Tommaso Zillio

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science of music

The target audience for this newsletter is rather particular: people who want to understand (some of) the science behind music theory.

If you are not interested in that - well, why are you reading this? Did you not see the title? Are you a glutton for punishment? (Not judging, but curious!)

But if you are interested, then keep reading!

So, first of all, let's make one thing crystal clear:

You do not have to understand a single iota of science (1) to make music

Or in other words:

  • You can write great music without knowing what overtones (or undertones) actually are

  • You can understand all the music theory you will ever need without knowing anything about the wave equation

  • You can understand every single nut and bolt of music theory without knowing even the tiniest bit of algebra or calculus.

So let me reassure you that you don't need this.

On the other hand... inquiring minds want to know! It is fun to know more!!

Buuuut... who can explain to you all that?

Why, with the magic of the Internet, you could probably find a lot of explanations out there from either musicians or physicists!

But perusing these videos for just a few minutes, you will sound find out that there are two problems:

  1. Most musicians-not-physicists really don't understand much physics. (I particularly bang my head on the wall in frustration when people talk about "undertones" as if they were a real, physical thing...)

  2. Most physicists-not-musicians have no idea how music actually works from the point of view of a musician (2). And do you really want an explanation that is 70% or more made of equations? (6)

Ah, if only there was somebody out there who knows a thing or two about physics AND about music theory...

... waaaait ...

... that would be me! (7)

So here's for you a calculus-free intro-level explanation of some of the science stuff underpinning music and music theory:

Now, if you want a much more practical thing to learn that will improve your guitar playing, how about improving your knowledge of chords? Check out my Complete Chord Mastery guitar course if you want to immediately improve your knowledge of chords on the guitar.


(we scientists like those...)

  1. Science = physics of sound, in this case. Not rocket science (either literal or metaphorical)

  2. Of course, since I am a physicist, I also know that physicists are supremely confident that they know how music works (3). I was too. Until I learned enough to write actual music, and I realized that... no,just because I knew physics that did not make me a good musician. Strange as it is to write it now, I was genuinely surprised.

  3. Incidentally, this does not seem to be a physicist-only problem. Everybody thinks they understand music. Mostly because they listen to music and they know their taste is better than others. Which is like saying that if you can catch a ball, you are an expert on resolving the equations of motion of a semi-rigid body in a gravitational field.(4)

  4. Which in turn is as false as saying that if you can resolve the equations of motion of a semi-rigid body in a gravitational field, then you are great at catching a ball (5)

  5. I think this email is setting the record on the number of footnotes

  6. If you do, you're better off reading a book (or ten) on Fourier series. Good luck ;-)

  7. Yeah, I do have a PhD in physics. Not bragging. Or at least, trying really hard not to.

Video Transcription

Tommaso Zillio 0:00
Hello internet, so nice to see you! people keep asking me questions about the science of music theory. So I'm going to I grab a comment that contains a lot of questions about that I'm going to answer now, before I even start, though, let me tell you one thing. In order to write music or to play guitar, you don't need to know the science of music theory.

You don't need to know the science behind it and the mathematics and stuff. You just don't. I know a lot of you out there want to know that and are curious about that. So I'm making this video because after all, I know my music theory really well and teach it. I teach music theory for a living after all. And I do have a PhD in physics, I confess.

So I'm pretty much the right person to answer this question. It's okay. At the same time, nothing of what they say today will make any difference in your music and just telling me that he's right now, if this is something interesting at the mainstream, because you guys like it, it's not gonna make a difference in the way you play. Okay?

I'm saying these also, because if you watch this video and you understand half of it, don't worry, you can still learn your music theory, you can still learn all the music theory you want and write great music. You don't have to understand all this. But hey, it's interesting. So let's go and say that I had a totally unrelated question. Yes, indeed, as I was saying is a totally unrelated question.

Kamil 1:25
When we play a note on the guitar or any other instrument has bass frequency acts, but also other frequencies such as times two times three times four, etc, are also still there, but in lower amplitudes compared to the bass.

