GIUSEPPE ARCIMBOLDO : the musical grey scale

A human likeness shaped from disparate objects

Illustration 1 : "VERTUMNUS",
Giuseppe Arcimboldo, 1590-1.

Whoever you may be, when you behold
This odd, misshapen picture which is me,
And there is laughter on your lips,
Your eyes are flashing with hilarity,
And your whole face is seized by mirth
As you discover yet another monstrous detail
In him who bears the name Vertumnus,
Being thus called in poems of the ancients
And by Apollo's learned sons;
Unless you clearly see that ugliness
Which makes me beautiful,
You cannot know that there's a certain
Ugliness more beautiful than any beauty.
There's diversity within me,
Though despite my diverse aspect, I am one.
That diversity of mine
Renders faithfully and truly
Diverse things just as they are...

"Vertumnus", by
Don Gregorio Comanini.

Giuseppe Arcimboldo served as court painter to the Holy Roman Emperors - Ferdinand I, Maximilian II, and Rudolf II - from 1562 until his retirement in 1587. After returning to his native Milan, Arcimboldo continued to paint for his Imperial patron and, by 1591, had dispatched two completed works to the court in Prague. One was "Flora", a portrait bust of the goddess of spring, composed entirely of blossoms. "Vertumnus" (above) was the other, an allegorical portrayal of Rudolf II, in the guise of the Roman god of vegetation and transformation. Fruits, flowers and vegetables, of various seasons and from different regions of Europe, were combined in the figure of the emperor. Each picture was accompanied by a lengthy poem by Arcimboldo's friend, the humanist Gregorio Comanini. In his puff pieces, the poet lauded the ingenuity of a painter who created in imitation of the gods. When Zeus had made the world, he separated the elements and their qualities from chaos; he wrapped them round the earth, then clothed it in vegetation. In emulation of Jove, Arcimboldo wove "this great product of creation into an artistic garland". Just as the sun's rays painted grapes in red and yellow, so too did the artist. His images, Comanini exclaimed, were neither gods nor plants but paradoxically both. The poet flattered the emperor Rudolf by styling him as Vertumnus - a male equivalent of the goddess Flora - and lavishing god-like attributes upon him. The fruits of the realm were his very substance, which, in turn, refreshed and sustained nymphs, soldiers, and peasants alike.

Arcimboldo was among the first artists to elevate humble subject matter over studies of the human form. (Still-life painting only emerged as a legitimate genre in the following century.) Still, he felt constrained to disguise the common objects as people, resulting in ambiguous images that were self-consciously clever. Sometimes, in a painting of a bowl of vegetables or a platter of meats, Arcimboldo would conceal a human resemblance, which could only be discovered by hanging the picture upside down. Then the puzzle picture gratified viewers who were smart enough to uncover its illusion. Since the 1560s, Arcimboldo had painted patchwork figures, assembling them from objects that best represented universal themes. For instance, fish, crustaceans, and other sea life, were shaped into the profile of a man; with a shark's mouth, coral in his hair, and a row of pearls around his neck, he represented Water, one of the four elements. The face of Winter, one of the four seasons, emerged from a gnarled tree-trunk, with fungus for a mouth and ivy in his hair. The Hapsburg emperors were delighted with such personifications - Maximilian II himself had dressed in the costume of Winter, for a parade in 1571. Multiple versions of Arcimboldo's Seasons and Elements were ordered, as presents for visiting dignitaries.

Surprising and amusing, 'serious jokes' of this nature were enjoyed in 16th century society. They were the table talk of 1536, when supper was served on a blue-and-yellow Majolica plate attributed to Francesco Urbini. It sported a composite bust anticipating those of Arcimboldo, except the head was made entirely of penises and surrounded by the motto "Everybody looks at me as if I were a dickhead". Though Arcimboldo eschewed this kind of ribaldry, he still portrayed the emperor as a collection of vegetables. Comanini clearly expected people would find the image hilarious, and made much of its ambivalence in the accompanying poem. While his Vertumnus declared, "though my aspect may be monstrous, I bear noble traits within", it is still a rather cruel caricature. Arcimboldo - perhaps emboldened by his distance from Prague - applied his talents to the royal personage, and Rudolf seemed pleased enough with the likeness. Other artists copied his style, and Arcimboldo earned the admiration of the prominent art theorist, Giovanni Lomazzo. Galileo, however, scorned the composite portraits for their oblique mentality, for taking an ad hoc approach instead of "looking at the great book of nature". Still, the court at Prague appreciated their artist's ingenuity, and employed his services as designer and director for important pageants. Yet another of Arcimboldo's antics was an intriguing exercise, intended to match painting with music, which Comanini duly described below.


