0.30 time zones are actually quite common, Newfoundland has one, also India,and South Australia, and there are even some 0.20 time zones, I believe Bhutanmay use one, also one of the small Caribbean states of South America.

Ideally local time is such that the sun is highest at noon, and most countries try to make this as true as possible. It happens that it's very nearly true in Santa Barbara, because we're located almost exactly at 120 degrees West, and that's exactly where the sun is highest at noon in the Pacific time zone. It would be true in the Mountain time zone for anyone exactly at 105 West, and so on - each of the 24 whole-hour time zones is 15 degrees wide.

You could try that as a class experiment - Is the sun directly south and highest at noon? And what happens in the summer? Some places are so far from the center of their timezone that the sun is very far from highest at noon - an example is Western China, which uses the same time as Beijing - so try figuring out when the sun comes up and goes down in Urumchi (do it for mid March or mid September, when the sun comes up and goes down at the same times, 6am and 6pm, everywhere in the center of any time zone).

People in South Australia or Newfoundland or India have simply argued that on average people in those areas see the sun highest nearer noon if they choose a non-whole-hour time zone. Of course it's a pain for TV programming (any Canadian is familiar with announcements like "10 o'clock, 10.30 in Newfoundland"), and for travelers from Melbourne to Adelaide("set your watches back 30 minutes").