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Hi there, everybody.
Welcome to this lesson on the golden ratio and music.
This is from our Compose and Create unit, Cosmic Program Music.
My name's Mr. Croughan, and together we are going to discover what is the golden ratio and what on earth does it have to do with music.
Are we ready?
Let's begin.
By the end of this lesson, you'll be able to say that you can structure a short musical piece by including the climactic point at the golden ratio.
Let's begin by looking at today's keywords.
Now, we're going to use the word Fibonacci.
As you can see, it has a capital letter.
Fibonacci was an Italian mathematician.
Fibonacci sequence, a number sequence where each number is the sum of the previous two, starting from zero and one.
Then there's the Fibonacci spiral, a visual pattern formed by drawing arcs connecting corners of squares whose side lengths follow the Fibonacci sequence.
And we'll show you some of that in a little bit.
The golden ratio, a special or perfect proportion, and you find this in nature, in art, in music and maths, and it makes things look and sound balanced and beautiful.
Then the climax.
This is a point of the most intensity, the peak of the music.
Then golden brick.
This is Eric Whitaker's term to describe a small musical idea that's the core building block on which the whole piece is built.
Before every music lesson, we know how important it is to warm up our bodies, our voices, and our minds.
If we're singing, we want it to be safe.
If we're playing, we want it to be accurate.
And if we're playing or singing together as an ensemble, we want it to be neat and focused.
When we're listening carefully, that's gonna help us learn new music and play together.
So I'd like you to begin by joining in with these rhythm and beat warmups.
You must listen and watch carefully each time.
The first one is meter mix-up.
Are you ready?
Here it comes.
<v ->Join in with me and watch for the changing numbers.
</v> I'll count with you to begin with and then I'll drop out.
One, two, three, four, five, six, seven, eight.
One, two, three, four, five, six, seven, eight.
Five.
Three.
Seven.
Two.
Six.
Eight.
Five.
Two.
Four.
Seven.
Three.
Eight.
Six.
Four.
<v ->Next, you're going to be echoing rhythms</v> on body percussion.
You'll watch first and join in second.
So pay attention and be ready to keep that steady pulse.
Here it comes.
<v ->Let's do some body percussion.
</v> We're going to clap for ta.
We're going to stomp our feet for ta-di, and we're going to pat our knees for ta-ka-di-mi.
Okay, so I'll start and then you join in with the echo.
♪ Ready, steady, off we go ♪ Ta, ta-di, ta-ka-di-mi, ta.
Ta, ta-di, ta-ka-di-mi, ta.
Ta-di, ta-ka-di-mi, ta, ta.
Ta-di, ta-ka-di-mi, ta, ta.
Ta-di, ta-di, ta, ta-ka-di-mi.
Ta-di, ta-di, ta, ta-ka-di-mi.
Ta-di, ta, ta-ka-di-mi, ta.
Ta-di, ta, ta-ka-di-mi, ta.
<v ->Now, I'm hoping very much that's helped you feel focused</v> and ready to work together as an ensemble.
And Jacob says, "Remember, we are keeping that steady beat and watching a leader carefully.
" If you'd like to play those again, you can pause here.
Otherwise, let's press on.
There are two learning cycles in today's lesson.
The first one is understanding the golden ratio, and then second, we're going to be inspired further by Deep Field listening to "Impossible Magnitude.
" Let's start with the golden ratio.
Here is a sequence of numbers that never ends, and it's called the Fibonacci sequence.
Here's how it begins.
Zero, one, one, two, three, five, eight, 13, 21.
And I'd like you to see if you can work out the next number in the sequence.
There's something about what happens with the previous numbers that gives us the next one.
Pause here, see if you can work it out, and see if you all come to the same or different answer.
Off you go.
I wonder if you got it.
If you did, you will have got 34, and I'm gonna explain how we get there.
In the Fibonacci sequence, each number is the sum of the two numbers that came before it.
