The Physics Of Dissonance

According to physics, this is the most dissonant chord.

I mean, look at this chart.

It's a graph showing the dissonance of all possible three note chords.

Peaks correspond to dissonant sounding combinations.

And the deep wells correspond to consonant combinations where the notes are in harmony

with each other.

And it seems to work shockingly well.

The different deep wells each correspond to fundamental chords in Western music theory. major,

minor, their inversions, suspended chords, and so on.

And the peaks are dissonant chords made from horribly out of tune notes.

Except with just a slight tweak to the sound,

those same fundamental chords suddenly sound dissonant.

And now they're at the tops of peaks instead of the bottoms of deep

wells.

And here's an octave.

Supposedly the most consonant and fundamental interval in western music.

And it sounds pretty badly out of tune.

That's um unexpected.

The fact that a graph like this exists was a big surprise to me.

I want to explain how it works, the amazing things it does,

and where it falls short.

Also, why the same chord can sound either dissonant

or harmonious with just a few small changes.

What a particular combination of musical notes makes you feel,

which combinations seem settled versus the ones

that have tension and discomfort seems like it should be mostly qualitative depending on

a person,

their culture, psychology, musical exposure, etc.

And there's undeniably a ton of psychology

and environment and culture in our experience of music.

But there's also a fair bit of quantitative analysis we can do.

Sounds start off as physical pressure waves in the air, after all,

and are then transmitted

as vibrations through flesh

and bone before entering the realm of neuroscience,

psychology, and beyond.

This video is about the physics of dissonance,

the role physics plays in why certain notes sound dissonant when heard together,

and some sound harmonious,

and how those differences led to different music in different cultures.

There are many, many different ways to define dissonance and harmony,

most of which we're not going to get into here.

We'll focus on the physical and physiological aspects of dissonance,

the slightly more quantitative parts.

This video will have four main parts.

The dissonances between sine waves, the dissonances between notes, how dissonance leads to scales,

and how dissonance leads to chords.

Also, this video contains lots of audio examples.

If possible, I recommend listening with headphones or decent speakers.

Part one, the dissonance of pure sine waves.

The stereotypical way to talk about the sound of a musical note is to

say that it has a fundamental frequency

that determines the pitch of the sound

and which is almost always the lowest frequency sine wave involved.

And then there are overtones

or higher frequency sine waves

which determine the nature

or character of the sound.

It's tambber.

So the explanation goes a violin

and piano can play a note with the same pitch

but sound like a violin

or a piano.

And while it's certainly true that overtones determine the tambber of a sound,

it's not the only thing they do.

They can also determine what musical scales

and chords the note should even belong to.

First though, we should remember

that this overtones interpretation is only possible

because sound waves,

at least in air, can be split apart into fundamental building blocks,

typically sine waves.

Like this sound is made up of multiple sine waves playing at slightly different

frequencies and different amplitudes.

Add the sine waves together and get the final sound.

Start with a sound

and you can split it apart through clever math into the different frequency sine

waves that make it up.

My friend Grant of Three Blue

and Brown has covered this very well in his videos on Forier transforms.

Because sounds can be broken apart into their component sine waves,

we'll first talk about the dissonance between individual sine waves

and then move on to dissonances between the combinations of sine waves

that make up more complex sounds,

which is what leads to tuning and scales and harmony.

So, how are pure sine waves dissonant?

If you play a constant sinewave tone

and then add on another

that starts at the same pitch

but gradually rises in frequency,

most people hear the sound start off as a single pitch with no dissonance,

then go through an increase in dissonance before it calms down

and separates out into two different pitches. there's zero dissonance

when the frequencies are the same.

Then dissonance rises sharply and gradually falls back down,

but not all the way to zero as the pitches get farther apart.

The shape of this graph is in fact borne out by controlled experiments on

real people.

The point is there's a kind of zone of discomfort

when two sine waves are close to each other in frequency

but not close enough to sound the same.

And the graph is actually two-sided.

If the pitch of the second note falls below that of the first,

the same thing also happens.

So why are pure signs dissonant?

The physics here is twofold.

First, there's the phenomenon called beating where two sine waves of almost the same

frequency alternate between constructively interfering

then destructively interfering etc.

which results in a kind of wo wo wo modulation of their amplitudes.

The closer the frequencies, the longer it takes them to get out of phase,

then back in phase, etc.

So, the longer the beats last, the farther apart the frequencies,

the faster they get in and out of phase,

and the faster the beating happens.

