Harmonic entropy: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 515672698 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 515672922 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 15: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 15:19:39 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515672922</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
| Line 46: | Line 46: | ||
A fairly typical choice of settings for a basic dyadic HE model would be: | A fairly typical choice of settings for a basic dyadic HE model would be: | ||
* The set | * The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as **N** and set to at least 10000. | ||
* The spreading function is typically a Gaussian distribution with a frequency deviation of 1% either way, or about s=~17 cents | * 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 "Vos function" of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows: | Other spreading functions have also been explored, such as the use of the "Vos function" of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows: | ||
| Line 55: | Line 55: | ||
[[math]] | [[math]] | ||
where //s// becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation. | where //s// becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation. We assume here that the variable x represents cents, and will adopt this convention for the remainder of the article. Note that in previous expositions on Harmonic Entropy, //s// was sometimes given in units representing a percentage of linear-frequency deviation; we allow //s// to stand for cents here to simplify the notation. To convert from a percentage to cents, the formula cents = 1200*log2(1+percentage) can be used. | ||
It is also common to use as a basis set all those rationals bounded by some maximum Weil height, with a typical cutoff for N set to at least 100. This has sometimes been referred to as seeding HE with the "Farey sequence of order N" and its reciprocals, so references in Paul's work to "Farey series HE" vs "Tenney series HE" are sometimes seen. | |||
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. | 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 226: | Line 226: | ||
<br /> | <br /> | ||
A fairly typical choice of settings for a basic dyadic HE model would be:<br /> | A fairly typical choice of settings for a basic dyadic HE model would be:<br /> | ||
<ul><li>The set | <ul><li>The basis set is all those rationals bounded by some maximum Tenney height, with the bound typically notated as <strong>N</strong> and set to at least 10000.</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><br /> | ||
Other spreading functions have also been explored, such as the use of the &quot;Vos function&quot; of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows:<br /> | Other spreading functions have also been explored, such as the use of the &quot;Vos function&quot; of a·exp(b|x|) rather than the Gaussian distribution. We will assume the Gaussian distribution as the spreading function for the remainder of this article, so that the spreading function for a dyad d can be written as follows:<br /> | ||
<br /> | <br /> | ||
| Line 234: | Line 234: | ||
--><script type="math/tex">s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | --><script type="math/tex">s_d(x) = \frac{1}{s\sqrt{2\pi}} e^{-\frac{(x-d)^2}{2s^2}}</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
where <em>s</em> becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation.<br /> | where <em>s</em> becomes the standard deviation of the Gaussian, being an ASCII-friendly version of the more familiar symbol σ for representing the standard deviation. We assume here that the variable x represents cents, and will adopt this convention for the remainder of the article. Note that in previous expositions on Harmonic Entropy, <em>s</em> was sometimes given in units representing a percentage of linear-frequency deviation; we allow <em>s</em> to stand for cents here to simplify the notation. To convert from a percentage to cents, the formula cents = 1200*log2(1+percentage) can be used.<br /> | ||
<br /> | <br /> | ||
It is also common to use as a basis set all those rationals bounded by some maximum Weil height, with a typical cutoff for N set to at least 100. This has sometimes been referred to as seeding HE with the &quot;Farey sequence of order N&quot; and its reciprocals, so references in Paul's work to &quot;Farey series HE&quot; vs &quot;Tenney series HE&quot; are sometimes seen.<br /> | |||
<br /> | <br /> | ||
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 <strong>domain-integral approach</strong>, works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the <strong>complexity-normalization approach</strong>, 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.<br /> | 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 <strong>domain-integral approach</strong>, works for arbitrary nowhere dense sets of rationals without any further free parameters. The second, the <strong>complexity-normalization approach</strong>, 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.<br /> | ||