"at which time" vs. "at what time"

Question

I am confused with the grammar here. Which is grammatically correct?

• At which time will you call me?

• At what time will you call me?

And why?

Answer 1

In this sort of context which usually refers to a choice between previously defined alternatives. For instance:

I have some time free at 9:30, at 11:00 and at 1:30. At which time will you call me?

What is used when there are no such previously defined alternatives.

I have no appointments tomorrow, so I can talk at any time. At what time will you call me?

Both, however, are quite formal. Most people would say, simply

When will you call me?

Answer 2

I believe that though at which time is not grammatically wrong, you should stick with At what time will you call me? or even better: What time will you call (me) (again)?

Another idiomatic phrase is: When will you call?

Generally, when you ask a which question, it implies a choosing from choices. Asking for time usually an open question, so using what is better.

Answer 3

I'll add that "at which time" can also be used to introduce a non-restrictve relative clause:

The museum will remain closed until 6:00am, at which time we will open the doors to visitors.

Substituting "at what time" in place of "at which time" would not be appropriate in this sentence.