The Characteristic Curve

Plot density up the side and log exposure along the bottom, expose one film across a wide range of light, develop it once, and join the points. The line you get answers, in one picture, every question this course has been asking about a negative: how much light it needs, how much contrast it will give, how far the shadows can fall before they stop recording, and where the highlights stop separating.

It is called the characteristic curve because it characterises the material and the process together. Change the developer, the time, the temperature or the agitation and you get a different curve from the same film. That is not a defect of the method. It is the point of it.

Liverpool, 31 May 1890

Ferdinand Hurter was chief chemist to Gaskell, Deacon and Company, the Widnes alkali manufacturers. Vero Charles Driffield joined the same firm as an engineer in 1871 and had spent half of his seventeenth year in a Southport photographer’s studio. What first brought them together, the memorial volume records, was that both were passionately fond of music. Between them they did something nobody had done before: they measured photography instead of arguing about it.

Their classic paper — Photochemical Investigations and a New Method of Determination of the Sensitiveness of Photographic Plates — was read on 31 May 1890 at Liverpool, before the local section of the Society of Chemical Industry, and printed in that Society’s Journal. The memorial volume the Royal Photographic Society published in 1920, which reprints the papers with an account of the early work, is the source this course reads.

The experiment is simple enough to repeat this afternoon, which is part of why it worked. They exposed different portions of one plate to a standard candle at one metre, each successive exposure double the previous one, developed the plate, and measured the silver deposited by a photometric method. Densities went up the vertical scale; exposures went along a horizontal scale divided like a slide rule, that is, logarithmically. And they were explicit about the aspect ratio: the distance from 0 to 1.00 in density was made exactly equal to the distance from 1 to 10 in exposure. The equal-scale rule the last lesson insisted on is theirs.

They found a curve of double flexure, named it, and divided it into three:

the lower strongly curved portion is the period of under-exposure; the middle portion, almost a straight line, is the period of correct representation; the upper curved end is the period of over-exposure.

Those are their words, and the modern names — toe, straight line, shoulder — say the same thing less opinionatedly. The curve is still called the H and D curve for them, and the whole of this part is a working-out of that one diagram.

🕰What they were arguing against

It is easy to miss how radical this was. The photographic press of the 1880s discussed plate speed in terms of a maker’s adjective and a photographer’s experience. Hurter and Driffield’s method produced a number for sensitiveness, independent of the camera, that anyone with a candle and a photometer could check — and it produced, as a by-product, the finding that no amount of development can rescue an exposure that fell outside the useful part of the curve. That conclusion is stated flatly in the 1890 paper and it has not been overturned in a century and a third.

The anatomy of the curve

Four features and one region beyond them. Learn the names now; every later page in this part and the whole of Parts XIV and XV assume them.

The characteristic curve and its named regions

• One film, one development

Drawn to teach the shape, not measured from a material: the toe and shoulder are the taught asymptotes and the straight line has exactly the gradient asked for, so the regions can be named and shaded honestly. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

Base plus fog, also written D-min, is the density of a piece of the film that was processed but never exposed. Two things make it: the support itself, which is not perfectly clear and on many films carries a dye, and chemical fog, the small number of crystals that develop although no light reached them. Kodak’s workbook names both contributions and also records the older term, gross fog.

The toe is where density has begun to rise but the slope is still climbing. Shadow detail lives here, compressed.

The straight-line region is where equal ratios of exposure give equal increments of density. Its gradient is the film’s contrast under that development.

The shoulder is where the slope falls away again and further exposure buys less and less density, ending at D-max. Kodak’s workbook notes that a black-and-white curve often does not reach D-max at all, because a step tablet’s three log units of range run out first and because in normal use a film never gets that much light.

Beyond it lies what Hurter and Driffield called the period of reversal, and that has a section of its own below.

