Harmonic entropy: Difference between revisions
Wikispaces>mbattaglia1 **Imported revision 515616378 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition. | The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition. | ||
The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad "matches" every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear "victor" basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant. A clear mathematical way of quantifying this is via <span style="line-height: 1.5;">the | The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad "matches" every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear "victor" basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant. A clear mathematical way of quantifying this is via <span style="line-height: 1.5;">the [[@http://en.wikipedia.org/wiki/Entropy_(information_theory)|Shannon entropy]] of the probability distribution:</span> | ||
[[math]] | [[math]] | ||
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where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found [[@http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf|here]]. | where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found [[@http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf|here]]. | ||
Musically speaking, the parameter //**a**// can be interpreted as the extent to which | Musically speaking, the parameter //**a**// can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be an intelligent, active process by which the auditory system actively analyzes the probability distribution and seeks out the "victor" rational with the greatest probability. 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. Increasing values of //a// correspond to a model that is more "active" in this way. | ||
This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in [[@http://users.cis.fiu.edu/~smithg/papers/qest11.pdf|this paper]] and [[@http://www.ietf.org/rfc/rfc4086.txt|this RFC]]. More precisely, the Rényi entropy of order ∞, also called the **min-entropy**, is used to measure the strength of the randomness used to define a cryptographic secret against a "worst-case" attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be "cracked" by an intelligent auditory system, we can consider that the analogous "worst-case attacker" would be a "best-case auditory system" which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational. | This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in [[@http://users.cis.fiu.edu/~smithg/papers/qest11.pdf|this paper]] and [[@http://www.ietf.org/rfc/rfc4086.txt|this RFC]]. More precisely, the Rényi entropy of order ∞, also called the **min-entropy**, is used to measure the strength of the randomness used to define a cryptographic secret against a "worst-case" attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be "cracked" by an intelligent auditory system, we can consider that the analogous "worst-case attacker" would be a "best-case auditory system" which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational. | ||
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The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition.<br /> | The original Harmonic Entropy model limited itself to working with dyads. More recently, work by Steve Martin and others has extended this basic idea to higher-cardinality chords. This article will concern itself with dyads, as the dyadic case is still the most well-developed, and many of the ideas extend naturally to larger chords without need for much exposition.<br /> | ||
<br /> | <br /> | ||
The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad &quot;matches&quot; every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear &quot;victor&quot; basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant. A clear mathematical way of quantifying this is via <span style="line-height: 1.5;">the | The general idea of Harmonic Entropy is to first develop a discrete probability distribution quantifying how strongly an arbitrary incoming dyad &quot;matches&quot; every element in a set of basis rational intervals, and then seeing how evenly distributed the resulting probabilities are. If the distribution for some dyad is spread out very evenly, such that there is no clear &quot;victor&quot; basis interval that dominates the distribution, the dyad is considered to be more discordant; on the other extreme, if the distribution tends to concentrate on one or a small set of dyads, the dyad is considered to be more concordant. A clear mathematical way of quantifying this is via <span style="line-height: 1.5;">the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Entropy_(information_theory)" rel="nofollow" target="_blank">Shannon entropy</a> of the probability distribution:</span><br /> | ||
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where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found <a class="wiki_link_ext" href="http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf" rel="nofollow" target="_blank">here</a>.<br /> | where the p_d(b) are the probabilities assigned by dyad d to each basis rational b. It is noteworthy that Rényi entropy converges to Shannon entropy in the limit as a→1, a fact which can be verified using L'Hôpital's rule as found <a class="wiki_link_ext" href="http://www.sonycsl.co.jp/person/nielsen/Note-HopitalRuleShannonRenyiTsallis.pdf" rel="nofollow" target="_blank">here</a>.<br /> | ||
<br /> | <br /> | ||
Musically speaking, the parameter <em><strong>a</strong></em> can be interpreted as the extent to which | Musically speaking, the parameter <em><strong>a</strong></em> can be interpreted as the extent to which the rational-matching process for incoming dyads is considered to be an intelligent, active process by which the auditory system actively analyzes the probability distribution and seeks out the &quot;victor&quot; rational with the greatest probability. 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. Increasing values of <em>a</em> correspond to a model that is more &quot;active&quot; in this way.<br /> | ||
<br /> | <br /> | ||
This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in <a class="wiki_link_ext" href="http://users.cis.fiu.edu/~smithg/papers/qest11.pdf" rel="nofollow" target="_blank">this paper</a> and <a class="wiki_link_ext" href="http://www.ietf.org/rfc/rfc4086.txt" rel="nofollow" target="_blank">this RFC</a>. More precisely, the Rényi entropy of order ∞, also called the <strong>min-entropy</strong>, is used to measure the strength of the randomness used to define a cryptographic secret against a &quot;worst-case&quot; attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be &quot;cracked&quot; by an intelligent auditory system, we can consider that the analogous &quot;worst-case attacker&quot; would be a &quot;best-case auditory system&quot; which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational.<br /> | This interpretation comes from the use of the Rényi entropy in cryptography as a measure of the strength of a cryptographic code in the face of an intelligent attacker, an application for which Shannon entropy has long been known to be insufficient as described in <a class="wiki_link_ext" href="http://users.cis.fiu.edu/~smithg/papers/qest11.pdf" rel="nofollow" target="_blank">this paper</a> and <a class="wiki_link_ext" href="http://www.ietf.org/rfc/rfc4086.txt" rel="nofollow" target="_blank">this RFC</a>. More precisely, the Rényi entropy of order ∞, also called the <strong>min-entropy</strong>, is used to measure the strength of the randomness used to define a cryptographic secret against a &quot;worst-case&quot; attacker who has complete knowledge of the probability distribution from which cryptographic secrets are drawn. In a musical context, by considering the incoming dyad as analogous to a cryptographic code which is attempting to be &quot;cracked&quot; by an intelligent auditory system, we can consider that the analogous &quot;worst-case attacker&quot; would be a &quot;best-case auditory system&quot; which has complete awareness of the probability distribution for any incoming dyad and actively chooses the strongest rational.<br /> | ||