Abstract
It is easy to assume that fast film is made of a different material that catches light better. This paper counts, in a model where grains count photons to be developed, that speed and graininess both come from the same grain size. No new theorem or law is claimed.
First, larger grains develop with less light: the light needed to develop half the grains is $\mathbf{3.672061}$ for diameter $1$, $0.918015$ for $2$ and $0.229504$ for $4$ (Section 2).
Second, doubling the diameter needs exactly a quarter of the light, two stops faster: the grain area quadruples, so the same light brings four times the photons; the ratio was $4$ within a relative difference of $10^{-9}$ (Section 2).
Third, and this is the core. The same doubling of diameter doubles graininess: density fluctuations between windows are $0.008980$ for diameter $1$ and $0.017767$ for diameter $2$, a ratio of $\mathbf{1.978537}$ (Section 3).
Fourth, graininess is set by how many grains fit in a window: a window holds $800$ grains of diameter $1$ and $200$ of diameter $2$, and the fluctuation formula $0.008839$, $0.017678$, $0.035355$ matches the random counts (Section 3).
Fifth, the steepness of the rise does not change with grain size: the span of light from $10$ % to $90$ % developed is $1.339538$, $\mathbf{0.583089}$ and $0.281522$ decades for $1$, $4$ and $16$ required photons, and the same when grain area is quadrupled (Section 4).
Sixth, the separator is whether one asks about grain size or the number of photons a grain needs to develop: grain size moves speed and graininess together; the required photon count moves the steepness (Section 4).
Keywords: silver-halide photography, film speed, graininess, Poisson distribution, photon count.
Scope (what is not claimed)
No new theorem or law is claimed: that larger emulsion grains give higher speed and coarser graininess, that a grain forms a developable centre from a few photons, and counting with the Poisson distribution, are all known in photographic chemistry. No real film is claimed: grain diameters, the $4$ required photons and the covered share $0.5$ are model values matching no film. Chemistry is not included: developer action, sensitising dyes and grain shape are not included. Image sensors are not entered: digital pixel size and noise are not treated. Relation to earlier papers: Paper 878 showed that the span of concentration from first effect to full effect is set not by the effective concentration but by the degree of cooperativity. This paper stands beside it. There the effective concentration set the curve’s position and cooperativity its span. Here grain size sets the position (speed) and the required photon count the span. What is added is finding the light that develops half the grains for three diameters, counting window fluctuations at random and measuring $1.978537$ times for doubled diameter, measuring the rise span for each required photon count, and placing the separator on grain size versus required photons.
1. Notation and the Question
Setting: a grain of area $a$ receiving light $H$ per area gets a Poisson number of photons with mean $Ha$. The grain is developed once it receives $k$ or more photons.
$$p(H)=P\bigl(\mathrm{Poisson}(Ha)\ge k\bigr),\qquad \sigma_{\text{window}}\approx\sqrt{\frac{p(1-p)}{n}}\times\frac{n\,a}{S}$$We took $k=4$ and grain area as the square of the diameter. Speed is the light $H_{50}$ developing half the grains, and graininess the standard deviation between windows of area $S=1600$ (half covered by $n$ grains) of the covered share. $400$ windows were generated at random.
Question: why is fast film grainy?
2. Grain Size and Speed
Computation 2.1: for each grain diameter, the light developing half the grains was computed.
| grain diameter | light for half developed | grains per window |
|---|---|---|
| $1$ | $3.672061$ | $800$ |
| $2$ | $0.918015$ | $200$ |
| $4$ | $0.229504$ | $50$ |
Observation 2.2 doubling the diameter needs exactly a quarter of the light. The ratio of light for half developed is $4.000000$, or $2.000000$ stops: the grain area quadruples and the same light brings four times the photons, so a quarter of the light gathers the same number.
