Wilson norm

The Wilson norm, otherwise known as Wilson height, is a measure of complexity for just intervals, similar to the Tenney norm, 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 norm is the sum of prime factors with repetition of the number pq, counting multiplicity. This function is often written sopfr(pq). This is called Wilson's complexity in John Chalmers's Divisions of the Tetrachord^[1]

Note that we have sopfr(pq) = sopfr(p) + sopfr(q), similar to logarithm – as a result, this function is sometimes even referred to as the "integer logarithm". So, equivalently, we can define the Wilson norm of a rational number p/q as the Wilson norm of p, plus the Wilson norm 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.

L₁ norm on monzos

The Wilson norm has a nice, simple definition on monzos. For a p-limit monzo m = [m₁ m₂ … m_{π (p)}⟩ (π being the prime-counting function), the norm is given by

$$ \begin{align} \left\lVert H \vec m \right\lVert_1 &= 2 |m_1| + 3 |m_2| + \ldots + p |m_{\pi (p)}| \\ &= \text{sopfr}(2^{|m_1|} \cdot 3^{|m_2|} \cdot \ldots \cdot p^{|m_{\pi (p)}|}) \end{align} $$

where H is the transformation matrix such that, for the prime basis Q = ⟨2 3 5 … p],

$$ H = \operatorname {diag} (Q) $$

This is almost exactly the same as the Tenney norm, except that the weighting on each prime is simply q instead of log(q). Like the Tenney norm, it is a scaled L₁ norm. Similarly, we get a dual norm on vals, which is an L_∞ norm, and where each prime is weighted by 1/q. 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 q and 1/q weighting, but change things so that we have a weighted L₂ norm instead of a weighted L₁. 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 variant of the Cangwu badness.

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 = 3⁴, so the Wilson norm is 3 × 4 = 12. Likewise, for the number "80", we have 80 = 2⁴ × 5, so the Wilson norm is 2 × 4 + 5 = 13. The sum of both is 25, which is the Wilson norm of 81/80 – also obtainable by saying that 81 × 80 = 6480 = 2⁴ × 3⁴ × 5, for which the sopfr is 2 × 4 + 3 × 4 + 5 = 25.

This measure can similarly be extended to JI chords, so that the Wilson norm 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.

Motivations

There are several benefits of using the Wilson norm, 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 norm 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 norm of 25
80/79 has a Wilson norm of 92
82/81 has a Wilson norm 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 norm of the ratios. In comparison, the Tenney norms 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 norm rectifies this by having the weighting roll off much more quickly. This is also related to the BOP and BE tunings.
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 norm of the entire scale will quantify in some sense how simple the chords of the scale are. This metric is not always perfect as it will even treat individual intervals as better if they fit into a lower prime-limit, but it is still something of a useful heuristic.
The interpretation of Wilson norm 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 norm 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 norm of 18.

Wilson norm and Tenney norm: a psychoacoustic comparison

Note that the Wilson norm does not 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 norm does.

For instance, you will note that 7/4 has a Wilson norm of 11 and 9/4 has a Wilson norm 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 norm 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 norm 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 norm 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 norm. However, they are also quite common because they often appear in fairly typical 5-limit extended harmony, such as a major 13 #11 chord (1–5/4–3/2–15/8–9/1–45/4–27/2), 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 norm 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:

1–3/2–7/4–9/4–27/8
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, does not 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 norms of the two chords are quite different, whereas the Wilson norm 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 is not 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 norm, 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)² = ~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 1–5/4–5/3–9/4–3 has an extra 4/3 between the 5/3 and 9/4 (tempered equal to 27/20), or how in porcupine the chord 1–6/5–3/2–9/5–9/4–27/10 has the ~27/10 equal to ~11/4. This "tempered compoundness" does not derive directly from prime factorizability in this way, and the Wilson norm will thus miss interesting relationships like this. (It would be very interesting to derive similar metrics from the Wilson norm 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 norm toward the Wilson norm 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 norm 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 norm 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 norm, as well as a note about the smallest prime limit each ratio fits into.

You can see that as the Wilson norm increases, you get ratios that generally get higher in Tenney norm, 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 norm, in comparison, would rank all of these commas purely monotonically in nd 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)

External links

Github | Parametric Tenney–Wilson norm · FloraCanou/temperament_evaluator Wiki – a unified, parametrized Tenney–Wilson norm used by Flora Canou's Temperament Evaluator.

References

↑ Division of the Tetrachord, page 55. John Chalmers.

[1] Division of the Tetrachord, page 55. John Chalmers.

[1]