Marcel Berger, on p. 696 of his Springer-book: [A panoramic view of Riemannian Geometry](http://www.amazon.com/Panoramic-View-Riemannian-Geometry/dp/3540653171), writes "Next we define the canonical musical duality", referring to a footnote: "These dualities are called *musical* because they are often written with symbols like ♭: V $\rightarrow$ V* and ♯: V* $\rightarrow$ V."

Even if this is not the first metioning of "musical", we can assume that Berger invented that expression, because in a *panoramic viev* of over 800 pages, a reference to another originator could have been expected.

Further [Rob Kusner's](http://www.gang.umass.edu/~kusner/class/704hw.2000) hint *here # or "sharp" is what Marcel Berger calls a "musical isomorphism"* and Olivier Bégassat's comment here seem to provide sufficient evidence to settle this question.