Measuring musical dissonance


An empirical approach

When we hear two musical notes played together (either in succession, or simultaneously), we often characterize those notes as “dissonant” or “consonant”. But instead of having a sharp dichotomy between dissonance and consonance, it might be more useful to speak of a spectrum between the two. Then, the question before us is how to quantify the dissonance of any pair of notes.

12tone is a cool music theory channel, and he recently published a video discussing the solution thought up by the 18th century mathematician Leonhard Euler. I include the video below, but be warned that I’m going to trash Euler’s answer. I believe that any measure of musical dissonance must, at some point, refer to empirical observations of dissonance. Euler’s answer relies on mathematical supposition, and thus I would deride it as numerology.

When I say that a theory of dissonance must refer to empirical observation, I do not mean that we simply observe what people say is dissonant, and take that answer as truth. After all, what you consider dissonant might be different from what I consider dissonant. We could instead take a representative sample of the world population and measure the average opinion of dissonance, but this also has problems. On average, the world population has been strongly influenced by the western tradition of music. What we want is a measure of dissonance that transcends individual variation and cultural influence.

To do this, we generate theories of why we have this sensation of dissonance, and use empirical observations to select the best theory. Then we use the best theory to generate a prediction of what the average person would experience as dissonant in a hypothetical world where they were uninfluenced by musical culture. This has been done in a 19651 paper, “Tonal Consonance and Critical Bandwidth” by Plomp and Levelt, which I encountered when I did research on microtonal music.

Elegant, but wrong hypotheses

It is generally agreed that the most consonant musical intervals are the ones that have “nice” frequency ratios–that is, ratios between small integers like 1:2 or 2:3. So that set me to wondering if there was a “most dissonant” interval. I had a theory that the most dissonant interval would have a 1:1.618 ratio–that is, the golden ratio. Roughly speaking, the golden ratio is the irrational number that’s hardest to approximate with ratios of small integers.

It is an elegant hypothesis, but alas, it is not true. Plomp and Levelt disprove it.  A graph showing how likely people are to judge a pair of notes as consonant/pleasant, as a function of how far apart the notes are. The main feature is a large dip in consonance/pleasant near the 15:16 ratio, extending out past the 8:9 ratio.

This is Figure 1 from Plomp & Levelt, but the data is sourced from an earlier work by someone else. They performed an experiment where they played a pair of simple tones, and asked subjects to judge which the tones were more consonant (solid line) and more pleasant (dashed line). As you can see, “pleasant” and “consonant” mean nearly the same thing among these experimental subjects.2 And the most dissonant interval is fairly small, near the 15:16 frequency ratio.3 Not the golden ratio.

I should mention that this experiment was performed on subjects without musical training. If we want to understand dissonance independent of cultural influences, then we had better use test subjects who aren’t deliberately trained in those cultural traditions. But this is something that most investigators before the 20th century didn’t understand. They were only interested in experiments on musically trained subjects.

As a result, they missed or ignored a significant fact: above about a 5:6 ratio, most intervals sound about equally consonant to untrained ears, whether they consist of “nice” frequency ratios or not. This conflicts with the traditional western understanding, and also with Euler’s theory. What gives?

A matter of timbre

Earlier, I referred to “simple tones”, and I’m going to have to explain that part. A simple tone is a sound produced by a perfect sine wave. Most musical notes, such as the notes played by a piano, are “complex tones” have many sine waves (harmonics) overlaid on top of each other. The frequency of these harmonics are in a 1:2:3:4:5:etc. ratio. Wikipedia has a nice demonstration of how simple tones can be overlaid to create a complex tone (caution: may be loud). The particular way that sine waves get overlaid is one aspect of the “timbre” of a note (pronounced “tamber”).

So what happens when we perform the same experiment using complex tones instead of simple tones? That is the subject of Figure 2.

A graph showing how likely subjects are to judge a pair of notes as pleasant, as a function of how far apart they are. The solid curve shows simple tones while the dashed curve shows complex tones. The complex tones show peaks around integer ratios, whereas the simple tones do not.

Figure 2 is based on another earlier study, which played pairs of tones for untrained subjects and asked them to judge pleasantness. (The curve looks more compressed compared to Figure 1 but that’s just because of the horizontal scale.) The solid curve shows simple tones, and the dashed curve shows complex tones. So when we switch from simple tones to complex tones, there appear peaks in pleasantness that correspond to “nice” ratios between integers.

Well isn’t that interesting? Using simple tones, untrained listeners do not much care whether tones have nice frequency ratios. However, when we use complex tones, nice frequency ratios are preferred. Plomp & Levelt argue that this supports a theory of dissonance that they credit to Hermann von Helmholtz. Fundamentally, the sensation of dissonance comes from tones that are close together in frequency. This includes not just the fundamental tones, but also the higher harmonic tones. So if we have two complex tones, the higher harmonics create dissonance, unless the two complex tones happen to have a nice frequency ratio with one another.

