Wilson norm: Difference between revisions

# 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.

The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by

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]].

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.

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.

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.