Eventually, the beating happens

so fast that we stop hearing it

as a wo wo wo type sound

and more as a general roughness.

Around this same point, another phenomenon starts to get involved,

the imperfect resolution of the human ear.

I don't want to get into the details

and there are still some details

that aren't entirely clear even to science.

But because of the geometry of the coccia

and the physics of the fluid

and hairs within it,

an external sound of a single frequency will stimulate not just the auditory nerve

fibers that correspond to

that frequency,

but also fibers for slightly higher and lower frequencies.

It's kind of like our ears blur or smear out the frequencies a little.

Our ears need glasses.

So if two frequencies are too close to each other,

the ear can't fully and faithfully distinguish them as two different sounds.

Instead, we hear the combination

as a single but dissonant note. until the frequencies get far enough apart

that we're able to hear them

as two separate sounds.

One really cool way to experience these physical effects is to wear headphones

and play one pitch in one ear

and a slightly higher pitch in the other ear.

Put on headphones and try it.

For me at least,

I hear two distinct but nearby pitches when I listen with headphones.

While if I take the headphones off or play both pitches in both ears,

which I'm doing now, it sounds like one single dissonant pitch.

The point is that taken together,

the physics of beating

and the physics of the ear explain the zone of discomfort

when two sine waves are close in frequency

but not close enough to feel like the same pitch

and why when they're far away from each other,

they just sound like two separate pitches.

There have been various attempts to quantify exactly how wide

and tall this zone of discomfort is,

but there doesn't appear to be anything fully agreed upon.

So, when looking at these graphs,

don't dwell too much on the exact dimensions of the zone of discomfort.

The overall shape is what's more relevant.

And the overall simplicity of the shape of this dissonance graph might be surprising

to you.

At least it was to me since there's nothing

that singles out any musically relevant pitches.

No octaves, fifths, major thirds,

or literally anything at all that we think of as musical.

It just says sine waves either want to be exactly the same,

or they want some personal space.

Musical harmony, it turns out, requires more than pure sine waves.

Harmony comes from overtones.

Actually, that last bit isn't entirely true

because there is another type of beating

that happens between sine waves whose frequencies are slightly out of tune with small

integer multiples or fractions of each other like twice the frequency,

three halves, and so on.

It's called secondary beating.

And the wo wo wo is only a partial reduction in volume

or just a phase shift rather than a change in volume.

And no one I found seems to talk about the secondary beating

as relevant for tuning.

And I find it much harder to reliably hear the wo wo wo sound

depending on what speakers I'm listening on

and whether I'm using headphones

and so on.

My best guess is

that maybe we should make a very very small adjustment to the single sine

wave dissonance graph,

but we're going to ignore it for now

and just use the simple graph for the dissonance of pure sine waves.

It's going to get complicated soon enough, don't worry.

Part two, dissonance in notes with overtones.

In order to have strong dissonance

or consonants like we showed in the graph at the beginning of the video,

solitary sine waves won't do.

You need to use sounds made up of combinations of sine waves.

Lucky for us, in our universe,

the different physics and shapes of sound producing objects result in different combinations of

sine waves.

The overtones of a vibrating string look and sound like this.

Overtones of pipes look and sound like this or this.

Circular drums like this, bars like this, bells like this,

a random discrete series of overtones like this, and white noise like this.

As a side note, you'll sometimes hear people refer to overtones as harmonics,

but that name is usually reserved just for the overtones

that follow the specific pattern of frequencies of a vibrating string

or fully closed or fully open pipe.

The physics of a string under tension means it vibrates with overtones whose frequencies

are integer multiples of the fundamental frequency 1 2 3 4 etc.

For various historical reasons, this particular pattern of overtones is called harmonics,

while the overtones of other instruments don't get special names.

All these overtones mean that when listening to a pair of real world notes,

we experience not just the dissonance between the two fundamental sine waves of each

note,

but also between the fundamental sine wave of one note

and all the overtones of the other

and vice versa,

as well as the dissonances between all the overtones.

Like before, we can play a constant pitch note,

but this time with the overtones of a vibrating string,

and then add another note that starts at the same pitch, but gradually rises.

You'll probably hear the notes alternate between sounding dissonant and then in tune.

Just to contrast, let's go back to the dissonance of pure sine waves.

This dissonance graph is built by simply adding up the various sine wave dissonance

graphs for each possible pair of fundamental

and overtone frequencies that are in the two notes with the positions of their

zones of dissonance adjusted according to the frequencies in question

and the heights adjusted according to the relative loudness of the overtones.

