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

1:  MUSICAL PRISMS

   In 1704, Sir Isaac Newton, Master of the Mint, President of the Royal Society and the most renown natural philosopher of Europe, condescended to reveal the secrets of light and colour to the world. What insights he had uncovered in forty years of experiment and analysis were laid forth in "Opticks: Or a Treatise of the Reflexions, Refractions, Inflexions and Colours of Light." Through its three Books, each of several Parts, the reader is privileged to an inside view of Newton's laboratory technique, starting with simple experiments with a glass prism that anyone might conduct at home. It is still commonplace for school children the world over to duplicate his procedures - shining a thin beam of sunlight onto a transparent, triangular glass and noting the standard set of colours that emerge from the other side. Taking the process one step further, the same colours (focused with a lens) can be sent through a second prism: lo and behold! the colours recombine into a beam of the same white light from which they originated.
   At its publication, "Opticks" was the most definitive study of light and colour yet produced, and was to remain so for the rest of the 18th century. It was Newton's most popular work, and subsequent editions, including French and Latin versions, placed his theories before a worldwide audience. Several concepts that originated there took root in the popular imagination, where they still remain. The prism's ability to produce a spectrum from a single beam of white light is perhaps the most durable idea to emerge from "Opticks": the rock group Pink Floyd used an image of the event, on the cover of their hugely successful album "Dark Side of the Moon". While many fans may have realized the graphic (by Hipgnosis) referred to Newton's work, not all would have noted that six colours emerge from the prism, not the usual five or seven found in "Opticks". (Orange is included, but indigo omitted.) Still fewer were likely to know this contradicted Newton's ideas on colour music.

   After a sunbeam passes through a prism, it may cast its light on a wall opposite. To describe any image of the sun, so produced, Newton coined the term 'spectrum' (from a Greek word for 'apparition'). The word is now applied specifically to that oblong streak of light, which shows a band of colours of infinite variety. To talk of this array, spread from sunshine, individual colours need be named. We may agree to call a specific location yellow, say, or green, but there are no clear boundaries along the spectrum's length; colours merge imperceptibly into one other. While there are minute black cracks (called Fraunhofer lines) that cross the spectrum at fixed points, these were not noticed for almost a hundred years after "Opticks" was published. In any case, they do not separate colours conveniently, according to common names or colour groups. Newton had to be content to describe colour with words and labelled drawings. "Opticks" is provided with four fold-out sheets of illustrations, containing some fifty-six clear, black and white diagrams to amplify the text. That none of these is coloured was due to the technical limitations of 18th century printing; metal plates or woodblocks could not supply the subtle blend of colours required, and hand-tinting would have been prohibitively expensive.
   So Newton needed standardized labels to separate colours, one from the other. In his earliest research, he often referred to the spectrum by its extremities only (the red and blue ends), but shortly rendered it consistently with a fuller description. Five colour terms - red, yellow, green, blue and violet - were employed, to designate successive bands of colour. In "Opticks", they first occur in Book I, Part I, Experiment III, and Newton uses the five-fold description at least a dozen times more throughout the treatise. It is handily applied, specifying which area is effected in what way by the experiment under discussion in the text. By Experiment VII of Part I, Newton was providing more detail, and added orange and indigo to his five colours, giving seven - red, orange, yellow, green, blue, indigo and violet. The seven-fold division, ROYGBIV for short, appears a dozen times or so throughout "Opticks", about the same frequency as the five-colour description. (There are also a few hybrid versions of the two, mostly of six colours, as well as important examples of fuller sets of colours.).

