Hopefully this will boil most of it down:
Disclaimer: throughout, I will use the misnomer "focal length". Well, technically you can't have a "focal" length with something that does not actually focus light. I use it as shorthand to describe the distance between the pinhole and the film. I suppose a more proper term would be "radius", but I'm not using that because it sounds confusing and is a bit harder to visualize.
#1: Pick a film/paper size. 4x5, 8x10, 2.25x3.25, etc. The possibilities are endless. (My "under destruction" one happens to be 6x20 inches.)
#2: Pick a focal length. This should be primarily based on angle of view that you want. Wide, normal, or tight. (Or super wide or super tight....)
This is easiest to visualize if you base it on the wide dimension of your film instead of the diagonal.
A 50mm lens on a 24x36mm camera (AKA 35mm in shorthand) sees about 40 degrees horizontally. An 80mm lens sees about the same on 6x6, like your Mamiya, or on 6x4.5. 100mm on 6x7. 127mm on 6x9. 180mm on 4x5. 250mm on 5x7. 360mm on 8x10.
These are the focal lengths that will give you an angle of view that is similar to a 50mm lens on a 35mm camera. For a given format, if you use a shorter focal length, you get a wider AOV, and vice versa.
Here is the first formula, used to figure out what your angle of view will be at a given focal length and a given film width: 2 x Arctangent (width of film / 2 x focal length) = Angle of view. Arctangent is usually notated as TAN-1 on calculators.
Make sure that the focal length and the film width are expressed in the same system of units.
So, you use this formula to make your second big decision: what focal length you want.
#3: Calculate for approx. vignetting, and make adjustments if you feel the need.
Basically, the image circle will be 3.5 x focal length.
This means that ultra wides will vignette worse (better?) than longer lenses. You can try to avoid this, just deal with it, or even exaggerate it by your choices of format and focal length.
#4: Calculate the optimal (sharpest) pinhole size for your chosen focal length. Use metric units for this formula.
optimal size = 1.9 x sq. root of (focal length x average wavelength of light by which the film/paper will be exposed).
Make sure that the focal length and the wavelength are expressed in the same exact units for this equation. Millimeters is probably easiest. 550 nanometers goes into the equation for the wavelength, and as just mentioned, is converted to 0.00055 millimeters so it jives with the focal length. 550nm is used because it is a wavelength (color) of light to which pretty much all black and white films are sensitive. It is almost yellow, with a hint of green; a fairly safe bet to use for panchromatic film (AKA plain-ol' black and white film like FP4, HP5, etc.). You might want to use a lower number for blue/green sensitive materials like photo paper or litho film.
So, now you have the size hole you need to get the sharpest picture you can, using a very old formula. There are many, many people since who have come up with different formulas that might be better, but I just use this one since I find it easy to remember.
If you don't want it to be as sharp as possible, then use a different size.
You can order a laser-drilled pinhole from Lenox Laser, make your own using a needle, or have a jeweler drill one for you. I'm sure you can look on the web and find various methods of making the holes, but I won't go into it because I am just trying to cover the basic calculations that you might find useful; not the actual construction of the camera.
#5: Figure out which f/stop your aperture is closest to.
You probably already know this one. Simply divide your focal length by the size of the aperture (AKA the pinhole). Make sure that both are expressed in the same exact units.
Here is a list of the 1/3 stops that you might end up using, starting at f/45. I start there because it is on most light meters: The first f/stop in each row is a "full" stop:
45, 50, 56
64, 72, 80
90, 100, 112
128, 144, 160
180, 200, 224
256, 288, 320
360, 400, 448
512, 576, 640
...and so on and so forth. You can figure out the rest given that information. Just look for the pattern and continue it if you really end up with an aperture that small.
The result of this equation will probably fall in between these f stops. In reality, your camera will not be accurate to the third stop anyhow, so close is good. But, to make it easy on yourself, just use one of these "actual" f stops. If using neg. film, use the next highest third stop. For pos. film, use the next lowest.
#6: Not really an equation, but look at the reciprocity chart for your film so you have an idea of a starting point for compensation. Of course, your results will vary a bit from the manufacturer.
#7: Don't expect anything to be spot on based on calculations alone. The fine tuning (if that's what you really want) comes with practice.
So; the final equation: 1 + 1 = 3.
