John Lightfoot on John 13:38

[The cock shall not crow.] We must not understand this as if the cock should not crow at all before Peter had denied Christ thrice: this had not been true, because the cock had crowed twice before Peter had denied him. But we must understand it, The cock shall not have finished his crowing, &c. Nor indeed was that time above half over before Peter had denied his Master.

The Jewish doctors distinguished the cockcrowing into the first, second, and third. The first they call the cockcrowing. The second, when he repeats it. The third, when he does it a third time. The distinction also amongst other nations is not unknown. When the time indeed was near, and the very night wherein this was to happen, then Christ saith, This very night the cock shall not crow his second time, &c. But here, two days before this night, he only saith, The cock shall not crow, that is, shall not have done all his crowing, before thou deny me. And thus our Saviour meets with the arrogance of Peter, foretelling him that he should not have the courage he so confidently assumed to himself, but should within the time and space of cockcrowing deny him thrice.