The Science of Music

Hi folks, welcome to Discerning Truth, the webcast of the Biblical Science Institute. I'm Jason Lyle. We've got a really neat one for you today. We're going to look at the science of music, try to make sense of music.

What is it? How do we account for it? Its beauty, its ability to sway our emotions. How do we make sense of these things? I want to answer some very basic fundamental questions like, why are there seven notes in a scale?

Which musicians label A, B, C, D, E, F, G, or if you prefer Do, Re, Mi, Fa, So, La, Ti, and then you go back to the original again, but one octave higher. Why seven? Why not like ten? Or fifteen? Or twenty?

Why don't we have like a metric system with music, you know, where it's based on ten notes instead of seven? That's weird. If you include sharps and flats, then you have twelve notes in a chromatic scale.

Now that's interesting because seven and twelve, those are both Biblical numbers. They occur throughout the Bible. They both seem to indicate completion or perfection. The seven days of the week based on God's creation and rest, the twelve months in a year, twelve apostles, twelve tribes of Israel that inherit the land, and so on.

So is there a connection there? Could music be a proof of the Bible? So if you're a musician, this one's for you. Or frankly, if you just like music and are interested in it. Music really is a universal phenomenon, isn't it?

It seems like pretty much everybody likes music, and it's a good icebreaker. You meet somebody for the first time, what kind of music do you like? And they'll give you an answer, but the answer is never, oh no, not music, I don't care for that.

It seems like everybody likes music. They might have their own personal preference as to what style they like, but music itself is universal. That's interesting. Now, of course, music involves sound. Sound is a compressional vibration in a substance like air.

So if I were to take a surface like this and push it really quickly, that would compress the air in front of that surface, and then it will re-expand, but then it will push the air in front of that, which pushes the air in front of that, and so on.

So you get this compressional wave that moves through space until it impacts a surface like your eardrum, and the air wiggles your eardrum, and through some very complex machinery that probably deserves its own webcast, that is transmitted via an electrochemical signal to your brain, and you hear a sound.

Neat. But not all sound is music, right? I mean, there are some nice sounds out there, the sounds of birds chirping. That's nice. Pretty. Or the sound of a waterfall. Oh, that's just soothing. Those are very pleasant sounds, but they're not music.

Music involves notes. A note is a sound wave of one definite frequency held for some duration of time. So what does that mean? What's frequency? Well, when we analyze the compressions of air produced by a musical instrument, a flute, violin, what have you, we find it forms a repeating sequence of waves.

Now, when we represent these waves, we represent them as transverse waves, up and down, up and down. In fact, sound waves are compressional. They're forward and backward, forward and backward, but it's difficult to illustrate that.

So the transverse waves that we use to illustrate sound, they represent compression. So the peak of a wave represents air that's very compressed, and then the trough of the wave represents air that's decompressed, rarified.

So the waves, sound waves don't actually look up and down like that. They're compressional, but that's how we represent them. Now, that sound wave is moving in space, and in air, sound waves move at about 700 miles per hour.

It depends on temperature and pressure, but around 700 miles per hour. Now, the frequency of a sound wave is the number of wave peaks that pass through a surface per unit time, or impact a surface per unit time.

And frequency is measured in hertz, like the car rental company, and hertz are the number of waves that pass through a plane per second. So for example, if a wave has a frequency of 262 hertz, you hear this.

And that means that basically there are 262 waves that impact your eardrum per second, or it's causing your eardrum to wiggle 262 times per second, which is quite fast. Now, if the wave has a higher frequency, like 436 hertz, then you hear a higher pitch like this.

So the higher the sound, the faster it's wiggling your eardrum. Now, human beings can detect sounds with frequencies between about 20 hertz, 20 vibrations per second, and 20 ,000 hertz, 20 ,000 vibrations per second.

And of course, it depends from one person to the next. There's a little bit of variation there. And so now, if outside that range, if something, if there's a vibration that vibrates your eardrum at 30 ,000 hertz, you hear nothing.

Your ear and brain are not designed to be able to handle frequencies outside the range of about 20 to 20 ,000 hertz. And by the way, as you age, the upper limit on frequency drops a little bit. And so the older you get, it drops to maybe 17 ,000 hertz, maybe down to 15 ,000 hertz for the elderly.

And so parents, you probably can't hear some sounds that your children can hear because they still have that full range. Different animals that the Lord created. Dogs have a range of about 67 hertz to 45 ,000 hertz.

