@sclay Good stuff Shane, and hopefully the next installment illuminates further. 
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The three triad shapes
Hopefully you have been able to take the time to read, scrutinise and digest the information and the diagrams above. From the triple neck diagrams we saw the shapes for Major triads and for minor triads. Here it is again.
The CAGED chord shapes are in the centre. These are the source shapes for the triads we will find within. On the left are the Major and minor triad shapes on the D, G and B strings. On the right are the Major and minor triad shapes on the G, B and E strings. The highlighting shows duplicate triad shapes. If we eliminate those duplicates we will see that there are only three triad shapes for Major triads and three shapes for minor triads on the string sets under consideration.
Major and minor triad shapes on the D, G and B strings
Major and minor triad shapes on the G, B and E strings
That is it.
For the top four strings, there are no other triad shapes on adjacent string sets.
Next we will start to see how these triad shapes lead to the shapes of the 3rds that we learned and began using in Part A of this topic.
Triads and thirds
A triad, within its formulation, will contain two intervals of a third. Let us return to the seven triads (chords) derived from harmonising the D Major scale. We can refer to the note circle once again to determine the nature of the two intervals of thirds within each chord formula.
Remember, four semitones is an interval of a Major third and three semitones is an interval of a minor third.
1] D - F# - A = D Major chord … D to F# = Major third then F# to A = minor third.
2] E - G - B = E minor chord … E to G = minor third then G to B = Major third.
3] F# - A - C# = F# minor chord … F# to A = minor third then A to C# = Major third.
4] G - B - D = G Major chord … G to B = Major third then B to D = minor third.
5] A - C# - E = A Major chord … A to C# = Major third then C# to E = minor third.
6] B - D - F# = B minor chord … B to D = minor third then D to F# = Major third.
7] C# - E - G = C# diminished chord … C# to E = minor third then E to G = minor third
Note the alternating pattern.
Major chord triad formulae contain Major then minor thirds.
Minor chord triad formulae contain minor then Major thirds.
The diminished chord, as usual, does its own unique thing!
This may be easier to see and recognise if the diagrams are placed in a grid pattern.
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From Thirds to Chords To CAGED to Triads to the Shapes of 3rds
We can now revisit the Major and minor triad shapes with the intention of selecting all pairs of thirds in groups of ascending notes that either go from Root to 3 or 3 to 5. For all such pairings the notes have been changed to blue and other notes have been greyed out. A few triad shapes contain two such pairs so all three notes are blue. What we will do here is look for any repeating patterns / repeating shapes on the guitar neck. We want everything to go ‘pair shaped’. 
Pairs of thirds on the D, G and B strings
Pairs of thirds on the G, B and E strings
There may seem to be many pairs of thirds (Root to 3 and / or 3 to 5) on adjacent strings within these triads. But, in fact, on close inspection, we will find that there are actually only two shapes for each pair on any given set of two adjacent strings. Just two.
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Shapes of 3rds on D & G then G & B then B & E strings
In previous diagrams, the notes were shown within the CAGED structure, and the triads were shown from within those chord shapes. So each was labelled with Root, 3 or 5. We arrived at a point where the triads were shown as having either one or two pairs of thirds within. Those pairs of thirds we now view as separate shapes. We also revert to calling them 3rds.
As just mentioned, there are only two shapes on any two adjacent strings.
On the D and G string there is one Major 3rd shape and one minor 3rd shape.
Two and no more. That is all.
On the G and B string there is one Major 3rd shape and one minor 3rd shape.
Two and no more. That is all.
On the B and E string there is one Major 3rd shape and one minor 3rd shape.
Two and no more. That is all.
The 3rds in the diagram shown here deliberately show no intervals. They are not marked as Root or 3 or 5. That is no longer a necessary detail. They are only for illustrative purposes to show shapes on the guitar neck. Shapes that you fingers can make to play and make music.
Also, there are no fret numbers or marks. These shapes are movable. What the notes are, and the nature of the two notes, whilst always being 3rds, will depend on where they are played. And their ‘sound’ will depend on what they are being played over … bass notes / chords etc. More of that to come.
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3rds & Thirds Part C : The Concept of 3rds as Partial Chords
3rds in the key of D … reprise
It is worth repeating … there are only two shapes for 3rds on any given set of two adjacent strings.
At the very outset of this topic, 3rds in the key of D were introduced step by step. Firstly on the B & E strings. Then on the G & B strings. And lastly on the D & G strings. I deliberately chose to miss any 3rds that use the Low E & A strings.
For ease of reference, here are those 3rds in TAB format again.