I will gloss over the part where Plomp & Levelt discuss other theories of dissonance, and why they fail to account for this data. Let’s just say that Euler didn’t have much of a theory, believing that the “unconsciously counting soul” simply likes concurrent vibrations.

Dissonance and the human ear

Plomp & Levelt perform additional experiments, mainly trying to understand how close simple tones must be in order to produce this sensation of dissonance. Earlier I said 15:16 was the most dissonant interval, but that’s not always true. In fact, it doesn’t just depend on the frequency ratio, but also the absolute frequency. In the lower frequency range, the range of dissonant intervals is larger.4 In the higher frequency range, the range of dissonant intervals is narrower.

They make a connection between the range of dissonance, and “critical bandwidth”. Critical bandwidth is an idea that comes from the study of human hearing. Basically, it’s how close two frequencies can be before one tone interferes with the perception of the other. Apparently when the distance between two tones is about 25% of the critical bandwidth, it produces this sensation of dissonance. When the distance is much greater than or much less than 25%, the dissonance is diminished.

Using this knowledge, Plomp & Levelt are ready to make predictions. The following figure shows the prediction of consonance between two complex tones, as a function of distance between the tones. These complex tones include only 6 harmonics, so the function lacks peaks corresponding to, e.g. the ratio 5:7. In the case of realistic musical instruments, the landscape of dissonance would have more of a fractal structure.

A plot showing the theoretical consonance of two complex tones, as a function of the difference in frequency between them. There are several peaks corresponding to nice integer ratios.

The takeaway is that musical dissonance does not just depend on mathematical frequency ratios, as Euler believed. It depends on the timbre of the note, the absolute frequency of the note, and the properties of the human ear. Last but not least, it depends on your culture and musical training.


Footnotes

1. The skeptical reader may ask whether a paper from 1965 still holds up today. I am not an expert in the field so it is hard to say. However, I found Plomp & Levelt because it was discussed approvingly in the introduction to a 2000 paper by Marc Leman. This suggests to me that Plomp & Levelt has held up fairly well. However, it seems that in modern scholarship, the term “roughness” is preferred over “dissonance”.  I stick with the word “dissonance” to be consistent with the paper under discussion. (return)

2. The subjects of this experiment were not musically trained.  Among musically trained people, there is more of a distinction between “dissonant” and “unpleasant”.   This suggests that “dissonant” is not the same as “unpleasant”, but rather a distinctive quality of sound that untrained listeners (in western culture) tend to find less pleasant.  (return)

3. For those who are wondering, the 15:16 ratio is close to the minor second interval. 8:9 is close to the major second, 5:6 is close to the minor third, 4:5 is close to the major third, 3:4 is close to the perfect fourth, 5:7 is close to the tritone, 2:3 is close to the perfect fifth, 5:8 is close to the minor sixth, and 3:5 is close to the major sixth. I say “nearly” because in modern western music all these intervals actually have irrational ratios. The funny thing is that because of my musical training, intervals with rational ratios sound slightly dissonant to me, because they’re out of tune relative to western tuning. (return)

4. This echoes conventional wisdom among musicians, that small intervals sound worse in the low registers. (return)

Comments

  1. consciousness razor says

    This conflicts with the traditional western understanding, and also with Euler’s theory. What gives?

    Basically, tonal theories (i.e., the “tradition” for most of the last few centuries) aren’t complete theories of musical harmony. They presuppose that certain scales and sonorities will be involved. (And that they have a tonal “function,” which is a bit of a mysterious term, but it’s the one we use.) Categorizing different chords/intervals/etc. according to their tonal function usually works pretty well, when describing how one chord should progress (or resolve) to another, in music which fits in this framework. Even then, it’s rather problematic, but it’s definitely not able to cope with anything beyond that.
    People typically think that once they’ve got a handle on that stuff, they have thus “learned music theory” or at least the basics of what it says about harmony. But that is incorrect. They learned a theory that was developed mostly during the Renaissance and Baroque periods to try to make sense of what they heard then, along with some input from older theories of counterpoint that remained relevant. It’s sort of like studying old alchemy texts and thinking that you have a decent grasp of modern chemistry.
    Anyway, going back to the issue with consonance and dissonance…. People have somewhat differing ideas about them, as you point out. But it is basically just the smaller intervals that are (more or less always) considered dissonant. It has nothing to do with the rational numbers, as Pythagoreans and Platonists used to think.
    When I said dissonant intervals are smaller, that also means their inversions. The “inverse” is the additive inverse modulo 12, when there are 12 equal divisions of the octave. For example, 1+11 = 0 (mod 12). The point here is that the interval 11 (major seventh) is not small like 1 (minor second) is, but it is a small distance away from being an octave. Octaves are consonant, and they are even equivalent (although not identical) in the sense of being the same pitch class. To some extent, especially with untrained listeners, people won’t notice as easily that there is dissonance between a large interval like 11 and the nearby octave. That octave (call it “12”) is present in the harmonic series of the lower pitch (“0”), but it generally won’t be as loud, making it less noticeable (at least to some). So, for reasons like that, [0E] is not quite the same as [01], but that’s the basic idea.

