MUSIC FOR MEASURE :
on the 300th Anniversary of Newton's "Opticks"

2:  THIN AND THICK PLATES


1:  MUSICAL PRISMS
2:  THIN AND THICK PLATES
3:  MIXING IT

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At the end of Part 2, Book 2 of "Opticks", Newton concluded that "the Science of Colours becomes a Speculation as truly mathematical as any other part of Opticks". Though never arriving at a satisfying and coherent theory of light, he constantly emphasized its mathematical behaviour. By dividing the spectrum into seven colours (within eight borders set by notes of the diatonic scale of D), he stood to supply eight sets of calculations for each experiment. Anyone who has tried to follow that procedure will understand how tricky it can be for one case only, let alone the multiple instances in "Opticks". So Newton adopted a series of approximations, outlined in "Lectiones Opticae", his Cambridge lectures on optics, where he issued the following warning:

"But indeed remember here that these determinations are not precisely geometric, but still as nearly accurate as practical matters of this kind require, and to attempt anything further would manifest, beyond the tedium of calculating, an affected and vain curiosity."

Despite his imprecation, it is worth understanding something of the generalities that Newton employed and, consequently, the details he overlooked. His measurements of the spectrum usually involved the extremes of visible colour, the red and violet ends. Frequently, the mid-point was noted, and found to be at the borders of either green and blue, or green and yellow. The most 'lucid' or 'luminous' colour was also incorporated in some calculations -either bright yellow, or the 'citrine' at the border of yellow and orange. Rarely were the dimensions of a full set of seven (or even five) colours taken.

Part of the reason seems due to the "tedium of calculating", but inadequacies in Newton's laboratory technique are also relevant. He struggled with the quality of lenses, as to the perfection of the glass, and the tooling of their surfaces. Helmholtz was to remark on the pitfalls of prisms in the second half of the 19th century; he blamed the Newton's enlargement of blue and violet on prism use, and advocated diffraction gratings as the most accurate method of viewing the spectrum. But neither the technique, nor its explanation, were available to 17th century investigators.

   The raw material of Newton's investigations - a ray of sunlight admitted through a hole in the window-shutter - was also constantly on the move. And the slightest movement of a prism, leading to seemingly insignificant angular errors, could result in disproportionately large mistakes. (As it is, Alan Shapiro, in his edition of the lecture notes, allows Newton a 7% margin of error - a pretty good outcome under the circumstances). Still, he took some notable shortcuts: differences in refractive indices, for instance, were conveniently swept aside. They measure the behaviour of an individual colour, as a ratio of sines of incidence to refraction, and Newton might have extrapolated each value. He justified their omission with a casual remark in "Opticks":

"I do not affirm, that this proportion...holds in all the Rays; for the Sines of other sorts of Rays have other Proportions. But the differences of those Proportions are so little that I do not here consider them."

It was precisely when he turned to the mathematical differences between each of the colours, that Newton brought his colour-music analogy into play. With only one, two or three real measurements to go by, the positions and sizes of the remaining colours could be roughly derived from the standard measure of the musical scale. Newton became increasingly confident of the procedure, which first appeared in 1675-6 as an "Hypothesis explaining the Properties of Light", published in the Philosophical Transactions of the Royal Society. There, a musical measure was applied to prismatic colour, and a similar technique was used for thin plates. Though the musical component was not directly acknowledged in the latter case, its application to thin plates became a proud example of musical colour twenty-nine years later, in both Parts 1 and 2 of "Opticks", Book 2. A similar approach was applied to colours observed with thick plates, in Book 2, Part 4, where the musical measure was used almost brutally, even though the experimental results could barely support the analogy. In the details, colour music had become a substitute for both measurement and mathematics.

Newton made a detailed analysis of colours from thin plates

Illustration 5 : CROSS SECTION OF THIN PLATES, from "Opticks", Book 2, Part 1, Fig. 3.

