
Originally Posted by
NewGuyonGuitar
If I start on the low E string, 5th fret. I'm on an A. I now know if I move down 2 strings and up 2 frets, I'm on another A that's an octave away from my starting point. When I alternate between these two notes, I hear a difference. Compared to each other, one sounds higher pitched than the other and the other one sounds deeper, lower in pitch. I'm having a heckuva time discerning that these two notes sound the same on some level. Those A's should sound similar in some way. I can only hear the difference.
Compare the lower A with other notes either side of the higher A - a fret or two higher or lower. You should hear that those notes (G, G#, Bb, B) are much
more different from the low A than the high A is.
So it's not so much that you need to hear the two As as exactly "the same thing"; just that - as two different notes - they are much
more similar than notes of different pitch classes.

Originally Posted by
NewGuyonGuitar
I don't know what word is used to describe this similarity. Is it Timbre?
As speechless says, both As are the same "pitch class". But timbre is actually a relevant term here. The timbre of a musical note is a result of its combination of frequencies.
The open A string has a "fundamental" frequency of 110 cycles per second, or Hertz. But it also contains many partial vibrations, most of them (its "harmonic series") multiples of 110: 220, 330, 440, 550, etc. These are all progressively fainter. But its not a gradual slope. A guitar string sounds the way it does (ie different from a piano A of the same frequency) because its frequency
spectrum is different.
When it comes to octaves on the same instrument, then the open 5th string has the harmonic series 110-220-330-440-550-660 etc, while the A an octave higher is 220-440-660-880- etc. So you can see both how different the two notes are, but also how closely related they are: the whole harmonic series of the higher A is contained
within the lower one. And that's kind of the feeling you should get from listening to it: there is no aspect of the higher A that is not part of the lower one. Obviously we can't hear the frequency figures (!), but they blend perfectly. It's like the higher A is the "offspring" of the lower one; but with the different emphasis caused by the fact that we hear the 220 as the strongest frequency in the higher one, while the 220 harmonic of the lower one is less obvious.
When it comes to perfect 5ths they relate by a factor of 3 to the lower note. So the E on open 1st string is 330 Hz, 3x the frequency of the open A (that interval is a 12th, octave + 5th). Its harmonic series will run 330-660-990- etc. As with A=220, all these harmonics are contained in A=110; but the relationship is a little more distant or removed. This is how we hear a 12th as just a fraction less consonant than an octave.
When we lower the 12th by an octave to form a perfect 5th with low A (E at fret 2 string 4), the ratio is 3:2. When A=110, E=165. This is a little less consonant again than a 12th, because 165 is not part of the harmonic series of 110. But it's still a very "pure" sounding interval, because a lot of harmonics are shared between the notes.
(BTW, to be precise, the E is not 330, it's 329.6. This is because, in order to make all half-steps equal, we need to "temper" the scale by retuning notes slightly. You can't hear the difference between 330 and 329.6, but the latter is necessary to make the math work. Google "equal temperament" if this interests you.)

Originally Posted by
NewGuyonGuitar
I then alternated between my starting A, and a perfect 4th, or the D on the string right below, same fret. I can also hear that these notes sound different, one is deeper/higher pitched then the other.
The A and the D should sound different which I do hear, but I can't describe how they sound different except in terms of highness/lowness of pitch to each other.
The purpose of the exercise is be able to hear better what's going in music, and to be able to reproduce certain sounds.
Eg, you need to recognise the sound of an octave, or perfect 4th, or whatever, in order to be able to
play it. You don't need to characterise it any way - you don't even need to know its correct name; although of course both things can help.
One of the most popular ways of training yourself to hear intervals is with well known songs which begin with the interval in question. Eg, for a perfect 4th, the wedding march ("here comes the bride..."), or Auld Lang Syne. For a perfect 5th, you could choose "Twinkle Twinkle". A good octave one is "Somewhere Over the Rainbow".
You can find various lists on line, such as:
http://www.earmaster.com/intervalsongs/
So when you hear an unknown interval, you can hopefully match it to the start of one of those tunes.

Originally Posted by
NewGuyonGuitar
So I found some online ear training web site. It will play any 2 tones and ask me to guess and figure out if how far apart they are in terms of intervals.....
Is this the way to train my ear? Listen to these tones over and over until something finally clicks?
That's one way, but I wouldn't say it's the best way. I recommend working with real music as much as you can - because that's the point of the whole exercise after all!
(If you were training to be a runner, you'd simply go running as much as you can. You might use a gym now and then, if you couldn't go on a run for some reason, or to focus on something specific; but you wouldn't regard the gym as the
best training.)
Lastly, here's a bit of fun:
http://www.youtube.com/watch?v=nl2d4zS56cY
"The most important rule in music is you must always avoid discord. Use dat chord instead."