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 c 2 is T/ρ for strings and B/ρ for wind instruments.

 Initial Conditions: u(x,0) = f(x) u t (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” + c 2 λT = 0  Using our conditions we get: and  Solution:

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

 Frequency v is called the fundamental.  The component nv is the n th harmonic, or the (n-1) st overtone. n=1fundamental1 st harmonic242 Hz n=21 st overtone2 nd harmonic484 Hz n=32 nd overtone3 rd harmonic726 Hz n=43 rd overtone4 th harmonic968 Hz

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

References  Music: A Mathematical Offering by David J. Benson  Elementary Differential Equations and BVP by W.E. Boyce and R.C. DiPrima