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Does filming in 30p as opposed to 60p provide a brighter video? Would it be better to film at perhaps 24p in darker circumstances?

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Yes and no. Shutter speed is what determines how long light is gathered for and is not the same as frame rate. Frame rate does place a logical limit on your shutter speed as you need at least one exposure per frame in most cases, but there is nothing preventing you from using much faster shutter speeds than your frame rate.

Some video cameras will even allow for shutter speeds slower than the frame rate and will simply record the same image on more than one frame, though this arguably lowers the meaningful framerate (though faster frame rates will result in a closer match up to the exposure time).

As a general rule of thumb, it's best to use a shutter speed at least twice that of your frame rate, so you would use 1/60 exposure or faster for 30fps and 1/120 exposures for 60fps.

This is partially a throw back to old mechanical film cameras where the film advanced during the time that the shutter was not open. They used a rotating plate and when the plate was over the lens, the film moved forward. This is why you will see shutter speeds on some professional cameras measured in degrees. This corresponds to the number of degrees of the circle that were open. ie, a 180 degree shutter speed means that the shutter speed is twice that of the frame rate. 90 degree shutter speed would be 4 times the frame rate.

As long as the shutter is timed properly to the frame capture, there is no reason you can't use longer shutter speeds, but it may increase the appearance of motion blur and give an odd feeling to the footage unless that is what you are going for.

Reasons to use slower shutter speeds are to let in more light or to increase motion blur, reasons to use faster shutter speeds are to let in less light or decrease motion blur.

Reasons to use lower frame rates are to save space and to give a more film like quality. Reasons for higher frame rates are for smoother motion and more video data to work with (better slow motion, stabilization, etc).

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