Mathematics

and Music

Christina Scodary

Introduction My history with music Why I chose this topic

Topics Covered Pythagorean scale The cycle of fifths Just intonation Equal temperament The wave equation for strings Initial conditions Wind instruments Harmonics

Wave Equation

Where c2 is T/ρ for strings and B/ρ for wind instruments.

2

22

2

2

x

uc

t

u

Initial Conditions: u(x,0) = f(x)

ut(x,0) = g(x) Boundary Conditions: u(0,t) = 0

u(L,t) = 0

Wind Instruments Boundary conditions depend on whether

the end of the tube is open or closed. Flute: open at both ends

Same conditions as string

Assuming that u(x,t) = X(x)T(t) Separation of variables gives us:

X” + λX = 0 and T” + c2 λT = 0 Using our conditions we get:

and

Solution:

)sin()(L

xnCxX n

)cos()(

L

tcntT

L

tcn

L

xnCtxu

nn

cossin,

1

Harmonics

The terms in this series are the Harmonics. The frequency of the nth harmonic is given by

the formula:

L

tcn

L

xnCtxu

nn

cossin,

1

L

cnv

2

Frequency v is called the fundamental. The component nv is the nth harmonic, or the

(n-1)st overtone.

n=1 fundamental 1st harmonic 242 Hz

n=2 1st overtone 2nd harmonic 484 Hz

n=3 2nd overtone 3rd harmonic 726 Hz

n=4 3rd overtone 4th harmonic 968 Hz

Piano Fact

Did you ever notice that the back of a grand piano is shaped like an approximation of an exponential curve?

http://www.zainea.com/pint.gif

References Music: A Mathematical Offering by

David J. Benson Elementary Differential Equations and

BVP by W.E. Boyce and R.C. DiPrima