Transcript Synthesizing a Clarinet Nicole Bennett

Synthesizing a Clarinet
Nicole Bennett
Overview
Frequency modulation
Using FM to model instrument signals
Generating envelopes
Producing a clarinet note
A-440 note
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))
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*fc*t +
I(t)*cos(2Π*fm*t + Φm) + Φc) ‫٭٭‬
‫٭٭‬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.
x(t) = A(t)*cos(2Π*fc*t +
I(t)*cos(2Π*fm*t + Φm) + Φc)
 A(t): amplitude envelope
Function of time
Allows sound to fade slowly or be cut off
quickly
 fc: carrier frequency
Frequency without any modulation
 fm: modulating frequency
Rate of modulation of the instantaneous
frequency
x(t) = A(t)*cos(2Π*fc*t +
I(t)*cos(2Π*fm*t + Φm) + Φc)
Φc and Φm: 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
Generating A(t) and I(t)
 WOODWENV2.m
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.m
Synthesizing a Note
Now have most of the pieces of x(t):
x(t) = A(t)*cos(2Π*fc*t +
I(t)*cos(2Π*fm*t + Φm) + Φc)
fc and fm: ratio is 2:3 for the clarinet
f0: frequency of the note – will be greatest
common divisor of fc and fm
Clarinet Function
 clarinet.m
Playing a 440 Hz note
 Play the A note
 Limitations of the
equation
Conclusion
Modeling instrument signals
Generating a clarinet note
Problems with modeling an instrument
using a mathematical equation