### MS. REBECCA SHEIR

00:00:08

Welcome back to "Metro Connection." I'm Rebecca Sheir, and today we are all about transformations. So far, we've heard about ecological transformations, transportational transformations, even geological transformations. But we're going to kick off this part of the show by addressing musical transformations. Or would it be mathematical transformations? No, no, it would music or maybe math. Actually, you know what? It'd be both, or so says this woman.

### MS. YVONNE CARUTHERS

00:00:34

I'm Yvonne Caruthers. I'm a cellist in the National Symphony Orchestra.

### SHEIR

00:00:37

Because the way Caruthers sees it, math and music are intimately and intricately connected. And on Monday, April 9, she and her fellow musicians Aaron Goldman...

### MR. AARON GOLDMAN

00:00:46

I'm the assistant principle flutist of the National Symphony.

### SHEIR

00:00:49

And percussionist Danny Villanueva...

### MR. DANNY VILLANUEVA

00:00:51

I'm a freelance percussionist in town, play extra with the National Symphony quite a bit.

### SHEIR

00:00:55

...will appear at the Smithsonian's Discovery Theater to demonstrate the many magical ways you can transform math into music and music into math. The trio has done several student programs on the subject, but Monday's event is for people of all ages. I recently visited Caruthers, Goldman and Villanueva at a rehearsal room in the Kennedy Center where the National Symphony Orchestra regularly performs.

### SHEIR

00:01:15

Caruthers brought her cello, Goldman brought his flute, and Villanueva brought his drum practice pad. And they told me more about Monday night's seminar titled "Math and Music: Closer than You Think."

### CARUTHERS

00:01:26

Well, I thought a good way to get started would be to show one of the most obvious examples of how math and music are similar. And I'm just going to show you this, and maybe you could read it out loud.

### SHEIR

00:01:38

Okay. It's a piece of yellow paper. It says A squared plus B squared equals C squared.

### CARUTHERS

00:01:44

And do you remember what that is?

### SHEIR

00:01:46

I want to say that's the Pythagorean theorem.

### CARUTHERS

00:01:49

Very good. You took algebra and geometry, but the point is that looks really simple, ABC. And then there's those little twos up there, but it's actually very sophisticated. These little mathematical symbols express really big ideas, and music is very similar to that. We use these simple symbols to express abstract ideas. So if I fold this down and I show these little symbols right here to Aaron and I say, Aaron, what is that? He'll say?

### GOLDMAN

00:02:16

Beethoven 5.

### CARUTHERS

00:02:17

'Cause if you know how to read music, that's obviously the opening to "Beethoven's 5th Symphony."

### SHEIR

00:02:22

Is that the one that goes (singing) ?

### CARUTHERS

00:02:23

That's right. That's what that -- that's the one.

### CARUTHERS

00:02:27

Then this is his really goofy little percussion notation.

### VILLANUEVA

00:02:29

That's a roll off.

### CARUTHERS

00:02:31

Can you play it for us?

### CARUTHERS

00:02:39

So both math and music use these symbolic languages, and out of those have grown these huge things. So from that, you know, then you can go on into all these other cool aspects of music or of math and see all these commonalities.

### SHEIR

00:02:54

What are some other examples?

### CARUTHERS

00:02:55

Well, another one is the idea of patterns. I think there was a program I was listening to not long ago, and this scientist was saying that one of the most important attributes for scientists or mathematicians, to be able to spot patterns really quickly. And we do that a lot in music. Danny, you probably deal the most with patterns 'cause of all these different rhythms. Do you remember that example you did for us with the paradiddles?

### VILLANUEVA

00:03:19

So a paradiddle is just like it's pronounced, para-diddle, which is right-left-right-right, and we reverse that, which is left-right-left-left. So, through different sticking combinations as drummers, we learn these patterns, and we use those in everyday playing, put them together. So a paradiddle would sound like this.

### VILLANUEVA

00:03:36

Right-left-right-right. Left-right-left-left. Left-right-right-left.

