Altered dominants, and the tension that strengthens the resolution
The dominant chord with altered extensions — flat ninth, sharp ninth, raised fifth — which increase the tension without changing the function, because each alteration is one more semitone pointing at the arrival chord.
A dominant already has tension: the tritone between the third and the seventh. It can have more. Lower the ninth, or raise it, or raise the fifth, and the chord gains a second note asking to move. These are the altered dominants, and the reason they exist is simple: each alteration is one more semitone pointing at the next chord.
Three common alterations
Over G seventh, the dominant of C:
- Flat ninth — A flat in place of A.
- Sharp ninth — A sharp in place of A.
- Raised fifth — D sharp in place of D.
exists too, with a lowered fifth, and it sounds close to the raised eleventh. All of them keep the B–F tritone intact, and that is why they are still dominants: the alteration touches the colours, not the structure.
Why the tension resolves better
A dominant's resolution is, at bottom, movement by semitone: B rises to C, F falls to E. Each altered note adds one more of those movements.
The A flat of the flat ninth is a semitone from the G, the fifth of the C chord, and falls to it. The D sharp of the raised fifth is a semitone from the E, the third of C, and rises to it. Where the plain dominant had two semitones resolving, the altered one has three or four, and the arrival sounds more inevitable.
Compare the cadence without and with the alteration:
Which alteration to choose
The practical rule is in the melody. If the melody, over the dominant, passes through the A flat, the chord takes a flat ninth; if it passes through the A sharp, or the B flat that is the same key, a sharp ninth; if it passes through the D sharp or E flat, a raised fifth. The right alteration is the one that fits the melody, and the wrong one fights it.
When no melody is dictating, the flat ninth is the most discreet and the most classical, the one of choro and of nineteenth-century harmony; the sharp ninth is the harshest, the one of funk and of rock with jazz in it; the raised fifth pushes upward hardest.
On secondary dominants
Alterations are not the fifth degree's privilege. Any secondary dominant takes the same ones, and in a minor key they are nearly compulsory: the dominant of A minor, E seventh, gains the flat ninth almost for free, because the F natural — E's flat ninth — is a note of the scale.
The V7 of II in C major, A seventh, appears with a flat ninth for the same reason: B flat, A's flat ninth, is a semitone from the A that the D minor chord has as its fifth.
What to take from here
An altered dominant is a dominant with the ninth or the fifth displaced by a semitone, keeping the tritone that makes it dominant. Each alteration is one more note a semitone from the arrival chord, which is why the resolution gets stronger, not weaker. You choose the alteration by the melody, you use it on any dominant, main or secondary, and in a minor key the flat ninth is nearly part of the furniture.