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How do you learn the fretboard?

Playing arpeggios and naming the notes as you play them (even better is singing the note names- gets the ear involved too and gets away from purely mechanical aspects). And learn some basic intervals- for example, octaves. Two frets higher/two strings higer is an octave so if you know you're playing C, then you immediately know where another C is as well. Extrapolate to fifths, fourths, thirds, etc.

MUCH more useful and practical than just trying to memorize all the notes in a specific span of the fingerboard.

John
 
I learned by playing the cycle of 4ths.

Start on the E string. Play C F Bb Eb Ab Db Gb B E A D G. I used a metronome on about 60 bpm. I started playing each note as whole notes, then half, quarter and eighth. Then I increased the tempo. Do this on all strings. I found this exercise on studybass.com.
 
Know all the open strings first; assuming its a 4 string; EADG
After that, its all in order; write it down. E F F#(Gb) G G#(Ab) A A#(Bb)B C C#(Db) D D#(Eb) E F, etc . . .Write this down for each string all the way to the 12th fret. Practice you scales while saying the notes of the fret. Do this until your sick of doin it then do it some more. Everyday! To get good at anything you sometimes have to practice stuff you don't really want to.
 
I learned by before practicing every day I would go up each string and say the notes as I played them ( e f f# g g# so on and so forth ) it may also help to remember that the note on the string you are playing is 7 frets towards the headstock on the string above and towards the bridge on the string below
 
Reading music! I had to learn to play the bass in about 6 weeks. I started in the jazz band at school. Out of nowhere the director calls me in his office and shows me a brand new Fender P and a combo amp. He says "Jazz band starts in 6 weeks. Good luck with that." I took the sheet music and a beginner technique book home and got started. The only thing I used the book for was to figure out where each string started (open strings). I started finding my way through the charts and sheet music. The rest is history. I pretty much HAD to learn it.
 
-Learn and know the alphabet from A to G . . .
-Pick any note from A to G, say you choose C, now say the alphabet from C to B (CDEFGAB - 7 notes) keeping in mind the alphabet from A to G and starting on C - skip one whole fret to the next fret with each letter said, "except" after B and E . . . do not skip a fret.

-C//D//EF//G//A//B . . . //represents a single fret-two fret wires

-D//EF//G//A//BC//

-EF//G//A//BC//D//

-F//G//A//BC//D//E

-G//A//BC//D//EF//

-A//BC//D//EF//G//

-BC//D//EF//G//A//

Those notes in-between (//) the 7 are sharps (or flats) - 5 of them.

Somebody double-check this, as I was attempting to make it very simple.
 
The way I learned was to visualize four important reference points: the open strings and the 5th, 7th, and 12th frets.

Open strings are obvious. You need to learn them. Duh.

12th fret is the octave, where the pattern repeats exactly. If you learn all the notes from frets 0-11, then you automatically know the notes from fret 12 and up, right?

7th fret is the octave of the lower string: 7th fret A string is one octave above open E string; 7th fret D string is one octave above open A string; 7th fret G string is one octave above open D string.

5th fret is the unison of the higher string: 5th fret E string is same as open A string; 5th fret A string is same as open D string; 5th fret D string is same as open G string.

Putting it all together: You are never more than 3 frets from an E, A, D, or G. For example, picking a note at random.... 9th fret E string. I know the 12th fret E string is E, so I count down: E, Eb, D, and the answer is Db (or C#). Another example: D string 8th fret, let's see, I know the 7th fret is A, so I count up, the answer must be Bb.

Learn those reference points (or whichever reference points work best for you) and then you can easily find any other note in relation to those references.