What Bottom bracket would I need.......

For the record, the Shimano BB will have the size stamped on the axle (drive side). Should be LL113 if you measured correctly.

Also
 

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M739 is 110/113.
M737 is 107/113 Unless an etype mech and then 110/113.

First is standard 47.5 chainline, good for 7/8/9 speed with normal to mid band front mechs.

Later is 50mm chainline good for 9 speed modern setups or earlier with mid to wider band front mechs.
 
As a perpetual mover of chainsets and ratios etc. chainline intruiges me. Wouldn't a 2.5mm shift in chainline simply mean a different ratio is the 'straight chain sweetspot'?
 
Maybe? I look at the chainline or chain its self looking down at it as being a right triangle where this chain is theoretically the Adjecent and the 2.5 mm distance of chainline at the BB being the Opposite. The Hypotenuse will be variable as a result of a shifting chainline. Without physically measuring anything, taking the Tan(gent) of O/A likely results in a (Tan) theta angle that is trivial over the length of the Adjecent.

Basically if you are in the 11/12 tooth out back and on the middle chainring up front and the chain is rubbing on on the inside of your largest chainring, you need to either shim or increase the bb width by a few mil on the drive side and adjust the front derailleur.
 
I'd always been taught to keep a straight chain on a 3×7/8/9 system i.e. that the inner ring was for rear gears 1,2,3, the middle ring for 3,4,5 and the outer ring for 6,7, etc.

I can see how it is really important on 1x systems.
 

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