Harmonic entropy: Difference between revisions

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A fairly typical choice of settings for a basic dyadic HE model would be:

~~<ul><li>~~The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as <math>N</math> and set to at least 10,000.~~</li><li>~~The spreading function is typically a Gaussian distribution with a frequency deviation of 1% either way, or about s=~17 cents.~~</li></ul>~~

* The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as <math>N</math> and set to at least 10,000.

* The spreading function is typically a Gaussian distribution with a frequency deviation of 1% either way, or about s=~17 cents.

Other spreading functions have also been explored, such as the use of the heavy-tailed [https://en.wikipedia.org/wiki/Laplace_distribution Laplace distribution], sometimes described as the "Vos function" in Paul's writings. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for an incoming dyad <math>c</math> can be written as follows:

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It is also common to use as a basis set all those rationals bounded by some maximum Weil height, with a typical cutoff for <math>N</math> set to at least 100. This has sometimes been referred to as seeding HE with the "Farey sequence of order <math>N</math>" and its reciprocals, so references in Paul's work to "Farey series HE" vs "Tenney series HE" are sometimes seen.

Lastly, the set of rationals is often chosen to be only those "reduced" rationals within the cutoff, such that <math>n/d</math> is in the set only if <math>n</math> and <math>d</math> are coprime. HE can also be formulated with unreduced rationals as well. Both methods tend to give similar results. In Paul's work, reduced rationals are most common, although the use of unreduced rationals may be useful in extending HE to the case where <math>N=\infty</math>.

Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the '''domain-integral approach''', works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the '''complexity-normalization approach''', has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below.

@@ Line 73: / Line 73: @@
 A fairly typical choice of settings for a basic dyadic HE model would be:
-<ul><li>The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as <math>N</math> and set to at least 10,000.</li><li>The spreading function is typically a Gaussian distribution with a frequency deviation of 1% either way, or about s=~17 cents.</li></ul>
+* The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as <math>N</math> and set to at least 10,000.
+* The spreading function is typically a Gaussian distribution with a frequency deviation of 1% either way, or about s=~17 cents.
 Other spreading functions have also been explored, such as the use of the heavy-tailed [https://en.wikipedia.org/wiki/Laplace_distribution Laplace distribution], sometimes described as the "Vos function" in Paul's writings. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for an incoming dyad <math>c</math> can be written as follows:
@@ Line 86: / Line 87: @@
 It is also common to use as a basis set all those rationals bounded by some maximum Weil height, with a typical cutoff for <math>N</math> set to at least 100. This has sometimes been referred to as seeding HE with the "Farey sequence of order <math>N</math>" and its reciprocals, so references in Paul's work to "Farey series HE" vs "Tenney series HE" are sometimes seen.
+Lastly, the set of rationals is often chosen to be only those "reduced" rationals within the cutoff, such that <math>n/d</math> is in the set only if <math>n</math> and <math>d</math> are coprime. HE can also be formulated with unreduced rationals as well. Both methods tend to give similar results. In Paul's work, reduced rationals are most common, although the use of unreduced rationals may be useful in extending HE to the case where <math>N=\infty</math>.
 Given a spreading function and set of basis rationals, there are two different procedures commonly used to assign probabilities to each rational. The first, the '''domain-integral approach''', works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the '''complexity-normalization approach''', has nice mathematical properties which sometimes make it easier to compute and which may lead to generalizations to infinite sets of rationals which are sometimes dense in the reals. It is conjectured that there are certain important limiting situations where the two converge; both are described in detail below.