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How to analyze triads and more advanced chords?

The starting point is the major triad, in the example in C major, but this concept is valid for all keys, as usual.

The C major chord is built with these three notes:

C  E  G    

As we said this triad is built with the Root (C) the 3^rd (E) and the 5^th (G) of the major scale. Also, if we calculate the intervals between the Root and the other two notes we notice that there is an interval of a major 3^rd between C and E and of a perfect 5^th between C and G.

So if I wanted to write a formula for the major triad I would write

C   E  G

1   3   5 (Root-Major third-Perfect fifth)

If now we want to find the chord C minor all we have to do is lower the 3^rd of the chord (E is lowered to Eb)

So now the triad for C minor is

C  Eb  G

1  b3  5  (Notice how the formula changes Root –Minor Thirds – Perfect fifth) 

From this I can tell that the difference between a major and minor chord is in the 3^rd.

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The diminished and augmented triads can be told from the 5^th.

If C major is C E G

C augmented is    C  E   G#  (I have raised the 5^th of a halfstep)

    Formula             1   3   #5   

C diminished is     C  Eb Gb   (a minor triad with the flattened 5^th)

    Formula             1   b3  b5