Wilson norm: Difference between revisions

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# 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 ===

== L1 Norm on Monzos ==

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

<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>

which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. 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 <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]].

== Wilson Height and Tenney Height: A Psychoacoustic Comparison ==

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.

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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 ==~~

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

== Superparticular Ratios ==

~~<math>\| |e_2 \~~, ~~e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(~~2~~^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})<~~/~~math>~~

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.

~~which is almost exactly the same~~ as the ~~Tenney~~ height, ~~except~~ that ~~the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the~~ Tenney height, ~~it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and~~ where ~~each~~ prime ~~is weighted by <math>1/p</math>. 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~~.

You can see that as the Wilson height increases, you get ratios that are generally get higher in Tenney height, 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.

~~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 Tenney height, in comparison, would rank all of these commas purely monotonically in n*d and size, so that 80/79 is ahead of 81/80, which is ahead of 82/81, etc.

~~==Superparticular Ratios==~~

~~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.~~

~~You can see~~ that as the ~~Wilson~~ height ~~increases~~, ~~you get ratios that are generally get higher~~ in ~~Tenney height~~, ~~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~~.

2/1: 2 (2-limit)<br>

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 # 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 ===
+== L1 Norm on Monzos ==
+The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by
+<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>
+which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. 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 <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]].
+== Wilson Height and Tenney Height: A Psychoacoustic Comparison ==
 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.
@@ Line 57: / Line 67: @@
 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 ==
-The Wilson height has a nice, simple definition as a norm on monzos, which we can call the '''Wilson norm'''. It is given by
+== Superparticular Ratios ==
-<math>\| |e_2 \, e_3 \dotso e_p \rangle \|_{\text{Wil}} = |e_2| + 3\cdot|e_3| 3 + \dotso + p\cdot|e_p| = \text{sopfr}(2^{|e_2|} \cdot 3^{|e_3|} \cdot \dotso \cdot p^{|e_p|})</math>
+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.
-which is almost exactly the same as the Tenney height, except that the weighting on each prime is simply <math>p</math> instead of <math>\log(p)</math>. Like the Tenney height, it is a scaled <math>\ell_1</math> norm. Similarly, we get a dual norm on vals, which is an <math>\ell_\infty</math> norm, and where each prime is weighted by <math>1/p</math>. 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.
+You can see that as the Wilson height increases, you get ratios that are generally get higher in Tenney height, 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.
-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 Tenney height, in comparison, would rank all of these commas purely monotonically in n*d and size, so that 80/79 is ahead of 81/80, which is ahead of 82/81, etc.
-==Superparticular Ratios==
-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.
-You can see that as the Wilson height increases, you get ratios that are generally get higher in Tenney height, 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.
 /1: 2 (2-limit)<br>