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!
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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, or in other words, how closely related the two pitches are to each other. 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.
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 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 perfect fifth apart are also fairly harmonically similar.
This analogy may even be deeper than it appears, as it may in fact provide some insight into what the brain is doing when comparing two pitches. It may be very easy for the brain to subconsciously slide pitches around by octaves for a similar reason that it's easy to fold something in half repeatedly, and the same logic may extend to other harmonically similar pairs of pitches.
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 hypothesis
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 hypothesis
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.
Other implications
In addition to providing an attempted explanation for the perception of octave equivalence, the theory of harmonic similarity provides explanations for various other noteworthy phenomena.
Neutrality of the major second
The major second is often regarded as a dissonant interval, but it seems rather unique in that it produces auditory roughness, but doesn't necessarily feel emotionally jarring and unstable in the way something like one of the various flavors of tritone does. This is easily explained by the fact that two notes a major second apart are somewhat harmonically similar, due to being related by two factors of 3 and three factors of 2, which gives the major second a more emotionally neutral flavor than other dissonant intervals.
Usage of the circle of fifths
In Western music theory, the circle of fifths is often touted as a tool for determining which chords and keys are related to each other. The circle or chain of fifths is essentially an abstraction of the idea of harmonic similarity: it first assumes octave equivalence by implying that notes within the same pitch class can be treated as equivalent, and then denotes which pitch classes are harmonically related to each other. Chord progressions often involve successive chords whose root notes are adjacent on the chain of fifths; this is viewed as a smooth harmonic movement, which can be explained by the fact that the two root notes are harmonically similar to each other. The full theory of harmonic similarity correctly demonstrates that an even smoother harmonic movement can be achieved by moving from one chord to another chord whose root note is an octave away from that of the first, although this of course results in a change of character that's so minimal that Western music theory wouldn't even consider it to be a different chord at all.
Interval quality
Although the circle or chain of fifths is commonly used to relate pitch classes to each other, I never see it used to relate interval classes to each other. What this would mean is that one pitch class would be fixed in place, while another pitch class would be moved around the circle or chain of fifths in order to generate a mapping of harmonically similar interval classes. For example, ascending the chain of fifths in this manner starting at the unison would produce the unison, then the perfect fifth, then the major second, then the major sixth, and so on.
As an important note, the concept of interval class is usually defined to refer to undirected interval class: for example, a major third and minor sixth would be considered the same interval class due to being octave complements. However, this theory requires establishing the concept of directed interval class, in which octave complements are separate classes. In order for this to work, we need to define intervals as being directed in general; in other words, that an upward major third and a downward major third be considered two separate intervals. The downward major third falls into the same directed interval class as the upward minor sixth, but the upward major third does not. If the upward major third can be thought of as the ratio 5/4, then the downward major third can be thought of as 4/5. In the absence of a specifier for upward or downward direction, it should be assumed that an interval points in the upward direction.