Efficient multicast congestion control (MCC) is one of the critical components required to enable the IP multicast deployment over the Internet. Previously proposed MCC schemes can be categorized in two: single-rate or multi-rate. Single-rate schemes make all recipients get data at a common rate allowed by the slowest receiver, but are relatively simple. Multi-rate schemes allow of heterogeneous receive rates and thus provide better scalability, but rely heavily on frequent updates to group membership state in the routers. A recent work by Kwon and Byers, combined these two methods and provided a multi-rate scheme by means of single-rate schemes with relatively low complexity. In this paper, we propose a new scheme called generalized multicast congestion control (GMCC). GMCC provides multi-rate features at low complexity by using a set of independent single-rate sub-sessions (a.k.a layers) as building blocks. The scheme is named GMCC because single-rate MCC is just one of its special cases. Unlike the earlier work by Kwon and Byers, GMCC does not have the drawback of static configuration of the source which may not match with the dynamic network situations. GMCC is fully adaptive in that (i) it does not statically set a particular range for the sending rates of layers, and (ii) it eliminates redundant layers when they are not needed. Receivers can subscribe to different subsets of the available layers and hence can always obtain different throughput. While no redundant layers are used, GMCC allows receivers to activate a new layer in case existing layers do not accommodate the needs of the actual receivers.