So this may or may not be the right place to talk about this, but it's in regards to a guitar technique, so I'm choosing here >.< I was hoping to find someone who understands the physics behind natural and pinch harmonics. Perfectly fine executing both in my playing, but we are covering waves and sound in physics and I'm noticing quite a few things that could be applied, and I remember that when I was learning the techniques, I started reading about nodes and whatnot but didn't understand any part of why doing these things changes the sound. So here's a couple questions.

Start simple...I think I already know the answer to this one, but when you just pluck a guitar string, is it oscillating in a sin/cos pattern?

2.) With standing waves(which I don't understand entirely, but I feel like they have a lot to do with this), you have this concept called the fundamental frequency where your wavelength, lambda, is essentially twice the length of your string, so you only have 2 nodes, which are the endpoints(I'd interpret this as being the first harmonic if there was a way to perform a NH on a completely open string. Moving on to the 2nd harmonic(1st overtone) you would have 3 nodes, one in the center and one on each end, essentially making it look like one cycle of a sin function. So I'm sitting in class and thinking wait...the 12th fret is about halfway down the string, and one of the 3(?) places you'll get clear natural harmonics on a guitar. It also happens to be half hte number of frets on most electric guitars(that I've used anyway)...I'm unclear whether this is a coincidence but I'm guesing not but it could be.

This is where I'm getting into speculation, and where I hope people can clear things up. The reason I'm guessing it's not a coincidence is that if you move on to the 3rd harmonic, you have 4 nodes, 2 in the middle, cutting the string into 3 segments. Dividing your number of frets by 3 gives you 8, 8 would essentially be the metal fret between 7/8, and 7 is again another place you'll get natural harmonics. 4th harmonic if you do this same algorithm, you find it takes place at 6 and again that would be the fret between 5 and 6 and 5 is another place for natural harmonics. Is this correct, or is there a flaw somewhere making me think this? If it is the case, why do you have to stop at 4th harmonic, is it just because those are the only ones divisible into whole numbers and we can't move the frets on a fretted instrument?

3.) How does all this apply to artificial harmonics?
I thought some more about this, and since the formula (i think its 2A sin(w+x)cos(k+t)) has sin being multiplied by cos, if you graph it as a function of time, it oscilates like a sin function being shrunk and stretched by whatever the cos function is evaluating to in that instance, (depends on t)and since cos will be 0 at pi/2 and 3pi/2 (amongst other places), the functiom has to be 0(the x axis) at some points, and those points are the nodes. Meaning nodes are the places the string isnt oscillating, and these are also the places you can put your finger on the string while it vibrates and it wont mute it. Its making more and more sense to me but i feel like im missing a piece of this.
Last edited by bloodandempire at Feb 19, 2015,
I'm sorry this post ended up so long, I don't really have the willpower to go and cut it down, though. Hopefully it's still useable.

When you play a string, the picking doesn't have a particular note. It creates waves along the string and these, reaching the end and reflecting, form into standing waves based on the frequencies which can sustain long enough to be heard as more than a "twang" (the frequency of the string and its harmonics, basically). Several harmonics will be audible, generally at decreasing amplitudes further from the fundamental. The guitar string, since it vibrates in a standing wave, isn't moving in a sin/cos pattern, but the amplitude of each standing wave, on a graph against time or an oscilloscope, would produce a sine wave.

Now the 12th fret is your halfway point because an octave interval is simply doubling the frequency of the lower note. So your low E is one frequency (I believe it's E3, which is 165Hz), then the E on the second fret of the A string is twice that (330Hz), and the high E string is twice that again (659Hz). The way to double a string's frequency is to halve its length. As it happens, the lowest integer you can multiply a frequency by is also two, and since harmonics need to produce a whole number of standing waves on the string that's the lowest possible harmonic of the fundamental - effectively halving the string length to double the frequency - and this harmonic, of course, produces an octave of the fundamental.

