What is the purpose or role of the bass guitar in a music group?
To provide a foundation for the band to build on.
Ever hear a band without one? It sounds “thin” and incomplete without the low foundation of a bass.
I think of the bass (in its simplest form) as the bridge between the drums and the guitar or other lead instrument. The bass can help the other musicians keep time, which one of the functions of drums, but it also supports the melody and helps the other players keep track of that part of the music.
The bass also thickens the sound and really gets the audience into the groove of the song.
The bass also helps to establish the key (e.g. E-flat) in which the song is played. I also agree that it “fills out” the music. This is why I don’t like listening to music on a poor-quality speaker system, such as a small portable radio, as the speakers do not have the ability to play the lower-register notes.
In a lot of cases, the bassist is more or less mimicking the rhythm guitarist, but there are bands where it’s used a bit more freely, even almost to the point of providing a third guitar line. Check out some recent Tool for an example.
Bass helps bands like The Cure sound “extra doomy.”
True, enough, but you don’t have to go to examples like Tool even. You’re making it sound (to me, at least) that it’s unusual for the bass not to follow the rhythm guitar. I would argue much more common, but this depends on the genres we’re talking. I mean, look at pretty much all of Motown. In Motown, the bass is groovin’ and the rhythm guitar is usually just providing the 2 & 4. Look at most pop music. Look at rock in the 50s and 60s. Look at Led Zeppelin (“Ramble On,” for instance, although JPJ will do the follow-the-rhythm guitar when it’s applicable). Look at Peter Hook’s melodic lines, or Kim Deal (who, while she is banging out root notes, often does not follow the rhythm guitar’s rhythms). Look at almost all of blues, country, etc…
The most important relationship for the bass in a pop/rock context is not with the rhythm guitarist, but with the drummer (most importantly, with the bass drum).
There are bands who successfully get away without having a bass (The Jon Spencer Blues Explosion, The White Stripes, The Black Keys, and Sleater-Kinney all come to mind). However, the bass is essential in most cases to developing a solid rhythmic and harmonic foundation. Sometimes, it even provides countermelodies. In my experience the key to a great band is a solid drummer and bassist who play with one mind. You can have sloppy guitars, but as long as the bass & drums are playing together, you can get away with a lot.