Tenney height is much better for that.
On the other hand, if you are more interested in a composite measure of some JI interval's psychoacoustic properties, as well as its "decomposability" as a compound interval that is formed from simple primes, then the Wilson height is very useful. This is useful for commas and temperament searches, as previously mentioned, although we may also ask if there is 'any' direct psychoacoustic or perceptual property that correlates with the Wilson height at all.
One thing that is clearly of interest pertains to JI intervals such as 45/32 (= 5/4 * 9/8) and 27/16 (= 3/2 * 9/8). These intervals are very complex if one adopts the purely psychoacoustic standpoint of evaluating how well they fuse into a single sound in isolation. Thus, they are ranked pretty highly in Tenney Height. However, they are also quite common because they often appear in fairly typical 5-limit extended harmony, such as a major 13 #11 chord (1/1 5/4 3/2 15/8 9/1 45/4 27/2), much more so than their neighbors 47/32 and 43/32 do. These intervals will also be reached quite easily if one tends to modulate successively via simple intervals like 3/2, 5/4, and 6/5. So one measure of the Wilson height is that it tells you how musically important such intervals are predicted to be, given a style of music in which these kinds of chords and modulations are prominent. And even in deliberate compositional use of higher limits, of course, where one embraces more complex ratios, intervals like 27/16 and 45/16 are still relatively useful for what they are, being made of simple compound 3/2's and 5/4's and 2/1's.
We get something similar with chords. In particular, the use of 27/16 as a natural 13 which is just a 3/2 above the 9/4, is quite common and musically useful. It can be instructive to compare the perception of this interval with the interval 13/8. For instance, we can look at the following chords:
- 1/1 3/2 7/4 9/4 27/8
- 1/1 3/2 7/4 9/4 13/4
In the first chord, the 27/8 forms lots of simple dyads with the other notes in the chord, leading to a sort of "kaleidoscopic" structure in which there are little "pieces of chord" everywhere: various subdyads, upper structure triads, and so on, which are quite simple and relevant. The second one, on the other hand, doesn't have quite as much of this going on; it is a much more focused "otonal" or 4:6:7:9:13 pentad in which all of the "pieces" are coherent and almost every subchord points to the same fundamental. The Tenney heights of the two chords are quite different, whereas the Wilson height takes these subchords into view and ranks them fairly similarly. We can say that the first chord is Wilson-simple, whereas the second one is both Wilson-simple and Tenney-simple, and if we replace 27/8 with (for instance) 29/8 or 31/8, we get something that isn't particularly Wilson-simple or Tenney-simple.
One important hypothesis is that some of these perceptions may be very dependent on learning, rather than some innate psychoacoustic thing. After much exposure to a tuning system, one begins to learn how the various notes and chords relate to one another in such a way that inferences about common modulations, subchords, and so on start to be important. One important limitation of the Wilson height, then, is that tempered systems often have many such relationships which simply do not exist in JI. For example, in sensi temperament we have that 9/7 * 9/7 = 5/3 - certainly not true in JI. Similarly, complex chords can have extra consonant dyads in tempered systems, such as in meantone, where the chord 1/1 5/4 5/3 9/4 3/1 has an extra 4/3 between the 5/3 and 9/4 (tempered equal to 27/20), or how in porcupine the chord 1/1 6/5 3/2 9/5 9/4 27/10 has the 27/10 equal to 11/8. This "tempered compoundness" does not derive directly from prime factorizability in this way, and the Wilson height will thus "miss" interesting relationships like this. (It would be very interesting derive similar metrics from the Wilson height that are applicable to tempered systems).
Lastly, while we will not get into the weeds of measuring the "learnedness" of some listener here - which is clearly extremely subjective and dependent on musical context - it would be a rather interesting hypothesis to see if some listener's musical perception of JI intervals moves from the Tenney Height toward the Wilson Height given additional ear training in JI, or exposure to a style of JI music that frequently uses simple upper structure ratios and modulations, etc. Or, if it moves back towards the Tenney Height if one embraces a style of perhaps more "spectralist" music in which one mashes huge harmonic series chords without caring much about simple JI subchord relationships.
Superparticular Ratios
Perhaps the most immediate use of the Wilson height is in subgroup temperament comma searches. To illustrate this, 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.
Note that the Tenney height, in comparison, would rank all of these commas purely monotonically in n*d and size, so that 80/79 is ahead of 81/80, which is ahead of 82/81, etc.
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)