So dogs can hear sounds that are too high for us to hear. And that's how a dog whistle works. A dog whistle makes a sound at, let's say, 30 ,000 hertz. You hear nothing, but your dog hears it. Cats, they can hear about 45 hertz to 64 ,000 hertz.

So cats can hear dog whistles too, they just don't care. Now, different musical instruments will produce a different sound, even at a given frequency. So for example, if I hit a piano note that produces a frequency of 262 hertz, it sounds like this.

But a flute at 262 hertz sounds like this. It's the same note, it's the same frequency, but they have a different texture or flavor to them or color to them. And that's due to the shape of the wave. A piano playing a tone of 262 hertz looks like this, whereas a flute playing the same note has a different wave.

Looks like this. Now, generally, the smoother the wave is, the softer the sound of the note that it's playing. Flutes have a fairly soft sound. Pianos are a little more harsh. The simplest possible wave, mathematically, is a sine wave.

That's the trigonometric function. You remember sine and cosine? Sine is the, if you take the angle in a right triangle that's not the right angle, you take the other angle, and the sine of that angle will be the ratio of the opposite leg of that triangle divided by the hypotenuse.

So that's sine. It goes between 0 and 1 in a right triangle. But you can go beyond that, and sine will go all the way up to 1, and then it'll drop down to 0, and then go to negative 1, and so on. And so if you plot that over time, you get this nice smooth wave.

So that's a sine wave. And it sounds like this. Now, for our purposes, I'm going to use a modified sine wave. The sine wave is so soft that it can actually be a little hard to hear over computer speakers, so I want you to be able to hear these tones.

So I'm going to modify it a little bit. We're going to use a wave that looks like this, and that'll make it a little easier to hear. Or I'll use a piano, depending on what issue we're exploring. Now, music contains only certain pitches, certain frequencies of sound, and not others.

And I want to explore why is that? Why don't we use every possible frequency? There are seven main notes that are used in music, and that would be seven notes in what we call a diatonic scale. And we name them after the first seven letters of the alphabet, A, B, C, D, E, F, G.

And then the notes repeat at a higher octave. Or if you're Sister Maria, Do, Re, Mi, Fa, So, La, Ti, and then back to Do. Now, on a piano keyboard, we like to start with the key of C, because, well, it's easier.

And the way you find C on a piano keyboard is, you'll notice that the black keys are grouped into groups of either three or two. You find a group of two, and the white key immediately to the left of those is always a C.

Now, the C that's in the kind of the middle of your piano, it's called middle C, and it has a frequency of 262 Hertz. It's vibrating your eardrum 262 times per second. Now, if you start with that C, and then you hit the next white key, and the next one, and so on, you get a C major scale.

And it sounds nice. Those eight notes go well together, don't they? There are seven notes, one, two, three, four, five, six, seven, and then the eighth note sounds very similar to the original. And we also label it a C.

We call it high C. So, my first question is, why does high C sound so similar to middle C? And the musicians out there are going, well, duh, they're both C. But that's not really... that's the effect, not the cause.

We name them both C, because they sound very similar. But they're not the same note. One is higher than the other, right? We name them both C, because they sound similar. Some people say, well, yeah, but it's exactly one octave higher.

Okay, octave means eight, right? It's the eighth note. You start with C as number one, you walk it up, number eight will sound very similar to number one. But why does note number eight sound similar to number one?

Why not note number 10? I mean, we humans prefer things in 10. That's why we like the metric system so much. Well, except in the United States, but anyway. Well, when we compare the wave of middle C and high C, the answer is obvious.

High C is exactly twice the frequency of middle C. Middle C has a frequency of 262 Hz. High C has a frequency of 524 Hz. It's wiggling your eardrum exactly twice as fast. And somehow your brain knows this and says, that's nice.

Those are kind of the same note, even though one's higher than the other. Interesting. Now, you'll notice that high C is half the wavelength. The peaks are closer together, right? And that makes sense.

The higher the frequency, the smaller the wavelength. In fact, the wavelength is the reciprocal of the frequency multiplied by the speed of sound. So if you double the frequency of a note, you reduce the wavelength by half.

So those two are just opposites of each other. Now, high C has twice the frequency of middle C, which is why it sounds similar and yet somehow higher, right? Because it's a higher frequency. So it's the same note in one sense.

It's a C, but it's a different note in another sense. It is a different frequency, and so it's a high C rather than a middle C. So apparently if you multiply the frequency of middle C by 2, you get high C.

