Females usually give birth every two to three years, because pregnancy lasts 11 months and is followed by up to a year of nursing and care for the developing infant. Although most females with calves don't necessarily participate in mating, Maui researchers find that some do, and males therefore court mothers with young as well as solitary females. But for all an escort's devoted attention, Jones rarely sees any cow showing the slightest interest in return. These slow-moving, undemonstrative females can be tougher research subjects than the splashy groups in which a female is pursued by a crowd of males. Jones spends hours trying to stay near a quiet female (and her calf and escort, if present) and document every move she can see. It's not uncommon for her subjects to slip away into the boundless blue before she can determine a pattern. More often, she will have almost pieced together a pattern when it is interrupted by the arrival of a new male or small band of them, which generally causes the female to flee. Either way, Jones has to start over.

Jones's typical day begins shortly after dawn. Several sunstruck hours later, everyone aboard is operating with a partly heat-melted brain. The trade winds have strengthened, and her little boat is sloshing around between white-capped waves. Time to turn for home, but Jones will be saying, "We were so close before those incoming males wrecked the last session. Let's try just one more two-hour follow." And on it goes until the team pulls into the harbor late again, fully roasted, basted in salt spray, and making plans to do it all again the next day, always in the hope of one shining moment of insight.

One afternoon, 50 feet (15 meters) deep in dazzling azure, a baby humpback rests tucked beneath its mother's flipper, then moves to nestle in its next favorite place: under her throat. A second adult hovers close by. Its darker skin, scratched and scarred, suggests that it is a male.

Jones cuts the boat engine and maneuvers her vessel into position above the whales and slightly to one side. Jason Sturgis of Whale Trust drops quietly off the transom with snorkel gear. A quarter mile (0.4 kilometers) away, a second boat lowers a loudspeaker into the water and begins to play a sample of the very unmusical noises—grunts, glubs, rumbles, sneers, and whines—made by males in competitive groups. Sturgis records the female's response with a video camera.

Since females without calves are the ones most likely to breed, they ought to react differently from females with calves, Jones theorizes. To prove it, she will need this combination of calm water, smoothly functioning equipment, and approachable whales for many more such experiments. During the few I witness, neither type of female seems inspired to approach the sounds. So what are the steps that lead to these standoffish females being chased by a horde of suitors? And where in any of these social sequences does actual mating take place? For now, the humpbacks' mating game remains an enigma—one almost as profound as their song.