User:Grady/Harmonic similarity: Difference between revisions

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== Motivation ==

I developed this theory to attempt to answer the question, "Why are two notes an [[~~2/1|~~octave]] apart perceived as equivalent?" The typical explanation given is that it has something to do with the fact that the upper note is prominently featured in the harmonic spectrum of the lower note, assuming a typical [[harmonic timbre]]. However, this completely fails to explain why the octave is special in this regard, and why a similar phenomenon seemingly never occurs with other harmonics (even if less commonly, under more specific circumstances, or in a less pronounced manner), most namely the [[3/1|tritave]].

I developed this theory to attempt to answer the question, "Why are two notes an [[octave]] apart perceived as equivalent?" The typical explanation given is that it has something to do with the fact that the upper note is prominently featured in the harmonic spectrum of the lower note, assuming a typical [[harmonic timbre]]. However, this completely fails to explain why the octave is special in this regard, and why a similar phenomenon seemingly never occurs with other harmonics (even if less commonly, under more specific circumstances, or in a less pronounced manner), most namely the [[3/1|tritave]].

Some people have claimed to hear tritave equivalence, but so far no one has purported to hear notes any number of tritaves apart as being within the same [[pitch class]], something many musicians simply take for granted with regard to octaves, and in my opinion, a necessary requisite for true equivalence perception.

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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 relations first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other, and said relation is stronger the lower the integer multiple is. 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. This would also explain why lower integer multiples correspond to stronger basic similarity relations, because the lower the integer multiple is, the more prominent the corresponding overtone typically is in those harmonic timbres.

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.

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 [[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.

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 == Motivation ==
-I developed this theory to attempt to answer the question, "Why are two notes an [[2/1|octave]] apart perceived as equivalent?" The typical explanation given is that it has something to do with the fact that the upper note is prominently featured in the harmonic spectrum of the lower note, assuming a typical [[harmonic timbre]]. However, this completely fails to explain why the octave is special in this regard, and why a similar phenomenon seemingly never occurs with other harmonics (even if less commonly, under more specific circumstances, or in a less pronounced manner), most namely the [[3/1|tritave]].
+I developed this theory to attempt to answer the question, "Why are two notes an [[octave]] apart perceived as equivalent?" The typical explanation given is that it has something to do with the fact that the upper note is prominently featured in the harmonic spectrum of the lower note, assuming a typical [[harmonic timbre]]. However, this completely fails to explain why the octave is special in this regard, and why a similar phenomenon seemingly never occurs with other harmonics (even if less commonly, under more specific circumstances, or in a less pronounced manner), most namely the [[3/1|tritave]].
 Some people have claimed to hear tritave equivalence, but so far no one has purported to hear notes any number of tritaves apart as being within the same [[pitch class]], something many musicians simply take for granted with regard to octaves, and in my opinion, a necessary requisite for true equivalence perception.
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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 relations first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other, and said relation is stronger the lower the integer multiple is. 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. This would also explain why lower integer multiples correspond to stronger basic similarity relations, because the lower the integer multiple is, the more prominent the corresponding overtone typically is in those harmonic timbres.
-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.
+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 [[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.