EDIT: Refer here: http://en.wikipedia.org/wiki/Harmonic and here: http://en.wikipedia.org/wiki/Overtone

More harmonic complexity just gives you a fuller sounding tone. For a great example listen to the tone on Talk to Your Daughter and Brother by Robben Ford.
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(512) Audio Engineering - Custom Pedal Builds, Mods and Repairs
Last edited by mmolteratx at Jun 29, 2009,
well with acoustic guitars that usually refers to the combination of various tone woods used to make the front, back, and sides of the body and the overtones that are generated because of them. Basically when you hit an E note you dont just hear that one E note. Let it ring out and decay for a while and you'll hear it better. There are several E notes and octaves ringing out at the same time and they all work together to form the "main" note/octave. Terms like sparkle, depth, tightness, and various other ones are often used to describe an instruments harmonic complexity as well.

Now let it be known that amps themselves ARE instruments. They add their own harmonic complexities to the tones and overtones that a guitar already generates. There are a lot of factors that go into the harmonic complexities of electric guitars as well as acoustic guitars, but with electrics the factors are a bit different. The wood the guitar is made from(a bit less important with electrics that acoustics because a typical electric is a solid block of wood), the pickups used, the location of the pickup on the body, the quality of cables used to go into the amp, effect pedals, the amp(there are countless sub-categories we could go into here), and finally the speaker. There are more factors in between all of those, but you get the point.

I hope I explained myself enough
What I listen for are pleasing sounding overtones that ring and decay almost separately from the note that you're playing, but still part of the overall sound, and just gives your sound this really live, round and 3-dimensional kind of quality. That's what I think about at least.

Take a listen to these videos. You can really hear the overtones ringing out in these and they almost always sounds like they're on the verge of decaying into feedback. Especially on those clean notes. It's kind of hard to really convey in a video though...


(I simply cannot put into words how badly I want the amp in this video...)