At the time, they were simply presented as nothing more than shapes to explore musically as the fancy took. They were not ‘rooted’ (excuse the pun) in any knowledge or theory. They were not connected to scales or chords. They seemed to have their own independent existence separate of anything else. They sound good and they are fairly easy to play. What’s not to like? Why bother any further?
Well …
Having explored a fair chunk of the theory that can explain how these 3rds are derived, and having already seen the seven chords that come from stacking thirds in the process of harmonising the D Major scale in Part B, it is time to root those 3rds within the context of the D Major scale and the diatonic chords that it gives rise to.
It is time to root those 3rds within the context of CAGED shapes.
CAGED shapes that become triad shapes.
Triad shapes that contain the magic intervals of thirds.
Thirds that we view as note pairs.
Note pairs that we call 3rds.
And because we are rooting the 3rds within the context of chords, we will see that they can be thought of as partial chords.
The character and name given to these partial chords will, hopefully, be completely obvious in the first instance. But, beyond the obvious lie some hidden facets that may surprise. The 3rds can be very ambiguous, fluid, lead double-lives in their make-up. They can take the character of more than one chord.
3rds as ‘partial chords’
We are going to see our, by now familiar, 3rds following the sequence of chords from the harmonised D Major scale
As we saw earlier, harmonising the D Major scale gives these chords:
D Major, E minor, F# minor, G Major, A Major, B minor, C# diminished.
We are going to see these chords charted only using the D-shape triad forms on the G, B and E strings for now.
Let’s remove one note from each triad and see what we have …
1] C# diminished triad → 3rd … … … … … … 2] D Major triad → 3rd
3] E minor triad → 3rd … … … … … … … … 4] F# minor triad → 3rd
5] G Major triad → 3rd … … … … … … … … 6] A Major triad → 3rd
7] B minor triad → 3rd … … … … … … … … 8] C# diminished triad → 3rd
9] D Major triad → 3rd
Wow. Oh boy. Those shapes are familiar!
See how by simply ‘removing’ the notes on the G string, we have laid out the 3rds that we began exploring way back at the start of the topic?
The note ‘removed’ is the 5 of each chord so we can look at these 3rds as a sequence of partial chords, containing just the root and 3, that track exactly the sequence of chords in the harmonised D Major scale.
Play these 3rds from the ‘tonic’ D at frets 2 & 3 to its octave and you will hear the do-re-mi-fa-so-la-ti-do with a harmony voice of thirds singing above.
Our original TAB of 3rds could be labelled:
Where the chord names are really partial chords, intervals between Root and 3, where the sound of the 3rd is suggestive of the chord.
This exact process of converting triads to 3rds labelled as partial chords (by removing a note on just one string for each) can be done for the sets of 3rds on the G & B strings, using A-shape triads.
Here are those triads as seen earlier.
Once again, by always removing the 5 from each triad (which we find on the D string), we will move from triad shape to 3rd shape.
1] G Major triad → 3rd … … … … … … … … 2] A Major triad → 3rd
3] B minor triad → 3rd … … … … … … … … 4] C# diminished triad → 3rd
5] D Major triad → 3rd … … … … … … … … 6] E minor triad → 3rd
5] F# minor triad → 3rd … … … … … … … … 6] G Major triad → 3rd
9] A Major triad → 3rd
Wow. Oh boy. Those shapes are familiar. Again!
The original TAB of 3rds could be labelled:
Where the chord names are really partial chords, intervals between first and third, where the sound of the 3rd is suggestive of the chord.
This exact process of converting triads to 3rds labelled as partial chords, by removing a note on just one string for each, can be done once again for the sets of 3rds on the D & G strings, using E-shape triads.
Here are those triads as seen earlier.
Once again, by always removing the fifth from each triad which this time is always found on the B string, we will move to 3rd shapes.
1] E minor triad → 3rd … … … … … … … … 2] F# minor triad → 3rd
3] G Major triad → 3rd … … … … … … … … 4] A Major triad → 3rd
5] B minor triad → 3rd … … … … … … … … 6] C# diminished triad → 3rd
7] D Major triad → 3rd … … … … … … … … 8] E minor triad → 3rd
So, so familiar.
The original TAB of 3rds could be labelled:
Where the chord names are really partial chords, intervals between first and third, where the sound of the 3rd is suggestive of the chord.
Putting all three sets of TABs together, with each 3rd labelled as a partial chord, we have:
B and E strings:
G and B strings:
D and G strings:
More on this concept of 3rds as partial chords to follow in part D.
Cheers
Richard
Questions / comments?
OMG my brain hurts, this is really good stuff to learn about!
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Crystal clear… 
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