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I'd like to give a rigorous analysis of the "dissonance" of musical intervals -- say, a perfect fourth versus a perfect fifth. I think one way to get "rigor" here is to consider the frequencies and amplitudes of the overtone series for both pitches, and then appeal to the physiology and psychology of acoustical processing.

One relatively parsimonious way of approaching this problem is to think about playing these intervals on a monochord. Contemporary "demonstration monochords" have two strings, in fact, not one. I can pluck one string without a bridge and use that as the fundamental, and then use the monochord's bridge to tune the second string to a perfect fourth or perfect fifth.

Throughout this thought experiment, assume I pluck all strings at the middle of their lengths. (The plucking location affects which harmonics are sounded, but I'm not sure how sensitive the resulting dissonances are to changes in plucking location.)

String 1

Say the fundamental frequency for string 1 is $f$. Because it is plucked in the middle, the even harmonics aren't sounded. The frequencies are thus odd multiples of the fundamental, with amplitudes decaying inverse to the square:

$$\begin{eqnarray} \text{frequency} &:& f, 3f, 5f, 7f \ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$

Perfect Fourth above

Now say I use the bridge to tune string 2 to a perfect Pythagorean fourth (frequency ratio 4 to 3) above the fundamental of string 1. I will assume the amplitude of the fundamental in this series is the same as the amplitude of the fundamental for string 1's series, but I'm not sure whether this is a safe assumption. Again, because the string is plucked in the middle, even harmonics are missing:

$$\begin{eqnarray} \text{frequency} &:& \frac{4}{3}f, 4f, \frac{20}{3}f, \frac{28}{3}f\ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$

Perfect Fifth above

This time, tune string 2 to a perfect Pythagorean fifth (frequency ratio 3 to 2) above the fundamental of string 1. Again I assume a common amplitude for the fundamental, and again, because the string is plucked in the middle, even harmonics are missing:

$$\begin{eqnarray} \text{frequency} &:& \frac{3}{2}f, \frac{9}{2}f, \frac{15}{2}f, \frac{21}{2}f \ldots \\ \text{amplitude} &:& 1, 1/9, 1/25, 1/49 \ldots \end{eqnarray} $$


Question: How can I use this simple analysis (if it is more or less correct) to explain the "dissonance" of the perfect fourth versus the perfect fifth?

In particular, how can I finish the explanation to arrive at the standard conclusion that the perfect fifth is more "consonant" than the perfect fourth?

Perhaps one doesn't need overtone series for that conclusion at all -- maybe the biology says it's enough to look at the dissonance of 4/3f versus f compared to 3/2f versus f, the former giving a smaller "amplitude fluctuation" -- but I thought I might as well look at the overtone series, too.

I know I now need to import some facts from biology, so the question isn't strictly physics.

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    $\begingroup$ Writing as a musician, IMO the notion of "dissonance" here is entirely a learned construct. If you look at the history of western music over the last 1000 years or so (i.e. from the point where writings about music, and musical scores themselves, are at least fairly understandable!) ideas about what was consonant and dissonant were very different at different historical periods. And non-western music also has very different ideas about this. $\endgroup$ – alephzero Sep 5 '18 at 18:26
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    $\begingroup$ … If you want to do some scientifically rigorous psycho-acoustic experiments, on the influence of overtones on the perception of intervals, then by all means do that - but starting with subjectively defined words like "dissonant" isn't a good place to begin IMO. And don't forget that any real-world monochord does not have harmonics at exactly integer frequency ratios - look up "inharmonicity" for vibrating strings. If you really want exact frequency ratios, generate them electronically! I have no idea if that effect is important to your experiments, but it needs to be controlled for. $\endgroup$ – alephzero Sep 5 '18 at 18:30
  • $\begingroup$ I understand the concept of dissonance can be cashed out more or less objectively; that's why I put scare quotes around it at the top. The point of this toy example is to see whether the theory -- the physics plus the biology -- can predict the familiar classification of intervals in Western theory, according to some psychoacoustically objective measure of dissonance (beats, etc.). Or, if it isn't strong enough to make such a prediction, to see how close we can get. $\endgroup$ – symplectomorphic Sep 5 '18 at 18:59
  • $\begingroup$ @safesphere: nah, that's not right. when you pluck in the middle, you kill all modes with a node at the center, and those are the even multiples of the fundamental frequency. only the odd multiples of the fundamental remain. $\endgroup$ – symplectomorphic Sep 5 '18 at 20:28
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    $\begingroup$ @alephzero: Writing as a musician, IMO the notion of "dissonance" here is entirely a learned construct. If you check out some of the references on the pages linked from my answer, you'll see that this is not really true. Some of it is learned, and some is innate, and hearing theorists actually have a pretty clear idea which is which. $\endgroup$ – Ben Crowell Sep 5 '18 at 20:28
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There is a course on this topic at Ohio State, and they have a page summarizing the main theories, with references to the primary literature. Some aspects of this are hard-wired in ways that relate directly to the kind of physics you're talking about, while others are a matter of culture and training. I made an educational video that gives a simple presentation of what the Ohio State materials refer to as the tonotopic model, and they give a detailed description of that model here.

