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Christina Scodary

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

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Christina Scodary

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  1. Mathematics and Music Christina Scodary

  2. Introduction • My history with music • Why I chose this topic

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

  4. Wave Equation • Where c2 is T/ρ for strings and B/ρ for wind instruments.

  5. Initial Conditions: u(x,0) = f(x) ut(x,0) = g(x) • Boundary Conditions: u(0,t) = 0 u(L,t) = 0

  6. 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

  7. Assuming that u(x,t) = X(x)T(t) • Separation of variables gives us: X” + λX = 0 and T” + c2λT = 0 • Using our conditions we get: and • Solution:

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

  9. Frequency v is called the fundamental. • The component nv is the nth harmonic, or the (n-1)st overtone.

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

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

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

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