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, 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, rarefied.

So the 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 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 ,000 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.

It 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 will drop down to 0, and then go to negative 1, and so on.

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 will 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, Sol, 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 3 or 2.

You find a group of 2, 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 Hz. 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, 1, 2, 3, 4, 5, 6, 7, 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 ten? I mean, we humans prefer things in ten.

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 is 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? 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 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 ratio of 2 to 1?

Maybe 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 1 ½?

That would be kind of in between those two, wouldn't it? And sure enough, if you multiply the frequency of C by 1 ½, 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. 1 ½ is the same as 3 ½, so that means for every three peaks of the

G, there are two peaks of the C, 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 1 ½, or 3 ½ if you prefer. What would be the next simple fraction?

1 ½, 1 ⅓ would be the next simple fraction. So let's try that. And sure enough, if you multiply the frequency of C by 1 ⅓, you get an

F. Pretty neat. What about 1 ⅔? That works as well.

That produces an A. I think we're on to something. The next simplest fraction would be 1 ¼.

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 1 ¼ gives you an E. Now the next would be 1 ⅔, but that's the same as 1 ½.

That's a G, so we've already discovered that one. If we try 1 ¾, interestingly it's not a note that we use.

But it's pretty close to a Bb. 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 Bb. 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 1 ⅙. And that does produce a note, but it's a flat.

It's called Eb. That's the note that's in between D and E. It's a little below E, so it's

Eb, or it's a little above D, so it's D sharp. Same note on our modern system.

Now 1 ⅔ does not produce a note, so again that's a little disappointing. It's close to Gb, but slightly below it.

1 ⅜ gives an Ab. So that works.

And then 1 Ⅴ 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 Eb and Ab.

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 Eb and Ab.

But if our hypothesis is true that simple ratios sound better than more complex ratios, then all that means is

Eb and Ab should sound better with C than D and B. And frankly they do.

If you play a C with an Eb and an Ab, it sounds nice. It's an

Ab 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

Eb and Ab, since they go so well with C? And the answer is, they don't go well with the other notes that we've discovered.

Eb and Ab 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 1 and 1 8th, or 9 8ths.

How do we get that? Well D must be based on a simple ratio with one of the remaining notes then.

Now recall the simplest ratio on the C scale, aside from the 2 to 1 octave, is the ratio of G to C, which is 3 halves, or 1 and a half.

That's called a perfect 5th, because G is the 5th note on the C major scale.

So if we start with G, which is already 3 halves, and find its perfect 5th, that would be a

D. You'd have to multiply that by 3 halves, so 3 halves times 3 halves would be 9 4ths, 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 2 and you get 9 8ths.

So that's why D has a ratio of 9 8ths. It's the perfect 5th of G divided by 2.

Now to get the B, we take the perfect 5th of E. That's 5 4ths.

We multiply that by 3 halves, which is 15 8ths, or if you prefer 1 7 8ths.

So the B and D form the highest ratio with C, and indeed they don't sound as good as the other 4 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 7 notes of the C major scale sound nice together because 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 7 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, D, and G. Sounds nice. 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 3 chords you use. OK, what about the black keys then?

We're going to need those if we want to play a song in a different key signature, or if we want to play something in C minor, or something like that.

So we've already discovered 2 of them, the Eb and the Ab, which sound very nice with C, because they form small ratios of their frequencies with C.

So that's 9 notes so far, but we know there are more, not just because we can see that on a keyboard that there are 3 more notes yet to be discovered, but also because every note has to have a perfect 5th.

You have to be able to go 5 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 5th as well, a note that when you multiply its frequency by 3 halves, you get the current note.

So for example, the previous 5th of G would be C, that is the 5th of C is

G. So for example, the previous perfect 5th, starting with the

Ab, if you walk it down 5 steps, you would get a

Db. And so the frequency of Db must be 2 3rds the frequency of Ab, and indeed it is.

It turns out the note that would be the previous perfect 5th is the perfect 4th of the current note.

For example, the perfect 5th of C is G, whereas the perfect 4th of G is

C. And divide by 2 to drop the octave. Now if you take the perfect 4th of F, that would be the 4th note up, which would be, that would be the note that when you take its perfect 5th, you get an

F. Lo and behold, you get a Bb at 16 9ths. And then if you take the perfect 5th of B, you get a

Gb 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 of our, all the frequencies we've discovered, 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 5th that's on this chart.