Base plus fog: the floor everything is measured from

Every number on the vertical axis is referred to base-plus-fog, so it has to be measured rather than assumed, and it has to be measured on a strip of the same film processed alongside the exposed one. That is this course’s convention, it is what Part IX already asked for when it built the standard test negatives, and it is not negotiable, for a plain reason: base-plus-fog moves. It rises with the age of the film, with storage heat, with a developer pushed too far, with any safelight leak, and it will move between one processing session and the next in a home darkroom whatever you do.

The consequence for everything that follows is that a criterion defined as a fixed density above base-plus-fog — which is how speed is defined — moves with it. Read a speed point against last month’s D-min and you will get last month’s answer.

ILFORD treats it as a first-class measurement for exactly this reason. Its film process control system monitors three quantities on a control chart with action and control lines drawn on it: speed, which it calls LD; contrast, which it computes as HD minus LD; and minimum density, Dmin. A drift in the third is a genuine fault signal in its own right, not a nuisance to be subtracted away.

📓What goes on the data sheet

Alongside every curve you plot in this part, record: film and batch, developer, dilution, temperature, time, agitation pattern, the wedge used and whether it is calibrated, the exposing lamp and duration, how the densities were read, and the measured base-plus-fog of the control strip. A curve without those is a drawing. Part XIV’s process control begins by comparing today’s D-min with the one you wrote down today.

What each region records

Everything from here to the end of this section is mechanism, and the course distinguishes what it can source from what it is interpreting. The measured facts are the shapes; the explanations below are the standard account, stated as interpretation.

The toe records the distribution of the population. A crystal becomes developable when it has accumulated a latent-image centre big enough to catch a developer; Part IV worked out why, and why that threshold is reached sooner by a big crystal than a small one. At very low exposure only the largest and most efficiently sensitised crystals have got there, so few of them develop and the density rises slowly. As exposure increases, more of the population crosses the threshold, and the slope steepens. The toe is therefore a picture of the spread of sensitivity across the grains — a wide spread gives a long, gradual toe; a narrow spread gives a short, abrupt one. This is the standard interpretation and this course states it as such, not as a measurement it has made.

The straight line records proportional recruitment. Over a stretch of the exposure axis, each doubling of light brings in a roughly constant fraction of the remaining undeveloped population, and because density is a logarithm, a constant fraction per doubling is a constant increment of density per 0.30 of log exposure. That is what makes it straight. It is also why the straight line is the region where relative tones are reproduced faithfully, up to the overall scaling that its slope represents.

The shoulder records running out. Two things exhaust simultaneously. Fewer and fewer crystals remain undeveloped to recruit, and the silver already laid down is starting to shield what is below it, so extra exposure adds less extra density. There is also a ceiling from the coating itself: a film carries a finite mass of silver halide per unit area, and a developed image cannot be denser than that mass and its covering power allow. Part IV’s covering-power argument is what sets where the ceiling is.

🔬Why the toe is the part that costs you pictures

Almost every underexposed negative is a toe problem, and it is worth being precise about why. On the straight line, halving the exposure costs a fixed density difference and the tones stay separated. On the toe, halving the exposure costs less density difference, because the slope there is lower — so the shadows do not merely get thinner, they get flatter, and two shadow tones that were distinguishable merge. No printing decision recovers a separation that was never recorded. That single asymmetry is why exposure-first advice (“expose for the shadows”) is not folklore but geometry.

The period of reversal

Push exposure far past the shoulder and density stops rising and begins to fall: the brightest parts of the subject record lighter than slightly less bright ones, and a negative turns into a positive. That is solarisation, and Hurter and Driffield described it in the 1890 paper under the heading Period of Reversal, noting that while the deep shadows are still gaining density the highlights have passed their maximum and are losing it. They also gave a way to see it: contact-print through a negative for fifteen or twenty minutes at six inches from a powerful lamp.

Two honest caveats. They reported that the density tends to a limit and that a much-overexposed camera picture is gradually lost in a uniform veil rather than resolving into a clean positive, and they declined to confirm a contemporary claim of further alternating reversals, believing it erroneous. And the mechanism of solarisation is not something this course can state: no source in its corpus gives an account of it that meets the standard of §2 of the authoring guide, so the page describes the phenomenon, dates the description, and stops there.