   The production of colours in rainbows is explained in Proposition IX, Book I of "Opticks" . Again, Newton employed the seven hues of ROYGBIV, this time caused by light refracted through raindrops rather than a prism. They constitute the first consistent colour list for remembering the fleeting spectacle of the bow. Even Rene Descartes, in essays attached to "Discours de la method" of 1637, treated its colours cursorily, though he once listed up to six - orange, yellow, green and blue, lying between red and violet edges of the bow. (He did, however, clarify the size of rainbows, and was the first to publish the sine laws which govern the way light bends on passing into transparent substances.) Commentaries on bows, from Aristotle on, had usually specified a mere four, three, or even two colours: arguments were advanced excluding others, such as yellow, as somehow unreal, which did nothing for the advancement of optics. And medieval painters had rarely portrayed bows realistically; any number of variously coloured and patterned parallel bands seemed to do. Today, some spectral arrays can be quite as lax: winning cyclists at the World Championships are awarded 'rainbow' jerseys, that have horizontal bands of (from the top down) blue, red, black, yellow and green. These are the Olympic colours in a 'rainbow' configuration, and nothing like the real thing.
   It is important that some definition of rainbow colours exists, though a set of five (red, yellow, green, blue and violet) might seem adequate for any casual observer. But ever since "Opticks", the colour initials ROYGBIV have been used as a mnemonic for teaching children the colours of the rainbow. NASA Space Agency maintains a web page to that purpose, hosted by a Mr ROY G. BIV; in Indonesia, a popular chant, me-ji-ku-hi-bi-ni-u, shortens the Malay words for Newton's colours. Folk riddles name the same seven hues in Estonia, where they have entered into the encyclopaedia's definitions of rainbows. Likewise, the Oxford dictionary (at least my 1976 edition) gives ROYGBIV as the 'conventional' description of rainbow colours.

   Since light behaves similarly, whether outdoors or inside, Newton could study rainbow-like behaviour in the controlled environment of his laboratory, applying the new experimental techniques of natural philosophy. The effect of adding or subtracting colours to or from the spectrum, the differential displacement of colours when viewed through a prism - these and other effects were described in Part I of the first Book of "Opticks". But, other than in the introductory Axioms, detailed calculations were avoided until the last two Propositions. Here, he described his famous reflecting telescope, that had gained him entry to the Royal Society in 1672. He set out to show (erroneously, as it happened) how improvements to refracting telescopes were limited, because of colour distortion from the lens. The previous prism experiments in "Opticks" provided Newton with the means for his proof.
   A quadrant was used to measure the angle of incoming sunlight, as well as that after it passed through a prism. Measurements were taken of the spectrum cast, so the angle it subtended at the prism's face could be calculated. From his data, Newton could give fixed figures for the different colours at either end and in the centre of the spectrum. At a given angle of incidence for the incoming light, whose sine was 50, he found the sines for angles of emerging colours, to range between 77 (red) and 78 (violet), at either end of the spectrum. According to the fifth Axiom, the sine law of refracted light was applied to colour, giving each a unique "degree of refrangibility". This indicated how much a coloured ray of light was bent away from the vertical, when passing from one medium to another (such as out of the glass of a prism into air). In modern parlance, each colour is refracted to a different degree - red the least and violet the most. The ratios of sines, of incidence to refraction, are set; for example, red's index of refraction remains at 50 : 77, no matter if the direction of sunlight changes, as long as the two mediums were glass and air.
   Newton's sums gave the first quantitative proof of his revolutionary theory of colour, in support of his rather more qualitative experiments. His lectures at Cambridge, around 1670, had covered similar ground, with a great theoretical emphasis only hinted at in "Opticks". But there were limitations to both mathematical and experimental approaches. Newton never settled on a single method to account for all the coloured rays; he also treated light as if it were only refracted once, not twice, on passing through two faces of a prism. Acknowledging the weakness of one such generalization, he wrote:

"Moreover, if all the other kinds of rays were considered simultaneously, that assertion, though no longer absolutely true, would still so closely approach the truth that it could be taken as true with respect to sense and a mathematical calculation. Consequently, since a geometric calculation of the refraction... can be undertaken rather difficultly, I will not be afraid to do it in a way which, however mechanical, is more suited to practice, being confident that no fault aught to be attributed to me if, when I perform computation in physical matters, I omit the minutiae, that entail burdensome and fruitless work."

"...in proportion to one another, as the Numbers, 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2, and so to represent the Chords of the Key, and of a Tone, a third Minor, a fourth, a fifth, a sixth Major, a seventh, and an eighth above that Key..."

Illustration 1 : MUSICAL DIVISIONS OF THE PRISM from "Opticks" of 1704, Figure 4.

