TITLE: The Origin of the Musical Isomorphisms
QUESTION [26 upvotes]: In Riemannian geometry, the "lowering indices" operator is denoted by $\flat:TM \to T^*M$ and the "raising indices" operator by $\sharp:T^*M \to TM$. These isomorphisms are sometimes referred to as musical isomorphisms, as stated on Wikipedia and in several other sources. Surely, the motivation for such terminology is clear. I would nevertheless like to know who decided to adpot these (rather amusing) notations, so here is a question:

What was the first paper / textbook that made use of the notations $\flat$ and $\sharp$?

and a possible follow-up question:

If such notations were not adopted widely after the first appearance, who popularized them?

REPLY [13 votes]: The musical isomorphisms already appear in 1971 on page 21 of BERGER M. et al, Le spectre d'une variete riemannienne, Lecture Notes in Math. 194, 1971, Springer, see http://ci.nii.ac.jp/naid/10003477917/
However, I am not convinced that this is the first place where they appear, or that Berger was the inventor.  One would have to trace German textbooks in Riemannian geometry from the 1960s or perhaps earlier.  Interesting question!