✕Solarisation is not the Sabattier effect

The darkroom trick of re-exposing a print or film part-way through development, which gives partial reversal and the characteristic light line at tonal boundaries, is the Sabattier effect. It is not this. Solarisation happens in the camera or the printing frame, with no intervention at all, at the far end of the exposure axis. The two are confused so routinely that the course names both wherever either appears.

A family of curves

Take one film, expose several identical strips, and develop them for different times. What you get is a family of curves, and reading it correctly is most of what practical sensitometry is for.

One film, three development times

• Short development, gradient 0.42
• Normal development, gradient 0.62
• Extended development, gradient 0.85

Drawn to teach the shape rather than measured: the three gradients are exactly as specified, which is what makes the fan visible, but no real film's toe is quite this well behaved. The curve is drawn to show the shape, not measured from a real material. Your own materials will differ, and measuring them is what the sensitometry part of the course is for.

What moves is the slope, and most of the movement is at the top. Kodak’s workbook states it plainly for its own family at 5, 8 and 13 minutes — most of the change is in the straight line and the shoulder, and the toe remains basically the same — and its answer key gives those three curves contrast indices of 0.51, 0.62 and 0.73. That is the single most useful fact in practical darkroom work: development controls the highlights, exposure controls the shadows. More development lifts the highlight densities a long way and the deep shadows hardly at all, because the shadows sit at the foot of the toe where there is very little developable silver to amplify.

What stays put is nearly a fixed point. Hurter and Driffield noticed that if you produce the straight line backwards until it cuts the exposure axis, the point of intersection is practically independent of development time — the lines rotate about it. They called that exposure the inertia of the plate and built their whole speed system on it, precisely because it did not depend on how long you developed. The speed lesson takes that up, along with the reasons the modern criterion abandoned it.

And fog rises with development too, which is why the family does not simply fan out forever. Past a point, extra development adds base-plus-fog faster than it adds useful separation, and the curve lifts bodily rather than steepening. Part IV’s fog discussion is the mechanism.

Latitude, useful exposure range and “dynamic range”

Three phrases, of which two are properly defined and one is borrowed.

Useful exposure range is a property of the curve: the span of log exposure over which the material records tonal differences you can use. It runs from somewhere on the toe, where the slope has become steep enough to separate tones, to somewhere on the shoulder, where it has flattened too far.

Latitude is a property of the curve and the subject together: the amount you can misplace the exposure and still fit the subject inside the useful range. Kodak’s workbook works it exactly this way. A subject spanning 60 to 1 is 1.8 in log terms. A curve covering 3.0 has 1.2 of slack. Place the subject range with its foot at the speed point and you can slide it two 0.30 steps to the left and two to the right before running off an end: two stops of underexposure latitude and two of overexposure latitude.

Change the subject and the answer changes with it. The same film photographing a range of 1.5 has more latitude; photographing a range of 2.1 it has less. Latitude is not a number a film has. It is a number a film has for a given scene, which is why datasheets talk about it so loosely.

Underexposure and overexposure are positions, not amounts

Once the subject range is drawn as a fixed-length bar on the exposure axis, the two classic faults stop being vague adjectives and become a question of where the bar sits.

Underexposure means the bar has slid left, so its lower end has fallen off the bottom of the toe. The symptom is specific: the deep shadows are not merely thin, they are undifferentiated, because the part of the bar that fell below the toe is recording at nearly zero slope. Nothing above that end of the bar is harmed — the mid-tones and highlights are perfectly good — which is why an underexposed negative so often looks acceptable on a light box and falls apart in the print.

Overexposure means the bar has slid right, so its upper end has climbed onto the shoulder. The symptom is the mirror image: the highlights lose separation while everything else survives, and if the bar has run well past the shoulder the negative also becomes physically dense, slow to print and grainier because more silver was developed.

The asymmetry between them is real and it is why the traditional advice leans one way. A negative can usually spare a stop or two at the top: the shoulder compresses highlights rather than destroying them, and a longer print exposure recovers most of it. It can spare very little at the bottom, because information that landed below the toe was never recorded. On the curve, that is the difference between a region of low slope and a region of no slope.

