Final answer:
Franklin's inability to remember how to play the trumpet but ability to recall musical note names suggests damage to the cerebellum and motor cortex, which are associated with procedural memory.
Step-by-step explanation:
Based on the information provided, Franklin, who can no longer remember how to play his trumpet but can still recall the names of the musical notes, has likely sustained damage to the brain regions involved in procedural memory, specifically the cerebellum and possibly the motor cortex. Procedural memory is responsible for knowing how to perform tasks and is different from declarative memory, which encompasses facts and information, such as the names of musical notes.
The case of Franklin mirrors instances like Clive Wearing and patient H.M., where despite damage to areas such as the hippocampus, they could still recall information from before their injuries. In Franklin's case, he can remember note names, indicating that his declarative memory seems intact, therefore, the damage is more likely in areas related to motor skills and learned behaviors,mainly the cerebellum or other motor areas, which are crucial for skills like playing an instrument.