Abstract:

More than 2,500 years ago, Pythagoras realized that musical instruments sounded well together if the frequencies they produced were in simple ratios (such as 2:1 or 3:2). In fact, the concept of frequency did not exist then, but Pythagoras measured other parameters of musical instruments (e.g. flute length or the tension applied to string instruments) where he did find those simple ratios. In the XIX century, a French mathematician called Jules Lissajous described a family of curves made by combining two sinusoidal oscillations. He used tuning forks to produce those oscillations, and a beam of light to draw the curves on a dark screen, but just as Pythagoras many years before, he also found that when the frequency of the oscillations had a simple ratio relationship, the result (in this case the graphic result) was harmonious. A few years after Lissajous´ experiments, a device called harmonograph was described and built. The harmonograph combined two or three pendulums to produce astonishing graphics related to Lissajous curves and it also showed that harmonious and symmetrical results were produced when the frequencies of the pendulums were in simple ratios. In order to explore, understand and explain these and other music related concepts in the classroom it would be interesting to simultaneously produce sonic and graphical results when combining sinusoidal curves. This has already been done by combining a traditional harmonograph enriched with a variety of sensors and contact microphones with a modern modular synthesizer. Although the results are remarkable, this solution is not optimal for classroom use due to complexity and cost issues. We propose the use of VCV Rack, a software modular synthesizer, to simultaneously produce sonic and graphic output and thus explore the above and other related ideas. In this paper we will use VCV Rack to illustrate concepts such as beat frequency, combination of sinusoidal waves in simple ratios with and without attenuation or the generation of algorithmic rhythms.

Keywords:

Lissajous curves, harmonograph, harmony, Euclidean rhythms.