    Plomp & Levelt argue that this supports a theory of dissonance that they credit to Hermann von Helmholtz. Fundamentally, the sensation of dissonance comes from tones that are close together in frequency. This includes not just the fundamental tones, but also the higher harmonic tones. So if we have two complex tones, the higher harmonics create dissonance, unless the two complex tones happen to have a nice frequency ratio with one another.

    You might be misinterpreting here. Just to clear up any potential confusion…. Like I said above, the “niceness” of it doesn’t need to be about rational numbers with very small integers. Some of those happen to be consonant, like 1:2, 2:3, 3:4, 4:5, 5:6 (and their inversions). But that’s just coincidence, and it is only a (beneficial I suppose?) side effect that it will not make baby Pythagoras cry about the number being irrational. The issue is that both fundamentals won’t create noticeable “conflicts” with each other in the lower parts of their respective harmonic series, since those pitches are also present in ordinary circumstances (as we know from physics). People can’t really tell that some proportion of frequencies is exactly 2:3 or what have you. It depends on the exact nature of the experiment, but generally their pitch discrimination is actually pretty lousy. So, any arbitrary irrational ratio of frequencies will suffice, if it is close enough to being some such interval with the property that the overtones don’t grind up against one another “too much” for most people to know or care that it is happening.
    There is another set of effects to talk about — not just the big dip around the small intervals and their inverses, but also the fluctuations in the higher parts of those curves. I think a lot of that just has to do with familiarity. Some intervals are heard a lot and are given a certain status in tonal music, while others are much less familiar. People who’ve listened to music like that their whole lives won’t necessarily forget that sort of thing when they have to be in an experiment.

  2. says

    @CR #1,
    Yeah, I agree with what you’re saying. I wish this whole thing would be taught as a standard part of music theory, because I always had the impression that we just didn’t know why certain ratios were more consonant than others.

    You are right that complex tones don’t need to make nice ratios in order to avoid dissonance, and I was speaking imprecisely. There’s a window around each nice ratio that still sounds consonant, and there may be other intervals where as it happens there’s just no clashing among the loudest harmonics.

    I also think that in some instruments that only have odd harmonics, maybe there are even more consonant ratios. I’m tempted to try programming such sounds, to hear what it sounds like.

  3. consciousness razor says

    I also think that in some instruments that only have odd harmonics, maybe there are even more consonant ratios.

    Well, that pretty much boils down to a question of whether you like certain timbres or combinations of timbres. Do clarinets sound good to you? Do you like them playing with some combination of other instruments/voices, especially when they are playing specific sets of notes? If not, do you like something else? And so forth.
    You could call that “consonance” if you wanted to. But then, doesn’t the context of the sound still matter? I mean, you can hear one thing and say it’s terrible, but you might change your mind about it in the right circumstances, if for example it’s set up in a particular way or leads somewhere interesting. Or, suppose you have the idea to compare it to your experience of a major triad on the one hand, or on the other to some dense cluster of what you can only describe as irritating noise. It might sound relatively consonant compared to the latter. Or it still might not sound so great even by comparison to that, but in some other context it would. It’s very easy to think of cases where some usually-pleasant sound is just plain awful because of what else is happening in the music (before, after, or at the same time).
    To be a bit more concrete about it…. People used to think the tritone was the “diabolus,” but I have no doubt that you like listening to them on a very regular basis. Indeed, they’re pretty much unavoidable in tonal music, not to mention other kinds of music. Part of it is they’re not used in the same ways they were centuries ago. If you don’t really know what to do with them or how to listen to them, as people didn’t, then it’s not hard to imagine that may be a problem for you. But that can certainly change if learn some new ways to think about harmony (or are simply exposed to it as a listener). The point is that it’s just getting started on the wrong foot to think of it as if there were some fixed property of a sound which you can understand in isolation, like a statue or an equation or I don’t know what, without thinking about how it relates to other sounds in the music (or in your memory, etc.) or without thinking about all of the things you’re bringing to the table whenever you listen to it.
    In any case, timbre does matter a great deal. Real life is obviously more complicated than the simplistic version of the story that is usually peddled. It’s easier to talk about it as if there’s just one frequency (the fundamental) — a note gets a number, and then you say there’s a proportion between that and some other number that is meant to describe another note. Seems like a nice idea, but the only problem is that the real world is not nice (or maybe it’s even nicer than your idea).
    Anyway, something like a 1D number line is simply inadequate for representing everything that plays a role in your listening experience. For that matter, we wouldn’t even have to bring up timbre, just plain old musical sets. They were given numbers (by Forte back in the 60s or 70s, according to some very arbitrary rules he dreamed up), but it’s just sort of meaningless to list them “in order.” You might be interested in Dimitri Tymoczko’s (and others’) work on geometrical music theories. He did some popularizing a few years back, which I think is a bit sketchy at times, but there is a more serious side to it. For example, the space of the 3-note set-classes is shaped like a cone. In those spaces, they do go from more densely-packed clusters on one end to more evenly distributed on the other (near the point of the cone in the 3-note case), but they form these big, complicated and rather interesting structures, rather than something simple like a line or a list of objects.

