The Short Version

Using barre chords, you can easily play any guitar chord just by memorizing a small number of them. With the barre chord, you simply mimick a chord higher up the neck by using your index finger as the nut by barring all the strings. Most commonly this is done with the E and A chords:

 E     A
-0-   -0-
-0-   -2-
-1-   -2-
-2-   -2-
-2-   -0-
-0-   -X-

By using your index finger as the nut, you can make this shape with your other fingers and basically shift the key up. For example, if you shift E up one, you get an F which looks like:

-1-
-1-
-2-
-3-
-3-
-1-

You can keep moving this up the fretboard, and the chord you get is whatever the bass note is (i.e. the note on the first string for an E shaped barre chord, or the second string for an A shaped barre chord). So, in order to play the chords, you'll need to know the notes of the first two strings, so let's just review (you do know the chromatic scale, don't you?) up to the 12th fret (they just repeat, and remember that a sharp is the same as a flat for the next note, i.e. A# - Bb).

A  A#  B   C  C#  D  D#  E  F  F#  G  G#  A
E  F   F#  G  G#  A  A#  B  C  C#  D  D#  E
0  1   2   3  4   5  6   7  8  9  10  11  12

Now comes the fun part. With the above information, you can play any major chord. But there's a lot more chords out there than that! So as an easy reference, I'll show some of the basic shapes for some of the more popular chords. These are just the shapes of the E and A chords, but I'll show them as F and Bb chords which are the first barre chord on the neck.

E Shaped Chords

 F    F7    Fm    Fm7   Fmaj7  Fsus4  F7sus4
-1-   -1-   -1-   -1-    -1-    -1-    -1-
-1-   -4-   -1-   -4-    -1-    -1-    -1-
-2-   -2-   -1-   -1-    -2-    -3-    -3-
-3-   -3-   -3-   -3-    -2-    -3-    -1-
-3-   -3-   -3-   -3-    -3-    -3-    -3-
-1-   -1-   -1-   -1-    -1-    -1-    -1-

A Shaped Chords

Bb    Bb7   Bm    Bm7   Bmaj7  Bsus4  B7sus4
-1-   -1-   -1-   -4-    -1-    -1-    -4-
-3-   -3-   -2-   -2-    -3-    -4-    -4-
-3-   -1-   -3-   -1-    -2-    -3-    -1-
-3-   -3-   -3-   -3-    -3-    -3-    -3-
-1-   -1-   -1-   -1-    -1-    -1-    -1-
-X-   -X-   -X-   -X-    -X-    -X-    -X-

Now, these are by no means all the chord shapes, but they are the most common ones. For the rest, you can get the shapes from The Everything Guide to Guitar Chords. Armed with this knowledge, you can now play any guitar chord your heart desires! As an added bonus, because the shapes for these chords are all similar and you can slide up and down the neck, it makes changing chords fast and easy. Enjoy!

The quickest way to start increasing the number of chords/inversions you have at your disposal is a simple two-step process. First, take any chord you already know (start out with simple chords, like a major triad), and have a look at each of the tones played when the chord is struck. For example, your standard first position C chord is composed of the following:

---
-1- C (R)
-0- G (5)
-2- E (3)
-3- C (R
---
C Major
Secondly, and lastly, slide each note up to the next highest tone in the chord (alternately, you can move them all down the to the next lowest tone -- but not if the chord is in first position as in our example). Now, aks yourself: can you string all those tones together into a chord and play it? If so, you've got yourself a new inversion for your chord! Do this as many times as are useful per chord. Sliding our C Major chord up the neck gives us the following inversions:

-----------------
-5---8----13--
-5---9----12--
-5---10---14--
-7---10---15--
---------------

C/E*  C/G   C
Repeat this for every chord you know, practice playing chord progressions with the new chords for an hour every day, then after a couple months go look up more complex chords and do the same thing with them.

* This notation refers to a C chord with an E in the bass. This can be used for inversions (as in this case), or for non-chord tones, such as in A/F#

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