User:Grady/Harmonic similarity: Difference between revisions
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=== Margin for error === | === Margin for error === | ||
Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact. | Since our ears are imperfect (and perhaps even because the overtones we hear that may have trained our mental map of harmonic similarity aren't perfect integer harmonics either), it makes sense to assign some margin for error to the notion of harmonic similarity by adding the assertion that two notes are harmonically similar if they're very close in pitch. This allows us to treat two notes that are an interval such as a perfect fifth apart in a tempered system like [[12edo]] to be harmonically similar, even if the ratios are inexact. | ||
== 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. | |||