When light shines on a thin film of air, sandwiched between a flat piece of glass AB and a curved lens CD, some of it is reflected and the rest transmitted. Their colours are listed at the top and bottom, respectively. On the surface AB, centred on E, rainbow-like circles are formed, much like an oil slick on a wet road. They are called Newton's Rings. The positions of red are marked by vertical lines (such as BD), which show the thickness of air that produces it. These increase in size in a progression of the odd numbers 1, 3, 5, 7, etc., as do the squares of the circles' diameters, measured through the centre E.

For observations of thin plates, two glasses were required - a slightly concave lens and a flat plate - that sandwiched a thin layer of air between them. The air was the thin plate referred to: it could just as well be water between the glasses, or indeed a thin film of water surrounded by air, as in the skin of a bubble. They all produced concentric, rainbow-like rings of colour (called Newton's Rings) when a light shone on them, as distinct from the rectangular spectrum cast on a wall by the prism. In a controlled way, he was mimicking the rainbow colours of oil slicks, or those seen on thin sheets of mica which he had read about in Robert Hooke's "Micrographia" of 1665. As early as 1665-6, Newton was writing of his experiments to produce rings, in his university notebook "Of Colours". For "Opticks", Newton found colours of thin plates mostly ran in the reverse order to those found with a prism - except when the thin plate was denser than the surrounding medium, as is the case with bubbles.

Newton would cast a monochromatic light over the glasses, to show but one colour at a time. Precision of observation was thereby improved, and the relative expansions and contractions of the coloured rings - as the lenses were tilted, or when the observer's eye was moved - more accurately estimated. By careful measurements and mathematical dissection, Newton was able to pick apart the coloured patterning to detect an orderly sequence of rings. He found each iris, or ring, contained a full spectral array, bordered by dark rings that reflected no colour. (All light was transmitted through a dark region, except when a neighbouring iris interfered to conceal the dark ring.) Near to the centre of the glasses, the rainbow patterning was most defined; it became more obscure towards the perimeter, where rings began to overlap each other and merge into white.

Newton found the sequence of coloured rings to be mathematical. From each of their mid-points, through the centre of the lens, the diameters of bright and dark rings were measured. Their squares were taken, and found to form odd- and even-numbered sequences, respectively. Due to the geometry of the lens, the squared diameter of a ring, reflected on the surface, was proportional to the gap between the glasses (the plate thickness) at its edge. So the depths of thin plates, under the mid-points of successive rings, followed an arithmetic progression of odd numbers, 1, 3, 5, 7, 9, etc. Likewise, plate thicknesses at the mid-points of dark bands (between irises) ran in a sequence of the even numbers, 2, 4, 6, 8, etc. The minute gaps, that produced the rings, directly follow the same arithmetic progression as the sizes (in diameters squared) of the rings themselves.

Having established an overall framework, Newton applied musical divisions to the irises' widths, since each contained all the colours. He used selective rays of prismatic light to view the extreme red and violet margins, and establish the breadths of the rings. The air gaps at these edges were calculated to be consistently in a ratio of 14 : 9 (as were the squares of their diameters) for all the rings. But there was a problem; his usual musical measure of an octave (ratio 2 : 1) was too large to fit. So Newton decided to reduced it by applying a formula - the cube-roots of the squares - to the size of the scale. Nowhere does he attempt to justify or explain its use: it is tempting to think we are being blinded by science, that Newton was hiding behind mathematics to avoid awkward questions about his shrinking scale. Or he may simply have assumed his readership erudite enough to understand the derivation of his formula, and willing to go along with the approximation found in "Opticks", Book 2, Part 1, Observation 14:

"..the thicknesses of the Air between the Glasses there, where the Rings are successively made by the limits of the seven Colours, red, orange, yellow, green, blue, indigo, violet in order, are to one another as the Cube Roots of the Squares of the eight lengths of a Chord, which found the Notes in an eighth, sol, la, fa, sol, la, mi, fa, sol; that is, as the Cube Roots of the Squares of the Numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2."

