Harmonic entropy: Difference between revisions

Line 11:

There has been much research specifically on the musical implications critical band effects in the literature (e.g. Sethares's work), which are perhaps the psychoacoustic phenomena that readers are most familiar with. However, the modern xenharmonic community has displayed immense interest in exploring the other effects mentioned above as well, which have proven extremely important to the development of modern xenharmonic music.

These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former.

These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former. However, it is useful to have a term that refers to the general presence of these types of effects. The term "consonance" has sometimes been used for this; however, there are many other meanings of consonance which may not be psychoacoustic in nature. Thus, we instead speak of a general notion of psychoacoustic '''concordance'' - the degree to which effects such as the above will appear when an arbitrary musical interval or chord is played - as well and psychoacoustic '''discordance'''. Timbral fusion, the appearance of virtual fundamentals, beatlessness, and periodicity buzz, can all be thought of as different aspects of psychoacoustic concordance.

However, ~~suppose we want~~ to ~~come up with a combined~~ measure ~~for~~ how ~~often effects~~ such as the ~~above tend~~ to ~~occur~~. It is ~~then useful~~ to ~~note that:~~

Harmonic Entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [1973 paper](https://asa.scitation.org/doi/10.1121/1.1914448) "An optimum processor theory for the central formation of the pitch of complex tones." It can also be thought of as an elaboration on similar research by Terhardt, Parncutt and others, which addresses some of the shortcomings suggested by Erlich in prior models. The model basically asks how "confused your brain is," in Erlich's words, when trying to match the incoming sound to that of one single harmonic timbre played on a missing fundamental. However, 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 more of a feature of how much the dyads exhibits some of the other psychoacoustic qualities spoken of previously, such as beatlessness, rather than how easy it is for your brain to choose one particular missing fundamental.

* In general, ~~effects such~~ as ~~these tend~~ to ~~appear most strongly for those chords with large subsets that correspond~~ to ~~simple chunks~~ of the ~~harmonic series~~

For dyads, the basic harmonic entropy model is fairly simple: it places the dyad we are trying to measure amidst a backdrop of JI candidates. Then, it uses a point-spread function to determine the relative strengths of the match to each, which are then normalized and treated as probabilities. The "entropy" of the resulting probability distribution is a way to measure how closely this distribution tends to focus on one possibility, rather than being spread out among a set of equally-likely possibilities. If there is only one clear choice of dyad which far exceeds all others in probability, the entropy will be lower. If, on the other hand, there are many equally-likely probabilities, the entropy will be higher.

* In general, the ~~effects produced exhibit some degree of tolerance for mistuning~~

~~This enables us~~ to ~~speak of~~ a ~~general notion~~ of the psychoacoustic ~~'''~~concordance~~'''~~ of ~~an interval~~ - the ~~degree to which effects such as~~ the ~~above will appear when an arbitrary musical~~ chord ~~is played. Additionally~~, chords ~~which are very inharmonic often exhibit~~ a ~~quality known as psychoacoustic '''discordance'''~~.

The basic harmonic entropy model can also be extended to modeling triads, tetrads, and so on; the standard way to do so is to simply look at the incoming triad's match to a set of candidate JI triads, and likewise with tetrads, and etc. 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).

While psychoacoustic concordance is not a feature universal to all styles of music, it has been utilized significantly in Western music in the study of intonation. For instance, flexible-pitch ensembles operating within 12-EDO, such as barbershop quartets and string ensembles, will often adjust intonationally from the underlying 12-EDO reference to maximize the concordance of individual chords. Indeed, the entire history of Western tuning theory -- from meantone temperament, to the various Baroque well-temperaments, to 12-EDO itself, to the modern [[Regular_Temperaments|theory of regular temperament]] -- can be seen as an attempt to reason mathematically about how to generate manageable tuning systems that will maximize concordance and minimize discordance.

~~The Harmonic Entropy model is a simple way of quantifying how much an arbitrary chord will exhibit psychoacoustic concordance.~~

Concordance has often been confused with actual musical consonance, an unfortunate fact made more common by the psychoacoustics literature under the unfortunate name '''sensory consonance''', most often used to refer to phenomena related to roughness and beatlessness specifically. This is not to be confused with the more familiar construct of tonal stability, typically just called "consonance" in Western common practice music theory and sometimes clarified as "musical consonance" in the music cognition literature. To make matters worse, the literature has also at times referred to concordance -- and not tonal stability -- as '''tonal consonance''', often referring to phenomena related to virtual pitch integration, creating a complete terminological mess. As a result, the term "consonance" has been completely avoided in this article.

