One stop, one doubling
A neutral density filter is a piece of grey glass that removes light evenly across the visible spectrum without shifting colour. Its whole purpose is to let you use a shutter speed or an aperture you could not otherwise use in the available light — a four-minute exposure at noon, or f/1.4 outdoors.
The arithmetic rests on a single definition. A stop is a factor of two in the amount of light reaching the sensor. Remove one stop and you must double the exposure time to compensate; remove ten and you must multiply it by 2¹⁰ = 1,024. That is the entire calculation: new time equals metered time times two raised to the number of stops.
What makes it awkward in the field is that the industry never settled on one way of writing the filter's strength. The same filter can be sold as a "10-stop", as an "ND1000", and as a "3.0 density" filter, and a photographer with two brands in the bag will have both notations in front of them. Converting between the three is the first thing this calculator does.
Three notations for the same piece of glass
Stops is the direct notation: the exponent itself. A 6-stop filter multiplies exposure by 64.
ND number is the exposure multiplier written out — the reciprocal of the transmission. ND8 passes one-eighth of the light and multiplies exposure by 8, which is 3 stops because 2³ = 8. To convert, take the base-2 logarithm: S = log₂(ND). Note that the round marketing numbers are rounded: a filter sold as ND1000 is really 2¹⁰ = 1,024, and one sold as ND64000 is 2¹⁶ = 65,536. The difference is well under a tenth of a stop and does not matter, but it explains why log₂(1000) comes out at 9.97 rather than 10.
Optical density is the logarithmic scale used in photometry and printed on Lee, Kodak Wratten and most cinema filters: D = log₁₀(1 ÷ T), where T is the transmittance. A density of 0.3 passes half the light because log₁₀ 2 = 0.301, so each 0.3 of density is one stop and S = D ÷ 0.30103. A 1.2 ND is four stops; a 3.0 ND is ten.
Stacking adds stops, it does not multiply them. Because transmissions multiply and logarithms of products add, a 3-stop over a 10-stop gives 13 stops, not 30 and not 9. In ND-number terms, ND8 × ND1000 = ND8000; using the exact factors those filters really have, 8 × 1,024 = 8,192 and log₂(8,192) = 13. This is the single most common mistake in filter arithmetic, and it always errs enormously.
The exposure compensation field feeds the same exponent. If you also close the aperture by a stop, or want the frame a third of a stop brighter than the meter suggests, add it in stops and the multiplier takes care of itself.
Worked example: a 10-stop and a 3-stop over a metered 1/60 s
You are photographing a waterfall. With no filter, the camera meters 1/60 s at f/11, ISO 100. You want the water to smooth out, so you fit a 10-stop filter and stack a 3-stop over it.
- Convert the base time to seconds. 1/60 = 0.016667 s.
- Add the stops. 10 + 3 = 13 stops.
- Find the multiplier. 2¹³ = 8,192.
- Multiply. 0.016667 × 8,192 = 136.5 s, which is 2 min 16 s. Set the camera to bulb and time it.
- Cross-check with ND numbers. ND1000 × ND8 = ND8000 ≈ 8,192, and 0.016667 × 8,192 = 136.5 s. ✓
Now suppose you are shooting film with a data sheet giving a Schwarzschild exponent of p = 1.25. The measured 136.5 s becomes 136.51.25: log₁₀ 136.5 = 2.1351, times 1.25 is 2.6689, and 102.6689 = 466 s, or 7 min 46 s. Reciprocity failure has more than tripled an already long exposure — which is exactly why long-exposure film photographers carry the data sheet and not just the filter.
Reading the result and what it will not tell you
Meter first, then fit the filter. Dense ND filters defeat both the camera's meter and its autofocus, so the correct sequence is: compose, focus, meter, switch to manual focus and manual exposure, then fit the filter and dial in the corrected time. Anything metered through a 10-stop filter is a guess dressed up as a reading.
Filters are not exactly what they claim. Manufacturing tolerance on a strong ND is commonly a fraction of a stop, and strong filters frequently pass a colour cast — most often magenta or blue — that has to be corrected in raw processing. Treat the calculated time as the starting point for a test frame, then adjust. The stronger the filter, the more likely you will need a third to a half stop of adjustment, which is what the compensation field is for once you have measured your own filter.
Beyond about half a minute, other things start to matter. Sensor noise rises with exposure time and with sensor temperature; long-exposure noise reduction doubles the time you stand there; and viewfinder light leaking backwards into the camera can fog the frame on a mirrorless or DSLR body with the eyepiece uncovered. Above 30 minutes most cameras will not time the exposure at all and you need bulb mode plus an intervalometer.
