The four poles (two north, two south), produce two "cycles" of the magnetic field when they move forward by that four-pole spacing.  Think of these "cycles" as 360 degrees each, for a total of 720 degrees.

Consider two coils at different positions along that path.  The phase angle between their output is equal to the fraction of that four-pole spacing that they are displaced from each other.

With three coils you equally divide the cycle.  First one is arbitrarily called zero degrees.  Next one is 2/3 of a cycle (240 degrees) delayed.  One after that is 4/3 (= 1/3, because whole cycles don't count) of a cycle (480 - 360 = 120 degrees) delayed.  Three phases, equally spread around the "circle", at zero, 120, and 240 degrees.

After the four poles and three coils the phase relationships repeat.

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It would be easier to see if you doubled up the number of coils, arranging the extras so that their sides "paved" the empty spaces in the middle of the first set.  Then it would be three coils for each two poles, and as you went forward they'd be at 0, 1/3 and 2/3 cycle (0, 120, and 240 degrees) before things start to repeat.

But paving the whole area means one of two things:

 - You have to open the gap to make room for the second layer of coils, lowering the field and still leaving half the space unoccupied by copper (though this time it's half the space vertically.

 - You can't use flat coils but have to "bend" the portion outside the magnet path (or wind the coils in a warped 3-d configuration) to keep the thickness between the magnet poles the same.

Winding warped coils is a real pain, and then you have to cast a stator with a groove under the magnet path and a fatter part where the outside-the-magnet-path portion of the coils sits.  Running the magnets in a groove like that inhibits cooling, too.

Weakening the field by opening the gap means you end up with less voltage per coil but the same resistance, which means a higher percentage of losses.  And that's forever.

Only putting in half the coils effectively means doubling the number of magnets for a given number of coils compared to a "paved" design and letting the magnets only work half the time.  But extra magnets are just a one-time expense.

So the N times (four pole, three coil) three-phase design is usually used, because it's so easy to construct an efficient alternator.

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