File:Just Intonation Lissajous Curves

Description
Just intonation is a musical tuning system that uses intervals based on rational numbers. In other words, the ratios of the frequencies of the two notes in each interval in just intonation can be represented by a fraction of two whole numbers.

The thing to notice here is that the curves created by the oscilloscope remain in the same place on the screen. (Note, however, that although they are stationary, they may flicker in interesting ways. This is because the frame rate of the camera and the period of the waveforms are not the same.) The oscilloscope keeps tracing over the same curve because the ratio of the frequencies of the two inputs is a rational number.

This video shows an oscilloscope in XY mode, where one input is a constant sine wave at 440 Hertz (the A above middle C) and the other input is a sine wave that starts out in phase with (hence, at the same frequency as) the first input, but abruptly changes frequency every three seconds. Specifically, the two tones start in unison, and at the end of every third second after the sound starts, the second input moves up in frequency to the next note in the chromatic scale until the interval between the two tones is an octave. Essentially, one input moves the green dot on the screen left and right (parallel to the x axis) while the other input moves the same dot up and down (parallel to the y axis) all at the same time to produce a two dimensional image. The inputs oscillate (go back and forth) so fast (at least 440 times a second in this video) that you see a single curve instead of a moving dot.