In the tonotopic model, dissonance results when two harmonics differ in frequency by more than about 1% and less than about 10%. The early paper by Kameoka and Kuriyagawa gives a formula that they constructed to make this into a numerical measure of dissonance, with parameters fitted to experiments using subjects who were not musically trained, but basically what matters is whether you have relatively strong harmonics that lie within this critical range of about 1-10%. I don't know much about the physiology, but I believe this is explained by the structure of the cochlea, and not by beats as was originally hypothesized by some people (e.g., maybe Helmholtz in the 19th century?).

By this definition, a tone can actually be dissonant with itself, since, e.g., the 13th and 14th harmonics will differ in frequency by 8%. However, those high harmonics are usually quite faint, and therefore they don't cause a strong sense of dissonance.

In your example, it's actually easier to analyze an octave plus a perfect fifth rather than just a fifth. An octave plus a 5th is actually equivalent to a single periodic function with the frequency of the lower note, because the harmonics of the 5th are exactly equal to a subset of the frequencies in the overtones of the root. So it could have some small amount of dissonance, but only for the same reasons that a single note can have some small amount of dissonance, as described above. A plain old fifth could also have some small amount of dissonance, but I think it would also be a very small amount, since most of the strong, low-frequency harmonics are either exactly the same, or fairly far apart.

In a perfect-tempered fourth, we have fundamentals $f$ and $g=(4/3)f$. Going up in the overtone series, which usually would mean going toward weaker and weaker harmonics, the first clashes you get that lie in the 1-10% range are $5f$ with $4g$ (about a 7% difference in frequency) and $7f$ with $5g$ (about 5%). These are quite high up in the overtone series, so they are unlikely to cause any significant amount of dissonance. If you're interested in doing an actual calculation using the K&K model, the Ohio State folks have an open-source program available for that.

You could probably quantify this more by looking at either Kameoka and Kuriyagawa's experimental data or their model, but I think the basic answer is that to an untrained ear, a perfect 4th is not more dissonant than a perfect 5th. We're taught that it is, but like many of the things we learn in music theory, this particular fact only applies to the culture of western music, and it's something that is learned -- not innate. Basically musicians tend to consider a 4th as dissonant because of the way it functions in voice leading. Typically if you want a sound of a nice solid consonance, like at the end of a piece, you expect to hear it in root position, not in the second inversion. In most classical music, if you hear a voicing like G C E (going from the bottom up), it's actually not a C triad, it's functioning as a G triad with two suspensions. The C is going to move down to a B, and the E to a D, so then you'll have the triad G B D.

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  • $\begingroup$ Thanks for this. It's something like the percentage range that I was looking for. $\endgroup$ – symplectomorphic Sep 5 '18 at 20:36
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    $\begingroup$ "In your example, a perfect-tempered perfect 5th is actually equivalent to a single periodic function with the frequency of the lower note, because the harmonics of the 5th are exactly equal to a subset of the frequencies in the overtones of the root." I don't follow you here. I gave the frequencies for the harmonics of the fifth in my post: $1.5f, 4.5f, 7.5f, 10.5f, \ldots$. Those frequencies aren't included in the series for the fundamental, $f, 3f, 5f, 7f, \ldots$. $\endgroup$ – symplectomorphic Sep 5 '18 at 21:59
  • $\begingroup$ You're right. I had in mind an octave plus a fifth, not just a fifth. I'll edit my answer. $\endgroup$ – Ben Crowell Sep 6 '18 at 19:51
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Accidentally I have been interested in musical physics for a few years, and have been interested in neuroscience for other few years and counting, so I am in a perfect position to answer your question. I think your simple analysis is correct. I make the following comments over my head from many years of intense book and scientific paper reading but good references may not be easily reached.

