Welcome to the second appendix to the Technical Field Guide for the Discerning Analog Photographer. In Appendix A I set out the push and pull development formulae used throughout the main field guide (which you may purchase here).

In the field guide’s chapters on multiple exposures, we suggested that beginners multiply their film’s ISO by the number of desired exposures in order to obtain the exposure index for each shot. However, we subsequently used a (sometimes infinite) mix of exotic EIs to obtain a perfect exposure for a given roll.

There are often times where having two exposures on the same frame at the same EI doesn’t best serve the image. For instance, when one wants to put a mountain into a silhouette, it might make sense to expose the silhouette one-two stops less than the mountain, relying on reciprocity failure in the personage in order to keep a clean “canvas” for the generic mountainscape.

Of course, for the sake of brevity nested in my desire to keep the guide “handy” at under 1000 pages, we swept the calculations for these EIs under the rug. So, in this article—Appendix B—I walk you through the formulae used to meter for both finite multiple exposures, and the formula for infinite, descending multiple exposures hinted at on pages 42, 334, 472, 574 and 786 of the guide.

Metering for finite multiple exposures

All of our examples for properly metering for finite multiple exposures use the formula:

EI = \frac{1}{\sum_{i=1}^n \frac{1}{EI_i}}

where EI is the speed at which you are shooting your film; n is the number of exposures to make a single image, p; and E_i is the exposure index at which the i^{th} exposure is metered.

This formula follows from the simple fact that ISO is linear.

Note*: as one lets n tend to infinity this formula still holds, allowing you to use this same formula to meter for infinite exposures with exotic mix of EIs.

*At some point you will need an infinitely long exposure to pull this off, as EI must get arbitrarily close to \infty.

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Metering for Infinite, descending Multiple Exposures

As an appeal to more expirimental photographers with considerable time on their hands, we proposed the following, simplified, multiple exposure metering formula, where we let n range from 1 to \infty.

EI_n := 2^nEI

where the photographer meters the first exposure down by one stop, and reduces by one stop each successive exposure until either taking infinite exposures or his or her aperture ceases to get smaller (at fastest shutter speed). We can prove this satisfies equation 0.3 as promised in p.344 in the text:

\frac{1}{\sum_{i=1}^{\infty} \frac{1}{2^i EI}} =\frac{1}{\frac{1}{EI}\sum_{i=1}^{\infty} \frac{1}{2^i }}

= \frac{EI}{\sum_{i=1}^{\infty} \frac{1}{2^i }}

Therefore, it suffices to show \sum_{i=1}^{\infty} \frac{1}{2^i }=1.

We leave this as an exercise to the reader.

When in doubt…

Just create a double exposure of a neon sign over someone’s face. Everybody loves that no matter what. Example images (devoid of neon), created with the techniques described here follow below.

I hope this is of help in clarifying some of the field guide’s assumed values. In the next appendix, we will cover accounting for lens movements when using splitting masks.

Until then, keep shooting to infinity and beyond!

~ David

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About the author

David Allen

Tried to use photography to replace mathematics. Was paid for mathematics.


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