Chord complexity: Difference between revisions

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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 is most true if the notes of the chord are played with sine waves, with the 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, ~~that~~ 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 see, we will depart from strict adherence to this criterion anyway.)

Strictly speaking, this is most true if the notes of the chord are played with sine waves, with the 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 see, we will depart from strict adherence to this criterion anyway.)

== Dirichlet Complexity ==

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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 is most true if the notes of the chord are played with sine waves, with the 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, that 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 see, we will depart from strict adherence to this criterion anyway.)
+Strictly speaking, this is most true if the notes of the chord are played with sine waves, with the 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 see, we will depart from strict adherence to this criterion anyway.)
 == Dirichlet Complexity ==