from Gregorio Comanini's dialogue on painting,
"Il Figino, overo del fine della pittura", 1591.
("On the Purpose of Painting")

"But nevertheless painting approaches music, just as poetry occasionally does. I would like to prove by means of the renowned Arcimboldo, who has located the tones, semitones, the diatesseron, the diapente, the diapason, and all the other musical consonances in colours, using the formulae which Pythagoras invented to define the same proportions in harmony. For he noticed in blacksmiths' forges that the blows of the hammers on the anvil produced consonances proportionate to their weight, and having noted the numbers from whose concurrent diversity the many consonances of ordinary melody are formed, hung corresponding weights from the same number of strings as the number of hammers seen in the forges, from which it appeared that a string which was longer than another in the sesquiottava proportion produced a tone against the other, i.e., a full and perfect sound in the proportion of nine to eight. Similarly, he covered a board with some particularly white colour, and darkening it a little, one part after another he drew from this the sesquioctaval and the tone itself.

['Colour' was once a general term, frequently used to describe tonal effects as much as hues. Here, it applies mainly to a grey scale - black, white, and all shades in between. It would hardly be called 'colourful' today. Comanini seems to suggest Arcimboldo used thinly applied washes, or glazes, of black over a white ground, to make his musical grey scale. He might also have used opaque paints, premixed to the required shade of grey.
The legend of the blacksmith's forge is apocryphal, as shown elsewhere. In most sources, the sizes given to hammers and weights were misleading: measurements of string lengths gave the only reliable guides to musical pitch.
The musical formulae of the Pythagoreans were outmoded by the end of the 16th century. Their chief intervals, however, remained in use, including the major tone with a ratio of 9 : 8. Comanini's Greek and Latin terms, for intervals and ratios, are set out in a table at the end of this section.]

He even exceeded Pythagoras in this area: i.e., the brilliant philosopher could not divide the tone into two equal semi-tones because the number nine resists division into two equal parts, but was able to find a semi-tone somewhat larger than the mean, and another somewhat smaller than it (commonly called 'diesis' by professors of music). This ingenious painter was able not only to find these larger and smaller semi-tones in his colours, but also the division of the whole tone into two equal parts, so lightly and deftly did he darken the white, always little by little, gradually ascending to greater darkness, just as from a low note one moves gradually to a higher one and then on to an even higher one.

[The division of the tone into exact halves, by taking its square root, was a classic problem. Early in the 4th century BC, Archytas had proved it was impossible for ratios such as 9 : 8. The root of nine is easily found to be three (despite Comanini's remark), but the root of eight includes an incommensurable number, the square root of two. Because it could not be expressed in whole numbers, Pythagoreans avoided it. Arcimboldo did not since he, apparently, was a greater genius.
The diesis was defined as the smallest melodic interval. Philolaus assigned that role to the Pythagorean semitone (ratio 256 : 243), early in the 5th century BC. At about 44% of a whole tone, the semitone was too large for later theorists. They called it a limma and devised a dozen or so smaller dieses, ranging from 41% to 15% of a tone. By Arcimboldo's time, a common diesis was the minor semitone, in the system of Just Intonation. At 25 : 24, and around 39% of the whole tone, it seems to fulfil Comanini's requirement - that a diesis be smaller than the major semitone of 16 : 15, at some 61%.
Musicians in the 16th century tuned lutes to a kind of equal temperament, so all semitones were as uniform as possible. Vincenzo Galilei described the method in 1581: frets for the semitones were fixed along the neck of a lute, so their distances from the bridge increased progressively in the proportion 18 : 17.]

