Seeing Tuning – with Lissajous Curves

Lissajous curves are a system of parametric equations describing complex harmonic motion. In very simple terms, they are composed of multiple sinusoidal motions where each axis represents one sinusoid.

In 2 dimensions, if we have a sinusoid on each axis with the same frequency and phase we end up with a straight diagonal line. If the two sinusoids are out oh phase by 90 degrees (like a sin(x) and a sin(y)), we end up with a circular motion in the 2 dimensions.


In my experiments I take multiple sinusoids, tuned to each other based on

  • 12 tone equal temperament (12TET) tuning, popularly used today in all synths/autotunes. (dividing the log spaced octave into 12 equal parts)
  • Just Intonation tuning. (using whole number ratios from the natural harmonic series)

Each tuning system has a different way of breaking down the octave into intervals. Tuning system differences, their historic placement and potential “pros and cons” are a constant topic of essay and debate. I would point you to any of the equal tempered vs just intonation articles online for historic context and music theory background.

In this article, I just visualize the difference between having a whole number rational ratio in just intonation to the transcendental number ratios. It would be great to explain the concept of different tuning systems through this demo (potential future article).


Intervals (2d):

Lets begin with 2d plots of different intervals.

The blue line indicates the lissaojous path for Just Intonation and

The red line indicates the lissajous path for 12 TET




Chords (3d):

The blue line indicates the lissaojous path for Just Intonation and

The orange line indicates the lissajous path for 12 TET























Chaos is a beautiful thing!

I do not claim that one tuning system is superior to the other, but these plots are really beautiful to look at and bring out another way of visualizing a chord.

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