So let's look at zero and one.
Zero plus one is one.
Move on along.
One plus one is two.
One plus two is three.
Two plus three is five.
Which two numbers are we going to add together next?
Three and five, which gives us eight, and then five plus eight is 13.
And then which two are we going to add next?
We're going to add together eight and 13, which will give us 21.
And then if you've got the right answer, what you will have done is added 13 and 21 together, and that will have given you the next number in the sequence, 34.
Now, the numbers in the Fibonacci sequence begin to form a spiral pattern.
And the spiral pattern, which I'll show you in a moment, can actually be found all over nature, and in music, and in art.
In fact, it's all around us.
The Fibonacci sequence can be shown as squares, and the size of each square comes from the sequence.
Let me show you the first one, one and one.
Add those together, we get two.
Two plus one gives us three.
Can you see the squares bigger representing the size?
Three plus two gives us five.
Five plus three gives us eight.
Then we add five plus eight will give us 13.
Then what we're going to add, eight plus 13 gives us 21.
And if we draw an arc around the side of each of those squares that we're creating, it creates a spiral shape, and this is called the Fibonacci spiral.
Does it remind you of anything?
A careful look.
Hmm.
Sam suggests it's a bit like a shell.
Now you can see that we've got some fossils there.
Ammonites.
Does look a bit like that, indeed.
We can find the Fibonacci spiral all around us.
If you look at something like a pine cone or even a pineapple, the clockwise and counterclockwise spirals as it goes out follow that sequence.
And the reason for that in nature is 'cause it makes sure that the cone is tightly packed and the seeds are protected.
"The spiral shape can even be seen in the formation of galaxies," Lucas adds.
So when you are out and about in nature, why not become a Fibonacci detective?
See what you can find.
Look for that Fibonacci sequence in nature, in plants, in humans, out in the world.
Okay, quick check in then.
The Fibonacci sequence, that creates a shape.
What shape does the Fibonacci sequence create?
Is it an oval, a circle, a spiral, or a star?
I'm pretty sure you're all going to get this one right.
Can you just check.
Fibonacci sequence, which shape can it make?
Oval, circle, spiral, star.
Call that letter out now.
Absolutely right.
It is c, it's a spiral shape.
Sofia shows us, "This is the Fibonacci spiral," created by those arcs drawn in each of the squares that represent the size of the number.
Look at these numbers in the Fibonacci sequence.
The ones that are in purple on the screen are three, five, and eight.
So we have five, we add three.
The sum is eight.
We have eight, we add five.
The sum is 13.
Okay, we have 13.
Let's add on eight.
The sum is 21.
If you look about the relationship between the three numbers each time, look at those on the number lines below, is there something you notice about that?
The color changes roughly at the same point each time.
And this is what we call the golden ratio.
Now, the golden ratio occurs just after the 2/3 point, So divided into three, first third, second third, 2/3 of the way in.
Three plus five is eight, five plus eight is 13, and eight plus 13 is 21.
So you can see the way, if you look at how those pictures are divided, 2/3 of the way each time.
If you look at the shaded part of the purple in the first one, 2/3 of that are shaded.
If you look at the the right hand side of the green one, 2/3 of that are shaded.
And if you look at the whole image there, 2/3 of that are shaded.
So what fraction is closest to the golden ratio?
I wonder if I might have given it away.
a is 1/2, b is 2/3, and c is 3/4.
Which fraction's closest to the golden ratio, please?
a, b, or c?
It's a brilliant way to check that you are listening and following.
It is b.
2/3 is the fraction closest to the golden ratio.
Now let's get into some music.
The golden ratio can be found in music.
Sometimes composers will use the golden ratio purposefully to create in their piece of music this feeling of balance and beauty.
Sometimes it occurs naturally, just creating music that reflects natural beauty of the ratio found in nature and art around them.
If we think of a pop song, the structure often changes roughly 2/3 of the way through the song.