You can go 5 notes up exactly and it will, it will sound nice together, okay?

So these are all the notes that exist. Our hypothesis is that simpler ratios of frequencies, like 3 to 2, 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 7 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 5 notes that we need so that every note has a perfect 5th.

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, let's say you play a

D major chord, it doesn't sound quite right.

And that's because the perfect 5th of a D should be an A, which means when you multiply the frequency of D by 3 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 3 halves, you should get a frequency of 27 over 16, which is about 1 .687.

But in fact, A is tuned to 5 3rds, 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 5th because it sounds like a wolf howling.

It's horribly out of tune. And so using the just temperament results in some 5ths that sound out of tune because they're not a ratio of 3 to 2 with the note that they're supposed to be a 5th from.

Now some perfect 5ths are. The C to G, perfect 5th. G to D, perfect 5th.

F to C, E to B, A to E, those are all perfect 5ths in this system, but some are not.

D to A, not a perfect 5th. D sharp to A sharp, not a perfect 5th. Is there a way to fix that?

Well, and there's another mystery too. In our C major scale, there are 7 notes, but 2 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 7 note scale involves two half steps. Why don't we just combine those into one whole step and use a 6 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 6 note scale of entire whole notes.

At least in places. Some of Debussy's music does that. And frankly, some in modern fantasy or sci -fi type movies that has a sound track, if they want it to sound spacey, a 6 note scale will do it.

And perhaps the reason it sounds spacey is because there is no definite key signature. You see, it doesn't matter what note you start on.

If you start on a C, 6 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 6 note scale doesn't sound as natural as a 7 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 7 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 6 note scale of whole steps, the 4th 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 7 notes of a diatonic scale, it sounds nice.

If you hold the C and you play the 6 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 7 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 5 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 2 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 1, 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 2, and you just divide by 2 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 3 halves, it's 1 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 2.

We're multiplying the frequency of C by 3 halves to get G. And if you multiply it again by 3 halves, you don't get 2, you don't get back to C, you actually get a

D. So in fact, the bottom point on our circle would be the square root of 2.

You'd have to multiply C, which is 1, by the square root of 2, and then by the square root of 2 again to get back to 2, which you divide by 2 to get to 1.

So the middle point is actually the square root of 2, not 1 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 1 and a half to get its perfect fifth.

Then you take that note, the G, and multiply its frequency by 3 halves to get its perfect fifth, which would be

D, and if it goes outside the octave, you just divide by 2 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 2, 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, too.

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.

And every fifth on this system will be perfectly in tune except the B to F, which is a wolf interval, because we cheated on that one.

Now notice not only does this explain why we have a seven note scale, but it also explains why two of the steps are half steps.

Notice that the distance between C and D, and D and E, is about twice that of E to F.

E to F is a half step. And then we have a whole step from F to G, from G to A, from A to B, and then another half step from B to C.

You see the math works out that way, where if you base everything on fifths, you'll naturally end up with two half steps in your seven note diatonic scale.

And if you tune your instrument using the Pythagorean system, the C major scale sounds like this.

Sounds pretty good. But what if we don't round down that B to F transition and we keep going?

What if we allow for the existence of a note between F and G? Let's call it F sharp. Then the fifth of B is actually an

F sharp. And the perfect fifth of the F sharp is a C sharp, which has a perfect fifth that is a

G sharp, then D sharp, then A sharp. And the perfect fifth of an A sharp, well it should be

F, but mathematically it doesn't quite align with our original F. And that little gap is called the

Pythagorean comma. So again, if we cheat just a little bit and round down, we close the system.

That will create a wolf fifth, but only for the A sharp to F transition. All the others will be perfect fifths.

And so that should sound really nice. And so you see, and that's our 12 note system, isn't it?

You can get from any note to any other note by perfect fifths and the occasional octave drop.

Notice that the distance between notes and frequency is about the same for all these notes. So these are the half steps.

That's why there are 12 notes in a scale. If we use half steps, this is the 12 step chromatic scale.

