The first step towards mastering a quadruple turn is having a clean, consistent triple pirouette under your belt. (If you don't, work on that first.) From there, figure out how much force you need to add a rotation without losing control. This will require some trial and error, and it may be much less than you think. As with any pirouette, you need to have solid core and upper back muscles, so that you can turn in one piece; I find that plank exercises help strengthen these areas. Practice balancing in passé, too, pulling up high into the supporting hip as you push down into the floor. And as you spot, try not to tense up. Keep your neck relaxed and allow the rhythm of the music to help you.
Speaking of rhythm, I'm concerned that you don't have time for five to six consecutive quadruple pirouettes during the Odalisque variation's famous diagonal of turns. If you listen to the music, there are only six counts for each set of turns: chassé into relevé arabesque (1-2), fourth preparation (3), pirouette (4-5-6). To squeeze in four rotations, you would need to cut your arabesque and preparation short and turn at a very high speed. You not only risk losing control, but getting behind the music—and, yes, that matters at competitions. Judges would rather see a clean, musical double or triple than a messy quadruple. A better plan might be to save your quad for the diagonal's last pirouette, for an extra-special finish.
Have a question? Send it to Pointe editor in chief and former dancer Amy Brandt at email@example.com.