Chord complexity: Difference between revisions

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== The Psychoacoustics of a Dyad ==

Consonance and dissonance are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.

Consonance and dissonance of tones and their combinations are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.

When we are only looking at dyads, many of the ~~purely~~ psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple frequency ratios. In general, for some ratio n/d, these qualities tend to decrease as n and d increase, unless n/d is a complex ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.

When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio n/d, these qualities tend to decrease as n and d increase, unless n/d is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.

If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for n/d, any function of n and d that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of n*d and log(n*d), and the [[Weil height]] of max(n,d) or log(max(n,d)), which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).

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 == The Psychoacoustics of a Dyad ==
-Consonance and dissonance are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.
+Consonance and dissonance of tones and their combinations are rather tricky and elusive phenomena to model, in part because the terms don't unambiguously refer to one thing. David Huron, for instance, lists at least 14 different types of dissonance [https://archive.md/ICWLQ here], some of which are psychoacoustic, some of which depend on some kind of larger musical or "tonal" setting, and some of which are clearly dependent on learned expectations. It is thus very likely that consonance is a multidimensional quantity that cannot be represented by a single scalar value.
-When we are only looking at dyads, many of the purely psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple frequency ratios. In general, for some ratio n/d, these qualities tend to decrease as n and d increase, unless n/d is a complex ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.
+When we are only looking at dyads made from harmonic sounds, many of the psychoacoustic qualities associated with consonance above simplify to the same basic metric, which is that they are strongest for dyads that are close to simple (numerically small) frequency ratios. In general, for some ratio n/d, these qualities tend to decrease as n and d increase, unless n/d is a complex (numerically large) ratio that happens to also be very close to a simple ratio. In that situation, the perception of the complex ratio per se starts to be eclipsed by the perception of it as a slightly-detuned version of the nearby simpler ratio.
 If we don't care about modeling the latter effect, and only care about modeling the complexity of a ratio directly, then for n/d, any function of n and d that is monotonically increasing in either variable will do. The [[Height|height]] functions on this Wiki are some simple examples of this. The two most commonly used are the [[Benedetti height]]/[[Tenney height]] of n*d and log(n*d), and the [[Weil height]] of max(n,d) or log(max(n,d)), which have the useful property that their logarithmic versions are norms on the space of [[monzos]] (in particular, the first is a type of L1 norm).