Mathematics and Music Christina Scodary. Introduction My history with music Why I chose this...
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Transcript of Mathematics and Music Christina Scodary. Introduction My history with music Why I chose this...
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