Several weeks ago I started experimenting with different tuning systems. Just about the same time, I happened upon some material on Lissajous curves: patterns which are formed by plotting two audio channels perpendicular to one another on the x and y axis. Part of the reason I find them so interesting is that they link in so directly with previous work I’ve done using co-prime polyrhythms as background rhythmic structures. By projecting low-integer tuning ratios such as 3/2, 4/3, 5/4, and so forth you can really see the relationship between the two frequencies reflected in a visual way, similarly to the sine-based polyrhythmic graphs I was making a few years ago to compose pieces like Border Sea and Perfect Information. Of course, the difference here is that you can see the resultant patterns of more complex ratios as they evolve in real time. In the video below you can see an X/Y plot of eight frequencies, triggered sequentially in pairs. The interval ratios used are all in 5-limit just intonation except for the major third, which is in ¼ comma meantone temperament. Interestingly, it is this one M3 ratio which causes the beautiful motion seen from the second set of pitches in the video:

eight frequencies.

Even more interesting: if you look closely, you notice that the motion of the graph is actually a 2D projection of what could be a slowly rotating 3D object. This realization lead me to the motion of double pendulums, which seem to have a similar ‘look’, aside from the fact that their motion is chaotic whilst these lissajous graphs are totally deterministic. In the next video, you can again see the X/Y plot of eight frequencies, this time with discrete amplitude modulations on the four pairs of two. The amp modulations are quite slow, at 0.11, 0.13, 0.14, and 0.17 Hz respectively for each pair of frequencies. Again, the beautiful, tangled rotating motion is caused solely by the presence of the one tempered ratio, the major third in ¼ comma meantone:

eight frequencies, modulating amplitudes in pairs.

Back in 2001, cyclo. (Ryoji Ikeda and Carsten Nicolai) did a record using lissajous curves as a focus point for their audio:

cyclo. id#00 (Carsten Nicolai & Ryoji Ikeda).