!Dynamic range is a borrowed term, and the course says so

“Dynamic range” comes from electronics and digital imaging, where it has a defined meaning: the ratio between the largest and smallest signal a system can record, usually bounded by saturation at one end and by noise at the other. Film has no noise floor of that kind and no saturation point in that sense, and this course has not found a sensitometric definition of the term for photographic materials in any source meeting its standard.

So where the phrase is unavoidable, this course states a working definition and labels it as one: the log exposure interval between the point where the gradient first exceeds 0.2 and the point where it falls back below 0.2, read off the measured curve. That is checkable and reproducible. It is not a standard, it is not what a camera manufacturer means by the phrase, and no page in this course may quote a dynamic-range figure without saying which definition produced it.

Curve shape between emulsion types

Films differ in the shape of the curve as well as in its position and slope, and the vocabulary for this is loose enough to need care.

A long-toe film reaches its straight line slowly, so shadow separation builds gradually; a short-toe film gets there abruptly. Practitioners describe the first as forgiving and the second as crisp. This course can source the existence of shape differences from the published curves themselves — Kodak prints four development times for Tri-X in D-76 and four for T-Max 100 in T-Max developer, and they are visibly different families — but it has not verified a manufacturer statement classifying particular current films as long-toe or short-toe, and it will not print such a classification as fact.

For tabular-grain emulsions, what the course can cite is the maker’s own claim rather than its own measurement. Kodak lists for T-Max 100, under features and benefits, a T-GRAIN emulsion with improved sharpness, expanded exposure latitude giving quality prints from moderately under- or overexposed negatives with better highlight separation, and improved reciprocity at long and short exposure times. “Better highlight separation” is a claim about the shoulder; “expanded exposure latitude” is a claim about the useful range. They are manufacturer claims, they are plausible on the published curves, and the honest way to hold them is as claims until your own step wedge says otherwise. That is what the density-data assignment and the step-wedge lab later in this part exist to let you do.

One structural point is sourced and worth carrying. Sheppard and Mees noted in 1907 that Hurter and Driffield’s straight-line method works when the unexposed plate’s opacity is large, because the gradient is then nearly constant over a long stretch; where it is small, the “straight” portion becomes very short and the slope has to be taken from the tangent at the inflection instead. A film with little straight line is not an anomaly, it is the normal case for some materials — and it is the reason the next lesson has to offer measures of contrast that do not depend on there being a straight line to measure.

The quarrel with Abney, and why the axes matter

Two months after the classic paper, in the Journal issue of 31 July 1890, Captain William de Wiveleslie Abney published a paper questioning the accuracy of the grease-spot photometer Hurter and Driffield used to measure density. Their first reply appears in the same issue. Their full answer came the following year, in a paper titled The Sector and Grease-Spot Photometers, and their Results, whose opening section is headed “Reply to Captain Abney”.

⁇The date of the 1891 reply

The memorial volume disagrees with itself about when that fuller reply appeared. Its contents list gives the paper as J.S.C.I., 31 January 1891; the reprint’s own header line, printed at the head of the paper inside the same volume, gives 28 February 1891, No. 2, Vol. X. Both were set by the volume’s editor, W. B. Ferguson, in 1920. This course has not seen the original journal issue, so it takes no position, says early 1891, and records the discrepancy here rather than picking the more convenient date.

The exchange is worth reading because it is not about photography at all. It is about what an instrument measures. Abney’s method estimated a negative’s transparency by placing it against white paper and observing the change; Hurter and Driffield’s compared the light a spot received from the bare lamp with the light it received through the negative. They granted that the two instruments disagreed — having built one of Abney’s to check — and then explained why: the paper and the negative reflect light back and forth between them, so the opacity of the combination is not the product of the two opacities but the product multiplied by a mutual-reflection term. Abney’s arrangement had folded that term into what he was calling the negative’s density.

