The flames considered are those which maintain vibration of the air in a surrounding “air tube.” Earlier theories are reviewed. Rayleigh's theory involves standing waves in the gas supply tube and leads to certain lengths of gas tube for which singing does not occur. It is now found that maintenance of vibration in the air tube may be excellent when the gas tube is so long that no appreciable standing waves are formed in it, and that under suitable conditions there are no lengths of gas tube for which a flame will not sing. Rayleigh's theory is modified by taking into account (1) the increase in viscosity which the flame brings about in the orifice from which it burns and (2) the progressive waves that run downward through the gas tube. These waves involve a maximum rate of efflux of gas near the phase of least pressure in the air tube, whereas Rayleigh's theory involves a maximum rate of efflux a quarter of a period earlier or later, depending on the length of the gas tube. If the most rapid efflux is near the phase of least pressure, and if the flame gases rise from the orifice to the top of the flame in approximately half a period, maintenance is possible. In thirteen cases where the flame sang, values obtained for the time of rise run from 0.6Tto 0.9T; and in sixteen cases where the flame was silent, they run from 1.2Tto 3.3T. In view of approximations made in reaching these results they are regarded as satisfactory. When the gas tube is of only moderate length, standing waves in it may be superposed on the running waves. This superposition leads to the possibility of lengths of gas tube for which the flame will not sing. Regions of silence are more likely to occur when the flame is small. The size of the flame is also a factor in determining the distance to which it must be inserted in the air tube before maintenance is possible. The distance is a minimum for flames of moderate size. The amended theory gives explanations of these facts and of several others. The initiation of the vibration is attributed to the slight disturbances that are always present in the atmosphere.