Black and white work frequently calls for two filters at once: a contrast filter to separate tones, plus a polariser to deepen a sky or kill reflections, or a neutral density filter to reach a longer shutter speed. Combining them is straightforward in principle, but the exposure arithmetic trips up anyone who treats it as addition, and the optical cost of the extra glass is real and quantifiable.

Factors Multiply, Stops Add

Every filter carries a filter factor: the multiplicative amount by which it reduces light reaching the film. A factor of 2 halves the light and costs one stop; a factor of 4 costs two stops; a factor of 8 costs three; a factor of 16 costs four. The relationship is logarithmic — stops equal the base-2 logarithm of the factor — which is why factors and stops behave differently when filters are combined. When filters are stacked, their factors multiply while their stop values add. The combined factor of two filters is the product of the individuals, and the simplest way to avoid a mental-arithmetic slip is to add the published stops instead of multiplying the factors.

The common daylight set, with Wratten designations and factors, spreads as follows:

Filter	Wratten	Factor	Stops
Yellow (K2)	No. 8	2	1
Yellow-green (X1)	No. 11	4	2
Deep yellow (G)	No. 15	~2.5	1⅓
Orange (YA3)	No. 21	4	2
Red (A)	No. 25	8	3
Deep red (F)	No. 29	16	4
Green	No. 58	~6	~2⅔
Blue (C5)	No. 47	~6	~2⅔

These are the figures that matter when you stack: a Wratten No. 25 over a polariser is not 8 plus 2.5, it is 8 times 2.5.

Two Worked Examples

A Kodak Wratten No. 25 red filter has a factor of 8, three stops. A polariser does not have one fixed factor — manufacturers vary, with B+W and Hoya specifying roughly 2.3 to 2.8 (about 1.2 to 1.5 stops) and Tiffen quoting 1.5 to 2 stops — but the commonly quoted nominal is a factor of about 2.5, one and a third stops. Stacked, the factor is 8 × 2.5 = 20, not 10.5. Twenty corresponds to roughly 4.3 stops, which is also three plus one and a third. Add the stops; do not multiply in your head.

A second pairing makes the same point with the tonal intent attached. An orange No. 21 (factor 4, two stops) plus a polariser (about one and a third stops) gives a factor product of 4 × 2.5 = 10, near three and a third stops. You do not buy those stops for nothing: a red No. 25 typically renders a clear blue sky two to three zones darker than panchromatic film records it unfiltered, separating cloud from sky; the orange-plus-polariser stack drops the sky around two zones with a gentler horizon-to-zenith gradient. A green No. 58 instead lightens foliage. Filters lighten their own colour and darken the complementary — that zone shift, not the light loss, is the reason to carry the glass. Ansel Adams sets this out in The Negative (1981); Kodak’s Photographic Filters Handbook (Publication B-3) is the manufacturer reference for the Wratten spectral and factor data.

Linear Versus Circular, and Why TTL Can Lie

A polariser complicates the figure because its effect depends on orientation: the nominal factor applies near the non-polarising angle, and rotating toward maximum effect against a clear sky shifts the apparent loss. Through-the-lens metering tracks that change — but only if the polariser is the circular kind. A linear polariser corrupts TTL metering and phase-detect autofocus on any camera that uses a semi-silvered beamsplitter mirror, because the intensity reaching the metering and AF sensors depends on the polarisation angle of the light hitting that mirror. A circular polariser places a quarter-wave plate behind the linear element, converting the output to circularly polarised light so the reflected intensity becomes independent of orientation. That is the entire reason circular polarisers exist. On a beamsplitter body, trust TTL only with a circular polariser; with a linear one, meter without it and add the factor by hand.

The Optical Penalty, in Numbers

Each filter adds two air-to-glass surfaces, and roughly 4% of the light reflects at each uncoated interface (the Fresnel result), so an uncoated surface transmits about 96%. Two uncoated filters present four surfaces: 0.96⁴ ≈ 0.849, around 15% of the light lost to surface reflection alone, before counting either filter’s own spectral absorption. Multicoating drops per-surface reflectance to roughly 0.2 to 0.5%; at 0.997 per surface, 0.997⁴ ≈ 0.988, about 1% lost. That difference is the practical argument for coating a stack.

Reflection that does not simply disappear becomes flare. The worst offender is the air gap between the two stacked filters: even well-coated outer faces leave two surfaces facing each other a millimetre or two apart, bouncing a bright source back and forth. A street lamp just outside the frame will throw a ghost — a faint inverted copy of the source — straight onto the negative, regardless of how good the coatings on the outward faces are.

Vignetting and the Long-Exposure Trap

The second mechanical penalty is vignetting. Standard filter rings measure about 5 to 7 mm; slim or low-profile rings run roughly 3.2 to 5 mm. Stacked, two rings extend the assembly forward and the front edge can intrude into the image circle. On full-frame (135), mechanical vignetting from a stack typically begins to bite below about 28 mm and becomes severe below 24 mm; step-up and step-down rings add their own height to the stack. The discipline on wide lenses is a single filter wherever the result allows.

Neutral density behaves more simply on the exposure side — its density is deliberately flat across the visible spectrum, so a 3-stop ND (ND8, factor 8) on a 3-stop red filter totals six stops. The trap is what those six stops do to shutter speed. Reaching multi-second exposures this way pushes many films into reciprocity failure: Ilford films want correction beyond about one second, and Fomapan 100 and 400 fail steeply, needing substantial added exposure for multi-second times. The combined metered factor is only the starting point; add the film’s published reciprocity correction on top of it.

Filter factors after Kodak’s Wratten/B-3 data; tonal and Zone System guidance after Ansel Adams, The Negative (1981).