Altered dominant chords
Altered dominant chords feature either an augmented or diminished fifth. Augmented fifths are indicated in analysis by “+” beside the Roman numeral. Diminished fifths are indicated by a “o” beside the Roman numeral.
Typically, raised fifths resolve upward by step, while lowered fifths resolve downward by step.
Note that the augmented triad is a symmetrical chord than can be interpreted in multiple ways, making it difficult to identify its root without proper surrounding context.
Extended dominant chords
Extensions can be added to dominant chords to create new and interesting sonorities. These chords are typically found only in root position.
When composing these chords in a four-voice texture, you need to decide which notes to leave out. These chords will always include the root and the chordal seventh.
The V9 chord replaces a doubled root with a ninth. The ninth should resolve down by step.
The V11 chord replaces the third with an eleventh. The eleventh “resolves” by common-tone. This chord typically includes both the ninth and the eleventh, and resembles a IV chord with scale-degree 5 in the bass.
The V13 chord replaces the fifth with a thirteenth. The thirteenth “resolves” by leaping down by third to scale-degree 1.