User:Grady/Harmonic similarity: Difference between revisions

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== Explanation of octave equivalence ==

Most musicians think of two notes any number of octaves apart as being equivalent to some extent; that is, being within the same [[pitch class]]. This obviously doesn't mean musicians are entirely unable to distinguish between notes that are some number of octaves apart, but it does mean that it's viewed as a type of [[wikipedia:Equivalence_relation|equivalence relation]], not just a similarity relation. That is, there is some property, namely pitch class, which is considered to be entirely [[wikipedia:Invariant_(mathematics)|invariant]] under transposition by octaves. This seems in direct contradiction to the theory of harmonic similarity, which posits that two notes separated by one or more octaves merely share a similarity relation, not an equivalence relation. I have two different hypotheses that attempt to explain this, which I'll detail below.

~~First, it's important to reiterate that the falloff in similarity from the octave to the tritave is assumed to be very prominent for most listeners.~~

=== Limited hearing range ===

We humans have a finite hearing range of only around 10 octaves, and only around six or seven octaves of that range is actually musically useful. Because of this, it may be impossible for any number of stacked octaves that fits within our musical hearing range to have a lower similarity than the tritave for most listeners, since the dissimilarity induced by each successive octave doesn't accumulate quickly enough. (Again, the tritave is assumed to have significantly higher dissimilarity than the octave.) This may mean that if we could experience sound perception with a wider hearing range, that it would be possible to hear two notes a very high number of octaves apart as no longer sounding very equivalent due to the accumulation of dissimilarity.

We humans have a finite hearing range of only around 10 octaves, and only around six or seven octaves of that range is actually musically useful. Because of this, it may be impossible for any number of stacked octaves that fits within our musical hearing range to have a lower similarity than the tritave for most listeners, since the dissimilarity induced by each successive octave doesn't accumulate quickly enough. (Again, the tritave is assumed to have significantly higher dissimilarity than the octave, due to the extremely fast falloff mentioned earlier.) This may mean that if we could experience sound perception with a wider hearing range, that it would be possible to hear two notes a very high number of octaves apart as no longer sounding very equivalent due to the accumulation of dissimilarity.

=== Octave reduction ===

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 == Explanation of octave equivalence ==
 Most musicians think of two notes any number of octaves apart as being equivalent to some extent; that is, being within the same [[pitch class]]. This obviously doesn't mean musicians are entirely unable to distinguish between notes that are some number of octaves apart, but it does mean that it's viewed as a type of [[wikipedia:Equivalence_relation|equivalence relation]], not just a similarity relation. That is, there is some property, namely pitch class, which is considered to be entirely [[wikipedia:Invariant_(mathematics)|invariant]] under transposition by octaves. This seems in direct contradiction to the theory of harmonic similarity, which posits that two notes separated by one or more octaves merely share a similarity relation, not an equivalence relation. I have two different hypotheses that attempt to explain this, which I'll detail below.
-First, it's important to reiterate that the falloff in similarity from the octave to the tritave is assumed to be very prominent for most listeners.
 === Limited hearing range ===
-We humans have a finite hearing range of only around 10 octaves, and only around six or seven octaves of that range is actually musically useful. Because of this, it may be impossible for any number of stacked octaves that fits within our musical hearing range to have a lower similarity than the tritave for most listeners, since the dissimilarity induced by each successive octave doesn't accumulate quickly enough. (Again, the tritave is assumed to have significantly higher dissimilarity than the octave.) This may mean that if we could experience sound perception with a wider hearing range, that it would be possible to hear two notes a very high number of octaves apart as no longer sounding very equivalent due to the accumulation of dissimilarity.
+We humans have a finite hearing range of only around 10 octaves, and only around six or seven octaves of that range is actually musically useful. Because of this, it may be impossible for any number of stacked octaves that fits within our musical hearing range to have a lower similarity than the tritave for most listeners, since the dissimilarity induced by each successive octave doesn't accumulate quickly enough. (Again, the tritave is assumed to have significantly higher dissimilarity than the octave, due to the extremely fast falloff mentioned earlier.) This may mean that if we could experience sound perception with a wider hearing range, that it would be possible to hear two notes a very high number of octaves apart as no longer sounding very equivalent due to the accumulation of dissimilarity.
 === Octave reduction ===