User:Grady/Harmonic similarity: Difference between revisions

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=== 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.

== String analogy ==

There's a very useful analogy I came up with to help understand the mechanism of determining how harmonically similar two pitches are. Imagine you have two strings such that the ratio between their lengths equals the ratio between the two frequencies you want to compare. (This is especially on the nose since affixing the ends of those two strings and plucking them would produce the musical interval in question.) In order to determine how harmonically similar the two notes are, ask yourself how easy it is to fold both strings to be the same length.

For example, if one string is four times the length of the other, simply fold the longer string in half twice to make it the same length as the shorter string. This is extremely easy to do, since folding something in half is extremely easy. Therefore, two pitches related by a [[4/1|factor of four]] are highly harmonically similar.

If one string is 3/2 the length of the other, this requires folding the shorter string in half and folding the longer string into thirds. This is arguably more challenging, as folding something into thirds requires a fair bit more futzing than folding it in half. However, it's still massively easier than folding something into fifths or sevenths, so it isn't a huge challenge to fold these two strings to be the same length. Thus, two pitches a [[3/2|perfect fifth]] apart are also fairly harmonically similar.

== Explanation of octave equivalence ==

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 === 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.
+== String analogy ==
+There's a very useful analogy I came up with to help understand the mechanism of determining how harmonically similar two pitches are. Imagine you have two strings such that the ratio between their lengths equals the ratio between the two frequencies you want to compare. (This is especially on the nose since affixing the ends of those two strings and plucking them would produce the musical interval in question.) In order to determine how harmonically similar the two notes are, ask yourself how easy it is to fold both strings to be the same length.
+For example, if one string is four times the length of the other, simply fold the longer string in half twice to make it the same length as the shorter string. This is extremely easy to do, since folding something in half is extremely easy. Therefore, two pitches related by a [[4/1|factor of four]] are highly harmonically similar.
+If one string is 3/2 the length of the other, this requires folding the shorter string in half and folding the longer string into thirds. This is arguably more challenging, as folding something into thirds requires a fair bit more futzing than folding it in half. However, it's still massively easier than folding something into fifths or sevenths, so it isn't a huge challenge to fold these two strings to be the same length. Thus, two pitches a [[3/2|perfect fifth]] apart are also fairly harmonically similar.
 == Explanation of octave equivalence ==