The YouTube video illustrates the effect of a series of five tones generated by oscillators ranging from 500 to 2500 Hz on the graph displayed. (The frequency range on the horizontal axis is from 0 to 3000 Hz.)
I understand about harmonics -- I did take complex analysis. :~) But your plot didn't have equally spaced single spikes, like a comb. It had clumps of about four consecutive large values, then a dip, then another clump. I'm trying to understand that. If it were a 12-string guitar, say, not exactly in tune (perhaps on purpose), then I'd expect wide peaks like that. For a single string, I expect sharp peaks.