Wilson norm: Difference between revisions

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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~~, whereas the neighboring ratio~~ 80/79 has a Wilson height of 92~~, and~~ 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 ~~seems~~ 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.

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:

1. 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.

# 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.

=== Wilson Height and Tenney Height ===

Note that the Wilson height doesn't really correspond directly to the psychoacoustic "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 crunchiness, 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 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 from a pure psychoacoustic "crunchiness" standpoint, if heard in isolation, and are thus ranked pretty highly in Tenney Height. However, they are quite common because they often appear in fairly typical 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 29/16 and 43/32 do. 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 upper structure triads and modulations by simple intervals like 3/2 are used frequently. This is albeit from a relatively simplistic prime-limit JI-only perspective, but is still an interesting starting point, and even in 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:

2~~. 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.

* 1/1 3/2 7/4 9/4 27/8

* 1/1 3/2 7/4 9/4 13/4

~~3. When used on JI chords~~, ~~this measures 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~~.

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 the various subdyads and upper structure triads 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. 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'''.

~~4. When used on JI intervals~~, ~~such as 15/8~~, ~~this measures~~ how ~~well~~ the ~~interval can 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 (~~which is not that much worse~~).

A important perceptual caveat is that this perception of the "compoundness" of an interval is probably not some innate psychoacoustic thing - rather, it is likely learned after much exposure to a tuning system, where one begins to learn how the various notes and chords relate to one another in this way in a lattice of upper structure triads and modulations. An important additional limitation is that, if one plays in tempered tuning systems, one can learn many important "compound" relationships that simply do not exist in JI, such as how in sensi temperament we have that 9/7 * 9/7 = 5/3 (which it doesn't in JI), or how in meantone the chord 1/1 5/4 5/3 9/4 3/1 has the interval between the 5/4 and 9/4 also equal to a perfect fourth, 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. So while the Wilson height is useful in elucidating on some of these relationships, it will "miss" many of these interesting relationships which exist in tempered systems (although could perhaps be the starting point to derive tempered metrics to measure some of those).

~~The common theme in #1, #3~~, and ~~#4,~~ on a ~~mathematical level, is that~~ the ~~sopfr function measures~~ in ~~some sense "how composite"~~ a ~~number~~, ~~ratio~~, ~~or chord is. This property is what makes~~ it ~~useful in indirectly measuring~~ the ~~subsets~~ of ~~the chord. Note that #2 is also a useful property that seems unrelated to this~~.

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.

== L1 Norm on Monzos ==

@@ Line 17: / Line 17: @@
 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, whereas the neighboring ratio 80/79 has a Wilson height of 92, and 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 seems 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.
+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.
+# 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.
+=== Wilson Height and Tenney Height ===
+Note that the Wilson height doesn't really correspond directly to the psychoacoustic "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 crunchiness, 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 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 from a pure psychoacoustic "crunchiness" standpoint, if heard in isolation, and are thus ranked pretty highly in Tenney Height. However, they are quite common because they often appear in fairly typical 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 29/16 and 43/32 do. 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 upper structure triads and modulations by simple intervals like 3/2 are used frequently. This is albeit from a relatively simplistic prime-limit JI-only perspective, but is still an interesting starting point, and even in 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:
-. 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.
+* 1/1 3/2 7/4 9/4 27/8
+* 1/1 3/2 7/4 9/4 13/4
-. When used on JI chords, this measures 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.
+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 the various subdyads and upper structure triads 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. 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'''.
-. When used on JI intervals, such as 15/8, this measures how well the interval can 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 (which is not that much worse).
+A important perceptual caveat is that this perception of the "compoundness" of an interval is probably not some innate psychoacoustic thing - rather, it is likely learned after much exposure to a tuning system, where one begins to learn how the various notes and chords relate to one another in this way in a lattice of upper structure triads and modulations. An important additional limitation is that, if one plays in tempered tuning systems, one can learn many important "compound" relationships that simply do not exist in JI, such as how in sensi temperament we have that 9/7 * 9/7 = 5/3 (which it doesn't in JI), or how in meantone the chord 1/1 5/4 5/3 9/4 3/1 has the interval between the 5/4 and 9/4 also equal to a perfect fourth, 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. So while the Wilson height is useful in elucidating on some of these relationships, it will "miss" many of these interesting relationships which exist in tempered systems (although could perhaps be the starting point to derive tempered metrics to measure some of those).
-The common theme in #1, #3, and #4, on a mathematical level, is that the sopfr function measures in some sense "how composite" a number, ratio, or chord is. This property is what makes it useful in indirectly measuring the subsets of the chord. Note that #2 is also a useful property that seems unrelated to this.
+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.
 == L1 Norm on Monzos ==