Tommaso Zillio 1:34
Okay, first thing, it's, as you're gonna see, in a moment, exactly what this sentence mean. But let me just put these in right now, it's not true that the bass frequency, which technically is the fundamental frequency, not the bass, but it's not true that the fundamental frequency have a larger amplitude than the RF frequencies. Okay? Did you have any situation where it's not? Okay, just putting it there? Because it's not true.

Kamil 1:57
Naturally created sound always seems to have this aspect.

Tommaso Zillio 2:01
Okay. So let's start immediately there. It's not a question of natural decay to sound artificially created sound or anything like that. There are two kinds of sounds, okay. That's what physics tells us. There are two kinds of sounds, periodic sounds and non periodic sounds, okay? Periodic sounds is when something repeats several times.

And it usually happens so fast that you can eat your ear cannot feel the repetition, it just happens too many times in a second, and no periodic sound where the sound does not repeat. We call the first kind harmonic sound, because when you hear a periodic sound, your ear interprets that as a note with a specific frequency, which is the frequency of repetition of the sound, okay? So for instance, you have something like a waveform, okay, you have something like that. And then the same thing repeats.

And I mean, if I work with that drawing, it will be exactly the same thing, if it is seven times. Now, if this happens 440 times a second. This will be one fun and 40th of a second disciples 440 times in a second, this will sound like an a note to you, and it will sound like this. And the exact shape of these wave and repeating will tell you what is the timbre of the note, then you have non periodic sounds, we call them noises in general, because your brain does not interpret that as a note, okay, so this happens when you struck what is called a non non tune percussion.

So let's say you hit a snare drum, for instance, or you hit, you hit the you hit the white wall, or anything else that's unknown, periodic noise doesn't sound like a note, it's a noise. Okay, this doesn't repeat the waveform, this doesn't repeat. So it doesn't create an okay, if you're following me so far, that's great.

Now, what happens is that when you have a periodic sound, so the known the thing repeats 440 times a second, okay, then, by a complex mathematical reasons, I'm not going to go into into into it here, but you can find everything you want by studying what is called the Fourier transform. Actually, in this case, the Fourier series. You can go there, you can Google that. And then if you know your math, you have hours of fun.

If you don't know your math, you will have to study three years of math before you understand that, but if this happens, then you can take your strange way even and rewrite it as one pure wave at the base frequency or the at the fundamental frequency. And now they're pure wave at twice that frequency. And this point is starts to be hard for my limited drawing abilities here, pretty much at that point.

Now the fundament another frequent another wave up three times the frequency 23123123. And fundamental frequencies are four times the frequency 12341234. Pretty much, okay? By changing the amplitude of all those waves, and so that is fundamental two times three times, four times and summing all of them together, you can obtain any shape here. Okay?

Doesn't matter, what's the shape here, you can always write as a sum of that putting together all those waves. So if you have enough for sine waves, if you have enough of those tuning forks, each one do the right way. And you can modulate the amplitude, you can recreate any kind of timbre, you can make the sound like a violin, like a trumpet, like any other synthesizer sound, okay? And that's what I'm synthesizers actually do directly some of these. Okay, additive generation of sound.

Makes sense so far, but this happens only if the sound is periodic, so that you perceive it as a musical note. Otherwise, you cannot do that. If you take a bass drum, okay, or something that doesn't sound like I know what to adjust these. And you cannot do the same thing because you don't have a bass frequency, you don't have a fundamental frequency. So you cannot multiply it.

So that you need to use something completely different. That is called the Fourier transform. Okay, I don't know why the guys decided the one that was a series when he was transforming. I know, but does have a very similar name, they should have you something more, more clear on for the distinction, because people just hear the name Fourier, and they think it's the same thing. Now it's the different procedures very different.

That's what creates sounds. So yes, everything is made everything you play on the guitar, this is just the sum of sin waves and things like the waves with the right proportion to create that. And the fundamental one doesn't have to be the one with the biggest amplitude. Okay? Indeed, frequency for us distortion is the typical is, okay. If you, if you if you're playing I don't know, a violin, violin or a cello, a big part of getting a good tone out of that is to try to make these waves bigger than the other waves.