And the nice thing about having those extra five notes is you can do some fun things with them and add a little bit of color to your music.

And that's what chromatic means. It means color. So the 12 note system allows us to play in any key, any key signature.

The disadvantage of the Pythagorean system is that it creates a wolf fifth between A sharp and F.

But all the other fifths are perfectly in tune. And depending on where you start, by the way, you can push that wolf fifth anywhere along the circle.

You don't have to start with C or F. You can start anywhere and then the last one is where you'll get the wolf fifth.

But you can push that wolf fifth anywhere along the circle. So after you tune your instrument according to this

Pythagorean method, you don't play in the key that has the wolf fifth. And the other keys sound pretty good.

There will always be one key signature that sounds out of tune and so you just don't use that one. And if you do want to play in that one, you pick a different starting point on the

Pythagorean circle and you tune the instrument based on that one. Now the Pythagorean system results in different ratios than the just temperament system.

But that's what we wanted because the temperament system resulted in a horrible wolf fifth between D and A.

The Pythagorean system has a perfect fifth between D and A. But the interesting thing is some of the ratios become quite large when we take a look at those ratios.

And this is if you start with C and you go both forward and backward an equal amount of times.

These are the ratios that you get and they're pretty high ratios. But a lot of those ratios, if you round them down or up just a little bit, they'll be very close to a very simple ratio.

So apparently the ratio of frequencies does not need to form an exact simple ratio as long as it's close to one.

Your ear can't tell the difference. Or it can, but it doesn't mind. So under the

Pythagorean tuning system, there will always be one key signature that has a wolf fifth and sounds badly out of tune.

And if you really need to play in that key, you just pick a different starting note on the circle of fifths to construct your fifths and that will push the wolf interval to a different location.

And then you tune your instrument based on that key. But today we don't want to retune our instrument when we want to change keys, right?

We want a system that doesn't have any wolf fifths and where we can play sort of equally well in any key.

And the way we do that is with what's called an equal temperament system. What we do is we take a look at those 12 steps on the chromatic scale and we force the notes on the

Pythagorean circle to be exactly equally spaced. How do you do that?

Basically you take the 12th root of 2. That's the number that when you multiply it by itself 12 times, you get the number 2.

The 12th root of 2 is approximately 1 .05946. You take that number, you multiply that by C.

That gets you the C sharp. You multiply it again, it gets you a D. Multiply it again, D sharp,

E, and so on. And you get all the way around the circle. And the nice thing about that is it forces the circle to close exactly when you get back to the

C again. So that's an equal temperament system. And under equal temperament, the

C major scale sounds like this. Sounds pretty good, doesn't it? It eliminates wolf fifths and it causes all key signatures to be equally in tune.

Pretty nice. The disadvantage is that all key signatures are equally out of tune as well.

You see, under the equal temperament system, none of the notes form a simple ratio with C except for the high

C which is the octave which is 2 to 1. All the other ones form an irrational number with C.

But apparently since they're close to a simple ratio, they still sound pretty good.

In equal temperament, the fifth is no longer exactly 1 .5 times the original frequency.

Rather, it's 1 .498307 etc. So it's actually an irrational number.

So rather than a perfect fifth, this is sometimes called a tempered fifth. It's not exactly 1 .5 but it's close and it causes all the keys to be equally in tune no matter what key you're playing in.

So that's nice. You don't have to retune your instrument if you want to play in a different key. Now if we compare the frequencies of the just temperament in the left column with the frequencies of equal temperament in the right column and look at the difference, the greatest difference is the

A which is nearly four hertz sharp on the equal temperament system relative to the just. But remember that's what we want because the just system, the

D to A transition sounded terrible. It formed a wolf fifth. So if we play the chord

C, F, and A on a just temperament and then an equal temperament, they will sound a little bit different.

See if you can hear the difference. Now the

Pythagorean temperament is even closer to equal temperament. If we look at the chart, the greatest difference occurs at G sharp and it's just 2 .3

hertz difference. Now the ear can hear that but apparently it's not enough that it sounds noticeably out of tune.

Now equal temperament is actually a very modern solution. It really wasn't used until the 20th century due to the technological limitations in measuring frequencies very precisely and to be able to do that with irrational numbers rather than simple ratios.

