Computer Music: Musc 216
Creating a Square Wave
Using Additive Synthesis Techniques

In this tutorial you will used SYD to create a SQUARE WAVE by combining 6 SINE waves. This is a demonstration of the technique described in the tutorial, Additive Synthesis. Using the application, SYD, you will need to add 7 operators: 6 Oscillator operators and a Mixer operator (1 operator).

If you have not already done so, please read the tutorials, Additive Synthesis.

A SQUARE WAVE can be created using the NATURAL HARMONIC SERIES. However, only the ODD HARMONICS ARE USED: The Fundamental, the 3rd harmonic, the 5th harmonic, the 7th harmonic, etc. In addition, the respective amplitudes are inversely proportional to their partial number. For example, if the Fundamental has an amplitude of 1, the 3rd partial will have an amplitude of 1/3, the 5th partial will have an amplitude of 1/5.... etc.

Using a Fundamental Frequency of 100 Hz, here is a formula for creating a square wave using 6 harmonics. Combine sine waves with these frequency ratios:

OPERATOR
Frequency
Amplitude
6th wave
1100
1/11
5th wave
900
1/9
4th wave
700
1/7
3rd wave
500
1/5
2nd wave
300
1/3
1st wave
100
1.0

Figure 1: Frequency Ratios of a Simple Bell


Here is how the SYD patch should be organized with 6 Oscillator operators and 1 Mixer operator. Set the Frequencies and amplitudes in the Oscillators according to the chart in Figure 1 above:

Figure 2: 'Recipe' for a Square Wave from 6 Sine Waves

Click the 'Synthesize' button to create the sound and then click 'Play' to hear it. To see a graph of the sound, choose GRAPH from the OPTIONS pull-down menu:

Figure 3: Graph of a Square Wave created from 6 Sine Waves

Here's a 3-D SPECTRUM of this synthesized SQUARE WAVE for 2 seconds with a FUNDAMENTAL and 5 OVERTONES:

Figure 4: Spectrum of a Square Wave created from 6 Sine Waves. Compare with the Spectrum of a REAL Square Wave.

The fundamental was 100 Hz; 2nd partial was 300 Hz; 3rd partial = 500 Hz; 5 partial = 700 Hz, etc. This graph has the frequencies marked in kHz, so 100 Hz = 0.1.

Here's a 3-D SPECTRUM of a 'real' SQUARE WAVE for 2 seconds. Note that the SPECTRUM is much more complex, showing many more OVERTONES:

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Figure 5: Spectrum of a 'real' Square Wave showing many more harmonics (partials)

Further Experimentation

Try adding more Oscillators to this Square Wave patch to extend its spectrum.

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