Before going into chords, it would be a good idea to go back (if needed) to intervals and scales.
Chords are notes being played at the same time, three notes or more. The simplest chords are chords built using 3rds (as in intervals), they are called triads. We are going to look into these first before getting deeper into chords and harmony.
Major Triad
The major triad consists of a root, a major 3rd and a perfect 5th. To build it, we consider the major scale rooted at the chord pitch and get the major 3rd and perfect 5th (just like we did when looking for intervals). A major triad is referred to by the name of the root.
Let's look at getting the F major triad:
1_2_3^4__5_6_7^8
F G A B♭ C D E F
The major 3rd is A and the perfect 5th is C, which gives us F-A-C for the F major triad (F).
Minor Triad
The minor triad consists of a root, a minor 3rd and a perfect 5th. A minor triad is referred to by the name of the root followed by 'm'.
If we look at how we obtained the F major triad, all we need to do is diminish the major 3rd by a half-step to get the minor 3rd, A♭. The F minor triad (Fm) is then F-A♭-C.
A minor triad could also be obtained using the natural minor scale with the same root as the chord. In that case, we would need to consider the root, 3rd and 5th degrees.
Diminished Triad
The diminished triad consists of a root, a minor 3rd and a diminished 5th. A diminished triad is referred to by the name of the root followed by 'dim' (or °).
If we look at how we obtained the F minor triad, all we need to do is diminish the perfect 5th by a half-step to get the diminished 5th, C♭. The F minor triad (Fm) is then F-A♭-B. Remember that there is a natural half-step between B and C, and therefore, if we diminish C by a half-step, we get B. We could have of course looked at the F major triad and diminish both the major 3rd and perfect 5th degrees.
Augmented Triad
The augmented triad consists of a root, a major 3rd and an augmented 5th. An augmented triad is referred to by the name of the root followed by 'aug' (or +).
Again, if we look at how we obtained the F major triad, all we need to do is augment the perfect 5th by a half-step to get the augmented 5th, C#. The F augmented triad (Faug) is then F-A-C#.
Suspended Triad
The sus4 chord consists of a root, a perfect 4th and a perfect 5th.
The sus2 chord consists of a root, a major 2nd and a perfect 5th.
As you have probably noticed, those are not really triads since they are not built using thirds.
Inversion
The way the notes composing the chord are arranged is called voicing. When the root of the chord is the lowest note, the chord is said to be in root position.
If the root is raised by an octave, the 3rd degree note (or 2nd note in the chord construction) becomes the lowest note of the chord. This is called first inversion.
If the root and 3rd are both raised by an octave, the 5th degree note (or 3rd note in the chord construction) becomes the lowest note of the chord. This is called second inversion.
A chord inversion is indicated by adding to the chord name a slash '/' followed by the lowest (bass) note.
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