In addition, Pythagoras, plucking a string which had double the weight of another and was thus doubly stretched, and at the same time comparing its doubled proportions to those of its companion, discovered the diapason, or octave as we chose to call it. The one string being stretched twice as much as the other because of the pull of its weights returned to its linear shape with double vigour and double speed resulting in a high pitch, while the other, stretched only half as much as its neighbour returned to its linear shape more slowly and produced a lower pitch (yet the two pitches were nevertheless so compatible that they seemed to be the same sound produced simultaneously by two strings; one more stretched and the other less so).
Similarly, Arcimboldo darkened white in double proportion compared to another and formed the same proportion as the diapason, ascending by eight degrees of darkness from the purest white and by increments of the same size.

[To raise pitch an octave, the weight stretching a musical string has to be increased fourfold (not doubled). The same effect is achieved by playing only half the string, after dividing its length into two with a finger or other device. The original note and its octave sound somehow similar, though the higher note vibrates at twice the rate of the lower.
With his grey scale, Arcimboldo mimicked the eight notes of an octave scale, beginning on white for the lowest note. Comanini is unclear where it ended; his 'doubly-darkened white' might be a mid-grey, even a black. Likewise, differences between successive greys may have looked the same size, or they might have corresponded to musical intervals. The former gives a smooth progression; the latter is more uneven.]

Further, just as Pythagoras discovered with strings the sesquiterza proportion and thus found the diatesseron, or the fourth, so Arcimboldo added black to white in sesquiterza proportion compared to another white and formed the ratio from which the diatesseron results, which is the proportion four to three in which four consists of three plus its third, which is one, since one multiplied three times is three. He did the same with the diapente, i.e., the fifth: for just as Pythagoras found it in the sesquialtera proportion which occurs when the larger number contains the smaller plus its half (as in the case of three and two, since three consists of two plus one, which multiplied by two is two). Similarly, Arcimboldo extended black from white in the same ratio and gave it five degrees of darkness corresponding to the five steps of the diapente, thus expressing the visual nature of the same consonance.

[Arcimboldo executed two more grey scales, of four and five steps each. Where they ended, and how he mixed his paints, remains a mystery. In this passage, Comanini repeats material found in "De institutione musica" of Boethius. Written in the early 6th century, it transcribed much of Greek music theory into Latin. In Book I, Boethius explained how Latin terms such as 'sesquitertia' defined a musical ratio in numbers, while the Greek 'diatesseron' described its sound or pitch.]

And what can be observed form this? Pythagoras saw that from the diatesseron and the diapente together the diapason results, and Arcimboldo utilized these two proportions and produced the octave in his colours. And from the triple proportion the diapason diapente results, which is the twelfth; and with this proportion he proceeded to assign the twelve steps from darkness to white. He did the same in forming the disdiapason, i.e., the fifteenth, which is generated by the 4:1 proportion.
Taught by this method, Mauro Cremonese della Viuola, musician to Emperor Rudolf II, was able to locate on the cembalo all the consonances which Arcimboldo had indicated in colour on a chart. Thus you can see, Guazzo, how the arts of painting and poetry proceed together by the same laws to form their images...

[For another grey scale of the octave, Arcimboldo combined the four and five step scales. This implies that one began where the other left off, around mid-grey. Further grey scales were produced, one of an octave plus fifth, and the other of two octaves. The latter consisted of two parts, musically identical, so the second half should be a darkened version of the first.
A musician playing a harpsichord, or a hammer dulcimer, put the artist’s precision to the test. After some coaching, Mauro Cremonese translated different greys as notes (reading from a chart of swatches, perhaps, rather than from a continuous grey scale). It is not stated whether his purpose was to educate or amuse, whether his audience included others besides Comanini and Arcimboldo, or if he picked out any tune.
For Comanini, the success of the enterprise depended on musical structure; Arcimboldo's grey values conformed somehow to Pythagorean ratios. The test instrument, either harpsichord or dulcimer, should accord with the tradition. But notes on keyboards were usually tuned to a meantone temperament, whose fourths and fifths were impure.]



minor semitone (diesis/limma) 256 : 243
major semitone (apotome) 2187 : 2048
Pythagorean tone (tonos) 9 : 8
= sesquiottava
fourth diatesseron 4 : 3
= sesquiterza
fifth diapente 3 : 2
= sesquialtera
octave diapason 2 : 1
octave + fifth
= twelfth
diapason diapente 3 : 1
double octave
= fifteenth
disdiapason 4 : 1