The dissonance graph reminds me of graphs of potential energy in physics.

Like when a ball wants to roll down a hill to be in the

minimum energy state or

when two hydrogen atoms are close together,

there's a point of minimum energy here where the hydrogen atoms want to form

a hydrogen molecule.

In a sense, it's their least dissonant position relative to each other.

Similarly, there's a sort of emotional pull towards two notes wanting to be in

the bottoms of the valleys of the dissonance graph.

These are places where the two notes sound in tune,

and they sound in tune precisely

because the combined dissonances of their fundamental frequencies

and all their overtones happens to be very low there.

The tops of the peaks corresponds to situations where the overtones of the two

notes line up so

that many of them are dissonant.

In particular, when the notes are very close to each other

or very close to the first overtone,

you get very big dissonance spikes

that are reminiscent of the dissonance graph for a pure sine wave.

The bottoms of the valleys correspond to situations where the overtones of the two

notes line up well enough

that very few pairs of them are dissonant with each other.

So, you get a low overall value for dissonance.

For example, this big valley right here corresponds to a place where the first

overtone of one sound exactly lines up with the fundamental frequency of the second.

And this one is where the second overtone of one sound lines up with

the first overtone of the other.

Here's where the second and third overtones are lining up really nicely.

The locations of the valleys are precisely determined by whatever ratios can be built

out of the overtone frequencies.

Like if you have an overtone with twice the fundamental frequency

and one at three times,

then the dissonance value they generate will be at three halves

or 1.5 times the fundamental frequency.

Here are all the valley bottoms played in sequence.

Part three, the relationship between dissonances and scales.

If these intune notes sound familiar to you,

it's because they're the fundamental intervals of Western music.

The octave, the fifth, the fourth, major 6th, the major 3rd, minor 3rd, etc.

So, are these musical intervals somehow pre-ordained from the fundamental fabric of our universe?

Not really.

These intervals sound in tune

because they are the intervals

that happen to sound the least dissonant

when using the overtones of a vibrating string

or pipe.

Sure, the physics of our universe does dictate what the overtones of a vibrating

string or pipe are.

And those do happen to be two fairly common

and simple ways of creating sound,

but they're not the only ones.

If we change to the overtones of say a bell,

then the dissonance graph changes

and the intervals important in western music sound dissonant

while there are new intune notes at the new valley bottoms.

Not our familiar intervals, but they're the ones that are in tune.

Now, if we play a perfect fifth,

which has a frequency ratio of 1.5,

it sounds more dissonant than this valley near it with a frequency ratio of

1.47.

The way the dissonance graph changes

when you change overtones illustrates part of why many bells

and drums don't sound quite

as melodic in Western music.

Their overtones don't match the overtones of the strings

and pipes on whose overtones western music is built.

I should also mention here

that the perfectly intune string

and pipe notes of the graphs are not in fact the exact notes we

use in western music anyway

because for various reasons some of

which I talk about in my video about the physics of piano tuning.

Modern music uses a series of equal spaced multiples of the fundamental frequency for

all of our notes called equal temperament

which closely approximates the most intune intervals according to the dissonance graph

but not perfectly.

And in fact, some of them result in dissonance peaks.

Like the major third in equal tempered tuning sits here at this peak rather

than the slightly lower pitched just inonation major third,

which is less dissonant, but introduces other problems when used in actual music.

If you start with just one overtone, then two, then three, and so on,

you can see how the graph of dissonance builds up

and where the most strongly in tune notes are.

First, it's just the root.

Then additional overtones add the zones of dissonance

that give you the octave.

then the fifth,

the fourth, the major 3rd and major 6th, minor 3rd,

subminor 3rd and tri-onee and minor 7th, etc.

You can use the buildup of intune notes to either study one of the

more important or fundamental intervals

or chords in a musical system,

or to build your own musical instrument and corresponding tuning system.

The first overtone you add will be the first intune note,

which is the equivalent of the octave.

Then once you add a second overtone,

you'll add two more intune notes. one at

that overtone and one where the first overtone of one note matches the second

overtone of the other

and so on.

The point is this,

the combinations of notes

that sound good together are due to whatever the overtones of the notes are,

not to some deep relationship between fundamental frequencies.

An A doesn't sound in tune with a D just

because the fundamental frequencies of the A

and D have a ratio of 3:2.

