Why are there multiple contradictory ways to control flash ratio?

I am reading the following introduction to flash photography:

http://www.cambridgeincolour.com/tutorials/camera-flash-2.htm

The "flash ratio" is an important way to describe the mix between ambient light and light from your flash. ...[T]he mix of flash and ambient light is adjusted using only two camera settings: (i) the length of the exposure and (ii) the flash intensity.

But the article also says:

The key to changing the flash ratio is using the right combination of flash exposure compensation (FEC) and ordinary exposure compensation (EC).

Don't these two statements say different things? Is the choice dictated by personal preference? Or do they mean the same thing?

EDIT: I just came across yet another suggestion to change the ISO instead (http://www.scribd.com/doc/60727727/42/If-You-Can%E2%80%99t-Find-the-Light-Create-the-Light):

See if you have a good blend. If the background is too dark, try moving up to the next level of ISO. Too bright a background? Lower the ISO.

This is really confusing for beginners like me!

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They both say the same thing, though in a different order. You control the length of the exposure using ordinary exposure compensation (assuming you are in Aperture Priority mode), and you control the intensity of the flash using flash exposure compensation.

Essentially they have described the general terms in the first part, and the specific settings in the second.

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Short and to-the-point. It took me a while to understand, but it makes sense now. – anon Oct 27 '11 at 3:26

This is odd on the face of it. The term "ratio" when used in reference to flash typically refers to the relative intensity of flashes in a given setup. So, for example, a 2:1 ratio involves two lights one at (say) f/8 and the other at f/5.6. Bear with me if you are unfamiliar with using f-stop settings to refer to light intensity. The point is, one light emits twice as much light as the other. In a typical portrait situation, a main light, positioned at 45 degrees to the right of the subject and a fill at 45 degrees and to the left might be set in a 2:1 ratio where the main light illuminates and gives definition, while the fill light softens the image by adding light in the harsh shadows.

Your example is about mixing a controlled light source (the flash) with an uncontrolled one (ambient). Clearly, the more flash there is relative to the strength of the ambient, the less the ambient matters. Expressing this as a ratio, however, is unusual. But, the "more flash" part can be achieved by using flash exposure compensation to increase the output of the flash while using exposure compensation to reduce the effective amount of light reaching the sensor.

However... The most important factor governing the mixing (or not) of ambient light is the shutter speed. The higher the shutter speed, up to the flash sync speed, the less the ambient light will factor into the exposure. Say you have determined that a correct flash exposure is whatever your flash puts out at f/5.6. It could be that a correct exposure for ambient light is 1/30 sec at f/5.6. So, following along this example, if you expose at 1/30 of a second, the ambient light will be a major factor in your image. However, if you expose at 1/60, it will be half the factor, at 1/100 a quarter (fudging because cameras normally switch from 1/60 to 1/100 instead of 1/120). Anyhow, by the time you reach sync speed at about 1/200 second (depending on your flash), you will have reduced the influence of the ambient light to a fraction of what it would have been at the "correct" ambient exposure.

Note: The reason I cited only an aperture for the correct flash exposure is that the flash duration is so short that the shutter speed is irrelevant -- only that you keep it at or longer than the flash sync speed.

Having gone through all of this, I would say two things:

1. Ratios among several lights can (are) important to understand. However, I don't think this is what you were talking about in your question.
2. The most important thing to understand about mixing flash and ambient light is that faster shutter speeds reduce the effect of ambient light.
3. How you reduce the shutter speed is up to you. My method is to set the camera on manual and decide for myself. This implies that I do one of two other things: Either meter the flash to help me decide (for studio strobes); or use TTL flash so a speedlight can adjust its output automagically (and, amazingly, it will!)

I hope I haven't overcomplicated this, but I would recommend not using FEC/EC just to jerk the camera into using different shutter speeds. That gives the camera more control than you want. Try M. You'll find yourself liking it.

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Sync speed does not depend on flash, it depends on shutter mechanism (located in body, or in lens as with leaf shutters). – Imre Oct 25 '11 at 21:16
I didn't mean to imply that sync speed depended on the flash, but it is irrelevant without one. That's why it is so often called "flash sync speed". The whole point of this is that there is a maximum shutter speed your camera/lens setup can achieve and still guarantee the flash will have discharged while the shutter is completely open. This is the maximum flash sync speed. Because it is about the speed with which the camera can sync with the flash. – Steve Ross Oct 25 '11 at 22:01
Using flash is implied for the whole paragraph, so as it stands the dependency statement is quite confusing. – Imre Oct 25 '11 at 22:09
Cameras substitute 1/120 with 1/125; 1/100 is only 2/3 steps faster than 1/60. – Imre Oct 25 '11 at 22:11
This is distracting from the original question but for some reference on why I used the term "flash sync", the original term was "flash X-sync". This is widely used in specs -- look here on dpreview: dpreview.com/reviews/Canoneos600d/page2.asp where the specs on the Canon 600D are listed. Anyhow, let's stay with the topic the OP raised, which is ratios. – Steve Ross Oct 25 '11 at 23:02

Which of these ways you use depends on metering mode for your camera and flash.

If you use both in manual mode, you can adjust aperture and flash intensity. The camera does not calculate anything, so "compensation" is meaningless.

If you use automatic exposure on body and flash (e.g. aperture priority mode and TTL flash), you use compensation values to change ratios of ambient and flash lighting. You can't enter flash power directly, and simple change of aperture will make camera adjust both flash power and shutter time, so you must use compensation values to actually change ratios.

Using body and flash in mixed auto/manual mode as is not very common and results might be unpredictable. However, there is still only one way that makes sense when adjusting exposure ratio - corresponding to what was previously outlaid for the flash/body in its mode.

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Actually, on my D80 and SB700 on aperture priority and TTL mode for the flash, I still can dial in compensation (as well as exposure compensation). Perhaps the behavior you describe (no ability to change flash compensation) is Canon-specific? – anon Oct 27 '11 at 3:24
@anon aperture priority and TTL flash is the example I brought under "when you use compensation", I don't see any contradiction here? – Imre Oct 27 '11 at 4:11