At the beginning of Comanini's panegyric to Arcimboldo, painting aspires to the elevated aims of speculative music, while poetry tags along behind. The author struts his musical erudition throughout, to parade novel ideas in a traditional garb. He gives painting a musical structure - which makes it a mathematical art, and equivalent to arithmetic, geometry and astronomy. By the end, the painter's method is vindicated, as attested in performance by a reputable musician. Painting and poetry finally walk away, arm in arm, with equal share in the laws that govern the arts. If the account is to be believed, Comanini was at Arcimboldo's shoulder to witness the entire process (though it must have been as tedious as watching paint dry). It is implied that several grey scales were produced, probably starting from white and ending with darker shades, if not black. Comanini assures us of some structured and intimate relationship, between the painter's method and the ratios of music, but few details are given. While musical numbers multiply together to construct a scale, it is unclear how the same numbers could generate a grey scale, or determine the corresponding amounts of black or white. It seems that darkness increased proportionately, in the manner of musical pitch, with the quantity of black accelerating at every step. Arcimboldo's exercise seems to be among the first attempts to quantify a grey scale, to regulate its appearance (if not its ingredients) by musical measures. His ideal grey scales might be reconstructed, as follows, from the outline Comanini supplied.


Grey scales ascending from white to black

On the top diagram, fifteen equal steps of grey (uppermost) are matched against a musically-based scale of two octaves (beneath). Both greys and notes form reversible scales: they can start from either end, though a musical mode - played on white notes of a keyboard - will be symmetrical only if it starts on D. Here, the high note d' is given black, the lowest D is white, and d at the centre is mid-grey. Other pitches in a Pythagorean scale were converted to cents, then calibrated for relative brightness (according to B values in Photoshop's HSB colour picker). There is little apparent difference between the two scales, except where half-steps are included on the lower scale for the first of the sharps and flats.

Single octaves are shown on the middle diagram: uppermost is the traditional Pythagorean array, and below it a tuning in Just Intonation, a system familiar to Comanini's 'professors'. The two scales still appear similar; indeed, they share the same consonances at D, G, and A, the first, fourth and fifth notes of the scales. Since the range of greys spans a shorter compass, the unevenness in the scales becomes apparent. Abrupt changes in grey occur between D and E, and G and A, where larger musical intervals cause corresponding differences in brightness.

Subtle differences emerge when greys are compressed even further. Four-step scales, the smallest attributed to Arcimboldo, appear at the bottom of the illustration. The F# in each case is divided in two, to compare a normal semitone (at top) with a note at the centre of the tone (below it). The Pythagorean semitone is located closer to G than to F; being sharp of mid-interval, its grey is slightly darker. The difference is barely visible, at about 2%, and would disappear entirely if the grey scale were any larger. The semitone of Just Intonation is flatter than a half-tone, and noticeably lighter by some 4%. But the difference becomes very faint on the octave grey scale, and could not be detected in an expanded double-octave.

With Arcimboldo's system, all consonant notes within two octaves may be articulated in shades of grey, each governed by a musical pitch. The resultant grey scale looks evenly spaced. On closer inspection, equal semitones can be subtly distinguished from normal ones, and the unevenness of the scale becomes apparent. (As displayed on an LCD monitor, the different greys in Illustration 2 will be clearer if viewed obliquely.) Distinctions in lightness are hard to see below a 2% threshold, so a normal eye can readily detect a mere fifty or so steps between black and white. While the progression of greys has some limits, vision is a very subjective faculty. In the controlled conditions of a laboratory, over 200 different greys may be noticed. By comparison, a fine ear can distinguish about fifty pitch differences between the notes D and F alone; two hundred or more can be heard in an octave, and twice that number in a double octave. Arcimboldo could not have hoped to indicate as many distinctions with his grey scale.