If you multiply the high C's frequency by 2, you get an even higher C, and so on. Now, if you start with middle C and you divide its frequency by 2, you get a low C. You divide its frequency by 2, you get a lower C still, and so on.

So all notes of a given name have a frequency that is a multiple of 2 from each other. You multiply by 2 to increase the octave by 1, you divide by 2 to decrease the octave by 1. So that's one mystery solved.

So that's why we have octaves. But that doesn't answer why there are 6 notes between the middle E and the high C. Why is it we have 7 notes in a scale? And C, D, E, F, G, A, B, and then you're back to C, so C is the 8th note.

Why is it those 7 notes? Why not 10? Right? And frankly the notes are not equally spaced. That might bother you too, because you'll notice that when you play a D and then an E, there's a note that's in between them that you could have hit.

That's a D sharp or an E flat. You're skipping a note. But then when you go from E to F, there's not a note in between those two. And that's because E and F are closer together in frequency than D and E.

So they're not even equally spaced. Why? Why not divide the octave into 10 equally spaced frequencies, so that there are 10 notes and then you hit the C again. So the next C would be the 11th note. Right?

I mean we're not changing the frequency of the C's, because the high C has to be twice the frequency of middle C. But we can divide those into any number of notes. So why don't we have, why isn't it C, D, E, F, G, H, I, J, A, B, C?

Why don't we have 10 notes instead of 7? And we can do that. And it sounds like this. It's kind of weird. Or if we hold that first C and then play our 10 pitches ending up at number 11, it sounds like this.

It doesn't sound right. But why? Is it because we're just used to having 7 notes in a scale rather than 10? If history had developed differently, if the first person to make a musical instrument, who was Jubal by the way, the Bible records that interestingly, if the first person who made a musical instrument had created it with 10 notes between octaves and we kind of got used to that, would that 10 note scale sound right?

Or 16 or 27? Is it just a matter of the way society developed that we have 7 notes in a scale and not 10 or 16 or 27? Now there are some cultures of the past and some of the present that use a 5 note scale.

It's called a pentatonic system. But those 5 notes are a subset of the 7 that we have in our scale. They use C, D, E, G, and A. So they're using the same notes we use. They're not using 2 of them. They're not using the F and the B.

So whereas we have all 7. And if you include the black keys on a piano, which are sharps and flats, that adds an additional 5. And we have 12 notes in what we call a chromatic scale. A chromatic scale is where you hit each and every note between middle C and high C, for example.

And as I mentioned earlier, 7 and 12 are numbers that are very significant in Scripture. They indicate completion. So that's why we have 7 days in a week, because God created it over 6 days and rested 1 as a pattern for us.

There are other patterns of 7 in Scripture. There are patterns of 12 in Scripture. There are 12 months in the year. There are 12 tribes of Israel that inherit the land, 12 sons of Jacob, 12 disciples.

So perhaps there's a fundamental reason why 7 and 12 are the numbers that we have in a diatonic and chromatic scale, respectively. But what could those reasons be? Well, we've already noticed that high C is exactly twice the frequency of middle C and that those two notes sound good together.

For high C, every other peak it lines up again with a peak of middle C. So those frequencies form a ratio of 2 to 1. Two peaks of high C for every one peak of middle C. What if the other keys in between them—D, E, F, G, A, B—what if they also form a nice frequency ratio?

With middle C, if you multiply its frequency by 1, you get middle C. If you multiply it by 2, you get high C. What if we multiply it by one and a half? That would be kind of in between those two, wouldn't it?

And sure enough, if you multiply the frequency of C by one and a half, you get G. And it sounds nice. In fact, G and C sound nice together, apparently because they have a frequency ratio of 3 to 2. One and a half is the same as three halves, so that means for every three peaks of the G, there are two peaks of the C, and so they line up periodically.

And apparently our brain likes that. So that's why we have a G. So we have C at 1 times the frequency of C, high C at 2, G at one and a half, or three halves if you prefer. What would be the next simple fraction?

One and a half, one and a third would be the next simple fraction. So let's try that, and sure enough, if you multiply the frequency of C by one and a third, you get an F. Pretty neat. What about one and two thirds?

That works as well. That produces an A. I think we're on to something. The next simplest fraction would be one and one fourth. And sure enough, that's an E. So apparently the notes that sound pleasant to us, that go well together, have frequency ratios that form a simple ratio, what we call a resonance.