  4. says

    @CR #3,
    There’s no doubt that the context matters, I was just curious what it would sound like. Not, “does it sound good?” (I don’t think dissonance sounds bad anyways) but rather, “does it sound different?” Would a major 7th interval sound significantly more consonant on clarinet vs other instruments? Or would my classically trained brain hear it about the same way?

    I seem to recall that Sevish programs music with inharmonic overtones (mentioned here), which he claims leads to a smoother sound. I’m not sure how well I notice it though.

  5. consciousness razor says

    Would a major 7th interval sound significantly more consonant on clarinet vs other instruments? Or would my classically trained brain hear it about the same way?

    I don’t know. Probably not very significant, since the odd partials still rub up against each other.
    A combination like that will sound different from other instruments, of course. Generally, I would not want to put something like that in terms of consonance versus dissonance. I rarely think that way in the first place, so maybe it’s just me. Often, I’ll use somewhat more descriptive language … metaphors like cool, warm, hollow, full, rich, and so forth. Or maybe something is a little spicy or pungent or whatever.
    I tend to associate “consonance” and “dissonance” with resolving a chord in tonality, such as a cadence with a (more dissonant) dominant triad resolving to a (more consonant) tonic triad. Notice that these are same types of chords (both major triads), so what people mean by it isn’t about the structure of that particular group of notes having a particular property. But the dominant contains the leading tone in that particular key which people think “pulls” toward the tonic note. Because you were prepared for it, you hear that pulling as if it were an intrinsic feature of the dominant triad (although it isn’t), and the whole bit of music doesn’t seem finished until the tonic.
    Or for another kind of example, there may be a note that “anticipates” the next harmony, which sounds dissonant at first but consonant once the other parts reach the second chord. In the dominant-tonic example, the anticipation could be the root or third of the tonic triad (the fifth is already in both, so it’s not a very helpful example).
    Anyway, if stuff like that isn’t happening, or if it’s not even tonal music let’s say, then those terms aren’t especially useful. But I’d still want to know something about what I’m hearing, so other concepts might still help.

    I seem to recall that Sevish programs music with inharmonic overtones (mentioned here), which he claims leads to a smoother sound. I’m not sure how well I notice it though.

    Well, he can claim that if he wants, but it sets off my bullshit detector. Too much woo out there about the golden ratio already. And I’m not really a fan of the way certain composers (especially in microtonal music) start with some random bit of math and treat it as if that does some kind of magic. Very generally, if your theory involves all sorts of complicated crap that has no impact on what real people can actually hear, it’s a bad theory.
    I have a good ear, much better than most, since I write/arrange all the time and have to use it constantly. But I’m sure I can’t tell the difference between 1.618 and the number phi, if that’s applied to something musical like a scale or whatever (and since we’re being realistic, computers also need to use approximations at some point, meaning we’d have to be very lucky and hope it just happens by accident). So it doesn’t really matter which number you call it, because they go into the same bucket for me. If you’re writing music for Superman, and if he can tell the difference, then I accept that he puts them in different buckets and that your theory may be able to account for that, when discussing Superman’s listening experience (but not anybody else’s). Otherwise, things like that just won’t be relevant.

  6. says

    @CR #5,

    Probably not very significant, since the odd partials still rub up against each other.

    I checked and yes, the 5th harmonic of one note rubs against the 9th of the other, fwiw.

    Well, he can claim that if he wants, but it sets off my bullshit detector. Too much woo out there about the golden ratio already.

    I don’t think Sevish is in that set. Anyway, it seems like electronically removing dissonant harmonics would have a real impact on what people hear–that’s basically what’s going on in Fig. 2 of the OP.

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