A turbomachine dual spool transmission system can include a transmission assembly configured to connect to a combination output of a dual spool differential at a transmission input to be driven by the combination output to turn a transmission output. The transmission assembly can be configured to provide a first output gear ratio in a first state and a second output gear ratio in a second state. The system can include the dual spool differential. The dual spool differential can include a gear assembly configured to combine a low pressure spool input and a high pressure spool input into a combination output to provide an output speed range smaller than a low pressure speed range alone.