The painter was able to draw a parallel between music and his grey scale, by selecting a limited number of notes. Comanini asserted the sounds were somehow identical to their colours, so musical progressions generated similar scales in grey. The most regular result, an evenly-spaced grey scale, would arise from an octave tuned in equal temperament, with twelve identical semitone intervals in sequence. While theorists knew of the scale in the 16th century, it was considered a mathematical curio and quite unmusical. Pythagorean theory, with its recent development into Just Intonation, was the convention. The antique traditions better served to showcase Arcimboldo's virtuosity, and to allow Comanini his rhetorical flourishes. Today, the uniformity of equal temperament is widely accepted as a musical standard. Modern grey scales are equally precise, relying on mathematical formulae - including those built into our computers. The result might seem banal to Arcimboldo and Comanini, but whether any 16th century artist could have worked so exactly, is another matter.

Traditional recipes for mixing paint were vague, and painters relied on the eye to guide them to the desired result. At the outset of the Renaissance, Cennino Cennini had occasionally specified measures of paint in "The Craftsman's Handbook". A 'bean' or a 'lentil' of pigment could be added to a mix, or a wash made with two drops of ink in "as much water as a nutshell would hold". A century later, Leonardo da Vinci wondered about the effect of adding an extra measure of black, to one ounce of black and one of white, though there is no evidence he decided the question. Nevertheless, Da Vinci was an acknowledged master of tonal effects, of the shading of forms from light to dark, known as chiaroscuro. The subtlety of his technique gave a smoky look, a sfumato, to his paintings. Leonardo's style was so influential that artists were indebted to him for centuries to come. The young Arcimboldo, at the start of his painting career in Milan, was undoubtedly impressed on seeing the notebooks left behind by the older artist. The way "Vertumnus" (Illustration 1) emerges from darkness into the light, the modelling of its fruits and flowers for three-dimensional effect, all are rendered after the example set by Da Vinci. Even in a technical exercise, Arcimboldo's skill in chiaroscuro enraptured Comanini, "so lightly and deftly did he darken the white".

No doubt the painter depended on his touch, and the judgement of his eye, as he delicately increased the amount of black. This he could achieve in several ways - washes or glazes of a very thin black could be added, one on top of the other; the density of black could be increased by adding more and more pigment to the wash or glaze; or black and white paints could be pre-mixed, for separate shades of grey. For the production of an oil painting, two or more of these techniques were likely combined. In the media of print, subtle greys arose from lines and dots, engraved in a metal plate or carved from the surface of a woodblock. When printed on paper, the small black marks were still visible to the naked eye, but their hatching and stippling produced the overall effects of shading. Control over technique guaranteed mastery of chiaroscuro. Whether painting or printmaking, some degree of conscious measuring was required of the artist. Arcimboldo's calculations of the grey scale, as described by Comanini, gave a rare insight into the intimate practices of art. In 1816, a German painter and lithographer, Mathias Klotz, left a more precise record. His method of a calibrating a grey scale seems similar to one that Arcimboldo might have followed. Klotz measured the pigment required for each successive shade of grey, and found the amount would double for each step of the scale. In passing, he noted a certain musical resemblance:

"Without thinking of the musical octave, I found that in successive grades of light-and-dark, so as not to be too little or too much different, the scale...has to be in an always doubling relationship, so that the resulting colors follow each other with the same degree of difference."

Table II from "A Thorough Colour Theory", Mathias Klotz, 1816.

Musical parallels to Klotz's grey scale

Mathias Klotz painted his grey scale (left) in nine vertical strips, shaded light to dark from left to right. Below it, Klotz drew a bar graph showing the amount of pigment used for each strip. Black was doubled from step to step - zero at the initial white stage, one unit for the second step, two for the third, then four and so on. Presumably, Klotz diluted the different amounts with the same volume of paint solvent each time. His simple sequence provided a recipe for the grey scale, a convenient rule-of-thumb for other painters, to ensure an even gradation.
Klotz was evidently aware that music progressed in similar ways - a succession of evenly-spaced octaves resulted when the frequency of a note was doubled and redoubled. (Comanini also recognized its pitch would rise by one, then two octaves, according to the doubling ratios of 2 : 1, and 4 : 1.) A blue line is superimposed on Klotz's bar graph (at right), to plot the correspondence between sequences of octaves and of greys. Low notes and light shades are located at the bottom left of the curve, with high notes and dark shades in the top right corner. By this analogy, Klotz's 9-step grey scale would span eight octaves of music - an entire piano keyboard, plus white protruding from the bass and black extending a little beyond the treble.
Vertical measurements on the diagram give objective quantities for both sounds and shades - the vibrational rates of notes, and volumes or weights of black pigment. More subjective qualities are represented on the horizontal axis - equal differences, as perceived, between notes and shades of grey. It is presumed that Klotz's choice of greys is generally acceptable, and that most people will recognize the spacing of notes an octave apart. Then both scales may be expressed in a single formula, as shown, and plotted as logarithmic curves.

