Harmonic entropy: Difference between revisions
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In recent years, it has become clearer that the model can also be very useful in modeling other types of concordance as well, particularly for dyads, where the same model does a very good job in also predicting beatlessness, periodicity buzz, and so on. In particular, Erlich has often suggested the same model, perhaps with slightly different parameters, can also be useful to measure how easy it is to tune a dyad by ear on an instrument such as a guitar, or how much of a sense of being "locked-in" the dyad gives as it is tuned more closely to JI. This may be less related to the perception of virtual fundamentals than it is to beatlessness and so on. | In recent years, it has become clearer that the model can also be very useful in modeling other types of concordance as well, particularly for dyads, where the same model does a very good job in also predicting beatlessness, periodicity buzz, and so on. In particular, Erlich has often suggested the same model, perhaps with slightly different parameters, can also be useful to measure how easy it is to tune a dyad by ear on an instrument such as a guitar, or how much of a sense of being "locked-in" the dyad gives as it is tuned more closely to JI. This may be less related to the perception of virtual fundamentals than it is to beatlessness and so on. | ||
However, it should be noted that the various aspects of psychoacoustic concordance tend to diverge quite strongly in their behavior for larger chords, and thus, when modeling different aspects of psychoacoustic concordance, different ways of generalizing the dyadic model to higher-cardinality chords may be appropriate. In particular, when | However, it should be noted that the various aspects of psychoacoustic concordance tend to diverge quite strongly in their behavior for larger chords, and thus, when modeling different aspects of psychoacoustic concordance, different ways of generalizing the dyadic model to higher-cardinality chords may be appropriate. In particular, when modeling beatlessness, Erlich has suggested instead looking only at the entropies of the pairwise dyadic subsets of the chord, so that the major and minor chords would be ranked equal in beatlessness, whereas they would not be ranked equal in their ability to produce a clear virtual fundamental (the major chord would be much stronger and lower in entropy). | ||
=== Concordance vs. actual consonance === | === Concordance vs. actual consonance === | ||
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An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of {{w|Entropy (information_ theory)|Shannon entropy}} by replacing it instead with {{w|Rényi entropy}}, a {{w|q-analog|''q''-analog}} of Shannon's original entropy. This can be thought of as adding a second parameter, called ''a'', to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval. | An extension to the base Harmonic Entropy model, proposed by Mike Battaglia, is to generalize the use of {{w|Entropy (information_ theory)|Shannon entropy}} by replacing it instead with {{w|Rényi entropy}}, a {{w|q-analog|''q''-analog}} of Shannon's original entropy. This can be thought of as adding a second parameter, called ''a'', to the model, reflecting how "intelligent" the brain's "decoding" process is when determining the most likely JI interpretation of an ambiguous interval. | ||
=== Definitions and | === Definitions and background === | ||
The '''Harmonic Rényi entropy of order ''a''''' of an incoming dyad can be defined as follows: | The '''Harmonic Rényi entropy of order ''a''''' of an incoming dyad can be defined as follows: | ||
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This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers. | This enables us to speak cognizantly of the harmonic entropy of an interval as measured against ''all'' rational numbers. | ||
=== Background: | === Background: unnormalized entropy === | ||
Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as ''Q''(''j''|''c''). For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as ''P''(''j''|''c''). | Our derivation only analytically continues the entropy function for the "unnormalized" set of probabilities, which we previously wrote as ''Q''(''j''|''c''). For this definition to be philosophically perfect, we would want to analytically continue the entropy function for the normalized sense of probabilities, previously written as ''P''(''j''|''c''). | ||
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Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times ({{nowrap|1 − a}}), rather than the UHE itself. The reasons for this will become clear later. If we do so, we get | Lastly, it so happens that it will be much easier to understand our analytic continuation if we look at the exponential of the UHE times ({{nowrap|1 − ''a''}}), rather than the UHE itself. The reasons for this will become clear later. If we do so, we get | ||
$$\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)$$ | $$\displaystyle \exp((1-a) \text{UHE}_a(c)) = \left( S^a \ast K^a \right)(-c)$$ | ||
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Note that this function is simply a monotonic transformation of the original, and so preserves the exact same concordance ranking on all intervals. | Note that this function is simply a monotonic transformation of the original, and so preserves the exact same concordance ranking on all intervals. | ||
==== Analytic | ==== Analytic continuation of the convolution kernel ==== | ||
The definition for ''K'' is: | The definition for ''K'' is: | ||
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Finally, it is noteworthy that for {{nowrap|''a'' > 2}}, we end up looking at slices of the zeta function for which {{nowrap|Re(''z'') > 1}}. This is where our original unnormalized HE function should converge as {{nowrap|''N'' → ∞}}, corresponding to the region where the Riemann zeta function Dirichlet series converges. For these values of ''a'', the exp-UHE ''is'' positive. So, we can take the log again and look at the usual UHE. This can be useful for plotting, since exp-UHE tends to "flatten" out the curve for high values of ''a'', whereas taking the log accentuates the minima and maxima (and more closely resembles the usual HRE). | Finally, it is noteworthy that for {{nowrap|''a'' > 2}}, we end up looking at slices of the zeta function for which {{nowrap|Re(''z'') > 1}}. This is where our original unnormalized HE function should converge as {{nowrap|''N'' → ∞}}, corresponding to the region where the Riemann zeta function Dirichlet series converges. For these values of ''a'', the exp-UHE ''is'' positive. So, we can take the log again and look at the usual UHE. This can be useful for plotting, since exp-UHE tends to "flatten" out the curve for high values of ''a'', whereas taking the log accentuates the minima and maxima (and more closely resembles the usual HRE). | ||
=== Interpretation as a | === Interpretation as a new free parameter: the weighting exponent === | ||
In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from (''nd'')<sup>0.5</sup> to some other (''nd'')<sup>''w''</sup>, with {{nowrap|''w'' > 1}}, for the sake of obtaining a series that converges. We then changed the exponent back to 0.5. | In our original derivation of the analytic continuation, we temporarily changed the weighting for rationals from (''nd'')<sup>0.5</sup> to some other (''nd'')<sup>''w''</sup>, with {{nowrap|''w'' > 1}}, for the sake of obtaining a series that converges. We then changed the exponent back to 0.5. | ||
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Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit. | Lastly, you will note that for the special value {{nowrap|''w'' {{=}} 0.5}}, corresponding to the usual <math>\sqrt{nd}</math> weighting, we end up dividing by the term ζ(1). This is the only pole in the zeta function, so we wind up dividing by infinity, making the entire function zero, as pointed out by Martin Gough. However, as we can get arbitrarily close to {{nowrap|''w'' {{=}} 0.5}} and still exhibit the behavior that the unreduced and reduced functions are scaled versions of one another, we can simply use the unreduced version of exp-UHE for {{nowrap|''w'' {{=}} 0.5}} and consider it equivalent to reduced exp-UHE in the limit. | ||
== | == Todo == | ||
There are a number of things that need to be added to this article. Below are listed some for reference: | There are a number of things that need to be added to this article. Below are listed some for reference: | ||
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[[Category:Consonance and dissonance]] | [[Category:Consonance and dissonance]] | ||
[[Category:Harmonic entropy| ]] <!-- main article --> | [[Category:Harmonic entropy| ]] <!-- main article --> | ||
[[Category: | [[Category:Essays]] | ||