A resonance is a ratio of two frequencies that's relatively simple, like 3 to 2, 5 to 4, something like that. So we're on to something, I think. So one and one fourth gives you an E. Now the next would be one and two fourths, but that's the same as one and a half.

That's a G, so we've already discovered that one. If we try one and three fourths, interestingly, it's not a note that we use. But it's pretty close to a B flat. If you're a jazz musician, it'd probably be close enough.

But for classical folks, it's a little bit flat. It's a little bit below B flat. So it's not going to work, interestingly. So that's kind of strange. But aside from that one anomaly, all the other notes we've discovered form a simple ratio with C.

3 to 2, or 4 to 3, 5 to 4, and what have you. So let's keep going and see if that trend continues. The next simplest fraction would be one and one fifth. And that does produce a note, but it's a flat.

It's called E flat. That's the note that's in between D and E. It's a little below E, so it's E flat, or it's a little above D, so it's D sharp. Same note on our modern system. Now one and two fifths does not produce a note, so again that's a little disappointing.

It's close to G flat, but slightly below it. One and three fifths gives an A flat, so that works. And then one and four fifths is again not a note. So apparently there's a little more to it than just forming a simple ratio, a simple resonance with C.

There must be something else. But that's at least part of the criterion, because we've gotten several notes that way. We've got four notes of the C major scale that are in resonance with C. And if you include the C itself, that's five.

So that means there's just two that have a higher ratio than E flat and A flat. So maybe naively you might be a little surprised that we get to the two of the black notes before we finish the C major scale, because that means that the D and B have higher ratios relative to C than E flat and A flat.

But if our hypothesis is true, that simple ratios sound better than more complex ratios, then all that means is E flat and A flat should sound better with C than D and B. And frankly they do. If you play a C with an E flat and an A flat, it sounds nice.

It's an A flat major chord. If you play a C with a D and a B, not quite as nice, because the ratios are higher. So it's actually consistent with our hypothesis. So why doesn't our C major scale include E flat and A flat, since they go so well with C?

And the answer is they don't go well with the other notes that we've discovered. E flat and A flat do go well with C, but they don't go well with E, F, G, and A. Those don't sound nice. So apparently the C major scale consists of notes that go well with C and also go well with each other.

And so the two remaining white keys in our C major scale, D and B, they don't have to harmonize quite as well with C, but if they harmonize well with one of the other notes, like a G for example, then that would explain why they're part of the C major scale.

Because G is the simplest ratio with C, other than high C, within our scale, right? And so what if we based our next notes on their ratio with G rather than C? It turns out the frequency for D is simply C multiplied by one and one eighth, or nine eighths.

How do we get that? Well, D must be based on a simple ratio with one of the remaining notes then, okay? Now recall the simplest ratio on the C scale, aside from the two to one octave, is the ratio of G to C, which is three halves, or one and a half.

That's called a perfect fifth, because G is the fifth note on the C major scale. So if we start with G, which is already three halves, and find its perfect fifth, that would be a D, you'd have to multiply that by three halves.

So three halves times three halves would be nine fourths, but that puts us outside of our octave. It's a high D. We want the middle D, right? And so to do that you just divide by two and you get nine eighths.

So that's why D has a ratio of nine eighths. It's the perfect fifth of G divided by two. Now to get the B, we take the perfect fifth of E. That's five fourths. We multiply that by three halves, which is fifteen eighths, or if you prefer, one and seven eighths.

So the B and D form the highest ratio with C, and indeed they don't sound as good as the other four notes of the C major scale when played with the C, but they do sound good with each other, and they do form nice simple ratios with G, and therefore D, G, and B sound good together.

That's a G major chord. So apparently the seven notes of the C major scale sound nice together because they're in resonant... they're either in resonance with C, or they're in resonance with G, which is in resonance with C.

So that's why apparently we have seven notes in the C major scale. The C major scale tuned this way sounds like this. Sounds nice, doesn't it? And the combination of notes that sound best together are those that form simple ratios of frequencies, such as C, E, and G.

That's a C major chord. Sounds nice. Or C, F, and A. That's an F major chord. That sounds nice. Or D, G, and B. A G major chord. And if you're a fan of modern worship music, those are the three chords you use.

Okay, what about the black keys then? We're going to need those if we want to play a song in a So we've already discovered two of them, the E flat and the A flat, which sound very nice with C because they form small ratios of their frequencies with C.

So that's nine notes so far, but we know there are more, not just because we can see that on a keyboard that there are three more notes yet to be discovered, but also because every note has to have a perfect fifth.