@@ Line 11: / Line 11: @@
 There has been much research specifically on the musical implications critical band effects in the literature (e.g. Sethares's work), which are perhaps the psychoacoustic phenomena that readers are most familiar with. However, the modern xenharmonic community has displayed immense interest in exploring the other effects mentioned above as well, which have proven extremely important to the development of modern xenharmonic music.
-These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former.
+These effects sometimes behave differently, and do not always appear strictly in tandem with one another. For instance, Paul Erlich has noted that most models for beatlessness measure 10:12:15 and 4:5:6 as being identical, whereas the latter exhibits more timbral fusion and a more salient virtual fundamental than the former. However, it is useful to have a term that refers to the general presence of these types of effects. The term "consonance" has sometimes been used for this; however, there are many other meanings of consonance which may not be psychoacoustic in nature. Thus, we instead speak of a general notion of psychoacoustic '''concordance'' - the degree to which effects such as the above will appear when an arbitrary musical interval or chord is played - as well and psychoacoustic '''discordance'''. Timbral fusion, the appearance of virtual fundamentals, beatlessness, and periodicity buzz, can all be thought of as different aspects of psychoacoustic concordance.
-However, suppose we want to come up with a combined measure for how often effects such as the above tend to occur. It is then useful to note that:
+Harmonic Entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [1973 paper](https://asa.scitation.org/doi/10.1121/1.1914448) "An optimum processor theory for the central formation of the pitch of complex tones." It can also be thought of as an elaboration on similar research by Terhardt, Parncutt and others, which addresses some of the shortcomings suggested by Erlich in prior models. The model basically asks how "confused your brain is," in Erlich's words, when trying to match the incoming sound to that of one single harmonic timbre played on a missing fundamental. However, 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 more of a feature of how much the dyads exhibits some of the other psychoacoustic qualities spoken of previously, such as beatlessness, rather than how easy it is for your brain to choose one particular missing fundamental.
-* In general, effects such as these tend to appear most strongly for those chords with large subsets that correspond to simple chunks of the harmonic series
+For dyads, the basic harmonic entropy model is fairly simple: it places the dyad we are trying to measure amidst a backdrop of JI candidates. Then, it uses a point-spread function to determine the relative strengths of the match to each, which are then normalized and treated as probabilities. The "entropy" of the resulting probability distribution is a way to measure how closely this distribution tends to focus on one possibility, rather than being spread out among a set of equally-likely possibilities. If there is only one clear choice of dyad which far exceeds all others in probability, the entropy will be lower. If, on the other hand, there are many equally-likely probabilities, the entropy will be higher.
-* In general, the effects produced exhibit some degree of tolerance for mistuning
-This enables us to speak of a general notion of the psychoacoustic '''concordance''' of an interval - the degree to which effects such as the above will appear when an arbitrary musical chord is played. Additionally, chords which are very inharmonic often exhibit a quality known as psychoacoustic '''discordance'''.
+The basic harmonic entropy model can also be extended to modeling triads, tetrads, and so on; the standard way to do so is to simply look at the incoming triad's match to a set of candidate JI triads, and likewise with tetrads, and etc. 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).
 While psychoacoustic concordance is not a feature universal to all styles of music, it has been utilized significantly in Western music in the study of intonation. For instance, flexible-pitch ensembles operating within 12-EDO, such as barbershop quartets and string ensembles, will often adjust intonationally from the underlying 12-EDO reference to maximize the concordance of individual chords. Indeed, the entire history of Western tuning theory -- from meantone temperament, to the various Baroque well-temperaments, to 12-EDO itself, to the modern [[Regular_Temperaments|theory of regular temperament]] -- can be seen as an attempt to reason mathematically about how to generate manageable tuning systems that will maximize concordance and minimize discordance.
-The Harmonic Entropy model is a simple way of quantifying how much an arbitrary chord will exhibit psychoacoustic concordance.
 Concordance has often been confused with actual musical consonance, an unfortunate fact made more common by the psychoacoustics literature under the unfortunate name '''sensory consonance''', most often used to refer to phenomena related to roughness and beatlessness specifically. This is not to be confused with the more familiar construct of tonal stability, typically just called "consonance" in Western common practice music theory and sometimes clarified as "musical consonance" in the music cognition literature. To make matters worse, the literature has also at times referred to concordance -- and not tonal stability -- as '''tonal consonance''', often referring to phenomena related to virtual pitch integration, creating a complete terminological mess. As a result, the term "consonance" has been completely avoided in this article.