1. Synthesizing a Clarinet Nicole Bennett

2. Overview  Frequency modulation  Using FM to model instrument signals  Generating envelopes  Producing a clarinet note  A-440 note

3. Frequency Modulation  Used to reproduce signals with frequencies that vary with time  General formula: x(t) = A*cos(ψ(t))  Oscillations of ψ(t) provide changes in instantaneous frequency - (ψ′(t))

4. Producing Instrument Sounds  ψ(t) must be sinusoidal in order to reproduce both the fundamental frequency and the overtones of an instrument  General equation: x(t) = A(t)*cos(2Π*f c *t + I(t)*cos(2Π*f m *t + Φ m ) + Φ c ) ٭٭ ٭٭John M. Chowning, “The Synthesis of Complex Audio Spectra by Means of Frequency Modulation,” Journal of the Audio Engineering Society, vol.21, no. 7, Sept. 1973, pp 526-534.

5. x(t) = A(t)*cos(2Π*f c *t + I(t)*cos(2Π*f m *t + Φ m ) + Φ c )  A(t): amplitude envelope  Function of time  Allows sound to fade slowly or be cut off quickly  f c : carrier frequency  Frequency without any modulation  f m : modulating frequency  Rate of modulation of the instantaneous frequency

6. x(t) = A(t)*cos(2Π*f c *t + I(t)*cos(2Π*f m *t + Φ m ) + Φ c )  Φ c and Φ m : phase constants  Set to Π/2 for this project  I(t): modulation index envelope  Used to vary the harmonic content of the sound  Produces the overtones

7. Generating A(t) and I(t)  WOODWENV2.m WOODWENV2.m

8. Scaling the I(t) Function  A(t) and I(t) are normalized by the WOODWENV function  I(t) must be scaled in order to produce a clarinet note  scale.mscale.m

9. Synthesizing a Note  Now have most of the pieces of x(t): x(t) = A(t)*cos(2Π*f c *t + I(t)*cos(2Π*f m *t + Φ m ) + Φ c )  f c and f m : ratio is 2:3 for the clarinet  f 0 : frequency of the note – will be greatest common divisor of f c and f m

10. Clarinet Function  clarinet.m clarinet.m

12. Playing a 440 Hz note  Play the A note Play the A note  Limitations of the equation

13. Conclusion  Modeling instrument signals  Generating a clarinet note  Problems with modeling an instrument using a mathematical equation