As we saw in module 6, the equation for an FM frequency waveform can be written as

\[ y(t) = A \cos( 2 \pi f_c t + I \sin(2 \pi f_m t)) \]

where

• f_c: The center frequency or "carrier frequency" of vibrato
• f_m: The modulation frequency, or how quickly we're going back and forth around the center
• I: Modulation index: the ratio of the modulation deviation to f_m, which can also be thought of as the amplitude of the inner sine wave
• A: The overall amplitude of the waveform

But we can actually make A and I functions of time, so that the time-varying inner and outer amplitude waveform can be written as

\[ y(t) = A(t) \cos( 2 \pi f_c t + I(t) \sin(2 \pi f_m t)) \]

As Chowning shows in his paper, choosing the right functions for A(t) and I(t) is crucial for creating different instrument sounds. In this module, you will warm up for assignment 2 by designing several such functions, referred to as "envelopes" in this context. Your job will be to look at the plots and to see if you can replicate them in numpy code, sampled at 44100hz. Remember, linspace is your friend!

Part 1: Wood Drum Envelope

The following envelope leads to a wood drum sound when applied both to A(t) and I(t)

Part 2: Brass Envelope

The following envelope leads to a brass-like sound when applied both to A(t) and I(t)

Part 3: Drum-Like Envelope

Chowning describes a "drum-like" sound when the following envelope is applied to both A(t) and I(t)

Part 4: Applying to FM Synthesis

For those who finish early, try to apply some of these envelopes to the FM synthesis equation to create sounds