Post by PeterW on Nov 21, 2006 19:22:29 GMT -5
Following the thread about fungus, and Ron Herron’s posting, I clicked on the link to his excellent HOW TO pages. Great stuff, Ron, and I liked your simple but accurate explanation of reciprocity. However, dealing with f-stops there’s a point that you seldom see mentioned anywhere, the breakdown of reciprocity at the maximum aperture of some lenses.
In many books on photography you will read that the f-number is the diameter of the aperture divided into the focal length of the lens. This is a reasonably accurate, guide but isn’t strictly true because the amount of light passed by an aperture depends on its area, which follows a square law whereas diameter is linear. So we ought really to compare aperture areas, not diameters to get reciprocity.
The principle of a stop passing twice the amount of light as the one numerically above it doesn’t necessarily hold good at the largest marked aperture, even on some famous lenses. For example, f/4.5 used to be a popular maximum aperture of a number of lenses in the 1930s, including the famous Tessar, but most makers continued the series with 5.6, 8, 11 and so on. F/4.5 doesn’t pass twice the light that f/5.6 does. If we square the numbers we get 20.25 and 31.36, so the increase in exposure from f/5.6 to f/4.5 is 1.548 to 1, not 2 to 1. Similarly, a number of lenses had a maximum aperture of f/6.3, but then carried on with 8, 11, 16 and so on. In this case, the increase in exposure between f/8 and f/6.3 is 1.61 to 1, not 2 to 1. I think it’s quite likely that these makers didn’t want to open the lens beyond its capability, but equally didn’t want to confuse photographers by using a completely unfamiliar series of numbers which didn’t appear on any light meters.
All very neat and tidy, but just to put the cat spitting and squealing among the pigeons, I have an old Goerz Double Anastigmat lens that has an f-stop series running 4.8, 5.5, 6.3, 11, 16, 22 and 32. If we square these numbers we get 23.04, 30.25, 39.69, 121, 256, 484 and 1024. There’s no way, even with rounding off, that these squares double as you go up the series. I don’t know how or why Goerz decided on them – unless it was some complicated formula that took into account dispersion, diffraction, the number of air-to-glass surfaces and possibly other criteria as well. Phases of the moon or the juxtaposition of planets perhaps?
A relatively unimportant thing, and of little practical consequence, but just out of curiosity I checked the two largest stops on a Canon f/1.8 lens. They are f/1.8 and f/2.8. Square these numbers and we get 3.24 and 7.84. So assuming the f/1.8 aperture is accurate the next smaller stop should be the square root of 6.48, which is 2.54. The next largest should work out at the square root of 12.96 which is 3.6. But it isn’t, it’s f/4. But then an f-stop series that ran 1.8, 2.5, 3.6 and so on would look very unfamiliar.
If Canon had continued the series numerically downward, the next larger stop to f/2.8 would be f/1.979 – near enough f/2, which it is on quite a few lenses. But if the lens could be opened up to f/1.8 without appreciable increase in aberrations, why not open it up, even if the maximum aperture doesn’t quite agree with reciprocity, and theoretically gives slight over exposure? It sounds much better for marketing, and with auto exposure, aperture priority and electronic stepless shutter speeds the exposure at maximum aperture is probably pretty accurate.
Just another unimportant detour into the by-ways of lenses, but I thought some of you might find it mildly interesting. If you're not already bored see the Camera Chit-Chat pages on my website for a more rambling discourse on the subject.
PeterW
In many books on photography you will read that the f-number is the diameter of the aperture divided into the focal length of the lens. This is a reasonably accurate, guide but isn’t strictly true because the amount of light passed by an aperture depends on its area, which follows a square law whereas diameter is linear. So we ought really to compare aperture areas, not diameters to get reciprocity.
The principle of a stop passing twice the amount of light as the one numerically above it doesn’t necessarily hold good at the largest marked aperture, even on some famous lenses. For example, f/4.5 used to be a popular maximum aperture of a number of lenses in the 1930s, including the famous Tessar, but most makers continued the series with 5.6, 8, 11 and so on. F/4.5 doesn’t pass twice the light that f/5.6 does. If we square the numbers we get 20.25 and 31.36, so the increase in exposure from f/5.6 to f/4.5 is 1.548 to 1, not 2 to 1. Similarly, a number of lenses had a maximum aperture of f/6.3, but then carried on with 8, 11, 16 and so on. In this case, the increase in exposure between f/8 and f/6.3 is 1.61 to 1, not 2 to 1. I think it’s quite likely that these makers didn’t want to open the lens beyond its capability, but equally didn’t want to confuse photographers by using a completely unfamiliar series of numbers which didn’t appear on any light meters.
All very neat and tidy, but just to put the cat spitting and squealing among the pigeons, I have an old Goerz Double Anastigmat lens that has an f-stop series running 4.8, 5.5, 6.3, 11, 16, 22 and 32. If we square these numbers we get 23.04, 30.25, 39.69, 121, 256, 484 and 1024. There’s no way, even with rounding off, that these squares double as you go up the series. I don’t know how or why Goerz decided on them – unless it was some complicated formula that took into account dispersion, diffraction, the number of air-to-glass surfaces and possibly other criteria as well. Phases of the moon or the juxtaposition of planets perhaps?
A relatively unimportant thing, and of little practical consequence, but just out of curiosity I checked the two largest stops on a Canon f/1.8 lens. They are f/1.8 and f/2.8. Square these numbers and we get 3.24 and 7.84. So assuming the f/1.8 aperture is accurate the next smaller stop should be the square root of 6.48, which is 2.54. The next largest should work out at the square root of 12.96 which is 3.6. But it isn’t, it’s f/4. But then an f-stop series that ran 1.8, 2.5, 3.6 and so on would look very unfamiliar.
If Canon had continued the series numerically downward, the next larger stop to f/2.8 would be f/1.979 – near enough f/2, which it is on quite a few lenses. But if the lens could be opened up to f/1.8 without appreciable increase in aberrations, why not open it up, even if the maximum aperture doesn’t quite agree with reciprocity, and theoretically gives slight over exposure? It sounds much better for marketing, and with auto exposure, aperture priority and electronic stepless shutter speeds the exposure at maximum aperture is probably pretty accurate.
Just another unimportant detour into the by-ways of lenses, but I thought some of you might find it mildly interesting. If you're not already bored see the Camera Chit-Chat pages on my website for a more rambling discourse on the subject.
PeterW
. I left design offices and went into full-time writing, and later photography, in 1964 so my maths is way out of date now, and very rusty. I never learned Boolean or Polish algebra, mine was based on a book 'The Algebra of Rhadakrishnan', first published in the early 1900s, and I always did get muddled up between logs to base 10 and logs to base e. 
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