You have to be able to go five notes up, and right now not every note does in terms of the ones we've discovered. And every note has to have a previous perfect fifth as well, a note that when you take when you multiply its frequency by three halves, you get the current note.

So for example, the previous fifth of G would be C. That is the fifth of C is G. So for example, the previous perfect fifth, starting with the A flat, if you walk it down five steps, you would get a D flat.

And so the frequency of D flat must be two-thirds the frequency of A flat, and indeed it is. It turns out the note that would be the previous perfect fifth is the perfect fourth of the current note. For example, the perfect fifth of C is G, whereas the perfect fourth of G is C, and divide by two to drop the octave.

Now if you take the perfect fourth of F, that'd be the fourth note up, which would be that would be the note that when you take its perfect fifth, you get an F. Lo and behold, you get a B flat at 16 ninths.

And then if you take the perfect fifth of B, you get a G flat or an F sharp at a ratio of 45 over 32 to C. Now that's a pretty high ratio. It's higher than any other that we discovered. In fact, if we take a look at all the frequencies we've discovered, and that's all 12 notes now.

So we've got all 12 of them. And that's the complete system because every note on this chart also has a fifth that's on this chart. You can go five notes up exactly, and it'll sound nice together, okay?

So these are all the notes that exist. Our hypothesis is that simpler ratios of frequencies, like three to two, sound better than more complex frequencies. And if that's true, then G and C should sound much better together than C and F sharp, because F sharp has the most complex ratio, 45 over 32.

And indeed, if we play a C and an F sharp, they don't sound very good together. So our hypothesis seems to be vindicated. The simpler the ratio, the better the two notes sound together. So somehow your brain is able to analyze the frequency of different notes, the frequency ratio.

And if it forms a simple ratio, your brain says, yeah, that's nice. If it forms a complex ratio, you say, I don't like that so much. So the seven notes of our C major scale are based on simple resonances with C or with G, which is based on a simple resonance with C.

And then we have additional five notes that we need so that every note has a perfect fifth. And this method of tuning is called just temperament. Just as in right. But this is not actually the system that we use today in modern instruments.

It's close. But we tune our instruments a little differently from the just temperament, because there's a problem with this system. Just temperament sounds great if you stay in the key on which you base the ratio.

In our case, the note C. If you stay in C major, it's going to sound really great. But if you try a different key, like let's say you play a D major chord, it doesn't sound quite right. And that's because the perfect fifth of a D should be an A, which means when you multiply the frequency of D by three halves, you should get the frequency of A, but you don't.

Okay. That's because, I mean, you should get, if you multiply D by three halves, you should get a frequency of 27 over 16, which is about 1 .687. But in fact, A is tuned to five thirds the frequency of C, which is 1 .666.

And so it sounds flat when you play it with a D. This is called a wolf fifth, because it sounds like a wolf howling. It's horribly out of tune. And so using the just temperament results in some fifths that sound out of tune, because they're not a ratio of three to two with the note that they're supposed to be a fifth from.

Now, some perfect fifths are. The C to G, perfect fifth. G to D, perfect fifth. F to C, E to B, A to E. Those are all perfect fifths in this system, but some are not. D to A, not a perfect fifth. D sharp to A sharp, not a perfect fifth.

Is there a way to fix that? Well, and there's another mystery too. In our C major scale, there are seven notes, but two of them involve what we call half steps. You see, on a piano, when you move from one note to another note and you skip a key in between, like when you go from C to D, but you skip the C sharp, that's called a whole step.

So C to D and D to E are whole steps. But then E to F is a half step, because there is no note between them. Then F to G, that's another whole step. G to A and A to B, those are all whole steps. And then we have to take another half step to get back to C.

So our seven note scale involves two half steps. Why don't we just combine those into one whole step and use a six note scale, right? Consisting entirely of whole steps. And you can do that. And it sounds like this.

Kind of weird, doesn't it? And even if you play the notes together, it doesn't sound bad, but it does sound kind of strange. And there are some musical compositions that use that six note scale of entire whole notes, at least in places.

Some of WC's music does that. And frankly, some in modern fantasy or sci-fi type movies that has a soundtrack, if they want it to sound spacey, a six note scale will do it. And perhaps the reason it sounds spacey is because there's no definite key signature.

You see, it doesn't matter what note you start on. If you start on a C, six notes and then back to C. If you start on a D, same thing. All notes are equal in that system because there's no half steps to tell you what key signature you're in.

