New moon and full moon are real things you can see, not an arbitrary mathematical division. And then the first / third quarter is just halfway between. Results in about 7 days per lunar quarter.
But it's really not even close to "about 7 days per lunar quarter," at least not close enough that using the calendar can tell you anything useful about when future new moons will be. Even if you're just looking ahead 4 weeks you'll already miss the new moon by a day and a half!
We can see the result of (not arbitrary, but) precise mathematical divisions. "Halfway between" is a mathematical division. I must be missing something huge here.