User:Grady/Harmonic similarity: Difference between revisions

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With this knowledge, we can use the pseudo-transitive property to discover higher-order similarity relations. For example, two notes a [[3/2|perfect fifth]] apart are perceived as harmonically similar, but not 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 [[2/1|octave]] apart ''are'' similar for that reason, and so are two notes a [[3/1|tritave]] apart, 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) by a basic similarity relation.

Something to note about the basic 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 [[4/1|double octave]] 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 relations.

Something to note about the basic 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 all this is that most listeners would rate the [[4/1|double octave]] to be a stronger similarity relation than the tritave, despite being a higher integer multiple. 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 relations.

=== Margin for error ===

Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like 12edo to be harmonically similar, even if the ratios are inexact.

Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact.

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 With this knowledge, we can use the pseudo-transitive property to discover higher-order similarity relations. For example, two notes a [[3/2|perfect fifth]] apart are perceived as harmonically similar, but not 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 [[2/1|octave]] apart ''are'' similar for that reason, and so are two notes a [[3/1|tritave]] apart, 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) by a basic similarity relation.
-Something to note about the basic 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 [[4/1|double octave]] 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 relations.
+Something to note about the basic 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 all this is that most listeners would rate the [[4/1|double octave]] to be a stronger similarity relation than the tritave, despite being a higher integer multiple. 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 relations.
 === Margin for error ===
-Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like 12edo to be harmonically similar, even if the ratios are inexact.
+Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact.