# 1 – Why There are Twelve Notes in Music

Hello guys and welcome to Music Minute, the

hot theory guide to learn those extra concepts the right way, brought to you by stevenjacks.com. My name is Steve and today we’re talking

about why there are twelve pitches in the musical language. First, let’s talk about how we get a pitch,

okay? Where do pitches come from? Okay, well let’s take a piece of string.

Let’s take this one on the right hand side here. And let’s pluck it and we get a certain

note. Okay? So that’s how you get a pitch. If

you were to cut that in half, if you were to take your string, whatever this string

length this is, and you cut it in half, you get this one right here. As you can see, we

have a two-to-one ratio here. The just cuts this one in half, okay? So if you do this, and you play this note,

you get a very interesting relationship. Listen to this one again, and then this one. Okay? If we cut it in half again, we get this

pitch. This is a very interesting relationship because if you were to play all these notes

in a normal scale, these would all be considered “Do” or they would all be “Re” or

“Mi” or whatever they are. So you’d go “Do”, “Re”, “Mi”, “Fa”,

“Sol”, “La”, “Ti”, “Do”,“Re”, “Mi”, “Fa”, “Sol”, “La”, “Ti”,

“Do”. Okay? So this two-to-one relationship, or, in this case, four-to-one, or eight-to-one,

or… (you just keep going in powers of two), it will always result in the same Pitch Class,

or “Do” “Do” “Do” “Do” “Do”. The interesting thing about the one-to-two

relationship is that it’s been found in nearly every single civilization. People that

had no contact with one another were able to take a string, cut it in half, and say

“Oh! These things are equal!” So the octave is kind of a universal concept

now. But for the sake of this video, and this lesson,

we will actually consider this one to be the lowest pitch. This one will be the middle

pitch, and this one will be the highest pitch. These are inverted. The way that we can make sense of this is

that this is the lowest frequency. This is twice as high in frequency, and this is four

times as high in frequency. So our notes are actually this one, this one, and this one.

Okay? So remember, this is now low, and this is

now high. Which is nice to know, because this is only a very low string, it only got about

there, and this one goes really high on the screen, right? Okay so the question then is “How do you

get other pitches to appear?” Well we just go around with a certain mathematical formula. We get this one by cutting this one in half,

and then multiplying by three. So here’s one, here’s two, and here’s three. This is called a two-to-three relationship.

A two-to-three ratio, if you will. Two-to-three, and this is two-to-three. And just like all the red ones are one-half

of each other going down or twice as high going up, the same happens with this blue

string. This is now half of this one. Therefor this

is twice as high as this one. And this continues for all octaves. Right now, we’re looking

at just two octaves, this “Do” to this “Do”, and this “Do” to that “Do”. But everything is consistent all the way through,

forever, extending in either direction. We’re simply going to see why we have twelve notes

per octave. The reason we use a two-to-three ratio is

because it’s the next simplest interval after one-to-two. So if you were to continue

this, your next one would be here. Now, you might be thinking well, shouldn’t this one

be on this side, or shouldn’t this one be where this one is? And you’re absolutely

right. But the reason that we’re able to put them here is we simply cut them in half.

So this one will also show up on the right hand side here, but we put it here.

And we put it here. Ta da. Thus we can fill in our octave. So if we keep going with this relationship,

we get these notes. If we stop at twelve, it looks really nice. We actually have pretty equidistant – I know

that this looks really short, this looks really long and that’s actually correct, that it

should be a little bit long, a little bit short in some places, because this system

is not perfect and there’s a reason why. The reason it’s not perfect is because you’ll

never actually end up coming back to this starting pitch. It’s impossible. The reason

it’s impossible is because powers of two, that we’re multiplying by, and dividing

by three, or multiplying by three and dividing by two, the powers of two and the powers of

three will never ever equal each other. The easiest way to argue this is that a power

of two will always be an even number and a power of three will always be an odd number. You can get really close, sure. If you continue

this pattern, you will actually be able to get really close to where we started. But

you’ll never end up exactly. So this is what twelve looks like. If we keep going, we see that our next pitch

is really close to our starting pitch, right there. And yes, if we went further, we’d

see one right here as well. Okay but if we keep going, you can see everything kind of

gets doubled for a little bit. And if we stop at twenty-four it looks kind of nice because

you have twenty-four different pitches each in pairs of two. But the problem is that nothing is really

equidistant anymore. You see the tiny gap between this one and the large gap between

this? That’s not really equal anymore, okay? If we keep going, we’ll eventually even

it out. And as I approach fifty three, you can see that it’s getting pretty good. If

we stop at fifty-three, it looks pretty fleshed out. This system does technically work. You have

equidistant pitches, you have fifty-three of them, but they are equidistant, they do

work out, it does look pretty nice. The problem is now you have fifty-three notes

in one octave. Arguably, fifty-three is a little too many

per octave. I mean, twelve is nice, fifty-three is nice, but it’s kind of a little too much,

don’t you think? If we keep going, we will see that this never,

ever stops. This pattern continues forever. At 306, it looks pretty good, but again you

have 306 notes per octave, so that’s kind of crazy. I’ll go back down to 300 just

so we can jump by hundreds easier. And you can see that this pattern completely

fleshes itself out into a full color spectrum. Okay if you look at this – it looks gorgeous,

right? But look. You have 7600 notes per octave. That’s a little too much. Okay let’s go onto another visual representation

of why we have twelve notes per octave. Okay so here we have the same thing. But instead

of a continuous line that we can only see two octaves of, this circle represents one

octave, so we can keep going around and around and around and the notes will just stack. So if we were to go to the next one, it would

be about here – about two-thirds trough. Again, these are the same two pitches we saw previously.

From there, our next stop will be about here, and then we can keep going, adding the next

notes. And again, we can get to twelve and it looks

pretty fleshed out. If you have a sharp eye, you can see that

these two notes are a little closer together than these two notes, and that’s because

the system is not perfect. It will keep going in this pattern forever, filling this entire

circle. Let’s continue and you can see all the notes

filling out. Again, if we stop here, it looks pretty good.

Everything is really quite equidistant, and it does work out. Again you have that fifty-three

notes per octave. Remember that this works out because we’re

comparing the octave relationship of one-to-two with the next ratio of two-to-three. If we were to continue, we’d get these notes. Notice that these are, again, really close

each other and everything will start pairing up. Here’s its pair, there’s a pair, there’s

a pair, and it keeps going. Okay, if we continue forever, we will never ever get back to where

we started, but we will fill out the circle indefinitely. Let’s go back to fifty-three. There are

certain problems to be found with fifty three notes per octave. The first problem is that

two notes might seem to almost sound the same. When you play one note, and you play another

note, they might be too close together to actually tell the difference. For a visual representation of that, if you

look at these two right here, you might argue that they are the same color. But based on

the algorithm, they are actually different in color. It’s very slight, and it’s very hard to

tell the difference, but they are actually different colors. If you only have twelve, you’re able to

see the colors quite vividly. Another problem with fifty-three is that the

possibilities for dissonance is incredible. When you only have twelve notes, you usually

don’t have too many problems for dissonant possibilities. But with fifty-three notes,

you usually will have a bigger problem to deal with when it comes to dissonance. The third and probably most practical reason

why fifty-three notes is so difficult to work with is because you would then have to make

instruments that were capable of playing fifty three notes in one octave. You would also then have to make music software

that would be able to support fifty-three notes per octave. These kind of reasons make it a little difficult

to make this a plausible system. So let’s go back to twelve. This system is great but if you were start

in the wrong place, certain things – for example, this note to this note would sound terrible.

This interval is called a Perfect Fifth, which sounds like this. And this interval is called a Wolf Fifth,

which sounds like this. This is because this not wants to go to the next note here. Because the pattern can never finish, we always

have to start in a certain place to make the notes we’re playing sound great. This is the problem found with Pythagorean

tuning, which is what this is, where you take a ratio of one-to-two and you compare it with

a ratio of two-to-three. To fix this problem, you can simply do something

called Equal Temperament, which was big in the 1600’s. The theorists of the 1600’s took all the

notes and made them equally spaced, like this. As you can see, our first note starts in the

same place. But now, when we go to our next stops, they get slightly more off tune. Our final stop looks really off right here,

but the nice part is all of these notes on the outside ring here are now equidistant.

They are evenly spaced, just as the numbers found on a clock. The pros found with this is that we can now

start anywhere and any relationship from any one note to any note following in its pattern

is the same everywhere. This is completely uniform. Thus you can change your keys, you can play

different scales, and everything and it all works, regardless of where you start. The

problem with this is that our perfect fifth of three-to-two is no longer correct. This

is slightly off to compensate. It’s okay, though, because human ears have

a hard time detecting the difference. After all, if we look at these two pitches right

here, these are very similar, just like we had the problem with the fifty-three note

system. These are too close together to actually hear

a difference, especially when played in context. Thus, our perfect system of twelve appears. So, overall, it’s because of that two-to-three

relationship mixed with that two to three relationship that has yielded twelve notes

per octave. If you wanted to make another system that compared different intervals,

you could do it with another ratio instead of that three-to-two. Instead of using a three-to-two or two-to-three

ratio, you could use the next step. The next simplest ratio is one-to-five. The reason the next step is one-to-five is

because one-to-four is just like doing one-to-two. Remember that all powers of two are equal. Instead of one-to-five, you could do four-to-five,

or even eight-to-five, these are all okay. The next steps after one-to-five are one-to-seven,

one-to-eleven, one-to-thirteen, et cetera. But the problem with using more and more complex

ratios like this, is that your initial interval doesn’t sound very good. Therefor all the

notes in this system will be based off that one really strange interval, and it might

make a really interesting sound. It certainly might be a great way to explore new territories

of music, but it might not sound very good. And that, my friends, is why we have twelve

notes in the musical system.