This is just ratios. Say you just have two of these, one that does 65 swings a minute, the other 70 swings a minute. Here a swing is just HALF of a period (one extreme to the other). When will they create some kind of synced pattern?

They'll make a noticeable pattern when both are at some outstanding position at the same time, probably when both are at an end. This happens when both have made an integer number of swings. After T seconds, the first will have T\*65/60 swings and the second will have T\*70/60 swings. Or, simplified, 13T/12 swings and 7T/6. When will both of these be integers? Whenever T is a multiple of 12, so every 12 seconds, they'll both line up and it'll look cool.

If the numbers were different, say 65 swings and 67 swings, then the times would be different. If you had more, then there are more combinations that can line up creating different patterns and, eventually, they would all line up and reset. This is what is in the clip.

Fourier Series are a more complicated combining of things like this, rather than just a comparison like this.