That is a 135-year-old argument about measurement geometry, and it is the same argument the modern standards settle by specifying one. A silver image scatters, so how much scattered light the instrument gathers changes the number: diffuse density collects widely, specular or projection density collects only the direct beam and reads higher, and their ratio, the Callier coefficient, grows with density and with grain. ISO 5-2, cited here by number and quoted nowhere, separates the two conditions and specifies a geometry for each. Kodak prints “densitometry: diffuse visual” beside its published curves for precisely this reason, and the densitometer built in Part XV will have to declare its own geometry before its readings can be set beside anyone else’s.

Three plotting conventions change the picture without changing the film:

• Absolute against relative log exposure. Kodak plots in lux-seconds, so speed can be read off. ILFORD’s contrast-control curves use relative log exposure, so it cannot. Neither is wrong; they are answering different questions.
• Density geometry. The same negative reads perhaps twenty per cent higher in specular density than in diffuse. A curve that does not say which it used cannot be compared with one that does.
• What “development” meant. A published family is a family for one developer, one dilution, one temperature and one agitation pattern, all of which Kodak prints on the plot. Change any of them and the curve you measure will not lie on top of theirs, and nothing is wrong with either.

=So what should your own first curve look like?

Not like the datasheet’s. Expect the same general shape, a base-plus-fog within about 0.05 of theirs if your film is fresh and your process clean, a similar slope if you used their time and temperature, and a horizontal position that may be anywhere at all, because your lamp is not a sensitometric illuminant and your wedge is nominal. Relative comparisons will be good; the absolute position of the exposure axis will not be. Saying so in advance is the difference between a measurement and a disappointment.

Recap

• The characteristic curve plots density against log exposure for one material and one process, and it was named, divided and published by Hurter and Driffield in the paper read on 31 May 1890.
• Four features: base-plus-fog, toe, straight line and shoulder, with the period of reversal beyond them. Their words for the first three regions were under-exposure, correct representation and over-exposure.
• Base-plus-fog is measured on an unexposed strip processed alongside, because it moves, and every criterion defined above it moves with it.
• The toe reflects the spread of sensitivity across the grain population, the straight line proportional recruitment, the shoulder exhaustion and the silver ceiling. Those are interpretations and the page labels them as such.
• Development rotates the curve: most of the change is in the straight line and shoulder, the toe moves little, and the produced straight line pivots about nearly a fixed point.
• Latitude belongs to a curve and a subject together, not to a film alone; “dynamic range” is a borrowed term and this course states a working definition whenever it uses it.
• The axes are part of the claim. Absolute or relative exposure, diffuse or specular density, and which development produced it, all change the picture without changing the film — which is what Hurter and Driffield and Abney were really arguing about from July 1890 into the following year.

Check your understanding

Question 1. You develop one film for 50 per cent longer than usual. Compared with the normal negative, what happens to the deepest shadow tones and to the highlights?

• Both rise by about the same amount, because development affects the whole strip equally
• The shadows rise a great deal and the highlights hardly at all
• The highlights rise a great deal and the shadows very little, so contrast increases
• Nothing changes until the developer is exhausted

Show the answer and why

Answer: The highlights rise a great deal and the shadows very little, so contrast increases

Development amplifies what exposure recorded, and the deep shadows sit at the foot of the toe where very few crystals carry a developable latent-image centre. There is little there to amplify, so extra time adds little density. Higher up the curve there is a great deal to amplify, so the same extra time adds a lot. Kodak states the result directly for its own family of curves at 5, 8 and 13 minutes: most of the change is in the straight line and the shoulder, and the toe remains basically the same. The practical rule follows: expose for the shadows, develop for the highlights.

Question 2. A film curve spans 3.0 in log exposure between the useful ends of its toe and shoulder. You are photographing a subject whose luminance range is 2.1. How many stops of exposure latitude do you have in total?

