Thank you very much for the Wiki article, Freddy Headey. It seems that tension is being used interchangeably with pitch or tuning. I see that now.

I think the article would be improved were it to address frequency and vibration because it make the understanding of why fretted instruments have various difficulties so much easier to get a grip on.

Let's take a string and tension (tune) it to A 440. That string will vibrate at 440 times per second thus producing an A note. The same thing will happen with anything that vibrates at 440 times per second. A column of air in a slide trombone, a drum head, a piece of metal, a violin string, a guitar string--virtually anything.

Suppose the guitar string in question is 24" in length from the nut to the top point of the saddle and is tensioned (tuned) so that it vibrates at 440 times a second. The note that sounds will be an A, no ifs, buts or maybes.

http://onlinetonegenerator.com/

That link is easy to use for this. When the frequency is 440/sec, an A note will be the result. If we half the string length which is what we do by playing the string at fret 12, we will get a frequency of 880/sec, and if we half it again (with say a steel knife), we will get a frequency of 1760/sec. They are all A notes. In theory we could do this to infinity, but because strings have a diameter, in real life we can't. Besides, we couldn't hear the results anyway. Now, set the frequency to 220/sec (that is, double the length of the string that produced an A 440) and it vibrates at half the rate but it still produces an A note. Keep in mind that calling a string that vibrates at 440/sec an A was a matter of choice. It's simply a convention used by musicians around the world. A at one time not so long ago was the note produced by a string that vibrated at 432/sec and that was accepted by many musicians around the world. Things have changed. Clear so far I hope. There are things that can alter the notes we hear: things that slow or speed up the sound wave--listen to a passing train when the whistle is blowing. The note it generates will drop as it speeds away from us. That's called the Doppler effect. Intrinsically, we know the train is making the same note, but because the sound waves that reach us take longer to reach us as the train moves on down the line, the note drops in frequency to our ear.

Frets simply make hitting notes easier. (It's hard to slip into a D13 b5 b9 on a fretless guitar.) Measure from the nut of your guitar to the saddle (that's the white piece of plastic, ivory or steel that sits on the bridge). Make note of the distance--lets say it's 24" (makes the math easier). Next, measure from the nut to the 12th fret. You should get a measure of 12". If the 24 inch string give an A note, so will the 12" string, but it will be an octave higher. (We accept that we haven't changed string tension/tuning in the interim.) Frets are placed where they are based on math.

The string tensioning when the fingers get into it is another potential problem source. Pressing a string harder results in increased note frequency—because the rate of vibration per second increases, thus making the note sound higher/sharper. A vibrato arm does exactly that by increasing or decreasing string tension. The neatest sound (it's so rare to encounter) is the sound generated by a high tension wire in 40 below zero. Make that wire vibrate and it will generate a frequency and we'll hear some sort of note. If the wire is vibrating at 440 time a second, we meet our old friend A again.

Anyway, I'll shut up now. Sorry to rattle on :-)