There is no such thing as a cover band. There are only bands that play music people like to hear again.
Like Mozart and Beethoven. There have been "cover bands" as long as there has been the ability to document a composition in writing, or pass music down as an aural tradition.
I'm in three cover bands. One is for worship service, and we cover inspirational pop and rock songs, such as Put A Little Love In Your Heart, Love (by Los Lobos), The Rising, Let It Be, What's So Funny About Peace Love and Understanding, etc.
The second is almost exclusively 60's covers, with a lot of Beatles and Beach Boys.
The third is just starting and will focus on 70s power pop.
It's all good, and I don't feel guilty for not being a writer.