That's not necessarily that the golden ratio has been applied purposely.
"Oh yes, I must now put the golden ratio into my song.
" But it does feel a very natural way of structuring a song.
We might have a verse, then a chorus we all know the words too.
A verse again, that familiar chorus, then something called the bridge, which is part of the song that sounds a little bit different.
Then we might have that familiar chorus again.
This is a really common structure of a pop song.
If you listen out for this structure in your favorite song, I want you to find out will that follow that structure each time.
And quite often it will.
The climax, the top of the hill, that build right at the top, that big intense part of a piece of music, often occurs at the golden ratio, roughly 2/3 of the way through the piece.
I'd like you to now pause and listen to this Prelude composed by Chopin.
And I'd like you to listen out, it's not long, I'd like you to listen out for the climax of the piece at the golden ratio.
So you could even suss out how long is that piece, work out in seconds how long is 2/3 in, and does that climax happen at that point?
Pause and have a listen now.
Very good.
So around 19 or 20 seconds in, we heard that climax of the piece.
And that was about 2/3 of the way through at the golden ratio.
Let's have a look at this rhythm.
The notations there on the board, and the rhythm's been inspired by the Fibonacci sequence and the golden ratio.
What do we notice?
Well, let's unpack it first and see if we can hear it and clap it.
It tells me at the beginning, it says two and a heart, so I know there's two beats in the bar.
That's gonna help me divide it up musically so that those bars are even.
I'm going to clap it and speak it first of all.
Then we'll see if we can discover anything else.
Ready, steady, off I go.
Ta, ta, ta-di, ta-ka-di, ta-ka-di-mi, ta.
Ta-ka-di-mi, ta-ka-di-mi, ta-ka-di-mi, ta.
Ta-ka-di-mi, ta-ka-di-mi, ta-ka-di, ta-di, ta, ta.
Another way, and this might give you a clue that we could think about speaking it and clapping it.
Have a listen this time.
1, 1, 1-2, 1-2-3, 1-2-3-4-5, 1-2-3-4-5-6-7-8.
Listen again.
1, 1, 1-2, 1-2-3, 1-2-3-4-5, 1-2-3-4-5-6-7-8.
A pause there.
Can you now see how this rhythm has been inspired by the Fibonacci sequence?
Let's look at it another way.
In a moment, we're gonna follow some body percussion.
And as we perform that body percussion, we're gonna start playing with our dynamics to show how we can build to the climax at the golden ratio on the 10th beat.
First, let's look how we're going to play the body percussion.
So we're following this part of the Fibonacci sequence.
One, two, three, five, eight.
And that's because two plus one is three.
Three plus two is five.
Five plus three is eight, okay?
When it's one, it's a ta.
We clap ta.
Let's do that together.
Ready, steady, off we go.
Ta.
Next one on our feet, ta-di.
Ready, steady, off we go, ta-di.
Now we're on three.
Number three in the Fibonacci sequence, which we'll do on our chests.
Ready, steady, off we go, ta-ka-di.
Lovely job.
Now as June points out, pattern five and pattern eight are both two beats.
So we have for starting on our knees, we've got ta-ka-di-mi, ta, for number five.
Are you ready, stead, off we go, ta-ka-di-mi, ta.
And then the last one where we are up to number eight in the sequence, all in our knees, two ta-ka-di-mis.
Here we go.
Ta-ka-di-mi, ta-ka-di-mi.
So you can pause if you need to practice each of those, follow that sequence.
And then in a moment I'm going to play the body percussion video for you to follow.
Pause if you need to.
Here comes the clip.
<v ->Ready, steady, off we go.
</v> Ta, ta, ta-di, ta-ka-di, ta-ka-di-mi, ta.
Ta-ka-di-mi, ta-ka-di-mi, ta-ka-di-mi, ta.
Ta-ka-di-mi, ta-ka-di-mi, ta-ka-di, ta-di, ta, ta.