So it's kind of weird. But somehow the six note scale doesn't sound as natural as a seven note, what we call a diatonic scale, which involves two half steps that are separated as far as they can be. I want to know why.

Well, one reason may be because when you start a scale, that starting note kind of sticks in your mind. You kind of remember it. And all seven notes of a diatonic scale go relatively well with that starting note because they form a fairly simple ratio of frequencies.

The worst being the B, but it's not too far off. But on a six note scale of whole steps, the fourth note, when you compare it with C, it forms that nasty devil's chord with the starting key. And that may be why it doesn't sound as natural.

If you continue to hold the C and you play the seven notes of a diatonic scale, it sounds nice. If you hold the C and you play the six notes of a whole note scale, you're going to hit that F sharp and it just doesn't sound right.

But there's another way to look at this that will help us answer the question of why are there seven notes in a diatonic scale? Why two of them are half steps? Why they occur at the locations they do?

Why the full chromatic scale has 12 steps? Why some cultures use a five note scale? And how to reduce the number of whole fifths down to one? Yes, there's a way of looking at it that will answer all those questions.

So I want to show you how this works. We can picture every pitch within an octave, every pitch on a scale, as being a location on a circle. Because eventually you go C, D, E, F, G, A, B, and then you're back to C.

So it circles back to itself. Now it's a high C, but you divide by two to drop the octave. And so every note on that circle going clockwise will be a specific pitch. It will be multiplied by C by some number.

So C itself is one. It starts at the top, and then you multiply that by something to get D, and then E, and F, G, and then A, B, and then C. The next C will be a factor of two, and you just drop, and you just divide by two to get back to the original C.

So that's a really neat way of looking at things. Now if we do this, G will occur at this location, because it's three halves. It's one and a half times C. You might be wondering, why isn't it exactly at the bottom?

Isn't G halfway between middle C and high C? Well, we're multiplying frequencies, not adding them. We're not adding a half, and then adding another half to get back to two. We're multiplying the frequency of C by three halves to get G.

And if you multiply it again by three halves, you don't get two. You don't get back to C. You actually get a D, right? So in fact, the bottom point on our circle would be the square root of two. You'd have to multiply C, which is one, by the square root of two, and then by the square root of two again to get back to two, which you divide by two to get to one.

Okay, so the middle point is actually the square root of two, not one and a half. G is a little bit past that. Now Pythagoras invented a neat way to tune instruments based entirely on perfect fifths. Basically, you start with one note, like C, and you multiply its frequency by one and a half to get its perfect fifth.

Then you take that note, the G, and multiply its frequency by three halves to get its perfect fifth, which would be D. And if it goes outside the octave, you just divide by two to bring it back to drop it an octave.

Okay, so we have to do that with the D. Then you take the perfect fifth of D to get an A. You take the perfect fifth of A to get an E. You'll have to drop it an octave, divide by two, and then you take the perfect fifth of E.

You get something that's pretty close to C, and so it's very tempting to just round that up a little bit and make it a C. Okay, and then that closes the loop. Now the cool thing about this system is you can get from any note to any other note by perfect fifths by doing enough of them.

You get all the way around the circle, and therefore all the perfect fifths in this circle, except the last one where we cheated a little bit, will be perfectly in tune. Now that last one, the E to C, that's going to sound terrible.

That's going to form a wolf fifth. It's not going to sound right. Now five notes, that's the pentatonic system, and indeed those are the five notes that are used in the pentatonic system. C, D, E, G, and A.

Or you can transpose those, of course. In fact, if you just use the black keys, that would form the pentatonic scale. Pretty neat. So that explains why some cultures use a five note system. The Pythagorean circle closes, or very nearly closes, after five steps, and then if you cheat a little bit, it does close.

But it was quite a stretch, really, to round E's fifth all the way up to C, when it really should be a B. So if we keep going, then that'll form a B. By the way, you can go the other direction you can find the previous fifth.

If we go to C and find its previous fifth, you do that by dividing by three halves, and then occasionally multiplying by two to pop it up an octave. That will give you the previous fifth, which is an F.

Okay, so there we have F to C, and so on, all the way around to B. And the perfect fifth of B is between F and G. And so once again, it's very tempting to round down a little bit and connect B to F. And now we have a complete system of seven notes.

And those are the seven notes that we use in a C major diatonic scale. So apparently, the reason why there are seven notes in a diatonic scale is because the Pythagorean circle nearly closes at seven steps, and does close if we cheat a little bit.