Those sine waves are far enough apart that they sound good together,

even if the frequencies shift around a bit.

If you don't believe me, I actually played that first A out of tune,

and you probably didn't even notice.

The notes A and D sound in tune

because when played on an instrument with overtones

that come from the harmonic series,

the overtones of the A

and D have less dissonance with each other

when the fundamental frequencies of A

and D are in a ratio of 3:2.

Here's the out of tune A.

A again, obviously out of tune due to its wo wo wo wobbling.

To show again how important the overtones are and which ones are important,

here are two pure sine waves played slightly out of tune from a perfect

fifth and then perfectly in tune.

Not that much of a difference.

Now the same pitches with the first overtone present.

Still doesn't sound that out of tune.

Now we add in the next overtone of a vibrating string.

And finally the out of tune perfect fifth actually sounds out of tune.

And correspondingly the in tune one sounds in tune.

But the perfect fourth doesn't yet sound in or out of tune.

Only when we add the next overtone does the fourth become well defined. to

belabor the point,

tuning comes from overtones.

So, how true is all of this?

This whole dissonance graph analysis is an explanatory theory developed over time by various

researchers.

It seems like a pretty decent explanation for why Western music works the way

it does,

though there are also other competing theories.

Maybe human brains naturally prefer notes that have frequency ratios that are small integers,

like 2:1 or 3:2.

Or maybe it's all just a coincidence.

But there are a few reasons I think the apparent usefulness of these dissonance

graphs is more than a coincidence.

For one, the zone of discomfort can help explain why two notes a half

step apart,

say an A and a B flat,

sound much more dissonant

when played in the same octave

as opposed to an A

and the B flat the next octave higher.

Their combined dissonance there is lower.

But to me, the real confirmation

that the physics and physiology represented in the dissonance graph is a major factor

in determining what notes sound in tune to us

and what musical scales

and intervals we use is what happens

when the instruments we use don't have overtones

that come from the harmonic series.

First off, did you know

that the higher notes on pianos are actually tuned slightly sharp

and the lower notes are tuned flat?

It's because piano strings are big

and stiff enough they don't behave perfectly like strings,

but also slightly like vibrating bars,

which shifts the overtones of the strings slightly upwards in pitch.

If you want the piano to sound in tune with itself,

you have to tune the higher notes up

so they match the stretched overtones of the middle notes.

And you have to tune the lower notes down

so their stretched overtones match the overtones of the middle notes.

The stretch is called a rails back curve

and is worse the shorter the strings,

which is one reason people like really big grand pianos.

As a second example,

you can more generally take our western musical system

and squeeze or stretch it

so all intervals are slightly flat

or sharp.

Instead of using the overtones of a vibrating string with the frequency multiples of

1 2 3 4,

etc., we can create a sound with overtones of frequency multiples 1 to the

0.95,

2 to the 0.95, 3 to the 0.95,

4 to the 0.95, etc.

This shifts the intune valleys slightly downwards

so that the most intune note isn't an octave up at double the frequency.

That's now a dissonance peak and sounds out of tune.

The intune octave is slightly flatter at 1.93 times the fundamental frequency

and sounds much more harmonious.

Similarly, the intune perfect fifth is no longer in tune at its normal position

of 1.5 times the fundamental frequency,

but is in tune at 1.47 times the fundamental.

Perhaps the best example of the explanatory power of dissonance graphs is non-western music.

For example, Indonesian gamalan music has many instruments in it

that use carefully shaped kettles,

gongs, and bars, which all have very non-string-like overtones in addition to singing

and other instruments with overtones

that more closely match those of a string

or pipe.

To oversimplify, if you make a dissonance curve comparing the sound from a pipe,

aka singing, with the sound from one of the kettallike instruments,

you end up with minimums

that almost perfectly match one of the main five note scales used in Gamalan

music.

And if you look at the dissonance between a sung note

and one of the bar instruments,

you end up with a dissonance graph whose valleys decently match one of the

non-uniform seven note scales used in Gammaan music.

Similarly, Thai classical music features instruments using bars

which when their overtones are combined with the sound from singing

or string overtones.

The dissonance graph looks very close to the seven note musical scale used in

the music.

Tuning comes from overtones.

You can also build all sorts of synthetic scales.

Like if you make a note with overtones

that are given by the prime numbers,

you end up with intune values at ratios

that can be built out of prime number fractions.

The point of all these examples is to show the usefulness of the dissonance

graph in explaining how

and why tuning comes from overtones.

