User:Grady/Harmonic similarity: Difference between revisions
m Minor changes to the introduction |
m →Room for error: Renamed section |
||
| Line 25: | Line 25: | ||
Something to note about the base similarity relations is that the falloff in similarity with increasing integer values seems to be extremely fast. It's not exactly clear why this might be the case, but it's one of the core underlying assumptions of the theory. For example, the octave is a much, ''much'' stronger similarity relation than the tritave to most listeners, hence the notion of octave equivalence. A seeming caveat of this is that most listeners would rate the double octave (an interval of 4/1) to be a stronger similarity relation than the tritave. However, this is only because the double octave can be decomposed into two octaves, meaning this strong similarity can be explained with the pseudo-transitive property. In other words, the note C4 is very strongly related to the note C5, and C5 is very strongly related to C6, therefore C4 and C6 are also very strongly related. However, C4 is only moderately related to G5 (the note a tritave above it), and the tritave can't be decomposed into any simpler relationships. | Something to note about the base similarity relations is that the falloff in similarity with increasing integer values seems to be extremely fast. It's not exactly clear why this might be the case, but it's one of the core underlying assumptions of the theory. For example, the octave is a much, ''much'' stronger similarity relation than the tritave to most listeners, hence the notion of octave equivalence. A seeming caveat of this is that most listeners would rate the double octave (an interval of 4/1) to be a stronger similarity relation than the tritave. However, this is only because the double octave can be decomposed into two octaves, meaning this strong similarity can be explained with the pseudo-transitive property. In other words, the note C4 is very strongly related to the note C5, and C5 is very strongly related to C6, therefore C4 and C6 are also very strongly related. However, C4 is only moderately related to G5 (the note a tritave above it), and the tritave can't be decomposed into any simpler relationships. | ||
=== | === 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. | ||