Reciprocity failure is a film-only correction. Digital sensors accumulate charge linearly and need no exponent. Film does not: at very low light levels, silver halide crystals lose sensitivity, so a doubling of time produces less than a doubling of density. The Schwarzschild relation tp models this with a single exponent from the emulsion's data sheet, and it only applies to exposures already long enough for the effect to appear — this calculator applies it above one second and leaves shorter times alone, since raising a fraction to a power greater than one would shorten it, which is the opposite of the physical effect.
ND notation conversion and corrected times from 1/125 s
| Stops | ND number | Optical density | Exposure factor | From 1/125 s |
|---|---|---|---|---|
| 1 | ND2 | 0.3 | 2× | 0.016 s |
| 2 | ND4 | 0.6 | 4× | 0.032 s |
| 3 | ND8 | 0.9 | 8× | 0.064 s |
| 4 | ND16 | 1.2 | 16× | 0.128 s |
| 5 | ND32 | 1.5 | 32× | 0.256 s |
| 6 | ND64 | 1.8 | 64× | 0.512 s |
| 7 | ND128 | 2.1 | 128× | 1.02 s |
| 8 | ND256 | 2.4 | 256× | 2.05 s |
| 9 | ND512 | 2.7 | 512× | 4.10 s |
| 10 | ND1000 | 3.0 | 1,024× | 8.19 s |
| 11 | ND2000 | 3.3 | 2,048× | 16.4 s |
| 12 | ND4000 | 3.6 | 4,096× | 32.8 s |
| 13 | ND8000 | 3.9 | 8,192× | 1 min 5 s |
| 14 | ND16000 | 4.2 | 16,384× | 2 min 11 s |
| 15 | ND32000 | 4.5 | 32,768× | 4 min 22 s |
| 16 | ND64000 | 4.8 | 65,536× | 8 min 44 s |
ND numbers on filter rings are rounded to marketing-friendly values; the true factors are the exact powers of two in the fourth column. The density column is stops × 0.30103, rounded to one decimal.
Mistakes that ruin a long exposure
- Multiplying ND numbers when stacking. Stops add. A 6-stop and a 10-stop give 16 stops, not 60. Convert to stops first, add, then convert back.
- Metering through the filter. Past about six stops, camera meters become unreliable and autofocus stops working. Meter and focus first, lock everything to manual, then fit the glass.
- Leaving the eyepiece uncovered. On a multi-minute exposure, light entering the viewfinder can reach the sensor and fog the frame. Most cameras ship with a small eyepiece cover for exactly this.
- Forgetting the aperture change. If you stop down from f/8 to f/16 after metering, that is two more stops of exposure. Put it in the compensation field rather than trying to hold it in your head.
- Applying reciprocity correction to a digital file. The correction models a property of photographic emulsion. Sensors respond linearly, so p = 1 is correct for digital.
- Trusting a strong filter's rating exactly. Tolerances of a third of a stop are common, and colour casts are near universal on 10-stop filters. Shoot a test frame and note your own filter's real offset.
- Assuming a longer exposure always smooths water more. Beyond the point at which the moving elements have crossed the frame, extra time changes the water very little and only adds noise and risk of movement. Test rather than reaching for the strongest filter you own.
Why every stop is a factor of two
The stop is a base-2 logarithmic unit for light. Aperture, shutter time and ISO all move in stops, which is what allows them to be traded against one another: one stop slower shutter, one stop wider aperture and one stop lower ISO all change the exposure by the same factor. Optical density is the same idea in base 10, which is why the conversion between them is the constant log₁₀ 2 = 0.30103, and why 0.3 density and one stop are the same thing.
Where the ND filter sits in the exposure decision
A neutral density filter is one of three ways to reach a long exposure, and the other two cost nothing. Lowering ISO to base and stopping the aperture down are always available first — though stopping down has an optical price, since diffraction begins to limit resolution past the point the hyperfocal distance calculator flags. The filter is what you reach for when ISO and aperture have run out and you still need more time.
Depth of field is usually the reason you cannot simply close the lens further: if a specific foreground has to be sharp, the aperture is already decided, and the depth of field calculator tells you where the limits fall. On the other side, exposures long enough to need an ND at night run into a different constraint entirely — star trailing — which the 500 rule calculator quantifies. And if the plan is a sequence rather than a single frame, the time-lapse interval calculator handles the relationship between shutter time, interval and clip length.
Graduated ND filters are a separate tool and are not covered by this arithmetic. A grad darkens only part of the frame to fit a high-contrast scene into the sensor's dynamic range; it does not lengthen the exposure of the whole picture, and applying a whole-frame correction to one would underexpose everything below the transition. Variable ND filters, which are two polarisers rotated against one another, do lengthen the exposure but their marked stop values are approximate and they can produce a cross-shaped darkening at the extreme end of their range. For both, meter a test frame rather than trusting the marking.