<v ->Great, so now you've watched that clip,</v> the notation and the guide to the body percussion is on the board in front of you.
It hasn't changed, it's the same.
But what you will notice is there are, at the very bottom, some dynamic markings.
They look like little bobby pins.
On the bottom there, there is this sort of alligator mouth shape getting bigger.
That's gonna show that at this point we are getting louder.
And then towards the end, so where we've got the ta-ka-di, ta-di, ta, ta, it's going from big to small, so we are getting quieter.
Hopefully you would've picked up that from the video.
I would like you to pause here and see if you can master playing that body percussion and that Fibonacci-inspired rhythm on your body percussion as an ensemble.
Off you go.
Fantastic.
Really well done.
You are playing in time as an ensemble.
You are following the notation.
You are translating that onto body percussion, and you're responding to the dynamic changes of getting louder and quieter.
Bit of feedback on what we've just done.
The rhythm patterns follow the numbers in the Fibonacci sequence.
And what I mean by that is to begin with, we've got one, one, then one-two, one-two-three, one-two-three-four-five, one-two-three-four-five-six-seven-eight.
Now of course the numbers would keep getting bigger, so you can see the repeat of five and eight.
That starts to make it clear that this is gonna be the climax of the piece.
We've repeated that five and eight sequence twice.
And the loudest part of the rhythm is at that golden ratio.
And the number sequences match that too.
There's eight bars of music, and there's two beats in every bar, so the bars are all even.
And we can fit the Fibonacci sequence into those even bars.
Let's now be inspired by "Deep Field: Impossible Magnitude.
" So Eric Whitaker's piece, "Deep Field," composed in 2015 is an impressive piece of program music that takes us on a journey through space.
And it has an accompanying film that shows images taken by NASA's Hubble Space Telescope.
So "Deep Field" can be divided into four movements, a bit like chapters in a book.
The first movement, "Cathedral of Reason.
" The second movement he calls "Pillars of Creation.
" The third movement, "Impossible Magnitude," which we're gonna be listening to more closely today.
And then the fourth movement is "Earth Choir.
" Now, Izzy explains magnitude, that word, it means the size, or the strength, or the importance of something comes actually from the word magnus, meaning great.
In this third movement, we reach the Deep Field at last.
And we can clearly hear the piece reach to its climax, the bold use of the golden bricks, those small musical ideas on which the piece is built, and a real build in the dynamics.
And then we hear these long, held notes and a crescendo of percussion crashing through.
And this is telling us we've reached Deep Field.
So I'd like you to listen to "Impossible Magnitude" and identify which bit when you think, "Yep, we finally reached Deep Field.
" Pause here and have a listen.
I wonder if you all felt that at the same time.
So where we're up to in the whole piece, if you had the timer on, it would say about 16 minutes 40.
But I imagine you're all starting to identify this feels like we're reaching Deep Field.
Now, the climax of the piece seems to last about a minute long.
And it's just after 2/3 of the way through the complete piece.
That's movement one, movement two, movement three and four.
So when it happens, Deep Field is reached in this moment.
In this particular movement, when we reach Deep Field with that celebratory percussion crescendo, that is exactly at the golden ratio of the individual moment in "Impossible Magnitude," 2/3 of the way through, bang on.
Pretty clever.
So the whole piece, it's about four minutes and four.
So the first part is the buildup, and then around 60 minutes 40, as we heard, Deep Field happens.
And then we've got the remaining part, which is the come down.
So 2/3 in, Deep Field's reached right at the golden ratio.
In "Impossible Magnitude," the golden brick, that musical idea that the whole piece is built upon, is described as struggle.
Now, on the screen it is drawn out the pitch rising, and that leads us up like climbing a ladder up to the climax.
I'd like you to have a listen to identify the golden brick.
So looking at that pitch rising, it's four notes that you can see on the screen going up each time known as struggle, have a listen to this part and can you identify the golden brick?