Newton, well aware that the colours did not always fill an octave, still preferred it as a standard measure throughout "Opticks". Any observed difference was quickly passed over, though he also noted in Observation 14 that the divisions of the spectrum "are to one another very nearly as the sixth lengths of a Chord which found the Notes in a sixth Major, sol, la, mi, fa, sol, la". He seems to be in error by nominating a major sixth, whose ratio of 15 : 9 in just intonation is too large for the observed results. Even the just minor sixth would be a tad bigger than 14 1/3 : 9, the maximum size he set for width of colour. To find internal colour divisions, Newton squared the geometrical, musical intervals, effectively doubling their sizes; an octave is increased this way to two octaves, or 24 semitones rather than 12. Taking a cube-root further divided the interval by three, so 24 semitones became 8, or a minor sixth.

The procedure deals with semitone intervals as if they were evenly tempered, and we find the tempered minor sixth, of 8 semitones, to have a ratio of 14.3 : 9. This is within the upper limits suggested in Observation 13, if the extreme rays were perfectly visible. But Newton's scale in "Opticks" makes for a larger sixth, with the different note sizes of just intonation, than the minor sixth of equal temperament. Moreover, there is no theoretical connection - via a factor of the cube-root of a square - between the octave and just sixths, major or minor, as there is in equal temperament. Despite the difficulties, the colours of thin plates were compressed. Newton composed a clever diagram of the arrangement of their rings, which showed precisely how their colours separated and blended. The irises were stacked above a line divided musically, as per instructions at the beginning of Part 2, Book 2, of "Opticks":

"...in proportion to one another, as the Cube-Roots of the Squares of the Numbers, 1/2, 9/16, 3/5, 2/3, 3/4, 5/6, 8/9, 1, whereby the Lengths of a Musical Chord to sound all the Notes in an eighth are represented..."
The colours from thin plates meet those from ordinary prisms

Illustration 6 : COLOURS OF THIN PLATES,
from "Opticks", Book 2, Part 2, Fig. 6.

At the beginning of Part 2, Book 2, of "Opticks", Newton dissected the colours of thin plates in Fig. 6. The slanted, solid lines (at top) represent each successive ring at its brightest. They are spaced apart, according to the odd and even numbers on the left vertical axis; colours are plotted along the horizontal axis, as shown. Newton used his musical yardstick to divide the colours in each ring. Developed for the prismatic spectrum, the measure was oversize for the colours of thin plates, and had to be compressed by a factor of the cube root of a square. An octave (12 frets on the big guitar) was reduced to a minor sixth (8 frets) to make a 'mini-octave' (12 frets on the small guitar).
In the 1675 "Hypothesis", Newton had avoided the problem, establishing the measure afresh - working directly from the small guitar, as it were. The text of "Opticks" stuck to the musical model derived from prisms and just intonation, and also misrepresented the colour span as a major (not minor) sixth. Each ring would be equivalent to the traditional hexachord and, in a superficial sense, the colours of thin plates were akin to the Guidonian Hand, with overlapping irises instead of interlocking hexachords. But the main import of the graph is the display of analytic and synthetic prowess by Newton. It demonstrates the mixing of coloured lights, where any horizontal line crossing the diagram (as is shown) will discover all components of light that the lenses reflect at a given point.

An earlier version of the above diagram had appeared of 1675. Here, he had used a graphical method to create the divisions, dividing a line at 14 and 9 units along its length, to represent the colour ratio of 14 : 9 and create the span of one octave. A reference point at 4 units provided a second octave, also 5 units long. From the reference point, string lengths for each note were measured, just as they had been in his original analysis of a prism's spectrum. Effectively, Newton was recreating his musical measure from scratch, but at a smaller scale (see small guitar in Illust. 6 above). Although the method using cube-roots of squares had been known - a letter to Hooke, unposted and unpublished, raised it in 1672 - Newton reserved its use for "Opticks". A mathematical device, it would directly connect experimental results to his musical measure, and immediately provide values for the horizontal axis of the colour-mixing diagram.

Apart from the prism measurements, Newton failed to mention the colour divisions in the "Hypothesis" were musical, nor give them sol-fa labels to indicate a key. But he effectively employed equal temperament (a system he had once described as 'unmusical'). Opting for twelve even divisions of the scale, Newton divided the 14 : 9 ratio of colour by "eleven mean proportionals" (or "twelve Geometrically progressionall parts" as his lecture notes had it) to create semitones. The musical notes he selected from these approximated the diatonic scale of D he used elsewhere, though each tone would be of uniform size, and a semitone exactly half that.

