From what I understand musical isomorphism is the formal way to speak about raising the index of a vector field (refer here). But what I don't understand is why is it called musical isomorphism! In the Wikipage, the following is said:

The exact origin of this notation is not known, but the term musicality in this context would be due to Marcel Berger

Was there any documentation on what exactly was musical in this to Marcel? BTW I am using the word musical in the following sense:

Musical, also called musical comedy, theatrical production that is characteristically sentimental and amusing in nature, with a simple but distinctive plot, and offering music, dancing, and dialogue.


2 Answers 2


The linked MathOverflow page (from the Wiki page) says that Berger himself doesn't remember how the name came about. The ♭: V → V* and ♯: V* → V notation uses musical symbols, which to me suggests an infinite time-travel loop between naming and selection of symbols :-) .


The $\flat$ map in music lowers the pitch of a note (by one half) and the corresponding map lowers indices, the $\sharp$ map raises it, as well as the indices. And of course adding a flat after a sharp goes back to the original pitch, i.e. gives you the identity map.

Isn't this all that should be said about it?

  • $\begingroup$ He would have been better using the words up or down, lowering a note by one half is lowering it by the octave! Halving a string length doubles the frenquency, doubling the string length halves the frequency, an inverse relationship! $\endgroup$ Commented Feb 6, 2022 at 11:51

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