When I take five and add six, I get eleven, but when I take six and add seven, I get one. What am I?

“A calculator”

A calculator provides straightforward arithmetic answers and does not exhibit the unique behavior of a clock where time wraps around after reaching a certain point.

“A calendar”

A calendar deals with dates and months, not with the cyclical nature of time in the way that a clock does, which is essential to the riddle.

“An hourglass”

An hourglass measures time but does not involve arithmetic operations like adding numbers, which is central to the riddle's premise.