Disclaimer: throughout, I will use the misnomer "focal length". Well, technically you can't have a "focal" length with something that does not actually focus light. I use it as shorthand to describe the distance between the pinhole and the film. I suppose a more proper term would be "radius", but I'm not using that because it sounds confusing and is a bit harder to visualize.
#1: Pick a film/paper size. 4x5, 8x10, 2.25x3.25, etc. The possibilities are endless. (My "under destruction" one happens to be 6x20 inches.)
#2: Pick a focal length. This should be primarily based on angle of view that you want. Wide, normal, or tight. (Or super wide or super tight....)
This is easiest to visualize if you base it on the wide dimension of your film instead of the diagonal.
A 50mm lens on a 24x36mm camera (AKA 35mm in shorthand) sees about 40 degrees horizontally. An 80mm lens sees about the same on 6x6, like your Mamiya, or on 6x4.5. 100mm on 6x7. 127mm on 6x9. 180mm on 4x5. 250mm on 5x7. 360mm on 8x10.
These are the focal lengths that will give you an angle of view that is similar to a 50mm lens on a 35mm camera. For a given format, if you use a shorter focal length, you get a wider AOV, and vice versa.
Here is the first formula, used to figure out what your angle of view will be at a given focal length and a given film width: 2 x Arctangent (width of film / 2 x focal length) = Angle of view. Arctangent is usually notated as TAN-1 on calculators.
Make sure that the focal length and the film width are expressed in the same system of units.
So, you use this formula to make your second big decision: what focal length you want.
#3: Calculate for approx. vignetting, and make adjustments if you feel the need.
Basically, the image circle will be 3.5 x focal length.
This means that ultra wides will vignette worse (better?) than longer lenses. You can try to avoid this, just deal with it, or even exaggerate it by your choices of format and focal length.
#4: Calculate the optimal (sharpest) pinhole size for your chosen focal length. Use metric units for this formula.
optimal size = 1.9 x sq. root of (focal length x average wavelength of light by which the film/paper will be exposed).
Make sure that the focal length and the wavelength are expressed in the same exact units for this equation. Millimeters is probably easiest. 550 nanometers goes into the equation for the wavelength, and as just mentioned, is converted to 0.00055 millimeters so it jives with the focal length. 550nm is used because it is a wavelength (color) of light to which pretty much all black and white films are sensitive. It is almost yellow, with a hint of green; a fairly safe bet to use for panchromatic film (AKA plain-ol' black and white film like FP4, HP5, etc.). You might want to use a lower number for blue/green sensitive materials like photo paper or litho film.
So, now you have the size hole you need to get the sharpest picture you can, using a very old formula. There are many, many people since who have come up with different formulas that might be better, but I just use this one since I find it easy to remember.
If you don't want it to be as sharp as possible, then use a different size.
You can order a laser-drilled pinhole from Lenox Laser, make your own using a needle, or have a jeweler drill one for you. I'm sure you can look on the web and find various methods of making the holes, but I won't go into it because I am just trying to cover the basic calculations that you might find useful; not the actual construction of the camera.
#5: Figure out which f/stop your aperture is closest to.
You probably already know this one. Simply divide your focal length by the size of the aperture (AKA the pinhole). Make sure that both are expressed in the same exact units.
Here is a list of the 1/3 stops that you might end up using, starting at f/45. I start there because it is on most light meters: The first f/stop in each row is a "full" stop:
45, 50, 56
64, 72, 80
90, 100, 112
128, 144, 160
180, 200, 224
256, 288, 320
360, 400, 448
512, 576, 640
...and so on and so forth. You can figure out the rest given that information. Just look for the pattern and continue it if you really end up with an aperture that small.
The result of this equation will probably fall in between these f stops. In reality, your camera will not be accurate to the third stop anyhow, so close is good. But, to make it easy on yourself, just use one of these "actual" f stops. If using neg. film, use the next highest third stop. For pos. film, use the next lowest.
#6: Not really an equation, but look at the reciprocity chart for your film so you have an idea of a starting point for compensation. Of course, your results will vary a bit from the manufacturer.
#7: Don't expect anything to be spot on based on calculations alone. The fine tuning (if that's what you really want) comes with practice.
So; the final equation: 1 + 1 = 3.
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