Sound Synthesis

The overall sound quality of an instrument, or timbre, is shaped by a variety of factors. These include the mixture of frequencies present in a sound wave, as well as the wave’s intensity over time.

Additive synthesis builds this mixture by adding pure sine waves together:

x(t)=A_1\sin(2\pi f t)+A_2\sin(2\pi(2f)t)+A_3\sin(2\pi(3f)t)+\cdots

The amplitudes $A_1, A_2, A_3, \ldots$ shape the harmonic content of the sound. The pitch of a particular note is largely determined by the fundamental frequency $f$ , but the higher harmonics (aka overtones) $2f, 3f, ...$ also effect the sound quality.

Below, you also have access to the sound’s ADSR envelope (attack, decay, sustain, release). Together, these shape the volume as a function of time for each note played.

Additive Synthesizer

Shape harmonic amplitudes, play chords on the piano keyboard, and compare the live waveform with a spectrogram.

What the Graphs Mean

The composite waveform shows one repeating cycle of the recipe you are shaping. The live waveform uses a fixed scrolling speed and fixed display scale, so pitch shows up as spatial spacing. The spectrogram turns that same sound into a moving frequency picture: lower notes sit near the bottom, higher harmonics stack above them, and brighter colors mean stronger frequency components.

Problem Solving

Harmonic Frequencies

Problem

A note has fundamental frequency $f = 220\ \text{Hz}$ . Find the first four harmonic frequencies.

Solve

Harmonics occur at integer multiples of the fundamental:

f_n = nf.

The first four are $220\ \text{Hz}$ , $440\ \text{Hz}$ , $660\ \text{Hz}$ , and $880\ \text{Hz}$ .

Reading a Spectrogram

Problem

A spectrogram has bright bands at $180\ \text{Hz}$ , $360\ \text{Hz}$ , and $540\ \text{Hz}$ . What fundamental frequency do these bands suggest?

Solve

The bands are integer multiples of $180\ \text{Hz}$ :

180,\quad 2(180),\quad 3(180).

The likely fundamental is $180\ \text{Hz}$ .

Sound Synthesis Checkpoint1 of 3

What does the fundamental frequency mainly determine for a musical note?