Efforts to reduce the noise from turbulent jets at fixed flow conditions, with aircraft noise as the principal technological motivation, have generally involved some degree of parametric empiricism often based upon a series of trial-and-error testing. As a result, it is unclear if the modest reductions found, in rare cases that do not greatly affect the flow field or incur prohibitive losses, are near the limit of what can be accomplished or if there are undiscovered opportunities for more substantive reductions with better designs or active control. We assess this using an adjoint-based optimization procedure in conjunction with an experimentally validated large-eddy simulation of a Mach 1.3 turbulent jet. The adjoint solution provides a definitive direction in which to adjust a model control actuation in order to reduce noise, providing guidance that seems lacking by any other current means. It is found that three conjugate-gradient iterations in the control space provide ∼3.5 dB of reduction, comparable to other reductions found empirically. The control seems to work by disrupting the coherence of acoustically efficient axisymmetric flow structures. The control and noise-reduction mechanisms are informative, but also suggest that any significantly quieter state would not be a simple perturbation from the uncontrolled jet. Additional iterations might reduce noise more significantly, but there might be only modest opportunities to reduce the sound from simple round turbulent jets without radical changes or relatively sophisticated controls. Though it is difficult to prove any behaviour in a global space of actuations, there does not seem to be a direct route based upon a local sensitivity gradient to substantially quieting a jet, even with an unrealistically flexible actuation. More complex jets or other noisy flows may be more amenable to control, in which case the adjoint-based optimization procedure demonstrated here could provide invaluable engineering guidance.