As I said in the "plot hole" thread, I place a range of no less than approximately three months, but definitely less than a year.
As for the short end - if Calms lasted less time than a pilgrimage took, it really wouldn't be worth it. You'd end up with a "summoner train". Spira would run out of people to save.
For the long end? All the reasons given in the other thread, by myself and others. If it was more than a year, then even fudging an estimate, you couldn't say Auron was with Tidus for "ten years". (And people in FFX2 would be skeptical. After only two years, NOBODY doubted it was an eternal calm.)
"As the good archmage often admonishes me, I ought not to let my mind wander, as it's too small to go off by itself." -Danilo Thann