Basic HD bridge and tuning issues.
The treb bridge is positioned so there is a 5th interval between the notes on each side. IE: D on the right and the A above D on the left. Things that can cause this to be off are the bridge out of position. Easy check is to check a course or 2 above and below the notes that are "off. If they're fine then the bridge is in the proper position. The reference line is the center of the rod on top of the bridge, some are Delrin and some are metal. The fifth interval works by having the string a constant mass from one side rail to the other. If there is a bit of the loop wrap past the side rails or some crap on the string then you wont ever get the pair tuned. If the notes one course above and below are right then the problem IS NOT a moved bridge. Look for trash or anything that is different where the wire crosses the rod or the side rails.
Its rare you will ever have to move the bridge but if yours is out of position its not hard to move it. Its held in position completely by string tension. On rare occasion the bridge may have "welded" to the top due to the pressure fusing the finish on the top to the bridge bottom. The easy way is to down tune all the strings a note or 2 to reduce the pressure. There is no need to actually tune exactly. Now tune only one wire at each marker dead on on the right side. Check the left side. If its sharp then the bridge needs to go right, flat-go left. To actually move the bridge use a new unsharpened pencil and put the eraser against the opposite the way you want to go against the bottom and GENTLY tap. A movement you cant even see will change the tuning more than you think. After each tap re-tune the right side and repeat the above as many times as it takes. If the bridge is welded as above you may have to tap it a bit harder to break that weld. Once you have one wire at each marker dead on on each side then tune the whole thing up and youre done. In most cases the bridge should be absolutely straight but Ive seen a few where the right spot will put a very slight curve to the bridge.
The higher courses will often stick a bit on the bridge rod throwing off the tuning. There are a number of tricks to deal with the stickeys. One is to lift the string clear of the rod then gently setting it back. All you need is a gap about as big as the thickness of a sheet of paper. Another one is to tune one side dead on then check the other side. If its flat then either wack the string on the other side or simply push that side down gently. If its sharp then do the wacking or pushing on the sharp side. GENTLY push it or tap it with your finger. Im so used to how mine acts I simply tune the side opposite a tad sharp and with a couple wacks things come right to tune.
As you dont use the right side of the Bass the position isn't as critical but in general shouldn't be more than a 1/4 inch off of where the maker put it. Theres braces under both bridges with the braces positioned over them and you need to stay close to where the maker put the bridge in relation to those braces.
"Floating bridges"?? never heard that before but there is basically 2 types of HDs as far as construction goes. The most common is box construction. Thats where ALL edges of the top are glued down. The other is whats known as a floating top. On these the top is held in place only next to the tuning pins. There will be a gap between the top and both rails. As to which is the best sounding is one of the hottest debates with the HD crowd...
There are bridges that are not one piece but is actually a very small bridge under each course not connected together. This is whats called chessmen bridges. Its the same deal setting the position the same but moving one wont have any effect on another course.
Hope that helps..