For Newton and his contemporaries, equal temperament was only one of several musical constructions available. In fact, one of the few items that interrupted the lengthy reading of his "Hypothesis" at the Royal Society in early 1676, was the presentation by a Mr. Berchenshaw of his 'new' system of music. (It was generally encouraged, even though many felt it smacked too much of the old Pythagorean modes.) Newton was a little old-fashioned in these matters himself. In an early treatise, "Of Music" in 1665, he had outlined the received wisdom of his time, arranging "ye 12 Modes in their order of Elegancy". He seemed to think his own arrangement, for both "Opticks" and the "Hypothesis", to be the standard for the first mode, with a minor third and a major sixth. Though Newton was never much interested in musical practice or taste - the word 'music' appeared only once in the "Hypothesis" - he admitted that fashion had changed and the second mode was preferred. It was equivalent to a white-note scale of G, and he dissected it systematically in his working notes.

For the public presentation of music in "Opticks", Newton opted for the syntonic diatonic on D, using conventional ratios of just intonation as the least likely to cause controversy. He approached music, on the whole, as a mathematical exercise (albeit with a philosophical dimension) rather than as a creative endeavour. On examination, his arrangement of notes was musically inferior to other possible just scales: two of the fifths it contained (E-B and F-C) were too small - howlers known as the 'wolf' - as were two of the thirds (E-G and A-C). In a system that prioritized the purity of thirds and fifths, Newton's scheme was unsatisfactory, and there were those who were not afraid to say so. But Newton was not intent on reforming either the practice or theory of music. Rather, he was promoting the notion that there is some cohesion to the world of nature, evident to those with the wisdom to see (and his subsequent reputation ensured many would take his claim seriously). When pressed upon the subject of music, Newton declared "these are some things which I cannot speak positively to for want of experiments and skill in musick". And he was just as self-effacing towards the world of painting in his lecture notes where giving reasons for preferring just intonation to equal temperament:

"...because it perhaps involves something about the harmonies of colours (such as painters are not altogether unacquainted with, but which I myself have not yet sufficiently studied) perhaps analogous to the accordances of sounds."

One of a series of six paintings of musicians, by Lely

Illustration 7 : "A MAN (possibly the artist)
PLAYING THE VIOLIN",

by Sir Peter Lely, c.1648.

Sir Peter Lely was the foremost painter of mid-17th century England, and depicted music from another perspective. Five of his surviving works from around 1650 showed single figures playing musical instruments - a violin, a recorder, a Jew's harp, and two kinds of lute - a popular motif throughout Europe and prominent in Dutch and Venetian painting for a century. The theme was commonly understood as an allegory of Love and Beauty and, at the least, these paintings demonstrate that music was considered a cultivated pastime. Its practice had survived recent Puritan opposition in England, where organ-playing in churches had been banned (and stained glass and other finery destroyed). Cambridge, a Parliamentary stronghold during the Civil War, had been particularly effected, and its Puritan legacy may have influenced Newton in his disregard for the practice of the arts. Some early writings, in a school notebook of 1659, had showed his intense interest in painterly effects, including the mixture of pigments. By adulthood, Newton's energies were redirected to the objective study of colour in light, and its illuminating effect on dry mixes of coloured powders - both topics covered in "Opticks".

In the small world of English high society, it was not unusual for artists and scientists to mix. Robert Hooke, for one, was apprenticed to Lely when first he came to London, and enjoyed the heady wine of his table as well as musical evenings in the taverns of the town. Painters and scientists shared the same patrons, too. Roger North, for example, had his portrait done by Lely and was also the executor of his estate, and it was he that approached Newton in 1677 for an opinion of his brother's book on music. It was typical that Newton directed his reply to physical matters, the mechanics of sound and the acoustics of hearing; he avoided the aesthetic innovations, of scale and key structures, that Francis North raised in his "Philosophical Essay of Musick". But despite their unique approach, natural philosophers like Newton still had much in common with artists. The musical instrument played for pleasure in the polite drawing room became an experimental tool in the laboratory, and music theory remained a cornerstone of the mathematician's education. The raw materials of painters and dyers, too, were subjected to chemical - or alchemical - processes, to be incorporated into scientific theories of colour and matter.

