I think it depends on what you want to get out of the bass.
IMHO, rhythm is by far one of the most important and first skill ALL musicians must learn.
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I played Trumpet for almost a decade before I ever picked up the bass. I know music theory, and can play bass charts, but not as efficiently as I can treble clef but what really helped me with bass was already having that fundamental knowledge of rhythm and how it fit within a group of musicians. Knowing about harmony/melody from trumpet parts also helped with my bass playing and finding the areas of music I needed to support with my bass.
Like domestique I came to the bass from seven years on the trumpet, with a stint of tuba and upright before I came to the bass guitar. He really expresses much of my feeling in the issue at hand. I feel I approach the bass guitar as a bassist, not as a guitarist. I think there's a difference.
I believe that having a solid background on a traditional melody instrument made me a more musical bass player than one who starts out on bass guitar alone in a strict genre specific guitar band only setting, at least when it comes to understanding performance of divergent music styles.
Contrary to what some have stated, I don't hold the opinion that the bass is primarily a rhythmic instrument. A friend of mine is a great drummer and his comment is that he hates playing with "lead bass players". My observation is that he's never played with anything other than rock bass players, as for almost 50 years he's been a rock drummer, no matter how good those bass players have been.
Being a good bass player means knowing when to be predominately rhythmic and when to be assertive and supportive in being melodic, even to the point of playing a melodic lead. That's what playing a melodic band instrument teaches you. You're sitting among dozens of players playing charts which demand that they count out rests for choruses at a time in some instances. You hear many more orchestrated textures and the use of different voices in different roles as each page of music passes before you. If you have musical sense you begin to appreciate this interplay in a way that a rock and roll quartet can't really bring to your experience.
While I also agree that it is critical to understand rhythm from a players' perspective for best ensemble playing, and best doing that takes formal training, I also believe that you can't make a silk purse from a sow's ear. If you don't have an innate sense of rhythm, no matter how much you're in love with the sound of music chances are you should take up woodworking, hunting, or the like. "It don't mean a thing if it ain't got that swing", but the stomp and swerve come from the heart and soul not the textbook, no matter how many pages of it you read.
Do you want to be a "bass player" or a "musician". If the latter knowing more and broadening your musical perspective is never a bad thing.
therhodeo makes an interesting and important distinction here. Please don't misunderstand me. There is absolutely nothing wrong with learning another instrument. It WILL make you a better MUSICIAN.
That said, the hours you spend woodshedding guitar won't make your bass technique automatically better. It will make you a better guitarist, and maybe even a better bass guitarist, but it won't make you a better BASSIST automatically. The bass is an instrument unto itself. Devotion to it is what you need to become a better bassist.
There are many players here from many types of music and conversations tend to be very egalitarian as to ignoring distinctions between styles practiced, which I think is great. However, no matter how much money (to use one measure) you might make being an amazing metal shredding bass player, you might forever be lost trying to have a sensitive spare interplay with a smokey vocalist. Proficient well roundedness as a musician is what I would say is the measure of what makes one a better musician. What I'm suggesting is that if your goal is to become a better musician, an expansion of your musical exposure is the path just as much or more so than simply an understanding of the guitar within the limits of the genre you are absorbed in at the moment.
So, looking exactly at the question the thread started with, "Should bassists learn how to play guitar as well?", I think the key word there is "Should". The answer there is, that's a personal choice, but it isn't so obviously yes as it might appear, at least to me.
I realize that the words of a friend about me ring true in this post. I don't have an opinion. I throw out the facts as I see them and live with them as my truth.
You get what you paid for.
iJazz