Part four, dissonance in chords.

Okay, we are finally ready to explain the 3D graph from the beginning of

the video with all those peaks

and valleys.

At this point, you may have already guessed

that it's the dissonance graph for a chord made up of three notes.

We pick a fixed frequency for the root note

and then the x-axis represents the relative pitch of the second note above the

first and the y represents the relative pitch of the third note.

And we add up all the dissonances between all the combinations of different overtones

of all three notes with each other to get a 3D version of the

dissonance graph.

As before, peaks represent highly dissonant chord combinations

and the bottoms of valleys represent combinations

that sound more in tune.

This particular graph is for the overtones of a string

or pipe and all the little valleys correspond to particular chords in western music.

Here's the major chord and its inversions, minor chord and its inversions,

suspended and augmented chords, etc.

When viewed from above, you may notice the graph appears to have horizontal, vertical,

and diagonal lines.

These lines are long values that cut across the whole graph.

The vertical lines are for

when the x-axis note is nicely in tune with the root note,

like when the x-axis note is a fifth

or fourth or whatever above the root.

Viewed from this side, the 3D graph looks like the 2D version.

After all, the horizontal lines are the same thing,

but for the y-axis note relative to the root.

Here it's a fifth.

Here it's a fourth, etc.

And the diagonal lines represent

when the x and y-axis notes are in tune relative to each other.

So naturally, the most intune chords are places where all three sets of lines

intersect,

where all three notes are in tune with each other.

For overtones of a string,

the most intune chords are the major and minor chords, a few inversions,

and a few suspended style chords.

And it's in the gaps between these lines that we find the dissonance peaks,

the most dissonant chords.

The big dissonance peaks near the corner come from the two higher notes of

the chord and their harmonics being too close in pitch to the root note

and its harmonics and too close in pitch to each other.

So there's a sense in

which the most dissonant chord is one

that sounds like this.

This dissonance is essentially just the dissonance of any pair of sine waves

and doesn't change much with different overtones.

Like here it is with the overtones of a bar

and of a circular drum.

So, in one sense,

this chord is the most universally dissonant chord on any type of instrument for

any type of tuning.

But because of that, it's also not that interesting.

It's just the dissonance of pure sine waves.

The same thing roughly applies to these ridges of dissonance

that extend out along the y

and x axis and on either side of the x equals y line.

They're chords where two of the three notes are very close in pitch to

each other.

Kind of boring.

The more interesting dissonant chords are the ones

that spring up farther out from the origin.

They have dissonance due to overtones

and harmony and they sound about how you'd expect.

Gross.

In this particular version of the graph, up to eight overtones,

the most dissonant chord according to the graph is this one.

Kind of partway in between a minor chord major chord in its first inversion

and suspended second chord.

To me, it just sounds like one of these chords badly out of tune,

which I guess it is.

Close behind is this chord

or this other one over here

or this other one over there.

There are a lot

and they're actually all pretty similar in how dissonant they sound.

I think to me what's most notable in clicking randomly around the space of

all possible three note chords is

that most possible chords sound pretty dissonant

and at a pretty similar level of dissonance.

And because of that, when you do happen upon an island without dissonance,

one of the deep valley bottoms.

It's such a change

and such a relief

that it's not at all surprising

that these islands would become the basis for musical harmony.

Where the islands of harmony are depends on, you guessed it, the overtones.

Because here's the 3D dissonance graph for a drum

and here's the graph for the prime numbers.

They have different harmony and tuning.

In summary, it appears if this theory of dissonance is at least somewhat correct,

that whether or not we humans think two notes are in tune depends on

how well the overtones of those notes line up.

And since the overtones of notes is different for different types of sound sources,

strings and pipes and bars and bells and drums,

different musical cultures can end up coming up with different musical scales

and chords that best fit the overtones

and sounds of their instruments.

It makes sense that a sizable proportion of human music, including western music,

is based at least approximately upon the harmony generated by the harmonic series of

overtones because that's the overtone series for strings

and pipes,

which are two pretty common sources of musical sounds.

But it appears that plenty of other human music is based on other harmonies

that come from overtones of other types of instruments.

In practice, music and dissonance

and harmony are a lot more complicated than we've begun to touch on,

with physical, biological, psycho acoustic,

and cultural factors well beyond the scope of this video.

But I at least found my mind opened by the idea

that dissonance and consonants

and the feelings of being in tune are

so well explained by physics

and so heavily influenced by not just the fundamental tone of a note

but the overtones as well.