Robert Hooke, as Curator of Experiments at the Royal Society, arranged many demonstrations to articulate musical sound in the 1660s. He was the first to show the gentlemen virtuosos there the direct relationship between pitch and frequency in 1681, with toothed wheels of brass that produced higher notes the faster they were turned. Independently of Newton, Hooke developed an ideal musical scale of his own; it was also based on just intervals formed from a symmetrical sequence of semitones. The two had other things in common, too - they both looked up to Robert Boyle, who mentored Hooke's experiments at the Royal Society. Boyle's scientific method was much imitated, and many of Newton's investigations into colour were adapted from his "Experiments and Considerations Touching Colours" of 1664. Even the prism experiment, on a single ray of sunlight let into a darkened room by a hole in the window shutter, is found there. Soon, both Hooke and Newton were to formulate their own theories of colour. Hooke outlined his in "Micrographia" of 1665, hypothesizing colours as modifications to pulse-like motions through the aether. Included amongst his many observations under the microscope, was the first description of the colours of thin plates, with rainbow hues seen on laminae of mica. Shortly after, Newton was noticing the same effect caused by surface irregularities on flat pieces of glass pressed together. But, unlike Hooke, he believed light was made of streams of corpuscles flying in a straight line. Boyle quickly deferred to both new theories, leaving the other two men to fight it out.

When Newton communicated his "New Theory about Light and Colours" to the Royal Society in 1672. Robert Hooke was among the first to read it and immediately saw a rival theory to his own. He repeatedly urged Newton to adopt a wavelength conception of colour, who objected in turn to any sweeping hypothesis. He only sought to point out that colours originated from white light, and were discoverable by their characteristic "refrangibilities". Anyway, light did not bend around obstacles, the way that waves of sound and water do. Although there was no mention of wave-like characteristics (let alone a likeness to music) in the "New Theory", Newton already had the basic ingredients for colour music - the general analogy between light and sound, the seemingly periodic nature of light in the colours of thin plates, and a symmetrical musical scale to structure the arrangement. By 1675, Newton was openly acknowledging some vibrational aspect to both sound and light in his "Hypothesis", including the first spectrum of colour and music. Shortly after, he left the field in a huff, but his eventual inclusion of the metaphor of colour music in "Opticks" may be traced, in large part, to Hooke's initial and persistent hounding:

"...as there are produced in sounds several harmonies by proportionate vibrations, so there are produced in light several curious and pleasant colours, by the proportionate and harmonious motions of vibrations intermingled, and as those of the one are sensated by the ear, so those of the other are by the eye."

Hooke and other critics could point to the colours of thin plates as evidence of wave motion: the same colour recurring at regular intervals suggested light was a wave motion like sound, and the depth of thin plate it crossed could measure multiples of the colour's wavelength, or distance between pulses. But Newton was loath to abandon a purely corpuscular theory of light. He attributed any wave-like characteristics to the surrounding aether, and surmised it was set in motion by the travelling light. The aether waves acted on light particles in turn, retarding or accelerating them according to where the wave pulse were dense or not. Vibrations caused at the surface of a medium, in advance of an approaching light particle, dictated whether a colour would be reflected or allowed to pass. To describe the state of a light particle at the interface of two mediums, Newton invented "Fits of easy Reflexion and easy Transmission" that determined its propensity to either bounce back or pass on through: the distances between these alternate states he called the "Intervals of Fits". By this means, Newton hoped to explain why coloured rays were selectively reflected, according to the thickness of the thin film they traversed.

