Tom i've watched that video before, the one thing i get confused about is when Garry takes his metering from the background, and then from the girls face, is the overall metering he uses of F/9 or F/10 for both background and the subject ? daft question i guess and one i should probably know, but i'm asking anyway
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The meter reading from the girl's face was f/10 so that is what I set on the camera. The meter reading on the background was around f/13, which means that the background was overexposed enough to make it white (because the aperture on the camera was set to f/10) but not sufficiently overexposed to cause the problems in these example photos (which have very nice expressions BTW).
As others have said, the trick is to light the background as evenly as possible (which is why I used background reflectors designed for the job) and to then overexpose it by the minimum possible amount that will get it white. If the lighting on the background is not even then the exposure needed to get the darkest parts white will create significant overexposure on the brightest parts, the light reflected from the background then lights the back of the subjects, causing the very common fault of the fine detail (hair etc) being eaten away. The only cure for this, other than getting the background lighting right, is to move the subject as far away from the background as possible so that the light reflected from the background has lost most of its power before it hits the subject.
If you don't have background reflectors you can improve things by crossing the background lights over - the one on the right is lighting the left hand side of the background and
vice versa - this helps to get the lighting even.