Once you've practised the E Minor chord a bit you can move on to another chord. One that goes well with E Minor is called A Minor.
This is the diagram for A Minor. To play it you need to try and get your two middle fingers (2 & 3) onto the second fret of the two middle strings (3 & 4). As if this isn't difficult enough, you need to then try and get your first finger onto the first fret of the second string (remember that the first string is the thinnest and the sixth is the thickest). Don't be put off if you can't do it straight away, the first few times are always impossible but it gets easier fast if you don't give up.
Pick through all the strings one at a time to make sure all the notes sound clear. If an open note sounds bad check that nothing is touching it, or if one of the fretted notes (the ones with fingers on) doesn't sound right try and get your finger closer to the fret in front of it and press a bit harder if you need to.
The quickest way to learn any chord is to make the shape over and over again, taking all your fingers off the strings and being careful that you're putting them back in the right place. If you try and do that for a couple of minutes, a few times a day, you should soon be able to make the chord quite quickly, without too many mistakes.
Once you can make the A Minor shape without too much difficulty try changing from A Minor to E Minor and back again. If you keep your second finger where it is (on the 2nd fret of the 4th string) it makes moving from one chord to the other a little easier. Remember to go really slow at first, and concentrate on getting it right instead of doing it quickly.
As unlikely as it might seem when you're sitting with a guitar trying to do this stuff, you'll learn much faster and better if you take it really, really slow. It's always tempting to play everything as fast as you can, but if you don't start off playing slowly and carefully you'll learn much slower and all the mistakes you're making - like missing the string or fret that you're aiming for - will become bad habits that are difficult to shake off.
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