Having measured the real colours of prisms and thin plates with his colour-music code, it was but a small step for Newton to apply the same method to an abstruse area of pure speculation - the Fits of easy Reflexion and easy Transmission. It was applied as for thin plates, squeezed down in size and omitting all mention of sines to which it was originally linked. The first, and simplest, example of musical division of the spectrum was connected to progressively more complex situations till, in one fell swoop, Newton had covered the distance between observed and hypothesized examples. In all cases, similar musical terms were used to describe the way colour was intimately divided. As Proposition 16 of Book 2, Part 3 had it:

"...the Intervals of the Fits of easy Reflexion and easy Transmission are either accurately, or very nearly, as the Cube-roots of the Squares of the lengths of a Chord, which found the Notes in an Eight, sol, la, fa, sol, la, mi, fa, sol, with all their intermediate degrees answering to the Colours of those Rays, according to the Analogy described in the seventh Experiment of the second Part of the first Book."

This was the closest Newton got to acknowledging that light had any wavelength characteristics of its own. But when it came to the aether, and the subjective sensation of colour, he had no need for obfuscation. Proposition 12 compared the aether vibrations to waves, caused by dropping stones on water, and to the sound vibrations in air coming from a percussion instrument. At the end of the third Book, Newton reiterated these remarks: in Query 13, red caused the largest vibration, violet the smallest; Query 17 repeated the analogy of aether waves to circular ripples in a pond, and to the spherical propagation of sound in air; and Query 28 stressed again that light was not a wave itself, since it did not bend round an obstacle into the region of shadow, as waves of water or sound did. Such observations were commonplaces of 17th century science, since Galileo first described the pulses of air that gave rise to sound. For natural philosophers, they were similar examples of movement in elastic mediums and, slowly, the principles of elasticity were being applied to sound and the known laws of music. Newton himself came to the fore in 1681, writing of the wave-form of sound in "Principia Mathematica". Other adventurous scientists tried to include light in the same category. Robert Hooke was one of the first; Christiaan Huygens also described waves for both sound and light, with beautiful mathematical models that Newton tore to shreds. Still, the formal idea persevered in the colour music of "Opticks" (perhaps against the wishes of Newton, or was he simply hedging his bets?), "for the analogy of nature is to be observed".

Two further examples of colour music followed in "Opticks", for the colours of thick plates. The plate, or speculum, was a concave dish of glass 1/4" thick and mirrored on the back, that reflected rings of colours when illuminated. Newton's first task was to establish the positions of the different colours, just as he had done for thin plates. However, he found the violet to be too faint to see clearly, and there was some confusion with the red, too. So instead of locating the outside edges of each ring of rainbow colours, he took inside measurements for the red-orange and blue-indigo borders. (As expected, their proportion of 9 : 8 was smaller than the 14 : 9 ratio of red and violet extremes, that would match results from thin plates.) His problem became how to find the other colour margins in a ring, using these two fixed points. Instead of reducing the musical measure by a factor of the cube root of squares, Newton decided to use fractional differences between string-lengths, and locate other colour edges proportionately. In Observation 5 of Book 2, Part 4, the differences in diameters of the margins, from outermost red to inner violet, were decided to be:

"...in proportion as the Differences of the Lengths of a Monochord which sounds the Tones in an Eight; sol, la, fa, sol, la, mi, fa, sol, that is, as the Numbers 1/9, 1/18, 1/12, 1/12, 2/27, 1/27, 1/18."

The theme of the painting may be that of Music, in the service of Love and Beauty

Illustration 8 : "THE CONCERT", (detail)
by Sir Peter Lely, late 1640s.
(The cellist may be another self-portrait by Lely.)

What follows from the pen of the ex-professor of Cambridge, was no less than a schoolboy exercise in arithmetic. It emerges, when the fractions are calculated out, that the sizes of blue and violet had been interchanged, along with their musical ratios of minor and major tones. At first sight, the rearrangement resembles earlier ones, from the mid-1660s, that Newton drew up in undergraduate notebooks. Both fourths and fifths were improved - one horrendous 'wolf' fifth is gone - though another third became impure. But rather than the search for an ideal scale, the likely explanation for the change seems a practical one. The shift of ratios simplified calculations of colour positions, by reducing the size of the common denominator of the fractions involved from 720 to 108.

