User:Grady/Harmonic similarity: Difference between revisions
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This page serves to document a personal theory of mine that attempts to serve as a generalization of octave equivalence, as well as the various implications of the theory. I've seen other people express similar ideas, but I'm not sure if the concept in this exact form has been articulated before. If anyone knows if it has, I'd love to know more! | This page serves to document a personal theory of mine that attempts to serve as a generalization of octave equivalence, as well as the various implications of the theory. I've seen other people express similar ideas, but I'm not sure if the concept in this exact form has been articulated before. If anyone knows if it has, I'd love to know more! | ||
If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]! | If you want to leave feedback about anything on this page, you can leave it on the Discussion page (see the tabs at the top) or [https://discordapp.com/users/241021202976473098 contact me on Discord]! Feel free to let me know whether you think these ideas hold up to scrutiny, or whether your own experience agrees or disagrees with what I've written here. | ||
== Quick aside: Naming == | == Quick aside: Naming == | ||
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== Motivation == | == Motivation == | ||
I developed this theory to attempt to answer the question, "Why are two notes an [[ | I developed this theory to attempt to answer the question, "Why are two notes an [[octave]] apart perceived as equivalent?" The typical explanation given is that it has something to do with the fact that the upper note is prominently featured in the harmonic spectrum of the lower note, assuming a typical [[harmonic timbre]]. However, this completely fails to explain why the octave is special in this regard, and why a similar phenomenon seemingly never occurs with other harmonics (even if less commonly, under more specific circumstances, or in a less pronounced manner), most namely the [[3/1|tritave]]. | ||
Some people have claimed to hear tritave equivalence, but so far no one has purported to hear notes any number of tritaves apart as being within the same [[pitch class]], something many musicians simply take for granted with regard to octaves, and in my opinion, a necessary requisite for true equivalence perception. | Some people have claimed to hear tritave equivalence, but so far no one has purported to hear notes any number of tritaves apart as being within the same [[pitch class]], something many musicians simply take for granted with regard to octaves, and in my opinion, a necessary requisite for true equivalence perception. | ||
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Of course, the knowledge that harmonic similarity should follow this pseudo-transitive property isn't very helpful in determining which notes are harmonically similar without establishing some basic similarity relations first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other, and said relation is stronger the lower the integer multiple is. The reason for this might be that we're highly accustomed to hearing harmonic timbres, so usually when we hear a note, we also somewhat prominently hear low integer multiples of that frequency as well, and over time this causes our brains to associate those higher harmonics as "similar" to the fundamental. This would also explain why lower integer multiples correspond to stronger basic similarity relations, because the lower the integer multiple is, the more prominent the corresponding overtone typically is in those harmonic timbres. | Of course, the knowledge that harmonic similarity should follow this pseudo-transitive property isn't very helpful in determining which notes are harmonically similar without establishing some basic similarity relations first. Essentially, two pitches have a basic similarity relation if the frequency of one is a low integer multiple of the other, and said relation is stronger the lower the integer multiple is. The reason for this might be that we're highly accustomed to hearing harmonic timbres, so usually when we hear a note, we also somewhat prominently hear low integer multiples of that frequency as well, and over time this causes our brains to associate those higher harmonics as "similar" to the fundamental. This would also explain why lower integer multiples correspond to stronger basic similarity relations, because the lower the integer multiple is, the more prominent the corresponding overtone typically is in those harmonic timbres. | ||
With this knowledge, we can use the pseudo-transitive property to discover higher-order similarity relations. For example, two notes a [[3/2|perfect fifth]] apart are perceived as harmonically similar, but not because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an [[ | With this knowledge, we can use the pseudo-transitive property to discover higher-order similarity relations. For example, two notes a [[3/2|perfect fifth]] apart are perceived as harmonically similar, but not because the upper note is present in the harmonics of the lower note (which it isn't, assuming a typical harmonic spectrum). Rather, they're similar because two notes an [[octave]] apart ''are'' similar for that reason, and so are two notes a [[3/1|tritave]] apart, and thus by the pseudo-transitive property, two notes a perfect fifth apart must be similar as well. For example, the note C4 is related to G4 (the note a perfect fifth above it) because both of those notes are related to G5 (or to C3) by a basic similarity relation. | ||
Something to note about the basic 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. | Something to note about the basic 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. | ||
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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. | ||
== Lattice visualization == | |||
A very good way to visualize a map of harmonically similar pitches is using a [[3-limit]] [[lattice]], with octaves on one axis and tritaves on the other. In order to demonstrate that octaves are a significantly stronger harmonic similarity relation than tritaves, one might consider spacing the notes much farther apart on the tritave axis than the octave axis. (No diagram yet, but that might be something I can add to this page later!) | |||
This essentially ignores the effects of any basic similarity relations beyond the third harmonic, which I believe is reasonable due to the extremely fast falloff mentioned earlier. | |||
== String analogy == | == String analogy == | ||
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Because 12edo has a perfectly tuned octave and near-perfectly tuned tritave, if any given interval is well approximated by the tuning system, then all the intervals that are harmonically similar to that interval will also be well approximated. Conversely, if an interval falls well outside of the tuning system, for example the [[7/4|subminor seventh]], then all the intervals that are harmonically similar to that interval, such as the [[7/6|subminor third]], will fall well outside of the tuning system as well. Because of this, not only do Western listeners not become acclimated to the subminor seventh itself, but they also don't become acclimated to other intervals that are similar in character, like the subminor third. If Western listeners regularly heard the subminor third but not the subminor seventh (which would be the case if [[9edo]] was the dominant tuning system instead, for example), then they might not find the subminor seventh as jarring, since they'd already be very familiar with a similar interval. | Because 12edo has a perfectly tuned octave and near-perfectly tuned tritave, if any given interval is well approximated by the tuning system, then all the intervals that are harmonically similar to that interval will also be well approximated. Conversely, if an interval falls well outside of the tuning system, for example the [[7/4|subminor seventh]], then all the intervals that are harmonically similar to that interval, such as the [[7/6|subminor third]], will fall well outside of the tuning system as well. Because of this, not only do Western listeners not become acclimated to the subminor seventh itself, but they also don't become acclimated to other intervals that are similar in character, like the subminor third. If Western listeners regularly heard the subminor third but not the subminor seventh (which would be the case if [[9edo]] was the dominant tuning system instead, for example), then they might not find the subminor seventh as jarring, since they'd already be very familiar with a similar interval. | ||
12edo even has a reasonable tuning of the [[5/1|fifth harmonic]], so it's arguably possible that this further contributes to the cohesion in harmonic similarity between the notes and intervals within the tuning system. However, it's debatable whether two frequencies separated by a factor of five have any non-negligible amount of harmonic similarity to each other, or whether 12edo's tuning of the fifth harmonic is accurate enough for this to take effect. | |||
=== The tritave versus the perfect fifth === | |||
Perhaps a more contentious implication of this theory is that it implies the tritave is a slightly stronger similarity relation than the perfect fifth. In Western music theory, they'd be thought of as equal in similarity, due to the abstraction of octave equivalence. However, this implication seems to generally align with the experiences of xenharmonic musicians, many of whom claim the tritave works better as an [[Interval of equivalence|equave]] than the perfect fifth. | |||