## Ask Professor Puzzler

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"I keep hearing about people who are good at math and music. But that seems so strange -- music is so artistic and flowery, while math is so technical and straightforward. How could either have anything in common with the other?" ~ Laura from Maine

Hi Laura, thanks for thre great question. I wish I could give you a solid scientific answer to this question, but the inner workings of the brain definitely are not in my field of study. Instead, I can give you some observations about both math and music, based on my own experiences as a mathematician and a musician.

First, I can confirm that what you said is typical in my experience - it is very common for people who are good at music to also be good at math (and vice-versa). Obviously, that's not a 100% rule (probably not even close!) but it is common enough that my high school math teacher used to refer to his best students as his "M & M students." Not because they liked chocolate in a candy shell, but because they excelled in both Math & Music.

So what's the connection? I'm going to suggest three connections: patterns, precision, and creativity.

**Patterns in Music**

Music is filled with patterns. That can be seen in the simplest of ways: a couple days ago I sat down with my five-year-old son and taught him a little bit about the piano. I showed him how to find the note C by looking for patterns. There are sets of three black keys, and there are sets of two black keys. C is any white key that comes right before a set of two black keys. From there, he could find any C on the keyboard. I showed him that the note names go from A to G, and then start over at A. From there, he could start at the low end of the keyboard and name every single white key on the piano. Patterns.

Eventually he will discover that there are certain intervals (major thirds, minor thirds, etc) that sound really nice together, and others that don't sound quite as nice. He'll discover that those intervals can be shifted up and down the keyboard by octaves, and then he'll discover that they can be shifted by other amounts, and the sound is just as nice.

Musical compositions are also filled wtih patterns. Just listen to Vivaldi's *Four Seasons*, and you'll hear the same "figure" repeated throughout a movement. Starting on one note, and then on another note, then with some variations to the pattern, but always with a pattern that your ear catches and hangs on to. Some of the most memorable classical compositions are the ones that have memorable patterns that appear over and over (think of "dum-dum-dum-DUM" in Beethoven's fifth symphony, or the themes of Rossini's "The Barber of Seville" and "William Tell Overture" - and if you're not familiar with those two compositions, think of Bugs Bunny as a barber, or the Lone Ranger).

Patterns crop up in more modern music as well - the pattern of verse-chorus-verse-chorus-bridge-chorus is one that gets repeated over and over again in popular music. Your brain processes those patterns even if you're not consciously aware of them.

As an added bonus: most of the patterns we notice in music have a mathematical reason why they exist. For example, an octave is an interval that sounds good because the frequencies of the notes are in a ratio of 1:2.

**Patterns in Math**

I'm not sure I even need to say that math is filled with patterns. It seems intuitively obvious. But let's consider just a couple things. First, I remember how fascinated I was when I first discovered the nine-times-table:

9 x 2 = 18

9 x 3 = 27

9 x 4 = 36

9 x 5 = 45...

...and so on. As the tens place increases by one, the ones place decreases by one, and in each case, the sum of the digits is 9. What a fascinating pattern.

Fibonacci was famous for discovering patterns, and exploring how they related to the natural world around us - for more on Fibonacci, you might want to consider this book, reviewed here by Book Scrounger: Blockhead: The Life of Fibonacci.

A couple days ago my son went for his yearly checkup, and his doctor commented that math learning at his age is all about finding patterns.

*People who like looking for patterns will naturally be drawn to either math or music, or both.*

**Precision in Music**

If you've ever taken music lessons, you understand this one. I remember, as a violin student, being expected to be very precise about a *lot *of things: how I hold the bow, how I hold the violin, where I put my fingers, how I draw the bow across the strings. And then I'd do scales over and over again, with my teacher telling me, "That C-sharp isn't sharp enough." or "Your bow is way down on the fingerboard." Getting good at playing a musical instrument requires a fairly stubborn and determined mindset to *get it right*. For a violin student, "getting it right" is sometimes a matter of a miniscule fraction of an inch one way or the other.

So when you said that "math is technical," my response to that is: so is music! Students who can't master the technical aspects will not excel.

**Precision in Math**

The same stubborn, determined mindset that makes someone decide they're going to practice a D-major scale until they get it right is necessary for a good math student as well. In elementary school it's memorizing those math facts. Then it turns into more and more complicated processes (arithmetic algorithms, arithmetic operations on fractions), and eventually the student is faced with complex algebraic and even calculus problems. And good math students practice these over and over again (it's why those cruel and nasty math teachers assign homework; doing 20 math problems is like practicing your D-major scale over and over and over again). You don't get that kind of precision any other way.

*People who have the determination to keep practicing things are more likely to excel in math or music, or both.*

**Creativity in Music**

Eventually, every violin student who sticks with it long enough gets to the point where their teacher says, "Okay, it's time to learn about vibrato." What is vibrato? Vibrato is, according to one dictionary: "a rapid, slight variation in pitch in singing or playing some musical instruments, producing a stronger or richer tone." In a sense, vibrato is a breaking of the rules, because you've been trying desperately to play in tune, and now your teacher is giving you permission to play *out of tune*. In fact, your teacher is *demanding* that you play out of tune. In my mind, it was at that moment that I moved from parroting to interpretation. You are no longer at the mercy of the musical composition; the notes on the page become a vehicle for your own personal expression. And vibrato is just one thing - whatever musical instrument you play, as you advance, you will discover more and more that there are ways to express yourself through creative use of the technical skills you've acquired. You've become creative.

And, do I need to mention? Writing musical compositions is a wonderful task in creativity!

**Creativity in Math**

Really? Math is "creative"? Sure it is! Math is problem solving! Math is curiosity, and exploring beyond what is known. When I graduated from high school, on Class Night, my math teacher presented me with a our school's highest math award, and at the same time presented me with a notebook which contained every math problem I'd solved during my independent study math course that year. What's the big deal about that (you might ask)? The big deal is that every one of the problems I worked on that year was a problem I made up myself, or decided to challenge myself with. Whether it was proving the quadratic formula, deriving a formula for the inverse of an N x N matrix, developing my own theorems, or just posing mathematical questions to myself and trying to solve them, I spent the entire year being creative. And that creativity is what makes a mathematician. Being a good mathematician doesn't mean being able to multiply 324 x 16.9; that's just the precision practice (like learning a musical scale); being a good mathematician means being able to take those skills you've practiced, and put them to use in *new * ways to solve problems you haven't solved yet.

When I have students who want to take my math classes as "honors level" classes, I expect problem solving from them; I expect them to take the things they've learned and put them to use in brand new ways that they've never thought of before. I expect them to combine two mathematical ideas that they've never used together before, and use them to develop a solution to a problem they didn't think they could solve. I expect them to get creative.

*People who are willing to use the tedium of repeated practice to explore new ideas and new ways of doing things will excel at either math, or music, or both.*

* "There is geometry in the humming of the strings, there is music in the spacing of the spheres."* ~ Pythagoras