Different spans of music can be fitted to the same grey scale, by dividing the eight octaves with a variable, 'n'. When n = 1, the blue curve results, and Klotz's greys represent nine notes at octave intervals. When n increases, each shade of grey signifies a smaller interval so the musical curve becomes shallower. By this method, the notes described by Comanini can be matched to greys (as shown in Illustration 2, above). A graph for one octave, drawn in red at right, is almost a straight line. Each shade of grey would represent three-quarters of a tone (an eighth of an octave, n = 8). The double octave, shown in green, is a little more concave (n = 4, for divisions of 1.5 tones each).
A further curve is added, in pink, to represent four octaves. That range of pitch was tested by W D Ward in 1954, using up-to-date electronic equipment. He confirmed a long-held belief - upper registers sound flatter than they ought. Subjective octaves, as heard by the ear, are slightly larger than either frequencies or traditional ratios would have them. Piano tuners compensate by stretching the high octaves: top notes are sharpened until the strings' overtones misalign, and their buzzing becomes more ugly than any overall flatness. The phenomenum, known as octave stretch, is plotted as a faint pink line, extending slightly above the rest of the graph.

Arcimboldo had observed a serendipity, some kind of match, between a visual scale and that of music. Even arrays of notes and greys could be produced by similar methods. The proportional increases he applied to paint mixtures, according to Comanini, were identical to the ratios that governed music. More than two centuries later, Klotz supplied some more credible measures for the ingredients of a grey scale. He noted their resemblance to musical ratios, over a span of eight octaves. It seems there was some truth to Arcimboldo's approach, though a match between auditory and visual scales was limited to the one example. For other musical intervals, any parallel to greys would be weakened by one-sided mathematical adjustments. The relationship of music to the grey scale then becomes less of a physical fact, and more a general analogy. But one similarity remains: to create the subjective impression of regular scales (with even steps between notes or greys), both physical stimuli must accelerate dramatically, and not proceed by equal amounts. Perhaps the same type of general equation, and the logarithmic curves it generates, could describe other kinds of sensory perception - the feelings of pain or muscular effort, for example. The physiologist Ernst Weber would observe as much, in 1834, and an experimental procedure was formalized by Gustav Fechner by 1860.

The Weber-Fechner law formed the basis for a new branch of science, with the attractive name of 'psychophysics'. Mental judgements were directly linked to physical causes and, for over a century, scientists and psychologists have monitored people's responses to a great variety of sensory stimuli. Among them, S S Stevens calibrated reactions to brightness and colour saturation, to loudness and pitch of sounds, to length and size, roughness, electric shock, and many other experiences. Moreover, he found that humans had an innate ability to translate an experience from one sense into another; vibrations felt by the fingertips could be accurately expressed in volumes of sound, while strength of a handgrip may be used to measure levels of pain. In the 16th century, something similar was attempted by Gioseffo Zarlino, the pre-eminent music theorist of Arcimboldo's day. He equated size to brightness - the bigger, the brighter. Since the largest bells produced the lowest sound, it followed that their notes were white. Like Arcimboldo, Zarlino suggested a grey scale for the musical gamut, starting from white at the lowest pitch and progressing to black, for the high tones of small bells. Comparisons are no longer quite so straightforward; Stevens was to postulate psychological units of sensation, which varied along with the physical stimuli that provoked them. His power law of 1957 overshadowed the old rules of psychophysics, including the logarithmic curves of Weber-Fechner (and the formula above) that lent all sensations a musical cast.