Here are some facts. Different tones stimulate different neuron groups in our brains (Tone "fields" have been found in a few animals' brain surfaces). Neurons in our brains associate features they observe when these features happen together all the time. Non-linearity happens in sound transmission in our ears, so overtones are produced in the ears and then observed by neurons. Neuron groups in higher level than the tone field neurons associate a tone (f) to its over tones (2f, 3f, 4f, ...) such that they are excited when any one of the base tone f and its overtones 2f, 3f, 4f, ... is heard. For example, some groups of neurons may be excited by tones close to f=440 Hz (or 2f=880 Hz, 3f=1320 Hz, ... etc) Other groups of neurons may be excited by tones close to f=660 Hz, or 2f=1320 Hz, ... etc).

Here are some reasonable speculations. Neurons do not like being over-stimulated all together (think of energy consumption constraint, oxygen consumption constraint, waste transportation constraint ... ). That's why noises (all tones sounding together) make people annoyed. On the other hand, neurons are unhappy with dull stimulus (This may be related to learning ---- animals are curious so they can learn and survive). That's why all musics we enjoy have all kinds of regularities with some surprises here and there.

Now we are ready to answer why perfect fifth is more "consonant" than perfect fourth. Let's list the tones and over tones of the base note and the perfect fifth note (shared overtones are in bold face),

base note: f, 2f, 3f, 4f, 5f, 6f, 7f, 8f, 9f, 10f, 11f, 12f ...

perfect fifth: $\frac{3}{2}$f, 3f, $\frac{9}{2}$f, 6f, $\frac{15}{2}$f, 9f, $\frac{21}{2}$f, 12f ...

Also let's list those for the base note and the perfect fourth note;

base note: f, 2f, 3f, 4f, 5f, 6f, 7f, 8f, 9f, 10f, 11f, 12f...

perfect fourth: $\frac{4}{3}$f, $\frac{8}{3}$f, 4f, $\frac{16}{3}$f, $\frac{20}{3}$f, 8f, $\frac{21}{3}$f, $\frac{32}{3}$f, 12f ...

The observation is that the perfect fifth share larger portion of overtones with the base note than the perfect fourth. We deduce from the brain biology mentioned above that the brain hears "cleaner" tones when the perfect fifth is played with the base note than when the perfect fourth is played with the base note. Of course, from similar reasoning, octave notes are more "consonant" than the perfect fifth. I think your question is answered.

Now let's talk about amplitude fluctuation. Amplitude fluctuation happens when your notes are not perfectly tuned thus interference happens. For example, if two strings of the same note are not perfectly tuned, and are played together, $$\cos(\omega t) + \cos([\omega + 2\delta\omega]t) = \frac{1}{2}\cos(\delta\omega t)\cdot\cos([\omega+\delta\omega] t)$$, where $\delta\omega$ is small compared to $\omega$, you hear the tone of $\omega+\delta\omega$ with a fluctuation frequency $\delta \omega$. Of course, when the two strings are perfectly tuned ($\delta\omega=0$) you do not hear fluctuation. You can definitely tune your perfect fifth or perfect fourth such that when they are played with the base note, their overtones do not interfere with the overtone of the base note. So this is not the reason why the perfect fifth is more "consonant" than the perfect fourth.

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  • $\begingroup$ What you are describing is only a minor part of the perception of the consonance. You can conceptually play music on an instrument without harmonics (e.g. tune a synthesizer this way). The music would not be as bright, but still very much recognizable and enjoyable. The critical part is the ratios of the main frequencies of notes, not of their harmonics. For example, the notes of a major chord relate to each other as the first three odd harmonics of the corresponding bass note, $1f,\,3f,\,5f$ or $1f,\,\dfrac{5}{4}f,\,\dfrac{3}{2}f$. $\endgroup$ – safesphere Sep 5 '18 at 20:24
  • $\begingroup$ @safesphere I think your examples (1f, 3f, 5f or 1f, $\frac{5}{4}$f, $\frac{3}{2}$f are consistent with my description. The specific ratios of the notes of a major chord are truly critical, and this can be readily explained by their overtone overlaps. Also to musics without harmonics, maybe they are enjoyable because they have other regularities (say, drum music), and not because they do not have harmonics. $\endgroup$ – verdelite Sep 5 '18 at 20:33
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The human ear is not an accurately calibrated instrument and no instrument has "exact" harmonics. Piano strings, for example, have a finite width this changes the harmonics. As does the shape of the horn, etc. What makes for dissonance is probably mostly "beats" between harmonics and / or notes (or harmonics) that are close enough in frequency, so that we can notice they are very close. There is lots of work on this, which (unfortunately) I don't immediately find

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