This way of thinking about consonants has its limits.

Like it doesn't necessarily give a lot of insight into why say certain chord

progressions work the way they do.

But to me it's an interesting

and useful way to think about the notes

that make up chords

and harmony.

To remember that a note is more than its fundamental sine wave in very

real and powerful ways.

My friend Otish has an interactive article on their website where you can play

around with these ideas.

It's where I first heard about this quantitative way to define dissonance

and I recommend you check it out.

And I've used screen grabs from the website for examples throughout the video.

There are also a bunch of caveats to the dissonance graph

that I want to mention.

But first, I want to say a big thank you to the Acoustical Society

of America,

which is literally the perfect sponsor for this video.

Almost all of the best references on this subject were in journals

or books published by the acoustical society.

There are so many cool careers possible in acoustics.

Music recording, forensic acoustics, rocket and aircraft noise research, speech pathology, underwater drone communication.

The list goes on and on.

Go check out the ASA's acoustics career toolkit at exploresound.org/acoustics careers.

It's an online resource designed to help you explore the possible career options in

sound,

science, and technology.

As the acoustical society says, if you're into science, engineering, healthcare, or the arts,

acoustics offers a career path for you.

Visit the acoustics career toolkit.

And thanks to the acoustical society for supporting this acoustics themed minute physics video.

And now, the caveats.

This isn't a caveat,

but we thought the dissonance graph was cool enough

that we should make a t-shirt out of it.

So, here it is.

You can get it at dftba.com/minutysics.

It's perfect for the nerds, the musicians, the music nerds in your life.

Again, dftba.com/minutics.

Now, back to the caveats.

One interesting feature of the dissonance graphs for strings

or pipes is that they include an intune valley bottom for a note between

the intervals of a fourth

and fifth,

which is called the trionee and considered the most dissonant interval in western music.

But according to the dissonance graph,

it shouldn't be any more dissonant than a minor 6th.

There's more to harmony than the dissonance graph.

You may have noticed

that we haven't really talked about whole steps

or half steps,

which are the smallest intervals in Western music

and sometimes thought of

as the building blocks of scales.

The problem is they're actually really dissonant.

And the same thing applies to the major 7th,

which is a half step below the octave.

If we start with a dissonance graph,

the place where something like a whole step really shows up is in the

differences between the notes

that make a fourth,

a fifth, and a major 6th.

And something like a half step shows up in the differences between the minor

3rd,

major 3rd, fourth, tri-onee, fifth, minor 6th, major 6th, etc.

Though the actual size of those steps is not at all consistent.

My personal theory is

that the origin of whole

and half steps comes from these interval differences rather than whole

and especially half steps being nicely intune intervals on their own.

because a half step never really sounds in tune.

The exact shape of the dissonance graph, even the simple one for sine waves,

depends on a lot.

Physics, human physiology, human psychology,

and probably even the type of musical harmony you've been exposed to in your

life.

Here are a few factors I know for sure that I glossed over.

Slight changes to the shape of the dissonance graph for pure sine waves result

in qualitatively similar but quantitatively different dissonance graphs for chords.

Like here's a dissonance curve drawn by Helm Holtz in the late 1800s that's

a lot curvier than the ones we've been using.

The overall graph also depends on whether

or not you take into account sensitivity of human hearing to different frequencies

and how loud the different overtones are relative to each other in the sound

you're listening to.

The overall shape of the dissonance graph also changes

if you include the dissonance contribution from all the overtones of one note with

each other before you even add a second note.

And the dissonance graph has been built by assuming

that dissonance of a sound adds up linearly in a nice clean superposition just

like the frequencies do.

That is, instead of calculating the dissonance directly,

we're assuming that just

because a sound can be broken apart into its component frequencies

and added back up together.

Similarly, if we break a sound apart into its components, calculate their dissonances,

then add the sound back up,

we're assuming the dissonances also add back up rather than being some complicated nonlinear

relationship.

Another way that the dissonance curve oversimplifies things is

that instruments don't just make one type of overtones.

Like a violin isn't just a bunch of vibrating strings.

It also has a wooden body that vibrates in ways similar to a drum.

And the body itself is an acoustic cavity for the air vibrating aside

and so on.

And because the shape of the violin body doesn't change even

if you change the note you're playing,

the overtones of a violin are actually a complicated combination of the body overtones

and the string overtones.

Thanks again for watching and go get your dissonance t-shirt.

It also comes in white.