"[O]ne of the first forms of melodic combination [was] a system in which one melody was paralleled by another melody removed from it by intervals, which were most frequently, but not exclusively, fourths and fifths."
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I decided to check this out. You can do it too.

Step up to a keyboard and put your thumb on C. (Use your right hand.) Put another finger on F. (This is a fourth apart.) Now keep your hand in the same postion and move it from note to note, making a simple tune. It will sound ancient. I found that I needed to play a fifth after a while, to end it.

I have heard this called "parallel fourths." Parallel fourths are illegal now. :)

Puttings a fifth apart. (C and G) wasn't as satisfying, but perhaps I need to make up some other tune.

Try it.