Do that now.
So again, if you did have that timer on and you could see the clock moving as the music was going along, it's about 14 minutes 38 in.
And what we hear is it's played slowly by string instruments and then it's repeated over and over using shorter notes.
And then the brass instruments play it slowly, getting louder and louder, building up until that climax begins.
So that musical idea, that golden brick, is already used in different ways, played in different ways, on different instruments.
So in your ensembles, you're now gonna compose your third movement for your cosmic program music piece.
So this is what you're gonna think about, how to plan to create a climax at the golden ratio of 2/3 in in this movement that says, "Hmm, we've reached Deep Field.
" Now if you'd like to, you can certainly challenge yourself.
Why not write out the Fibonacci sequence, zero, 1, 1, 2, 3, 5, 8, 13, 21, and then see if you want to use any of those numbers in sequence to form some of your rhythm patterns?
Up to you.
Here's some ideas to get you started.
This is our "Impossible Magnitude" ideas spidergram.
So first of all, for your buildup, you could use your golden brick from the second movement, and then compose a new golden brick to show your climax.
Start your piece and end your piece quietly.
That allows you space to build your dynamics for the climax.
If you start off loud, you've got nowhere to go.
Start off softly and gently and build.
You can combine your ideas that you used in your first and second movement.
Take ideas you've already got.
We don't need to work from scratch.
A really good way of structuring this third movement is have three clear chunks.
So you have your climax after the second.
So you've got a soft build, and then the climax, the end, back to soft, okay?
See it as those three sections.
And you can use some rhythms inspired by the numbers from the Fibonacci sequence if you would like to.
So pause here and work on your third movement in your ensembles using the golden ratio, and be inspired if you'd like by the Fibonacci sequence.
Off you go.
Wow, such creative ideas.
Lots of thoughts everywhere.
And you might want to get these written down.
So here's an example.
I'm going to play the track in the moment and you can follow along.
See if you can follow how this map works.
So from the beginning it's got the buildup, which is the first 2/3.
Then we've got our Deep Field moment, and then that calm come down at the end.
If I look towards the top, I can see that the rhythm notated says ta, ta, ta-di, ta-ka-di, ta-ka-di-mi, ta, ta-ka-di-mi, ta, ta, ta, ta.
So some of that is using elements of the Fibonacci system.
They've suggested the rhythm's gonna be played on claves.
It gets louder at the crescendo and quieter at the diminuendo.
They've got their golden brick, they've marked out that pattern.
That was the one they used last time, which repeats.
And then the new one.
They suggest in their notes that the golden brick repeats faster and faster and it's played in the glockenspiel.
And then with a new golden brick, we add tambourine shimmer, drum roll, and it gets louder and louder.
And towards the end, it's slower and quieter.
Have a listen and see if you can follow that.
And then perhaps to finish, you'll be able to jot yours down too.
Here it comes.
Super, I really hope you were able to follow along as you were listening.
Now, if you've not done so already, now is the time in your ensemble to jot down your composition as that kind of graphic score so you've got it for next time.
Remember, it's something that you need to be able to interpret as your ensemble.
Use the example on the board if it's useful to you.
So if you need to write that down ready for next time, do pause here.
And that's about all we have time for for today.
We've been inspired by the Fibonacci sequence from that Italian mathematician, Leonardo Bonacci, who we know as Fibonacci.
His sequence can be shown in a spiral shape, and we can find this sequence or it's Fibonacci spiral in nature and art.
And I do hope you go and look and discover it more.
The sequence and the spiral create a ratio that's approximately 2/3 to 1/3, and this is called the golden ratio.
The golden ratio can often be heard in music, with the climax of the music at that golden ratio point.
And a piece of music can be inspired and built from one small musical idea, which Eric Whitaker calls a golden brick.
Really well done for your composition today, fabulous ensemble playing, and I will look forward to seeing you next time.
Bye for now.
