Harmonic entropy: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 515668950 - Original comment: ** |
Wikispaces>mbattaglia1 **Imported revision 515669022 - Original comment: ** |
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<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 13: | : This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2014-07-06 13:32:19 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>515669022</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> | ||
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[[math]] | [[math]] | ||
p_d(b) = \int_{\ | p_d(b) = \int_{\cancel{c}(b_l)}^{\cancel{c}(b_u)} s_d(x) dx | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\sqrt{n_b \cdot d_b}} | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\max(n_b,d_b)} | ||
[[math]] | [[math]] | ||
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[[math]] | [[math]] | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\|b\|} | ||
[[math]] | [[math]] | ||
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The use of a=∞ min-entropy would reflect this view. In contrast, the use of a=1 Shannon entropy reflects a much "dumber" process which performs no such analysis and perhaps doesn't even seek to "choose" any sort of "victor" rational at all. As the parameter a interpolates between these two options, it <span style="line-height: 1.5;">can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be "intelligent" and "active" in this way.</span> | The use of a=∞ min-entropy would reflect this view. In contrast, the use of a=1 Shannon entropy reflects a much "dumber" process which performs no such analysis and perhaps doesn't even seek to "choose" any sort of "victor" rational at all. As the parameter a interpolates between these two options, it <span style="line-height: 1.5;">can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be "intelligent" and "active" in this way.</span> | ||
Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as "dumb" processes whereby nothing in particular is being "chosen," but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Different values of a can differentiate between the predominance given to these two types of effect in the overall construct of psychoacoustic concordance. | Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as "dumb" processes whereby nothing in particular is being "chosen," but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Different values of a can differentiate between the predominance given to these two types of effect in the overall construct of psychoacoustic concordance. | ||
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<!-- ws:start:WikiTextMathRule:2: | <!-- ws:start:WikiTextMathRule:2: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
p_d(b) = \int_{\ | p_d(b) = \int_{\cancel{c}(b_l)}^{\cancel{c}(b_u)} s_d(x) dx&lt;br/&gt;[[math]] | ||
--><script type="math/tex">p_d(b) = \int_{\ | --><script type="math/tex">p_d(b) = \int_{\cancel{c}(b_l)}^{\cancel{c}(b_u)} s_d(x) dx</script><!-- ws:end:WikiTextMathRule:2 --><br /> | ||
<br /> | <br /> | ||
where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &quot;cents&quot; function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span><br /> | where s<span style="vertical-align: sub;">d</span>(x) is the spreading function associated with d<span style="line-height: 1.5;">, b</span><span style="line-height: 1.5; vertical-align: sub;">l</span><span style="line-height: 1.5;"> and b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;"> are the domain lower and upper bounds associated with basis rational b, and ¢(f) = 1200·log2(f), or the &quot;cents&quot; function converting frequency ratios to cents. Normally, b</span><span style="line-height: 1.5; vertical-align: sub;">l </span><span style="line-height: 1.5;">is set equal to the mediant of b and its nearest lower neighbor (if it exists), or -∞ if not; likewise with b</span><span style="line-height: 1.5; vertical-align: sub;">u</span><span style="line-height: 1.5;">.</span><br /> | ||
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<!-- ws:start:WikiTextMathRule:3: | <!-- ws:start:WikiTextMathRule:3: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\sqrt{n_b \cdot d_b}}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cancel{c}(b))}{\sqrt{n_b \cdot d_b}}</script><!-- ws:end:WikiTextMathRule:3 --><br /> | ||
<br /> | <br /> | ||
where the q<span style="font-size: 12px; vertical-align: sub;">d</span>(b) now represent the unnormalized &quot;probabilities&quot;, and n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.<br /> | where the q<span style="font-size: 12px; vertical-align: sub;">d</span>(b) now represent the unnormalized &quot;probabilities&quot;, and n<span style="vertical-align: sub;">b</span> and d<span style="vertical-align: sub;">b</span> are the numerator and denominator, respectively, of basis rational b. Again, the set of basis rationals here is assumed to be all of those rationals of Tenney Height ≤ N for some N.<br /> | ||
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<!-- ws:start:WikiTextMathRule:4: | <!-- ws:start:WikiTextMathRule:4: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\max(n_b,d_b)}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cancel{c}(b))}{\max(n_b,d_b)}</script><!-- ws:end:WikiTextMathRule:4 --><br /> | ||
<br /> | <br /> | ||
where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.<br /> | where this time the set of basis rationals is assumed to be all of those of Weil Height ≤ N for some N.<br /> | ||
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<!-- ws:start:WikiTextMathRule:5: | <!-- ws:start:WikiTextMathRule:5: | ||
[[math]]&lt;br/&gt; | [[math]]&lt;br/&gt; | ||
q_d(b) = \frac{s_d(\ | q_d(b) = \frac{s_d(\cancel{c}(b))}{\|b\|}&lt;br/&gt;[[math]] | ||
--><script type="math/tex">q_d(b) = \frac{s_d(\ | --><script type="math/tex">q_d(b) = \frac{s_d(\cancel{c}(b))}{\|b\|}</script><!-- ws:end:WikiTextMathRule:5 --><br /> | ||
<br /> | <br /> | ||
where ||b|| denotes a complexity function mapping from rational numbers to reals.<br /> | where ||b|| denotes a complexity function mapping from rational numbers to reals.<br /> | ||
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<br /> | <br /> | ||
The use of a=∞ min-entropy would reflect this view. In contrast, the use of a=1 Shannon entropy reflects a much &quot;dumber&quot; process which performs no such analysis and perhaps doesn't even seek to &quot;choose&quot; any sort of &quot;victor&quot; rational at all. As the parameter a interpolates between these two options, it <span style="line-height: 1.5;">can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be &quot;intelligent&quot; and &quot;active&quot; in this way.</span><br /> | The use of a=∞ min-entropy would reflect this view. In contrast, the use of a=1 Shannon entropy reflects a much &quot;dumber&quot; process which performs no such analysis and perhaps doesn't even seek to &quot;choose&quot; any sort of &quot;victor&quot; rational at all. As the parameter a interpolates between these two options, it <span style="line-height: 1.5;">can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be &quot;intelligent&quot; and &quot;active&quot; in this way.</span><br /> | ||
<br /> | |||
Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as &quot;dumb&quot; processes whereby nothing in particular is being &quot;chosen,&quot; but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Different values of a can differentiate between the predominance given to these two types of effect in the overall construct of psychoacoustic concordance.<br /> | Some psychoacoustic effects naturally fit into this paradigm, such as the virtual pitch integration process, which actually does attempt to find a single victor when matching incoming chords with chunks of the harmonic series. Other psychoacoustic effects, such as that of beatlessness, may instead be better viewed as &quot;dumb&quot; processes whereby nothing in particular is being &quot;chosen,&quot; but where a more uniform distribution of matching rational numbers for a dyad simply generates a more discordant sonic effect. Different values of a can differentiate between the predominance given to these two types of effect in the overall construct of psychoacoustic concordance.<br /> | ||
<br /> | <br /> | ||