User:Grady/Harmonic similarity: Difference between revisions

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Harmonic similarity is a measure of how similar in harmonic function two pitches are. This is in contrast to consonance or concordance, which is generally defined as a measure of how harmonious or stable two or more pitches sound when played together, or perhaps less commonly a measure of how much timbral fusion occurs when playing them together.

Often, harmonic similarity between two notes correlates very strongly with consonance, ~~but~~ this is not always the case. For ~~example~~, two notes a major second apart (an interval of 9/8) are more harmonically similar to each other than two notes a major third apart (an interval of 5/4), but typically less consonant when played simultaneously. I'll be elaborating more on what I mean by this, but it aligns with how Western music theory would approach the question: two notes a major second apart are more closely related via the circle of fifths.

Often, harmonic similarity between two notes correlates very strongly with perceived consonance: for example, two notes an octave apart are both highly consonant with each other and highly similar to each other. However, this is not always the case. For instance, two notes a major second apart (an interval of 9/8) are more harmonically similar to each other than two notes a major third apart (an interval of 5/4), but typically less consonant when played simultaneously. I'll be elaborating more on what I mean by this, but it aligns with how Western music theory would approach the question: two notes a major second apart are more closely related via the circle of fifths.

== Definition ==

It should be intuitive that any equality or equivalence relation follows the transitive property. For example, if x = y and y = z, then x = z. Or in geometry, if A is congruent to B and B is congruent to C, then A is congruent to C. For a more practical example, if Alice's car is the same model as Bob's car, and Bob's car is the same model as Charlie's car, then Alice and Charlie also have the same model of car.

An extension of this is that any similarity or relatedness relation sort of follows a pseudo-transitive property. For example, if x ≈ y and y ≈ z, then you can probably say that x ≈ z, but the similarity in quantity between x and z might be less strong than that between x and y, or y and z. To extend the car example, if Alice's car is a similar color to Bob's car, and Bob's car is a similar color to Charlie's car, then Alice's car is probably a similar color to Charlie's car as well, but it depends on what your threshold is for defining two colors as "similar".

(Note that despite the choice of word, the concept of "similarity" in geometry is actually an equivalence relation by this definition, not a similarity relation: it means both objects have the exact same shape, not that they're close in shape.)

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 Harmonic similarity is a measure of how similar in harmonic function two pitches are. This is in contrast to consonance or concordance, which is generally defined as a measure of how harmonious or stable two or more pitches sound when played together, or perhaps less commonly a measure of how much timbral fusion occurs when playing them together.
-Often, harmonic similarity between two notes correlates very strongly with consonance, but this is not always the case. For example, two notes a major second apart (an interval of 9/8) are more harmonically similar to each other than two notes a major third apart (an interval of 5/4), but typically less consonant when played simultaneously. I'll be elaborating more on what I mean by this, but it aligns with how Western music theory would approach the question: two notes a major second apart are more closely related via the circle of fifths.
+Often, harmonic similarity between two notes correlates very strongly with perceived consonance: for example, two notes an octave apart are both highly consonant with each other and highly similar to each other. However, this is not always the case. For instance, two notes a major second apart (an interval of 9/8) are more harmonically similar to each other than two notes a major third apart (an interval of 5/4), but typically less consonant when played simultaneously. I'll be elaborating more on what I mean by this, but it aligns with how Western music theory would approach the question: two notes a major second apart are more closely related via the circle of fifths.
+== Definition ==
+It should be intuitive that any equality or equivalence relation follows the transitive property. For example, if x = y and y = z, then x = z. Or in geometry, if A is congruent to B and B is congruent to C, then A is congruent to C. For a more practical example, if Alice's car is the same model as Bob's car, and Bob's car is the same model as Charlie's car, then Alice and Charlie also have the same model of car.
+An extension of this is that any similarity or relatedness relation sort of follows a pseudo-transitive property. For example, if x ≈ y and y ≈ z, then you can probably say that x ≈ z, but the similarity in quantity between x and z might be less strong than that between x and y, or y and z. To extend the car example, if Alice's car is a similar color to Bob's car, and Bob's car is a similar color to Charlie's car, then Alice's car is probably a similar color to Charlie's car as well, but it depends on what your threshold is for defining two colors as "similar".
+(Note that despite the choice of word, the concept of "similarity" in geometry is actually an equivalence relation by this definition, not a similarity relation: it means both objects have the exact same shape, not that they're close in shape.)