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, 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~~

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.

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

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:

* 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'''.

* 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

* 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'''.

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.

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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, 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
+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.
-* effects such as these tend to appear most strongly for those chords with large subsets that correspond to simple chunks of the harmonic series
+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:
-* 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'''.
+* 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
+* 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'''.
 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.