Water leaks in kitchen ceiling when it rains. There is an old gas vent pipe on the roof in about the same location as where it leaks into the ceiling

Pitch is the slope of the roof usually expressed as rise/run, but could be expressed as degrees.

The flashing or boot around the vent pipe may need to be replaced.

As a roofer I would try to explain to customers this,

A roof leaks where objects go through the shingles.The shingles don't leak(for 99% of cases)it's the pipes,the flashing,the valley's,etc. only where there are breaks in the shingles(or a VERY low pitched roof and the shingle wasn't properly installed/shingle wasn't appropriate for that application.. )

If the leak is near the pipe,it most likely IS from that pipe.

An improperly installed pipe boot could leak from day one of installing, or not for years.. even a properly installed boot can eventually leak, time and elements will break down just about anything.

Have a roofer or handyman who is experienced in pipe boot replacement,come and install a new one(the cost of the boot is less than $20.00 ANYWHERE I have even been) and to install(by an experienced person) isn't a difficult job.

I wonder why you asked about the pitch,though.. it there a real high pitch to your roof?Say a 12/12?(12" of rise to 12" of run)