Nhbenz is correct, and so is HomeGuardDan's more-specific answer. It is the brass "end-cap" known as the sword-scabbard's "drag." Even more specifically, it is part of the drag from a US Model-1840 NCO (No-Commisioned-Officer) and Musician sword's scabbard. (I should mention, that model was still being manufactured during the civil war.) Although it's true that many Fraternal swords and their scabbards more-or-less copied the US Army Model-1840 sword's form, your brass scabbard-drag is an EXACT match for the US Military model-1840s drag. Note the shape of the flat "side-ridges," the width of each of the flat ridges, and especially the location of the little valleys in the ridges. Unfortunately I don't have a close-up photo of a Model-1840s scabbard-drag, but perhaps you can see its exact form by enlarging this photo.
I've seen the narrow brass "double-loop" used as an adjuster on civil war era swordbelts. Sorry, I don't know which specific model of swordbelt. (I must mention, it may have also been used on other types of belts.) Perhaps it was on the belt your scabbard-drag was used with.