Indirect relation of pigment & light levels

Colours are specified within the printing industry by measured amounts of standard inks. Swatches, printed by proprietary concerns such as Pantone, give ink mixes for a large range of colours. The system includes an eight-step grey scale, of spot colours made from different combinations of black and white. The percentages of black ink in every mix are traced by a pink curve, to the left. Like Klotz, Pantone doubled the amount of black from one grey to the next. Blacks mix with certain volumes of white, which reduce dramatically after the first two greys. Pantone's curve moves away from that of Klotz - from eight octaves (blue in Illustration 3, above right) to close on seven octaves.
In 1958, R L Williams tested dot screens of different grades, to make a useful grey scale for mapmakers. The dot screens are commercially available as tack-back transparent sheets, covered with fine patterns of dots or lines. For each shade, Williams measured the area covered by black ink when the dots were printed to paper. The red curve (left) plots increasing percentages of black, as required for an even gradation of greys. Despite its serpentine shape, the graph is concave overall; it conforms to the general shape of a single musical octave (red curve in Illustration 3, above right), by crossing it at the exact midpoint.
A simple grey scale can be improvised at home, using watercolour washes. White paper may serve as the starting point, and a first wash, of diluted black paint, can represent the initial shade of grey. The same paint is applied, layer upon layer, to build up a grey scale. The following numbers of washes seem adequate to construct eight successive steps: 0 for blank white paper, 1, 2, 4, 7, 11, 16, and 22 as the final black. If each wash deposits the same amount of pigment to paper, the increase in black can be traced on the orange graph (left). And, to extend the musical metaphor, the curve for watercolours is near to that of three octaves.

Pigment particles will absorb and scatter any light that falls on them. The consequent light coming from a painted surface can be found, using the Kubelka-Munk formulae of 1931. For each of Klotz's greys, the amount of reflected light is calculated, and subtracted from the maximum light value (of white). Measures of light absorption (or relative darkness) are so obtained, which yield the dotted green line to the right. It is convex in shape, quite different to the concave curves for pigments.
To calibrate steps in a grey scale, scientists usually measure human responses to increases in light. Psychophysical experiments helped establish the Munsell Color System on this basis. In 1933, people were asked to grade swatches of grey paper of known reflectances, as well as to adjust the luminance of a frosted glass target by moving the globe behind it. A convex curve once more results, as shown by the purple line at right. It would likely be even more curved (thin purple line), if people had been instructed to judge shades of darkness instead of light. In all cases, darkness curves fall toward the top of the graph, well away from the region occupied by pigment and musical curves.

Musical measures give only rough guides to the amounts of pigment needed for grey scales. The curve for watercolours, for example, is better described by a power curve, with the formula y equals x squared. (To put it another way, its degrees of darkness are proportional to the square roots of black pigment used.) It is also apparent that amounts of pigment - in weight, volume or area - are not optical measures of darkness. Across a grey scale, black paint and the resultant darkness grow at different rates, shown above as concave or convex curves respectively. To understand the reason, we can imagine a diluted layer of black paint as a filter. If it blocks 20% of the light reflected from white paper, 80% percent will be transmitted. An additional layer of the same paint will stop another 20%, but only of the remaining light - that is, 20% of 80%, a reduction of 16% from the total. Increases in dark pigment give diminishing returns in measured darkness, so much so that accelerated additions of pigment (concave curves, above left) will still result in a decelerating growth in darkness (convex curves, above right). To check the validity of any painted grey scale, each grey can be measured with a photometer. The amounts of reflected light should equal those used to establish corresponding values on an ideal darkness curve. Otherwise, the gap between pigment and darkness curves may be bridged with Kubelka-Munk equations. They are applied in forensic analysis, including the dating of paintings, though artists and designers often find it easier to rely on the judgment of their eyes. After all, pigments and darknesses are simply different ways of measuring the same artifact, a grey scale as perceived by human observers.

