Wilson norm: Difference between revisions

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.

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.

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).