User:Grady/Harmonic similarity

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!

Quick aside: Naming

I wasn't exactly sure what term to use for this concept, but I settled on harmonic similarity for now, at least for the purpose of deciding what to name this page. Some other terms I've considered are harmonic relatedness and harmonic affinity. I'm sure all of these terms have been used before, and some of them have certainly been used to express a similar concept as I'm putting forth here. If you have any opinions on what the most suitable name would be, I'd be glad to hear them!

Motivation

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 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.

Overview

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.

The concept of harmonic similarity is a generalization of octave equivalence: two notes that are some number of octaves apart will be considered highly harmonically similar, but there are other pairs of notes that are harmonically similar to a lesser degree, such as two notes a tritave apart.

Often, harmonic similarity between two notes correlates very strongly with perceived consonance: for example, two notes an octave apart are both extremely consonant with each other and extremely similar to each other. However, this is not always the case. For instance, two notes a major second apart are more harmonically similar to each other than two notes a major third apart, 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

Pseudo-transitivity

It should be intuitive that any 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.)

It follows from this that if we want to define a notion of two auditory pitches being similar or related in their harmonic function, said notion should also follow this pseudo-transitive property as well. This is the core principle that differentiates the notion of harmonic similarity from that of consonance, since it doesn't necessarily follow that if note X and note Y are consonant together, and note Y and note Z are consonant together, that notes X and Z ought to be consonant together to some degree as well.

Basic similarity relations

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 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 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.

A seeming caveat of all this is that most listeners would rate the double octave to be a stronger similarity relation than the tritave, despite being a higher integer multiple. 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 relations.

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.

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 equivalence relation, not just a similarity relation. That is, there is some property, namely pitch class, which is considered to be entirely 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 hypotheses that attempt to explain this, which I'll detail below.

Limited hearing range

We humans have a finite hearing range of only around 10 octaves, and only around six or seven octaves of that range is actually musically useful. Because of this, it may be impossible for any number of stacked octaves that fits within our musical hearing range to have a lower similarity than the tritave for most listeners, since the dissimilarity induced by each successive octave doesn't accumulate quickly enough. (Again, the tritave is assumed to have significantly higher dissimilarity than the octave, due to the extremely fast falloff mentioned earlier.) This may mean that if we could experience sound perception with a wider hearing range, that it would be possible to hear two notes a very high number of octaves apart as no longer sounding very equivalent due to the accumulation of dissimilarity.

Octave reduction

Because there's such a wide range of possible pitches to work with in music, it's often convenient to mentally reduce them down to the span of a single octave. In this sense, there actually is a property of notes that's truly invariant when transposing them by octaves, and that property is the note within any given one-octave span that they're most similar to. For example, the note within octave 3 (in other words, the notes C3 to B3, assuming a 12edo system) that the note E7 is most similar to is E3 (which is four octaves away). E7 might also be moderately similar to other notes within octave 3, such as A3 (which is two octaves and one tritave away), but it's most closely related to E3. As for why a span of one octave is chosen as opposed to any other size, that's because there isn't any other possible combination of window size and equivalence interval, besides one octave and one octave respectively, for which this property holds (but this remains to be rigorously proven).

Why this causes octave equivalence

Presumably, for one of these reasons or the other, or perhaps both, our brains create the abstraction that notes any number of octaves apart fall within the same pitch class. I believe this is essentially something our brains do just because they can, at least after receiving some musical training, since it creates a useful layer of abstraction without any real downsides.

It's only a useful layer of abstraction to do this with octaves only, because if we attempted to do it with both octaves and tritaves simultaneously, then every possible pitch would fall within the same pitch class, making the abstraction useless. This is because any two notes relate to each other by some combination of octaves and tritaves (within some arbitrarily small margin of error), but most pairs of notes cannot be related to each other by octaves only. In more technical terms, this means any possible interval is arbitrarily close to some 3-limit interval, but not arbitrarily close to some 2-limit interval.