Recommended Reading
I think a strong bass player has and develops some very basic things that allow for the music to happen. Playing in tune, with a good sound and good time. It's good to learn how to read music too. These are really simple fundamental attributes, but they are vital to the inner workings of a band. Above and beyond that I think to bring an openness to the music is very important, an ability to not hold back, and let the music go in many directions. I am aware of the "responsibilities" of the bass player but I also believe in trying to go beyond and push these expectations to help take music to another place.
From a beautiful vantage point as a bassist, nestled in the middle of the band, you're able to see that a great "solo" is really a collaborative event. An improvisation really happens when the soloist and the rhythm section become one, each inspiring each other, with open and sensitive ears.
I recommend listening to many different bass players and trying to hear how they approach their role as the bass player.
These are some bass players i am heavily influenced by:
Charlie Haden: http://dothemath.typepad.com/dtm/interview-with-charlie-haden.html (a great interview with Charlie)
Gary Peacock: http://www.youtube.com/watch?v=sx5gjOdh3Po (great video of Gary playing Keith Jarrett's music)
Larry Grenadier: http://www.youtube.com/watch?v=3udQCpHFjGk (great walking line bass solo)
Jimmy Blanton: http://www.youtube.com/watch?v=8xJlV0Agi1w (some great time and groove playing, strong sound)
