If you want to play bass, you need to know your bass neck thoroughly, and you need to be able to identify key changes and chord changes by ear and react instantly.
Let's take a look at the bass neck. In the diagram below, the red notes depict open strings; the blue numbers depict the frets, and of course, the black notes are what you are playing.
Here's the bass neck showing only the major notes from the nut to the 12th fret:
If you move up the neck from left to right, you call the "in between" notes sharps (#) and if you move down the neck (from right to left), you call the in between notes flats (b).
Here's the bass neck, showing the sharps between major notes.
Let's get oriented. If you are playing a right-handed bass, you play the above notes with your left hand and pick the strings with your right hand. Looking down at your bass neck, the bottom string in the above graphic is actually the top string, the thickest of the four strings. That's the E string (so called because when you play that string "open," without any left hand fingers touching it, the note is E). The string below it is the A string, then the D string and finally, the G string.
Here's the bass neck, showing the flats between major notes.
Moving your left hand fingers up or down the neck will enable you to play the notes pictured above.
Here's where it gets a little confusing. There are two notes that never have a sharped version. They are the notes B and E. When you advance one fret up from B, you do not get B sharp, you get C. When you advance one fret up from E, you do not get E sharp, you get F.
Similarly, there is no C flat (because it is B) and there is no F flat (because it is E). A bit confusing at first, but don't blame me, I didn't invent the system.
Now let's top it off with a useful YouTube video that demonstrates the above information on an actual bass guitar:
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