"Whatever" vs. "Whichever"

The same criteria for choosing between whichever and whatever apply here as for choosing between which and what. This is a question that has been asked several times on ELU, including :

"Which" vs. "what" — what's the difference and when should you use one or the other?

In summary, which is generally to be preferred when choosing from among a restricted set of options known to both interlocutors. So in the nursery rhyme about the aggressive fish, it is:

Which finger did it bite? and not What finger did it bite?

In the exchange about music, the answerer is not restricting the choice of music to a limited number of options, so whatever is the natural choice.

It is possible, however, to conceive of a context in which whichever fits:

Is it OK if we listen to Blood on the Tracks or the new Van Morrison?

Sure. I'll listen to whichever you'd prefer.

We can think about the W-ever words semantically as the 'W' word + the quantifier 'any'. The trick to understanding these terms is to realize that they apply to any single one of the referents, and at the same time refer to all of the referents.

Whatever - Any thing (This could also be every thing)
Whenever - Any time (This could also be 'every time')
Wherever - Any 'where' (Anywhere or everywhere).
Whoever - Anyone (Any person or every person, or sometimes used to refer to a person unknown to the speaker)
Whichever - Any 'which' (Choice between a group or set).
However - Any 'way' (In any manner or way, regardless of how).

Basically each one means:

"It does not matter what / when / where etc." OR

"An unknown thing / time / place etc."