And when it doesn't happen, when this is not big enough, you are screeching with the bow over the string and you go, Okay, what's happening is you're not making the fundamental wave big enough. If you want to have the scientific reason of that, then explaining exactly what you do with your bow. Completely different problem.

Kamil 7:46
Okay, but what about like tune percussion? What if a drummer is going to tune their bass drum to B flat or something like that?

Tommaso Zillio 7:52
Good question. So what happens if you have a drum and you're doing the drum, while you're doing what is called the Fourier transform, we had that before, when you have a free transform, you cannot separate these in frequency one frequency to frequency three, but you're using all the frequencies so you have less frequency.

Okay, that's amplitude. And you hit the whiteboard, okay? And what happens is that you look at your EQ, okay? And there is something going like this, okay? Right. But they're not those are not separated in like the first frequency. The second frequency, there's a you're using, you're using now, all the frequency, what you're doing when you're tuning the drum, is that you are trying to modify the tension of the ground or the diameter or other things, so that you get a peek somewhere.

So when you hear this noise, it's a noise because you have all this frequency, and he goes to Pang or something like that. But there is also a big component close to a specific note. So you are hearing that as if there was a note hidden in there. Okay, but it's not a real note is just as a peak in the spectrum.

You noticed that if you record these and you use either Logic or Cubase, or certainly put an EQ, you're gonna see the waveform, okay, you're gonna see Sorry, them the frequency spectrum, that's what it is, if fully transformative will take you from the sound to the frequency spectrum, okay. And you're gonna see a peak somewhere that's what you are moving you're trying to move these up and down to find the right

Kamil 9:34
Note an octave higher will contain frequencies times two times four, etc.

Tommaso Zillio 9:38
Well, that too fast. Okay, so what these are not the Okay. That's an interesting thing. And not David, is what happens when we have two notes. And the frequency of the one is twice the frequency of the other. So if this guitar is still in standard tuning, my A are the 12th fret of the fifth string, it's at 440 hertz for another 40 years.

My open string is at 220 hertz, okay? So open string string 220 hertz, hertz means vibrations per second, this thing goes up and down 220 times in a single second. It's very, very, very fast at the 12th fret and I have 440 Hertz. exactly twice. I mean, that's assuming you get the guitar is absolutely perfect. The string isn't is not rigid and all this kind of thing.

But let's, let's assume it okay. For some reason, our ear does not perceive those two things as two different notes, but it perceive it as something of the same two different level. Why would I play the note, an a and an E, you think that different notes when I play an AMA, you think those are clearly two different notes, but they're related somehow. So that's the same note higher. why this happened, this is not a property of the physics.

This is not there is nothing in the physics or mathematics of this. That tells you that your brain has to perceive things this way. But since you can hear around 10 octaves of sound, if you're very young, and your ear is perfectly pristine, you can hear 10 octaves of sound, it seems like your brain at a certain point evolved or decided that to do something and to make things easier for your brain to understand by thinking a I'm just repeating the same thing over and over. So that note, and that note, and sorry.

So that note, and that note, and that note and that note are the same note, at a different level, we call those octaves, okay, so kinda, you're really starting to compress the information rather than thinking that this is an order, that's a completely different Northern, that's a completely different note, your brain started to make sense of all this big input with different octaves. And we deal with the topic a lot of different frequencies by thinking okay, every time you double the frequency is the same note.

And for some reason, it seems to make sense to your ear. But lots of music theory is built on this perception thing, which has nothing to do with the physics of thing. There is nothing in the physics that says besides that, if you double the note, it's the same note, it's just something that we found out our brain does.

Kamil 12:42
For some reason, is it because of the overlap, we regard them as the same note?

Tommaso Zillio 12:46
Yeah, it's not a question of overlap, it's just simply that your brain thinks every time you double the frequency, it's the same note. And it doesn't matter if about the overlap of the spectrum. Now, those two notes will have a similar spectrum. What do we mean with that? Well, the low frequency here 220.

So they'll say that the bass frequencies 220 Okay, so you have the first note here, the low note as 220, and then let's say twice that so 440 and then say three times that this is zero, okay? Two times that three times then 123 and four times that and five times that and six times So, you have all the frequency to 22 times is 443 times is 664 times four 880 etc, etc, okay. The not the high note starts at 440.