Quote by bloodandempire
This is where I'm getting into speculation, and where I hope people can clear things up. The reason I'm guessing it's not a coincidence is that if you move on to the 3rd harmonic, you have 4 nodes, 2 in the middle, cutting the string into 3 segments. Dividing your number of frets by 3 gives you 8, 8 would essentially be the metal fret between 7/8, and 7 is again another place you'll get natural harmonics. 4th harmonic if you do this same algorithm, you find it takes place at 6 and again that would be the fret between 5 and 6 and 5 is another place for natural harmonics. Is this correct, or is there a flaw somewhere making me think this? If it is the case, why do you have to stop at 4th harmonic, is it just because those are the only ones divisible into whole numbers and we can't move the frets on a fretted instrument?

I think you've made a bit of a mistake here. Harmonics have nothing to do with frets, except that they're similarly positioned. It is the case that frets have to be at certain fixed distances from the nut, proportional to the length of the string, and harmonics likewise. An octave being twice the frequency of the fundamental, it's fretted halfway along the string, and the second harmonic, that octave, also uses a node halfway along the string because it's also doubling the frequency. The perfect fifth, the interval of a power chord, has a frequency one-and-a-half times that of the open string, so it's fretted at the 7th fret, because that's a third of the way along the string, effectively dividing the length by 1.5. Three is, of course, a whole number, so there's a harmonic there too. Then you have the perfect fourth on the 5th fret, where the frequency is divided by 1.25, and which position gives you the two octave harmonic, with five nodes and four anti-nodes (i.e. points at which the wave reaches maximum amplitude).

I know I'm rambling a bit so I did a quick MSPaint diagram to show the relation a bit:

The question about the 4th harmonic is based on the assumption that harmonics stop at the 5th fret - they don't. If you have enough gain, and preferably a bridge pickup, you can find a harmonic at the fourth fret and then in an increasingly tight cluster between the 3rd and 2nd fret. Anything much below that and the harmonic isn't loud enough to be used. There are, theoretically, an infinite number, though. As many harmonics as there are whole numbers to divide the string length by.
Quote by bloodandempire
3.) How does all this apply to artificial harmonics?

An artificial harmonic is just a harmonic on a fretted note, i.e. you've shortened the string length. If you experiment, you'll see that the harmonics still occur 12 frets above the fretted note, then 7 frets, then 5, etc.

Quote by bloodandempire
I thought some more about this, and since the formula (i think its 2A sin(w+x)cos(k+t)) has sin being multiplied by cos, if you graph it as a function of time, it oscilates like a sin function being shrunk and stretched by whatever the cos function is evaluating to in that instance, (depends on t)and since cos will be 0 at pi/2 and 3pi/2 (amongst other places), the functiom has to be 0(the x axis) at some points, and those points are the nodes. Meaning nodes are the places the string isnt oscillating, and these are also the places you can put your finger on the string while it vibrates and it wont mute it. Its making more and more sense to me but i feel like im missing a piece of this.

I'm not entirely sure what the question is here, but if I'm understanding right the point is basically that every harmonic occurs when you play a string, which is why a guitar doesn't just give a single, clear tone. Playing a harmonic is just muting the fundamental and the harmonics louder than the one you want to play. The nodes are, indeed, where a given harmonic isn't oscillating, so you touch the string on a node so that the harmonic isn't muted but the lower harmonics and fundamental are.

That's probably a bit of a TL;DR but here are a couple of useful diagrams off of wikipedia:
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Youre officially uber shit now.

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3d9310rd is far more upset than i

I'm a moron tho apparently and everyone should listen to you oh wise pretentious one
Awesome...I read this through once but itll take a couple more reads Thanks for taking the time to go through all that!

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http://en.wikipedia.org/wiki/Harmonic

There you go, I saved you 10 seconds by Googling it

Nice man Ill alert the media we can close down all the schools because google and wikipedia have it covered.