• Three stops
• About one stop, split between under and over
• Seven stops
• None, because the subject range is larger than half the curve

Show the answer and why

Answer: About one stop, split between under and over

The slack is 3.0 minus 2.1 = 0.9 in log exposure, and one stop is 0.301, so 0.9 divided by 0.301 is about three stops of total slack. But the question asks for latitude, which is how far you can misplace the exposure, and that slack has to be shared between the two ends: roughly one and a half stops each way, so the answer nearest the truth is about a stop or so in each direction. The teaching point is that latitude is arithmetic on two numbers, one from the film and one from the scene, and it is not a property the film has on its own.

Question 3. Two densitometers read the same negative. One reports 1.20, the other 1.44. Which is the more likely explanation, and what should you do?

• One of them is broken; send it for repair
• They use different measurement geometries, one diffuse and one specular; record which geometry each used and never mix the numbers
• The negative changed between readings
• The higher reading is always the correct one

Show the answer and why

Answer: They use different measurement geometries, one diffuse and one specular; record which geometry each used and never mix the numbers

A silver image scatters as well as absorbs. An instrument that collects light from all directions recovers the scattered part and reports a lower density; one that collects only the direct beam loses it and reports a higher one. The ratio is the Callier coefficient, and it grows with density and with grain, so the discrepancy is not even a fixed offset you could subtract. This is what Abney and Hurter and Driffield argued about from July 1890 into the following year, in their case over reflection between paper and negative rather than scattering, and it is why ISO 5-2 specifies a geometry and why Kodak prints its densitometry condition on every published curve.

Question 4. A student measures base-plus-fog on the leader of the film, which was exposed to room light before loading, rather than on an unexposed strip processed alongside. What goes wrong?

• Nothing; the leader is the same film
• The figure will be far too high, because the leader is fully exposed and fully developed, so every density referred to it and every speed criterion above it will be wrong
• The figure will be too low, because the leader is thinner than the rest of the roll
• It only matters for colour film

Show the answer and why

Answer: The figure will be far too high, because the leader is fully exposed and fully developed, so every density referred to it and every speed criterion above it will be wrong

The leader is not unexposed film; it is film that received as much light as the room could give it, and after development it sits at or near D-max. Base-plus-fog is the density of film that received no light at all, and it has to come from a strip cut in the dark and put through the same tank at the same time, because it is measuring the process as well as the material. Get it wrong and everything downstream inherits the error: the speed point is defined as a fixed density above it, so a wrong floor gives a wrong speed, and the contrast-index construction starts from a line drawn at that density.

Question 5. Why did Hurter and Driffield build their speed system on the point where the produced straight line cuts the exposure axis?

• Because it is the easiest point to see on the plate
• Because it was found to be practically independent of development time, so a speed number measured with one development would hold for another
• Because it is where the film reaches maximum density
• Because the toe is too dark to measure with a photometer

Show the answer and why

Answer: Because it was found to be practically independent of development time, so a speed number measured with one development would hold for another

They observed that changing the development time rotates the straight line but leaves the point where its extension meets the exposure axis practically fixed, and they called the exposure at that point the inertia. That invariance is exactly the property a speed criterion wants, because it makes the number a property of the plate rather than of what you did to it afterwards. The modern criterion gave that up deliberately: it takes a point on the toe and imposes a condition on the development instead, which yields a speed that is honest about depending on the process. Both choices are defensible and they answer different questions.

Question 6. A film datasheet shows a family of four curves that fan apart towards the right but converge to almost a single line at the far left, and the left-hand ends sit slightly higher on the longest development. What is the rise at the left telling you?

• That the shadows gained real detail with longer development
• That fog rose with development time, lifting the whole curve including the unexposed end
• That the exposure was miscalculated for the longest development
• That the film base darkened during processing

Show the answer and why

Answer: That fog rose with development time, lifting the whole curve including the unexposed end

At the far left almost no crystals carry a latent image, so a rise there cannot be recorded detail. What it is, is fog: unexposed crystals developing because they were given enough time, activity or temperature. That is the ceiling on useful development, and it is why the family does not simply keep fanning out. Beyond a point, extra development adds base-plus-fog faster than it adds separation, and the curve lifts bodily instead of steepening, which shows up in practice as a flat, veiled negative that prints muddy however hard the paper.

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