The ratios reverted to the usual symmetrical scale in Observation 8, following complex calculations to find Intervals of Fits of easy Reflexion and easy Transmission for the bright yellow of the same thick plate. Once more, the compression formula was applied to the octave, as in the previous example of thin plates, and as specified for Fits. But it was reduced to cube-roots only rather than cube-roots of squares. (The simplification seems to arise from diameters of some colours were being compared to square roots for others, so there is no real economy of calculation.) Newton was trying to prove how the method of Fits could produce "pretty nearly" the same result as the more orthodox approach in Observation 5. So doing, he hoped to add experimental weight to his flimsy hypothesis of Fits.

At the end of Observation 5, the extent of colours for thick plates had been compared to that found on thin plates. The former proportion of almost 3 : 2 from red to violet, Newton claimed, "differs not much" from that for thin plates. But he was exaggerating. In truth, the earlier results were always above that figure, at 14 : 9 from red to violet. Where they had been likened to a musical sixth, the colours of thick plates would not have spanned a fifth, falling short of thin plates by a semitone or more. That thick plates should even behave as thin ones did was brought into doubt by Thomas Young, commenting on Newton's experiments for the 1817 Encyclopaedia Britannica, and noting "the analogy ... between these colours and those of thin plates, is in fact very far from amounting to identity". Perhaps, towards the end of "Opticks", Newton was getting sloppy, making wild claims in his old age. It would be wise to heed the warning of D.T.Whiteside, in his preface to the facsimile edition of the Cambridge lecture notes, that we must:

"...be ever watchful, for the optimistically rounded off numerical result and the interpolated 'observed' measurement by which Newton often seeks, consciously or unconsciously, to cut corners in justifying his explanatory models and interpretations."

Newton's experiments with thin plates - even more than those with prisms - had provided him physical evidence on which to base a musical analogy. But the breadth of each coloured ring immediately suggested an interval of a sixth rather than the prism's octave. Thomas Young, in his 1801 Bakerian lecture, this time agreed that "the whole visible spectrum appears to be comprised within the ratio of three to five, or a major sixth in music". When he alluded to a sixth in passing, Newton may have had the traditional hexachord in mind, the old Guidonian Hand from which the scale was derived. It still had some currency in the 17th century, and provided the basis for the sol-fa - ut, re, mi. fa. sol, la. (The letter names for notes, A to G, were not yet in use, nor ut replaced by do, and the seventh note had not been given its later syllable of ti or si, though Newton called it a second fa.) The syllables from sol to la, supplied for a sixth in Observation 14 of Part 1, Book 2 of "Opticks", found a hard G hexachord, or the notes from G to E in Newton's white-note scale on D. Its standard format, a semitone with two whole tones on either side, could have accommodated the five-colour division of the spectrum that Newton often used - of red, yellow, green, blue, and violet. If he had so wished, Newton might have used thin plates for his musical standard, based on a sixth. The dark (or light) spot in the centre of the lenses gave a tangible reference point from which spectral rings could be measured. The areas bound by each ring (proportional to squares of diameters) and the thicknesses of air producing them, both followed the same orderly progression. The ratio of 14 : 9 for each ring's colour borders, from outer red to inner violet, gave Newton his most conclusive data.

The more arbitrary set-up for the octave had been established by eye, dividing the rectangular spectrum cast by a prism. The musical notes there were offset from a reference point whose location was almost capricious, a spectral length away in empty space. Based on unreal dimensions, the match of prismatic colour distribution to an octave was, from the outset, a physical fiction. But Newton stuck with the less convincing octave arrangement, as "it agrees something better with the Observation" - which is only true if one ignores the equal tempered minor sixth, the unorthodox basis of his calculations for thin plates. Moreover, green would have had to occupy the smallest division in a hexachord, the central semitone, and observations and preference might not allow it such a minor role. The octave, on the other hand, allowed the central green to be of maximum size, just like the red and violet at either end. Not only was symmetry reinforced this way, but the triad of rainbow colours favoured by the Aristotelians were subtly emphasized. At the same time, the octave's extended range spaced the important primaries of painters and dyers to occupy positions aligned with the musical triad based on the key note D. Overall, the sixth may have been just too antiquated for Newton's taste, even though it was supported by the data, and its use could have invoked derision from musicians and scientists alike. The octave would be a better understood reference, more in keeping with the formal conventions of gentlemen virtuosos.

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