\begin{definition}[Definition:Musical Isomorphisms]
Let $X$ be a smooth vector field.
Let $\omega$ be a smooth covector field.
Let $X^\flat$ and $\omega^\sharp$ denote the index lowering and index raising of $X$ and $\omega$ respectively.
Then the mappings $\flat$ and $\sharp$ are called the '''musical isomorphisms'''.
\end{definition}