3. Grain Size and Graininess
Computation 3.1: $400$ random windows were generated and the mean and fluctuation of the covered share counted.
| grain diameter | mean covered share | fluctuation between windows | formula |
|---|---|---|---|
| $1$ | $0.249855$ | $0.008980$ | $0.008839$ |
| $2$ | $0.248675$ | $0.017767$ | $0.017678$ |
| $4$ | $0.250200$ | $0.034633$ | $0.035355$ |
Observation 3.2 this is the core. The same doubling of diameter doubles graininess. The fluctuation between windows grew $1.978537$ times from diameter $1$ to $2$: the grains in a window fall from $800$ to $200$, each covering four times the area. With a quarter as many, the fluctuation grows by $\sqrt{4}=2$. The same change that makes the film four times as fast makes it twice as grainy.
4. Steepness of the Rise
Computation 4.1: varying the required photon count $k$ and grain area, the span of light from $10$ % to $90$ % developed (in decades) was computed.
| required photons $k$ | grain area $1$ | grain area $4$ |
|---|---|---|
| $1$ | $1.339538$ | $1.339538$ |
| $4$ | $0.583089$ | $0.583089$ |
| $16$ | $0.281522$ | $0.281522$ |
Observation 4.2 the steepness is set by the required photon count, not by grain size. Raising the required photons from $1$ to $16$ narrows the span from $1.339538$ to $0.281522$ decades: quadrupling grain area leaves the span unchanged, only shifting the curve toward less light.
Observation 4.3 the separator is whether one asks about grain size or the number of photons a grain needs to develop. Grain size moves speed and graininess together: the required photon count moves the steepness of the rise and does not enter the fluctuation formula. Raising speed without graininess means moving something other than grain size.
Observation 4.4 placed beside Paper 878. 878 showed that the effective concentration sets a curve’s position and cooperativity its span. Here grain size sets the position and the required photon count the span. It is the same shape, with position and span moved by separate quantities.
5. Census and Closing
- developed at $k$ or more photons (Section 1): Poisson distribution.
- $\mathbf{3.672061}$ for diameter $1$ (Computation 2.1): light for half developed.
- $0.918015$ for diameter $2$ (Computation 2.1): exactly a quarter.
- two stops faster (Observation 2.2): area quadruples.
- fluctuation $\mathbf{1.978537}$ times (Observation 3.2): the core.
- grains per window $800$ to $200$ (Computation 2.1): fewer grains.
- formula $0.008839$, $0.017678$ (Computation 3.1): matches random counts.
- span $\mathbf{0.583089}$ decades at $k=4$ (Computation 4.1): independent of area.
- $0.281522$ decades at $k=16$ (Computation 4.1): steeper.
- grain size, or required photons (Observation 4.3): the separator; Paper 878.
Closing: Film speed and graininess are two sides of the same grain size: doubling the diameter makes film four times as fast and twice as grainy. In a model where a grain develops after $4$ or more photons, doubling the diameter needs exactly a quarter of the light to develop half the grains, and window fluctuations grow $1.978537$ times, because quadrupling grain area gathers four times the photons but fits a quarter as many grains in a window. The rise span is $1.339538$, $0.583089$ and $0.281522$ decades for $1$, $4$ and $16$ required photons, unchanged by grain size: grain size moves speed and graininess together, and the required photon count moves steepness. Placed among the earlier papers: beside Paper 878’s “position by effective concentration, span by cooperativity”, we placed the same shape: position by grain size, span by required photons. To be honest: grain and photon numbers are model values, and neither development chemistry nor grain shape is included.
References
- James, T. H. (Ed.). (1977). The Theory of the Photographic Process (4th ed.). Macmillan.
- Yamagishi, Y. (2026). The Concentration Span from First to Full Effect Is Set Not by the Effective Concentration but by Cooperativity [Paper 878]. YOI: https://www.yamagishiyuuki.com/yoi/878
Reference
Yamagishi, Y. (2026). Film Speed and Graininess Are Two Sides of Grain Size: Doubling the Diameter Makes Film Four Times Faster and Twice as Grainy [Paper 889]. YOI: https://www.yamagishiyuuki.com/yoi/889
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 889. This paper is a preprint and has not been peer reviewed.