Thanks kecked. I saw laserium the first time in Denver in 1976 and a second time in OKC in 1977 at 28 years old. Seeing visuals set to music became my favorite thing, and having a background in music, art and audio made it a perfect fit for me.
Using the 4 XR2206 waveform generators in a phase-locked-loop configuration really opened my eyes for how two XY sine waves with integer harmonic relationships, with specific phase angle differences, combine to form visual harmonies.
When discussing harmonics, there must be a reference frequency that we use as the 1st harmonic. Further, lets limit the harmonic range from the 1st to the 10th harmonic. Lets set the 1st harmonic to be 50 Hz, send one phase version of this harmonic to the X axis scanner and another 1st harmonic to the the Y axis 1st harmonic that is exactly 90 degrees out of phase with the X axis 1st harmonic. The scanned image will be a perfect circle. (With a phase-locked-loop oscillator, its phase angle can be set by a 10-turn potentiometer to any desired angle between 0 and 360 degrees and it will lock on that phase angle. If the X and Y signal amplitudes are identical, the scanners will "draw" a perfect circle, 50 times per second.
Now lets add another independent PLL XY signal pair that can be "added" to the 1st XY signal pair such that the 2 X signals are added together and the 2 Y signals are added together to form a single composite XY signal. This is where it gets very rewarding and the harmonic math is quite simple. And lets limit our 2 XY signal amplitudes to be either full scale or half-scale in any combination. e.g. X1Y1 is full scale, X2Y2 is full scale, or X1Y1 is full scale and X2Y2 if half scale, or visa-versa. Lets assume all signals are sine waves.
Here are the most significant X1Y1+X2Y2 summed harmonic relationships that occur within a 1:10 range (these also occur in nature, such as in flowers).
1:2, creates a 2 petal flower or two pointed star
1:3, creates a 3 petal flower or pointed star
1:4, creates a 4 petal flower or pointed star
2:3, creates a 5 petal flower or pointed star
3:5, creates an 8 petal flower or pointed star
7:10, creates a 17 petal flower or pointed star.
Here are some examples from my PLL mixing/modulation console.
What is so awesome now days is that are numerous ways to this and far more complex harmonic signal generation via computer and software.