And then it has twice that, which is these, it does not have these, he does not have these, and he has three times that which is this and he does not have these, five, six, so the not at that higher octave as only the even components of the lower octave. So there is an overlap. But the overlap per se doesn't tell you anything on how your ear perceives that respect to the octaves. Different quantity of the spectra in the spectrum will give you different timbres. But you don't need that if I had a pure sine wave.

So a tuning fork and then a tuning fork on octave higher you will still perceive those as different topics and the tuning fork contains only this frequency, or only this frequency and he won't contain on the other frequencies at least an idea to the fork. Make sense? So the spectrum has nothing to do with the perception of octaves as far as we know.

Kamil 14:54
If music was made with pure sine waves would we still have the same naming system would chord inversions really be considered the same chord

Tommaso Zillio 14:59
Let me stress one thing, the naming system has absolutely nothing to do with the physics of it here that we call those notes, ABCDE, F, F, G, sharps flats, they did that we have keys and chords, etc, as practically very, very, very little to do with the physics of things. It's more like what your brain likes to listen to.

Okay, at least, this is the point where I get people like, yes, but the perfect fist is exactly three 1.5, the frequency of the low one, and the perfect four does that and the major third, and then guys, you've been listening to music composed in the 12, don't equal temperament forever. And all those all the notes on the temperament are completely out of tune, if you think about simple ratios between the notes.

Not only that, but it appears that if you return the music, if you take it and put everything in a computer, and that is a computer to play only the exact precise interval so that the fifth is exactly 1.5, the frequency, which is the perfectly tune, the just fifth, and the major third is exactly the perfectly tune, just major third, etc, etc, then the music kind of feels kind of empty, because at some level, we do like that the frequencies are not exactly exactly exactly where they should be.

And as likely out, it kind of gives the music a little bit more of a thickness of a rod, between the notes that wasn't there before. People have made this experiment. And by and large when you have a blind experiment, okay, so people don't know, which is in tune, and which is not people prefer if the North are slightly out of tune, it just sounds better.

Okay, to just saying that, there's really, you've been listening to major thirds, and minor thirds, that are horribly out of tune, if we think only about the spectrum. Okay. And yet, not only that, but sometimes even if I make them even more out of tune shape, but then one on one. If I just play these, you think it's a major third, even if the major third is here, I think you can hear a difference if I move the knob.

But if I just go there, or there, you don't hear much of a difference, it sounds exactly the same thing. So your ear automatically takes the pictures coming in and goes like, okay, with good approximation. What is this a major third minor third or something in between, and your brain? Pitch corrects everything you hear, okay, that's what people don't understand. That's why it's really hard for a beginner musician to know if they're out of tune or not, because their ear automatically pitches correct everything.

And a good part of becoming a good musician is to learn to disable the automatic pitch correction and actually listen and see if you're out of tune or not. Okay, so it's not a question of the content of the spectrum, if there's not conflict with each other if the spectrum exactly overlap, or know, the cetera, et cetera.

Okay, it's a pure question of, What does your brain hear? And how does your brain can interpret what you play, which means that even if you're a slightly out of tune, if you're 1%, out of tune, on paper, all this frequency goes completely wrong, and it should be horrible, but if you're 1% out of tune, most people will barely hear it. Okay?

So we are not relying on the perfection of peaches ratio to understand music. Examples like that are plenty. When you study Musicology, there are plenty of population in the world that did use scales, or sets of pitches that are completely out of tune with each other. If we think about them, only in the sense of frequency ratio, there are populations in Africa that use five notes equal temperament.

So they take the octave splitting five, the resulting intervals make no sense to a Western ear and yet if you listen to music made that way, it kind of makes sense. Okay? Your brain somehow adapted that cannot interpret them in the sense of there are 36 There's something in between, okay. But somehow it makes sense.

And also even if you play those notes, you cannot play from coordinate system, any any combination of notes, even all five of them is in time. Seems to sound good. As the moment you round up. So it's so it's not a question of pitches ratio.

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