James Chance and The Contortions
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One of the most important (if not THE most important) figures of that submovement born in New York between the '70s and '80s known as No Wave: James Chance and his Contortions, dedicated to total dissonance, redefine in what is effectively their only official album the concept of funky. Free wave, jazz, fusion, and noise are just some of the ways through which Chance's sax draws the undrawable against a backdrop of dry, abstruse, and torn guitars, a bass that seems like a drunken bum, and a drum that barely restrains an uncontainable sound.

And the most unthinkable thing is that all this strangeness is danceable, it's impossible to stay still. Examples like Contort Yourself and Designed To Kill suffice.
Indispensable and perfectly unknown to most, like almost every great band worth its name.