If I want to learn Clojure, should I start by learning Scheme or Common Lisp?
Or is Clojure different enough from both of these, that I should just start learning Clojure by itself?
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It would be to your benefit to learn all three, if only so you can pick which one is best for your needs. All three have their own strengths and weaknesses.
Clojure is vaguely like Scheme in that it’s a mostly-functional language and is a Lisp1. Clojure also borrows things from Common Lisp, like multimethods and macros, and people are always porting cool Common Lisp things to Clojure as libraries. The creator of Clojure was himself a Common Lisp hacker before writing Clojure. Clojure borrows a lot of terminology and conventions from Scheme and CL both (but also has its own flavors in many areas).
There is not a lot of literature for Clojure right now, it being such a new language (there is only one Clojure book so far). But there are loads of good Scheme-oriented books, like SICP and The Little Schemer / The Seasoned Schemer. There are also good CL books, like PCL, and many others.
Lisps also have a lot of history and it is to your benefit to understand the history, to see where and why Clojure deviates from it if nothing else.
I’d recommend starting with Scheme because it’s the simplest language of the three and therefore easiest to learn. Then dabble in CL and Clojure until you have a handle on things, then go full-steam in whichever of the two you gravitate toward.