User:Grady/Harmonic similarity: Difference between revisions

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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.

=== Limited hearing range ===

=== Limited hearing range hypothesis ===

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 ===

=== Octave reduction hypothesis ===

Because there's such a wide range of possible pitches to work with in music, it's often convenient to mentally reduce them down to the span of a single octave. In this sense, there actually is a property of notes that's truly invariant when transposing them by octaves, and that property is the note within any given one-octave span that they're most similar to. For example, the note within octave 3 (in other words, the notes C3 to B3, assuming a 12edo system) that the note E7 is most similar to is E3 (which is four octaves away). E7 might also be moderately similar to other notes within octave 3, such as A3 (which is two octaves and one tritave away), but it's most closely related to E3. As for why a span of one octave is chosen as opposed to any other size, that's because there isn't any other possible combination of window size and equivalence interval, besides one octave and one octave respectively, for which this property holds (but this remains to be rigorously proven).

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 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.
-=== Limited hearing range ===
+=== Limited hearing range hypothesis ===
 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 ===
+=== Octave reduction hypothesis ===
 Because there's such a wide range of possible pitches to work with in music, it's often convenient to mentally reduce them down to the span of a single octave. In this sense, there actually is a property of notes that's truly invariant when transposing them by octaves, and that property is the note within any given one-octave span that they're most similar to. For example, the note within octave 3 (in other words, the notes C3 to B3, assuming a 12edo system) that the note E7 is most similar to is E3 (which is four octaves away). E7 might also be moderately similar to other notes within octave 3, such as A3 (which is two octaves and one tritave away), but it's most closely related to E3. As for why a span of one octave is chosen as opposed to any other size, that's because there isn't any other possible combination of window size and equivalence interval, besides one octave and one octave respectively, for which this property holds (but this remains to be rigorously proven).