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How to transpose from E major down to B major with a 4 string?

It depends on how melodic the bassline is. If the line has a strong counter-melody, or a lot of movement up and down a scale, then you move the whole thing up a fifth. If the line mostly just moves the root notes of the chords, with octaves and fifths and passing tones, then it will probably sound better to pick those notes out in the lowest register possible.
 
If it’s just one song transposed like that, why make a mountain out of a mole hill? Buy a good capo and slap it on for that one song. Sure it will feel a little weird playing that high on the neck but probably something you can put up with for 3 to 4 minutes.

Or just learn the song in the new key, no reason to get wrapped up in note for note cover especially since your band is already changing the key so drastically.
 
Maybe I'm misunderstanding the OP's question, but...

On a four string without retuning, with the song transposed I'd just play the root notes of the chords (and any fills or connecting runs) down in the lower register. IOW, if in the original key (E) there was a I to IV move and the recorded bass went up, in the new key (B) I'd go down--because I could. Displacing the octaves of the bass line as the four strings allowed, IOW.
 
Something went wrong with the post #19 and I can’t delete it…anyway, what I wanted to say is, that’s a situation I’ve been in several times playin in cover band with a female singer covering also male singer’s songs. I always figured out a new bass part, as “functional ” as possible with the original, but able to retain some low end.
 
changing keys doesn't have to mean globally shifting every chord up by the same amount
context matters of course, it depends on the song
rarely you get a song with a signature bass line which demands a specific octave
in that case you may well be stuck "moving everything up/down a fifth"

otherwise, most styles allow for variation in which octave you support the chords from
if you play an E major scale and a B major scale in one position you'll see the notes are very close
in fact only 1 note is different, this holds true for scales and arpeggios for the chords of the key
Instead of moving the same pattern up or down the neck
learn to play the chord progression with a different pattern
many chords and chord relationships will change shape, the root movement going up instead of going down for example
in many styles an genres this is fine and no one cares if you change the bass line as long as the style is tight.

Figuring out different ways to support the harmony in a stylistically appropriate manner
is the kind of problem solving that makes applied theory so fun
 
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As other have already pointed out, and as ALWAYS with music, it depends, no set in stone rules that works universally and equally great for everything.

An approach that works great in one song might sound horribly off in another.

Use your ears and your imagination.
 
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Don't tune down because it would sound too heavy if doing so for that pop song. Just play it in B and make sure you use your lower notes for the other chords if need it.
Also play it as much as you can on the E string for a fatter sound. No one will notice the transposition ;-)
 
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