There is no fixed calibration factor for meters - it varies from manufacturer to manufacturer. The ISO for light meters only gives a suggested range of calibration factors, and manufacturers don't even stick to that.
It's my understanding that ISO 2720:1974 gives a range for K of 10.6 to 13.4, but that was for meters in general. ISO 2721:1982 defined exposure for in-camera meters, and supposedly doesn't even mention a K parameter. Exposure is described in terms of
H. The definition can be used to work out an average reflectance, which works out to be 12.7%.
Here's the quick answer:
This is in the film section, so my remarks were about metering for film.
You are correct about the range of K given in 2720 - 10.6 to 13.4. I suggested that not all manufacturers used the same calibration constant, and that some went outside that range. Gossen uses 11.4*, Sekonic, Canon and Nikon use 12.5, Pentax and Minolta/successors use 14.
It's actually difficult to tie that K to a fixed reflectance value - there lies the long version of the answer - but because of the variation in K factor we can say for sure that one cannot fix 12.7% as the design reflectance, so there is no real need to go into it, but we can if you wish. There is a case to suggest that K=12.5 corresponds to 15% and K=14 corresponds to 18%, but it isn't rock solid - no correlation is.
There's a further complication in Kodak's use of Aim Density as a working film speed determination method. Kodak have given aim densities for an 18% grey card exposure for their colour negative films for a very long time. Here's the quick version of how it works:
You take a series of meter readings off a grey card, changing the EI over a two-stop range centred on the ISO speed, at 1/3 stop intervals and make a set of exposures of the grey card (in practice you take one reading and work out the set of exposures). Then you read the density of the negative using Status M filtration and select the neg which is the best match to Kodak's numbers. Other methods of determining working film speed also use an 18% card. They all have the same purpose - to calibrate your equipment and your method to enable you to get the results you desire. It's really the photographer's choice, and the type of film will also have an effect on what the optimum 'standard reflectance' or adjustment to an 18% reading should be.
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There's another way of looking at the standard reflectance thing: how it all came about.
It had a lot to do with the invention and use of Kodachrome pre WW2. Kodachrome needed fairly accurate exposure for best results, and exposure meters were the answer. Simple averaging reflected light meters. The meter manufacturers did their best to set the meter up so that if you pointed it at a nice scenic landscape it showed you an exposure that would be a good one. Although photographers learned how to cope with different mixes of light and dark - where to point their meter - there was a desire to produce a standard reflectance that could be carried in the field. Various densities of grey were tried, and the one that was successful in practice with Kodachrome was a 14% reflectance card in the hands of an experienced Kodachrome user. Metering with a 14% card gave the same results as metering what was regarded as a 'standard Kodachrome scene for metering purposes'. This tied in with a lot of other work that was done separately to study the range of average reflectance of real scenes. Therefore it could be said that those meters were 'calibrated to 14% grey' even though they weren't designed that way originally - it just happened that that is what worked well.
The danger with Kodachrome tends to be overexposure rather than underexposure - a slightly underexposed picture generally looks better than a slightly overexposed picture. Therefore some meter manufacturers might want to build in a slight amount of underexposure. Imagine two meters that give identical readings when meter A is pointed at a 14% reflectance and Meter B is pointed at a 12.5% reflectance (in the same illumination). You could say that Meter A is calibrated for 14% and Meter B for 12.5%. What happens if Meter B is pointed at a standard Kodachrome scene of 14% average reflectance? It gives slight underexposure, so it lies on the safe side. It's 'calibrated for 12.5%' but it is meant to be pointed at 14% because we've decided that the pretty scenes have 14% average reflectance - it is really calibrated for 14% with slight underexposure built in deliberately to protect the highlights. It's not a card that sets the value, it is the best estimate for the real world.
This general practice remained, because reversal film is usually the most demanding in terms of accurate exposure, and it is best to protect the highlights. Experienced photographers could still set their own adjustments based on their experience and/or formal film testing.
The use of an 18% card as a reference had a lot to do with Ansel Adams - 18% is used as a guide to what we perceive as mid grey. It is perfectly valid to use with the Zone System, because the 18% reflectance is taken into account. It is perfectly valid to use it with any methodical approach to exposure that requires a standard reflectance. So blame St Ansel.
*Some Gossen meters may use a higher value - I am not certain. The practical differences I have noted between Sekonic and Gossen meters (I have five Sekonics and four Gossens) suggest that the Gossen K factor is higher than Sekonic's.