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What do you do with notes below the E on a 4 string?

For general purposes, it depends on the song for me. Noting that I am typically a 5 string player, but have run into this playing a 4 or doubling tuba parts on a 5 (years ago when I was still in school).

For sight reading, I would octave up as a rule. Generally ends up being my solution anyway, but with sight reading you need to just know what you are going to do so you can keep going.
 
Bass is a transposing instrument that sounds 1 octave lower than written, so when I see the Eb one ledger line below the staff, I figure it's probably a 78hz Eb2 and the original bassist on the album played it at the 1st fret of the D string (or 6th fret A string, or 11th fret E string). I'm talking like if I'm sight-reading from a mass-market Pop Singer's Greatest Hits vocal/piano/guitar songbook, that type of sight-reading situation.

For example I recently saw some pro-published Johnny Cash sheet music with tons of low C's (2 ledger lines below the staff). Do you think Johnny's bassist used a 5-string on "Walk the Lilne"? Nope. I watched the concert on YouTube, he played all those C's at the 8th fret of the E string. Pop keyboard sheet music is swimming with low C's and D's, but you don't need a 5-string bass to play those, because like I said, bass is a transposing instrument. Those are just our 'normal' C's and D's.

Finally a completely different situation: If the guitarist I'm playing with down-tunes, for example Hendrix and SRV songs in Eb, then I down-tune to match.
 
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I'd probably think about getting a new bass and maybe a new overdrive pedal. Or a preamp with a more gritty option. Just something to add hair but not super dirty. And I'd just play the note in the position that is available. Even on a 5 there are notes below B. Tuning per song is why you need a ton of basses but I've worked with singers who had us all drop half step. That's not so tough. But it doesn't stop me from wanting more pedals.
 
I don't play 4 strings. Sometimes, rather than go below E, a root+5th works well. Has a subtle effect with some extra heft. Definitely a different vibe than playing those lower notes, but in a good way. It fills sonic space in a way that I like when it's appropriate.
 
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Find one of these. It lets you detune for one note.
 

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Smartass answer: use a 5-string.
Practical answer: for Eb or D, a lot of my 4-strings have a detuning lever.
If you're sight-reading from a piano part or are given those extra low notes for some other reason but the setting clearly calls for a standard 4-string, just transpose. (Can scrawl in "8va" or even write a note where you want it if you are permitted to mark up your part.)
 
If you must remain in standard tuning, try notes below standard E an octave up. Sometimes it's not noticeable. Other times it sticks out like a sore thumb. In that case your options become:

Manual or Quick D-Tuning the 4th string

Permanently tune BEAD

Get yourself a decent entry-level 5 string and tune BEADG

I have not tried a pedal to achieve this, but I have done all the above. Current choice for bands/playlists that require more than a couple of songs below standard E is a five string bass. I have a lot of nice four-bangers, and if I don't need extended low range notes, I like playing them. But in a few projects, the fiver has been the staple.

It's also worth mentioning that some songs where the guitars and bass tune to Eb (Eb-Ab-Db-Gb), I like to put the bass in an Eb tuning which allows access to open notes. Some songs just play easier and sound better when you can use the open strings. I recall taking two four-string basses to some gigs tuned standard E and Eb.
 
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For example here is a professionally-engraved Hal Leonard arrangement for the classic "Earth Angel." Notice all those low Eb's and C's just in the first measure. If you didn't know bass was an octave-transposing instrument, you'd totally think you'd need a 5-string to play all those old doo wop hits of the 1950s. My first teacher totally dropped the ball on teaching me this, and it totally blew my mind when I realized it on my own, after struggling with sheet music like this for years. My friends would bring songbooks to band practice, and I'd be like, uh-oh, sorry guys, I don't have a 5-string, but I'll do my best!

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