There was another system that was in use during sort of the Middle Ages called the quarter comma mean temperament and the reason it was devised is the

Pythagorean temperament system which was used since very ancient times. When you do that, the fifths are perfect but the thirds are not.

The third would be like a C, D, E. So E and C would not sound quite right together.

The thirds will be slightly sharp using the Pythagorean system. So the quarter comma mean fixes that by tuning the fifths slightly flat such that when you eventually get to the third, the thirds are exactly right and that system was used until about 1650.

Then composers started using what we might call well temperament. This was the system whereby the fifths were modified slightly by various amounts so that first of all there were no wolf fifths and second no matter what key you play in, all the notes sound reasonably in tune.

There were several different well temperament systems that were in use but the notes are not equally spaced in well temperament as they are with our modern equal temperament.

So they're not the same and there's some confusion on this because in the past some people called what we call well temperament, they called it equal temperament but it's not the same as our modern equal temperament.

They called it that not because the spacing between the frequencies is equal because it wasn't but because you can play equally well in any key.

You don't have to retune the instrument so that's sometimes called equal temperament. But to distinguish that we'll call it well temperament and it does not use equal spacing between all the notes as our modern equal temperament does.

Therefore when you play in different keys using well temperament they will have a slightly different flavor or color to them.

Certain chords will sound better in certain keys than other chords or in a different key and this system was the main one that was used during the baroque and classical eras.

And composers during those times would compose their music in a particular key to take advantage of the unique characteristics of that key.

Pretty neat. Now that's all lost in our modern equal temperament system which is kind of a shame really.

But there are folks who will have their piano tuned to one of these old well temperaments so that they can play the original compositions with the flavor that the composer really intended.

Bach wrote a piece of music called the well tempered clavier in which he demonstrated the beauty of well temperament by constructing music in all 12 major and 12 minor keys and they all sounded good.

But our equal temperament is not the only solution to forcing that pythagorean circle to close.

For example what if we used the pythagorean fifth method but instead of cheating by forcing the fifth of A sharp to be an

F we just continued and added in another 12 notes. That would produce a 24 note system.

It doesn't close. You get a horrible little fifth if you try to close that that second version of the

A sharp back to the original F. That's not going to work. But if you're only modulating once this would work.

How would this work practically? Well one way to think of it would be to have a piano that's tuned to the first circle and then have another piano just kind of above it that's tuned to the second circle.

And so when you're playing and you're modulating from one key to the next when you go from A sharp you go from A sharp on the first keyboard to F on the second keyboard it will sound nice.

And then you can modulate from that F to other keys on the other piano and it'll sound nice and you'll have key signatures that are in between the key signatures that we normally use in our 12 note system.

And it'll sound nice as long as you don't try to modulate again because you don't have a third keyboard to go to. And there's some music that's been constructed that uses that 24 note system.

But it doesn't close. Well if we keep going around the circle does it ever close?

If we add more notes it turns out it comes really close to closing at 53 notes.

And lo and behold some music has been constructed using that system and it does work. Now you'd only use about seven notes at a time just like in our modern 12 note chromatic scale we tend to use only about seven in a given key signature and if we modulate then we'll then we'll drop some keys and pick up some others but you only use about seven at a time.

Likewise with the 53 note system you would use seven or so notes at a time it's just that when you modulate you have some other keys you can modulate to that don't exist in our system.

You certainly would not want to play two of the 53 notes right next to each other that would sound terrible.

So it really seems like our 12 note system is best because if once you start adding in more notes than 12 and you hear one note next to the other your brain tends to assume that rather than it being a distinct note it's an out of tune version of the original.

So the limitations on our ability to perceive frequencies causes the 12 note system to be probably the best.

There have been other solutions that involve 19 notes and 31 notes but these require using fifths that are tempered and not perfect and and they really never caught on because frankly they're complicated and again once you get beyond 12 notes in any scale any new notes start to sound like out of tune versions of previous notes.

So we've solved a lot of our dilemmas the reason we have a seven note diatonic scale with two half steps and a 12 note chromatic scale is because our brain prefers simple ratios like three to two and the pythagorean circle of fifths nearly closes at seven steps and again at 12 steps.