Chord complexity: Difference between revisions

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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 {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|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).

If we don't care about modelling the latter effect, and only care about modelling 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 {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|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).

Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.

If we do care about ~~modeling~~ the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.

If we do care about modelling the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.

== Some Caveats in Expanding to Chords of Arbitrary Size ==

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Now, of course, we have cheated somewhat—note that we have extended the chord in such a way that the differences between each frequency ratio are 2, making this an isoharmonic chord, which are known to strongly exhibit "periodicity buzz." Still, though, this general principle seems to hold to some degree, even if some of the notes are moved around by 1 here and there to form a non-isoharmonic chord, and it works well enough as a basic guiding principle to be viewed as significant, at least in the view of this author.

So we would at least like some kind of reasonable starting point in ~~modeling~~ this phenomenon, so that we can compare chords of different sizes.

So we would at least like some kind of reasonable starting point in modelling this phenomenon, so that we can compare chords of different sizes.

=== A simplified, but useful criterion ===

One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:

1. Given some fundamental frequency ''f''~~, an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6~~:~~7. This is just a restatement of our definition of the simple complexity above.~~

Given some fundamental frequency ''f'':

2. ~~Given some fundamental frequency~~ ''f'' ~~and~~ a chord built from ''f''{{`s}} harmonics, ~~adding~~ ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

1. An ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an ''N''-note chord built from ''f''{{`s}} lower harmonics. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of simple complexity above.

2. Adding ''another'' note from ''f''{{`s}} harmonics to a chord built from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.

The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.

Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good-enough" rule of thumb which is simple enough to be worth ~~modeling~~. (As we will see, we depart from strict adherence to this criterion anyway.)

Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good enough" rule of thumb which is simple enough to be worth modelling. (As we will see, we depart from strict adherence to this criterion anyway.)

== Dirichlet complexity ==

@@ Line 26: / Line 26: @@
 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 {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|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).
+If we don't care about modelling the latter effect, and only care about modelling 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 {{nowrap|log(''n''*''d'')}}, and the [[Weil height]] of {{nowrap|max(''n'', ''d'')}} or {{nowrap|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).
 Note that the Benedetti height and Tenney height are basically the same thing; it is fairly common when talking about height functions to equivocate between the logarithmic and non-logarithmic versions of the same function, as they rank rationals the same either way. We will sometimes equivocate between the two names, but in general the name "Benedetti height" has been given to the non-logarithmic version and the name "Tenney height" to the logarithmic version.
-If we do care about modeling the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.
+If we do care about modelling the aforementioned detuning effect, then [[Harmonic entropy]] is one way to model this, which has a free parameter determining how "tolerant" the listener's auditory system is to perceiving slightly detuned versions of simple ratios as slightly-off versions of themselves, rather than perceiving them as other, more complex ratios. Tenney and Weil height can also be used to seed the Harmonic entropy calculation to begin with, so that they can be thought of as "primitives" from which increasingly sophisticated models can be built.
 == Some Caveats in Expanding to Chords of Arbitrary Size ==
@@ Line 67: / Line 67: @@
 Now, of course, we have cheated somewhat—note that we have extended the chord in such a way that the differences between each frequency ratio are 2, making this an isoharmonic chord, which are known to strongly exhibit "periodicity buzz." Still, though, this general principle seems to hold to some degree, even if some of the notes are moved around by 1 here and there to form a non-isoharmonic chord, and it works well enough as a basic guiding principle to be viewed as significant, at least in the view of this author.
-So we would at least like some kind of reasonable starting point in modeling this phenomenon, so that we can compare chords of different sizes.
+So we would at least like some kind of reasonable starting point in modelling this phenomenon, so that we can compare chords of different sizes.
 === A simplified, but useful criterion ===
 One possible way forward is to imagine that the incoming JI chord as a set of upper harmonics of some fundamental frequency—the GCD of the notes of the chord—and we want to quantify how strongly the chord matches that virtual fundamental. We can make some very basic assumptions:
-. Given some fundamental frequency ''f'', an ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an N-note chord built from lower harmonics of ''f''. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of the simple complexity above.
+Given some fundamental frequency ''f'':
-. Given some fundamental frequency ''f'' and a chord built from ''f''{{`s}} harmonics, adding ''another'' note from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.
+. An ''N''-note chord built from very high harmonics of ''f'' will be a weaker match than an ''N''-note chord built from ''f''{{`s}} lower harmonics. In other words, 4:5:6 matches "1" better than 5:6:7. This is just a restatement of our definition of simple complexity above.
+. Adding ''another'' note from ''f''{{`s}} harmonics to a chord built from ''f''{{`s}} harmonics always ''increases'' the strength of the match to ''f''. In other words, 4:5:6:7 matches "1" better than 4:5:6.
 The second proposition is the interesting one. It means that the chord 1:2 evokes "1" less than 1:2:3, which is less than 1:2:3:4, and so on, so that the chord 1:2:3:4:... evokes the frequency "1" most strongly.
-Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good-enough" rule of thumb which is simple enough to be worth modeling. (As we will see, we depart from strict adherence to this criterion anyway.)
+Strictly speaking, this phenomenon—the reinforcement of virtual pitch—is most evident if the notes of the chord are played with sine waves, with volume decreasing as you get higher into the harmonic series. In that situation, the chord 1:2:3:4:5:6:7:... is basically something like a sawtooth wave. It isn't quite so apparent that if you instead have all harmonics at equal volume, the resulting "delta comb" should really be viewed as more "consonant" than a sine wave in an absolute sense. This is even more true if, instead of sine waves, all of the notes are being played with some arbitrary harmonic timbre! Still, though, we still view the basic spirit of this as a "good enough" rule of thumb which is simple enough to be worth modelling. (As we will see, we depart from strict adherence to this criterion anyway.)
 == Dirichlet complexity ==