Chord complexity: Difference between revisions

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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.

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 ''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 the simple complexity above.

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.

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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 modeling. (As we will see, we depart from strict adherence to this criterion anyway.)

== Dirichlet complexity ==

@@ Line 72: / Line 72: @@
 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'', 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 the simple complexity above.
 . 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.
@@ Line 78: / Line 78: @@
 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 modeling. (As we will see, we depart from strict adherence to this criterion anyway.)
 == Dirichlet complexity ==