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 ~~competing~~ hypotheses that attempt to explain this, which I'll detail below.

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.

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=== Octave reduction ===

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).

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).

=== Why this causes octave equivalence ===

Presumably, for one of these reasons or the other, or perhaps both, our brains create the abstraction that notes any number of octaves apart fall within the same pitch class. I believe this is essentially something our brains do just because they can, since it creates a useful layer of abstraction without any real downsides.

It's only a useful layer of abstraction to do this with octaves only, because if we attempted to do it with both octaves and tritaves simultaneously, then every possible pitch would fall within the same pitch class, making the abstraction useless. This is because any two notes relate to each other by some combination of octaves and tritaves (within some arbitrarily small margin of error), but most pairs of notes cannot be related to each other by octaves only. In more technical terms, this means any possible interval is arbitrarily close to some [[3-limit]] interval, but not arbitrarily close to some [[2-limit]] interval.

@@ Line 38: / Line 38: @@
 == 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 competing hypotheses that attempt to explain this, which I'll detail below.
+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.
@@ Line 46: / Line 46: @@
 === Octave reduction ===
-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).
+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).
+=== Why this causes octave equivalence ===
+Presumably, for one of these reasons or the other, or perhaps both, our brains create the abstraction that notes any number of octaves apart fall within the same pitch class. I believe this is essentially something our brains do just because they can, since it creates a useful layer of abstraction without any real downsides.
+It's only a useful layer of abstraction to do this with octaves only, because if we attempted to do it with both octaves and tritaves simultaneously, then every possible pitch would fall within the same pitch class, making the abstraction useless. This is because any two notes relate to each other by some combination of octaves and tritaves (within some arbitrarily small margin of error), but most pairs of notes cannot be related to each other by octaves only. In more technical terms, this means any possible interval is arbitrarily close to some [[3-limit]] interval, but not arbitrarily close to some [[2-limit]] interval.