Painters are acknowledged experts in the construction of grey scales. As late as 1873, J Plateau enlisted eight independent artists to paint mid-greys, in the name of scientific research. He found their results remarkably consistent, and went on to lay the basis for the first power laws that govern perception of light. Artists have occasionally summarized their own conclusions: Giuseppe Arcimboldo framed his thoughts in a theoretical music, acceptable in the intellectual climate of 17th-century Prague. His final presentation seemed to be a courtly ceremony, with musical accompaniment, though his ideas live on in the writings of Don Gregorio Comanini. In the Romantic era, Mathias Klotz would manage to publish his own findings. He hand-coloured the illustrations in all sixty copies of his book, when he could spare time from duties at the Bavarian court. Instinctively, the painters began their scales with white, the background shade of paper, and measured alterations to it by amounts of added black. A 'negative' scale of darkness resulted, though modern scales generally run in the opposite direction, as 'positive' scales increasing in lightness. (The latter can be seen in Illustration 4, above, by viewing the darkness scale upside down.) Ordinary mortals will judge positive and negative grey scales somewhat differently (thin purple line), though artists may be able to produce more consistent results.

Arcimboldo's self portrait

Illustration 5 : "THE MAN OF PAPER", Giuseppe Arcimboldo, 1587.

Arcimboldo was sixty years old when he drew his self portrait, with pencil, ink and washes on paper. Even in such a private moment, the artist did not restrain his taste for a telling metaphor. His expert draughtsmanship at first conceals the fact that this is no naturalistic depiction. The robes and ruff, as well as the beard and hair, appear made from rolls and folds of paper - the very material on which the drawing is made. The same technique is used for the ear, and perhaps for other facial features as well. A model or mask, seemingly of paper, simulates the appearance of flesh, hair and cloth.

The first specific details on construction of grey scales came from by an English physician, Francis Glisson, in 1677. He was concerned that names of colours were inadequate to describe appearances in the natural world, and devised a system structured around a grey scale with twenty-four gradations. Glisson started from a shade midway between black and white, mixed from one part carbon black and fifty parts lead white by weight. His numbers approximate recent figures, of 1 : 40 for the black-to-white ratio, though it varies with different modern pigments. For zinc white, with less covering power, the ratio for mid-grey is about 1 : 60, while for dense titanium white, the ratio is closer to 1 : 25. A 50-50 mix of black and white will not make a mid-grey; its shade is much darker, only a step or two away from solid black. Even greater variation comes about in different mediums, employing other materials and techniques (see Illustration 4, above left). The German Romantic painter, Philipp Otto Runge, noted the dissimilarities. If areas of black and white were painted in segments of a circle, they would optically blend to grey when the circle was rotated rapidly. Paints gave quite another result, as Runge wrote to Goethe in 1808:

"I noticed that when I mix carbon black and white on the palette very little black is required to obtain middle grey. But in case of the spinning disk much more black is required. In disk mixture, white appears to be three times as strong as black."

Spinning disk became the preferred method for mixing shades and colours during the 18th century. The scientifically-minded believed it yielded more reliable results, and every reputable laboratory housed a collection of disks by 1900. Artists were not to be left out: the American painter and educator Albert H Munsell took the idea into three dimensions, producing a Balanced Colour Sphere. When spun, its surface colours merged into a grey scale. (The concept of a sphere of colours had also occurred to Runge, another painter, ninety years previously.) The Munsell system helped form the foundations of modern colour science, where light is measured by machine, under strictly controlled conditions. After extensive testing, CIE (the Commission Internationale de L'Éclairage) has arrived at a formula for most purposes: perceived lightness is to be considered proportional to the cube root of the actual luminance. Science, with all technical aid and certainty of method, can finally tell us something of what we see and how we see it.

And what happened to music along the way? Scientists most likely consider it a hindrance, as they probe the grey scale for physiological and neurological truths. Indeed, the rules of psychophysics specifically deny a connection between lightness and pitch, the two essentials of Arcimboldo's musical grey scale. A comparison of light and loudness is preferred: greys differ only in the quantity of light given off, so they are like the same note played at different volumes. Musical pitch, on the other hand, is considered a variation in kind, similar to the sensations of different colours. Still, there is some utility in the mathematics of music. Though its arcane ratios are too clumsy for psychophysics, octave measures can give a general idea of various grey curves. People make such associations quite readily, with little regard for the strictures of science. More or less consciously, our eyes, ears and brain will interpret the surrounding world: measured scales, of sound and light, are wilful attempts to understand and regulate these sensations.

Illustration 6 : WHITE'S ILLUSION

The appearance of a grey is altered by its surroundings

All greys are equal.

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