How To Play All Of The 7th Chords
First, let's look at the 7th chords as they are typically shown in a chord chart. 
Figure 1.  
All of the 7th chords
Now take a closer look at the C#7, D7, and Eb7 chords.  You can see that they are really the same pattern, just shifted up the fretboard, one fret at a time.  This is a pretty simple concept, but it's a really key piece of information, so make sure you've got it. 
IMPORTANT POINT:  If you shift a chord pattern up or down the fretboard, you'll form a new chord.  Of course, the pattern needs to use all four strings, so that the pitch of each string changes by the same amount at the same time.
There is one more chord that's using the same pattern.  Do you see it?  It's the C7 chord.  What makes it look different is that you don't really need to place your fingers on the 2nd, 3rd, and 4th strings the nut on the ukulele (the nut is that slotted piece near the end of the neck that guides the strings onto the fretboard) does that for you!  Here's another way of looking at the C7 chord that makes the pattern a little more obvious: 
Figure 2.  
One pattern used to play 
four different chords
Now take a look at the Bb7 and B7 chords.  It's a different pattern than the one in Figure 2,  but you can see the same principle at work:  a pattern shifted up the fretboard to form a series of chords.  The A7 chord is played using this same pattern. 
Figure 3.  
Another chord pattern
Let's keep going.  The G7 and Ab7 chords show us yet another pattern. 
Figure 4. 
Yet another chord pattern
Finally, look at the E7, F7, and F#7 chords.  Another pattern!  The E7 chord doesn't need a finger on the second string because the nut takes care of that.  The F7 chord's first string (remember, the "first" string is the one on the right in the diagram) is unfretted for a different reason:  you can play the F7 chord with the first string either unfretted or on the third fret.  If the first string is left unfretted, the note you're playing is an "A", which is one of the notes in an F7 chord.  If the first string is fingered at the third fret, you're playing a "C" which is another of the notes in an F7 chord.  Either way works.  Here, we'll fret the first string. 
Figure 5. 
Still yet another chord pattern
IMPORTANT POINT:  All the 7th chords are formed from only four patterns.
If you can memorize those four patterns, and remember where they go, then you can play all twelve of the 7th chords.  Remembering where they go isn't hard we'll show you some easy and practical ways to do that in a few minutes.  But for now, let's continue looking at these patterns. 

Remember we said earlier that there are 72 different 7th chords.  So far, we've got 12 of the 72 chords taken care of.  What about the other 60?  No problem. 
Take a look at the pattern in Figure 2.  You can see that each time you move the pattern up the fretboard, you create a chord one-half step higher up the scale.  C7 to C#7, then C#7 to D7, then D7 to Eb7.  Well, what would happen if you kept on going?  Move the Eb7 chord up one fret, and voila! you've got an E7 chord.  It's formed in a different way than the E7 chord in Figure 5, but it's still an E7 chord:  this is what is meant by a different "position" for a chord. 
Figure 6. 
The E7 chord in two  
different positions
And you can keep going:  move that pattern up one more fret and you'll have an F7 chord, then an F#7 chord, then a G7 chord, and on and on until you eventually run out of frets on your ukulele.  So you see that you can play a 7th chord in any key using this pattern, if you place the pattern at the correct fret. 

Just as you can create a 7th chord in any key using the pattern from Figure 2, you can also create a 7th chord in any key using any of the other patterns.  For example, if you continue to move the pattern from Figure 3 up one more fret past the B7 chord, you'll create a C7 chord. 
Figure 7. 
The C7 chord in two  
different positions
Now, watch this:  take this same pattern (the pattern from Figure 3) and keep going.  Move it up four more frets:  C#7, D7, Eb7, E7.  We've now found a third way to play an E7 chord! 
Figure 8. 
The E7 chord in three  
different positions
Okay, now grab the pattern from Figure 4, and move it up to the ninth fret.  We now have a fourth position for our E7 chord. 
Figure 9. 
The E7 chord in four  
different positions
IMPORTANT POINT:  Any chord can be played using four different fingerings.
Well, we've used all four patterns, so do you think we've formed all the E7 chords?  No way!  If your fretboard is long enough (if you have a tenor uke or a baritone uke), move the first pattern in the series up 12 frets (exactly one octave) and you've got a fifth position for the E7 chord.  Move the second pattern in the series up 12 frets and you've got a sixth position. 
Figure 10. 
The E7 chord  
in six different 
If you had an infinitely long fretboard, we could keep doing this forever, with the set of four patterns repeating every twelve frets (every octave). 

So when you see an E7 chord on your sheet music, you can play any of the chords from Figure 10, or you can even move up and down the fretboard, playing all the different positions while everyone else is strumming the same old dull first position, staring in amazement at the virtuoso that you've become. 

An easy way to visualize how to move among the different positions for the chord is to consolidate everything from Figure 10 onto one fretboard.  To make it clear which pattern is which, let's connect the dots for each position. 
Figure 11. 
All the E7 chords
What you can see from the fretboard on the right in Figure 11 is how these patterns fit together how far you need to move from the first position to get to the second position, etc.  Note that, with the exception of the pattern from Figure 2, there is no space (and also no overlap) between patterns.  That is, the Figure 2 pattern has one blank fret above and below it, and all the other patterns are exactly adjacent to each other.  This sort of observation can help you place your fingers as you move among the different positions. 
TIP: To help remember the pattern sequence, note how far apart the patterns are:  Is there any overlap between adjacent patterns?  Is there any gap between them?
Recall that as you move any pattern up the fretboard, you move the chord up the scale.  Same thing with the whole series of patterns.  Watch: 
Figure 12. 
All the F7 chords
It's the same set of patterns, in the same sequence, just one fret higher up.  And you can keep on going: 
Figure 13.
The 7th chords in different keys
IMPORTANT POINT:  Any of the four patterns can be used to form a chord in any key. 

FOOTNOTE TO IMPORTANT POINT:  It may look like the last three Important Points say the same thing.  That's almost true, but not quite.  They're all looking at the same basic principle, but from a slightly different perspective.  Re-read them and make sure that you understand the differences.

In Figure 13, you can see that, as the pattern sequence moves higher up the fretboard, space opens up at the end for another pattern from the sequence.  You can see this happening as you move from the F#7 chord to the G7 chord, and again as you move from the Ab7 chord to the A7 chord.  But the sequence of patterns is the same, it just starts from a different point within the sequence. 
TIP:  Pick a 7th chord (A7 is a good one to start with), and then practice going up and down the fretboard playing A7 chords in different positions (you'll find the patterns to use in Figure 13).  Once this starts to feel easy, practice with a different 7th chord.  This will give you a feel for how far apart the patterns are, so that you can just jump right in and do this on a real song.  After all, that's the goal here:  to give you tools to enhance your playing in the real world.
Now if you've got sharp eyes, you may have noticed that at least one of the dots in each pattern is highlighted with a white center.  The dot that's highlighted is the root note of the chord.  (The "root note" is the first note of the scale in any key.  So for an E7 chord or an E major chord, or an E minor chord, etc. the root note is an E.  For a C7 chord the root note would be a C.)  You already know, for example, that the second string played open (unfretted) is an "E".  Looking at the E7 chord, you see from the first pattern that the second string open fret is highlighted, indicating that this is the root note of the E7 chord.  Same thing with the second pattern:  the third string, fret 4 is an "E".  This will be useful later when we start talking about where to find any chord on the fretboard.  But for now, let's talk some more about patterns. 
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