Wilson norm: Difference between revisions

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# 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.

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 # 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.