### VILLANUEVA

00:03:57

Paradiddle, and we can take further. And we can create a paradiddle-diddle, which adds another left-left at the end of that. So it's right-left-right-right-left-left, another combination. So it's right-left-right-right-left-left.

### VILLANUEVA

00:04:22

It's basically just right and left and what can we do with the right and the left and create these complex rhythms or patterns. And, as drummers, we learn to recognize those patterns as we're playing. It just it becomes innate, becomes part of us. And we just see it, and we just play it as goes.

### SHEIR

00:04:38

I would love some more examples. What else can we address here?

### CARUTHERS

00:04:41

Ratios. We love talking about ratios. Remember that guy Pythagoras?

### CARUTHERS

00:04:48

The story is he's walking down the street, and he passes the blacksmith shop. And he hears the guys banging on these big iron bars, and he sees that the guy with the little hammer hitting the iron bar makes one pitch and the guy hitting the iron bar with the huge hammer makes a different pitch. And so he goes home and scratches his head and comes up with these ratios that he worked out. And he did a lot of experiments with strings, and what he discovered was...

### CARUTHERS

00:05:16

There's a vibrating string. Now, if we divide that string in half, we get a pitch which is an octave higher.

### CARUTHERS

00:05:24

Same pitch, octave higher. This is known as a two to one ratio because the full string is the full length, and this is half the length. So the full string is twice as long. Now, we can take that part of the string, this half of the string, we could divide that in half.

### CARUTHERS

00:05:40

Another octave higher. We can divide that in half. So every time we divide the string in half, the pitch goes up an octave. And so our strings vibrate, and Aaron's flute, it's a -- what is it vibrating in the flute?

### GOLDMAN

00:05:52

The way it works, at least on flute, is, when I blow across the embouchure hole, my air is split by the far edge of the hole here. And we get pressure differences, both on the top and the bottom. So the air stream goes up out of the flute a little and then down into the flute a little bit, and up and down. It happens very quickly, depending on what note I'm playing. So when I play that A at 440 Hertz, the air is going up and out of the embouchure hole 440 times a second, and I get a standing way which re-enforces itself at a 440.

### GOLDMAN

00:06:28

Now, the same thing can happen when I do the octave leap like we did with Yvonne, is that all of a sudden now I'll be at 880. Though I don't do it with moving anything, I just increase my air a little bit and aim a little bit differently with my lips. And I get...

### GOLDMAN

00:06:49

And those are the same doubling and halving of the cycles per second or Hertz.

### SHEIR

00:06:54

Do ratios play out anyway with percussion, with drums?

### VILLANUEVA

00:06:58

Well, the same thing applies to mallet instruments. We have xylophones and, depending on how they're tuned, same thing. It's the bar length varies, and that determines the register, the octave, as well. And even a regular drum could be tuned with pitch. I could tune it to an A, and the way it's vibrating also by the tension of the heads, the length of the shell, all these things come into play with the vibrations. We can get pitches just the same. Unfortunately, with the practice pad, I can't do that with this but, yes.

### SHEIR

00:07:22

So when you've spoken with students, what has their reaction been? What have been some interesting responses they've had?

### GOLDMAN

00:07:27

When we have the kids here, they're about middle school-aged kids, right?

### GOLDMAN

00:07:31

And it was fascinating how much of the math that they really knew, the math that we talked about. We would post pictures up of these sequences, and they would all know the sequences, which was great. And many of them were also musicians.

### CARUTHERS

00:07:43

Actually, I always feel like the adults maybe get more out of it than the kids do. I went in and did a talk for teachers one time, a teacher workshop and these were music teachers. And I started talking about how music notation is a lot like a math graph, and these people went, wow, I never thought of that. I got to go talk to the math teachers. So sometimes it's just a matter of pointing things out.

### SHEIR

00:08:05

That was cellist Yvonne Caruthers, flutist Aaron Goldman and percussionist Danny Villanueva, who will be appearing in "Math and Music: Closer than You Think" at the Discovery Theater on Monday, April 9 at 7:00 pm. For more information, and to get a sneak peek at some of the mathematically musical and musically mathematical visuals the trio will share during the presentation, visit our website, metroconnection.org.

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