Harmonic entropy: Difference between revisions

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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. 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.

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.

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.

Harmonic Entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [https://asa.scitation.org/doi/10.1121/1.1914448 1973 paper] "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.

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.

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 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. 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.
+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.
-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.
+Harmonic Entropy was originally intended to measure, in particular, the "virtual fundamental" aspect of psychoacoustic concordance, being modeled on J. Goldstein's [https://asa.scitation.org/doi/10.1121/1.1914448 1973 paper] "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.
 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.