User:Grady/Harmonic similarity: Difference between revisions

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Of course, the knowledge that harmonic similarity should follow this pseudo-transitive property isn't very helpful in determining which notes are harmonically similar without establishing some basic similarity relationships first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other. The reason for this might be that we're highly accustomed to hearing harmonic timbres, so usually when we hear a note, we also somewhat prominently hear low integer multiples of that frequency as well, and over time this causes our brains to associate those higher harmonics as "similar" to the fundamental. The lower the integer multiple, the more prominently we hear it in those harmonic timbres, which corresponds to the fact that lower integer multiples represent stronger harmonic similarity relationships.

With this knowledge, we can use the pseudo-transitive property to discover higher-order harmonic similarity relationships. For example, two notes a perfect fifth apart (an interval of 3/2) aren't perceived as harmonically similar because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an octave apart (an interval of 2/1) ''are'' similar for that reason, and so are two notes a perfect twelfth~~, or tritave~~, apart (an interval of 3/1), and thus by the pseudo-transitive property, two notes a perfect fifth apart must be similar as well.

With this knowledge, we can use the pseudo-transitive property to discover higher-order harmonic similarity relationships. For example, two notes a perfect fifth apart (an interval of 3/2) aren't perceived as harmonically similar because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an octave apart (an interval of 2/1) ''are'' similar for that reason, and so are two notes a tritave, or perfect twelfth, apart (an interval of 3/1), and thus by the pseudo-transitive property, two notes a perfect fifth apart must be similar as well. For example, the note C4 is related to G4 (the note a perfect fifth above it) because both of those notes are related to G5 (or to C3).

Something to note about the base similarity relations is that the falloff in similarity with increasing integer values seems to be extremely fast. It's not exactly clear why this might be the case, but it's one of the core underlying assumptions of the theory. For example, the octave is a much, ''much'' stronger similarity relation than the tritave to most listeners, hence the notion of octave equivalence. A seeming caveat of this is that most listeners would rate the double octave (an interval of 4/1) to be a stronger similarity relation than the tritave. However, this is only because the double octave can be decomposed into two octaves, meaning this strong similarity can be explained with the pseudo-transitive property. In other words, the note C4 is very strongly related to the note C5, and C5 is very strongly related to C6, therefore C4 and C6 are also very strongly related. However, C4 is only moderately related to G5 (the note a tritave above it), and the tritave can't be decomposed into any simpler relationships.

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 Of course, the knowledge that harmonic similarity should follow this pseudo-transitive property isn't very helpful in determining which notes are harmonically similar without establishing some basic similarity relationships first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other. The reason for this might be that we're highly accustomed to hearing harmonic timbres, so usually when we hear a note, we also somewhat prominently hear low integer multiples of that frequency as well, and over time this causes our brains to associate those higher harmonics as "similar" to the fundamental. The lower the integer multiple, the more prominently we hear it in those harmonic timbres, which corresponds to the fact that lower integer multiples represent stronger harmonic similarity relationships.
-With this knowledge, we can use the pseudo-transitive property to discover higher-order harmonic similarity relationships. For example, two notes a perfect fifth apart (an interval of 3/2) aren't perceived as harmonically similar because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an octave apart (an interval of 2/1) ''are'' similar for that reason, and so are two notes a perfect twelfth, or tritave, apart (an interval of 3/1), and thus by the pseudo-transitive property, two notes a perfect fifth apart must be similar as well.
+With this knowledge, we can use the pseudo-transitive property to discover higher-order harmonic similarity relationships. For example, two notes a perfect fifth apart (an interval of 3/2) aren't perceived as harmonically similar because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an octave apart (an interval of 2/1) ''are'' similar for that reason, and so are two notes a tritave, or perfect twelfth, apart (an interval of 3/1), and thus by the pseudo-transitive property, two notes a perfect fifth apart must be similar as well. For example, the note C4 is related to G4 (the note a perfect fifth above it) because both of those notes are related to G5 (or to C3).
+Something to note about the base similarity relations is that the falloff in similarity with increasing integer values seems to be extremely fast. It's not exactly clear why this might be the case, but it's one of the core underlying assumptions of the theory. For example, the octave is a much, ''much'' stronger similarity relation than the tritave to most listeners, hence the notion of octave equivalence. A seeming caveat of this is that most listeners would rate the double octave (an interval of 4/1) to be a stronger similarity relation than the tritave. However, this is only because the double octave can be decomposed into two octaves, meaning this strong similarity can be explained with the pseudo-transitive property. In other words, the note C4 is very strongly related to the note C5, and C5 is very strongly related to C6, therefore C4 and C6 are also very strongly related. However, C4 is only moderately related to G5 (the note a tritave above it), and the tritave can't be decomposed into any simpler relationships.