Definition and purpose of chords
A chord is a harmonic structure, a set of two or more intervals. Two notes sounding simultaneously make an interval so a chord consists atleast of three different notes. Any three-tone chord is called a triad. The three tones of a triad are the root, the third and the fifth. The root is the main chord tone. It is the note on which the triad was built. The third determines the quality of the chord, so does the fifth. The function of a triad depends on the scale degree on which it was built. So the triad built on tonic will represent the tonic function. If the root of the triad is dominant it will represent the dominant function etc.
There are four quality types of triads. They are major, minor, diminished and augmented. Chords also vary by their stability. The major chord is the strongest and most stable because of the major third and the perfect fifth. The minor chord is quite stable because of the perfect fifth. However, the minor third is less stable interval than the major third. The diminished and the augmented chords are weak and unstable because of their fifths. Instead, of perfect consonances they are dissonant: diminished fifth and augmented fifth.
The major chord includes the major third and the perfect fifth: C + M3 + P5 = C maj.
The minor chord includes the minor third and the perfect fifth: A + m3 + P5 = A min.
The diminished chord includes the minor third and the perfect fifth: B + m3 + d5 = B dim.
The augmented chord includes the major third and the augmented fifth: G + M3 + A5 = G aug.
Natural C major chords
There are three major triads, three minor triads and one diminished triad in natural major.
Natural A minor chords
Natural minor also has three major triads, three minor triads and one diminished triad.
Harmonic A minor chords
But in harmonic minor, there are two diminished triads, three major triads, and two minor triads.
If you add another third above a fifth of a triad you will get a seventh chord. Seventh chords also have several quality types which are determined by their seventh. Here are some of the most useful of them: dominant (it is the most commonly used seventh chord), minor, diminished and half-diminished.
The dominant seventh chord includes the major third, the perfect fifth and the minor seventh: G + M3 + P5 + m7 = G7
The minor seventh chord includes the minor third, the perfect fifth and the minor seventh: D + m3 + P5 + m7 = D min7
The half-diminished seventh chord includes the minor third, the diminished fifth and the minor seventh: B + m3 + d5 + m7 = Bø7
The diminished seventh chord includes the minor third, the diminished fifth and the diminished seventh: G# + m3 + d5 + d7 = G#o7
Roman numerals and inversions
In harmonic analysis chords are marked by Roman numerals to distinguish on which scale degree the chord was built. Capital Roman numerals symbolize major chords. Lowercase Roman numerals symbolize minor chords. Lowercase Roman numerals with o - diminished chords. Capital Roman numerals with + - augmented chords. Roman numerals with 7 - seventh chords. If the root is not the lowest pitch of a triad it is called an inversion. When the lowest pitch is the third, the chord is in the first inversion. And when the lowest pitch is the fifth, the chord is in the second inversion. Seventh chords have also the third inversion. It appears when the seventh is the lowest pitch of a chord. Chords inversions have superscripts in their Roman numerals. For the first inversion, it is 6. For the second inversion, it is 64. For the first inversion of the seventh chord, it is 65. The second inversion of the seventh - 43. The third inversion - 2.
C major chord (C maj), C maj6, C maj64
A minor chord (A min), A min6, A min64
B diminished chord (Bo), Bo6, Bo64
G augmented chord (G+), G+6, G+64
Inverted dominant sevent
G dominant seventh chord (G7), G65, G43, G2
Inverted minor seventh
D minor seventh chord (D min7), D min65, D min43, D min2
Inverted half diminished
B half diminished seventh chord (Bø7), Bø65, Bø43, Bø2
Inverted diminished seventh
G# diminished seventh chord (G#o7), G#o65, G#o43, G#o2