   The colour-music layout, shown above, delineated notes as vertical lines, with the key note on the left at AG. The chord, or string length, that gave every note its pitch, ran from G to M at the other end of the spectrum (covering the whole octave, or eighth), and was extended the same distance to X, providing a second, reference octave from M to X. AG gave the lowest note, with GX as the chord of greatest length; at the other end, the note FM determined a string half the length, MX, sounding a full octave higher. (By musical convention, the interval between the two is given by the ratio of their string lengths, GX : MX, or 1 : 1/2 = 2 : 1, the ratio of an octave.) Other notes are found by the fractional distances marked along the spectrum: the third note from the left, for instance, was given as a third minor, and lies at 5/6 of a string-length from X. (Its measured distance from the key note AG is decided by the ratio 1 : 5/6 = 6 : 5, which is the accepted standard for the minor third, in a musical system known as just intonation.)
   The seven colours red, orange, yellow, green, blue, indigo and violet (ROYGBIV), fill the seven intervals between the eight notes, starting from the highest note on the right. In this instance, they form a scale running downwards from the high red, the least refracted light, to the deep violet, which is refracted the most. The colours are more cramped at the red, right-hand end, as are the musical notes. If one considers that the ratios for musical notes are really the same size at either end - both are major tones, of 9 : 8 in just intonation, as is the central green - the relative compression of red and expansion of violet are evident. The compression of higher intervals is a characteristic of musical scales; it occurs because their various ratios are successively multiplied, rather than added, together. (From the half string-length at extreme red, the next lowest note is obtained thus: 1/2 x 9/8 = 9/16, while 9/16 x 16/15 = 3/5 gives the following note, etc. To calculate notes from the low, violet end, the string-length need be divided by ratios, i.e., invert them and multiply: 1 x 8/9 = 8/9, and 8/9 x 15/16 = 5/6 give the two next ascending notes.) The compression is obvious by the spacing of frets on a guitar's neck, which grow closer together with ascending notes. The somewhat similar appearance of the spectrum may have influenced Newton, to suggest its likeness to music.

   But in other experiments, with combinations of lenses or prisms, Newton got the opposite results. Tilting the lenses made the spectrum shrink into white: further tilting caused the colours to re-emerge, but in reverse order. (The spectrum could be likened to a fan, with each blade a different colour, that becomes white when closed, and reveals the reversed colours by opening in the opposite direction.) As early as 1665, Newton had recorded in his student notebook, "Of Colours", how "Prismaticall colours appeare in the eye in a contrary order", when using a tank of water to bend the light. In "Opticks", Experiment 7 with the prism was the only instance, using the musical analogy, where violet was more expanded than red, and so occupied the lowest musical interval. In the other six occurrences of colour music - with a colour-mixing disc, another mixing diagram, experiments with thin and thick plates, and the calculations of 'Fits' for both - red occupied the low end of the scale.
   When red became more expanded than the violet, the intermediate colours swapped sizes too - orange with indigo, yellow with blue. The behaviour of coloured light still suggested a non-linear structure, similar to the known, geometric order of musical sound. But the whole colour music array - established in the diagram of the prism above - needed to be reversible, to accommodate colours in the opposite order, with red at greatest size. This Newton achieved with a symmetrical musical framework, selecting ratios of notes accordingly. They remained stationary, while the colours could then run either up or down the scale, as the case required.

   Why did Newton decide to use musical measurements? The most succinct explanation is they filled a mathematical gap. As soon as he had established the arrangement in Experiment 7, Newton paralleled the fractions of the string lengths to sine values, thus eliminating the need to make tedious and repeated calculations for the sine values of every colour margin. Assigning 77 to red and 78 to violet, he divided their difference with the same fractions by which the spectrum was divided. ( The red -orange border, for instance, occurs at 9/16, which lies 1/16 from red at 1/2. This amounts to 1/8 of the spectral length, and of the 1/2-string length. Its sine value is therefore 77 1/8: successive fractions were added to gain sine values for all coloured borders.) Similarly, Newton manipulated the musical fractions in later experiments - to compare the colours proportionately, to scale the spectrum to a different size, to establish relative positions of colours, and to otherwise insert them in more complex calculations. It is noteworthy that he nowhere mentioned the musical ratios themselves, preferring to deal with abstract numbers for string-lengths, as if they were experimental measurements. So no idealized musical scale was posited directly - perhaps in the hope of avoiding the ire of speculative theorists in the field.
   Whether the colour-music equation works for all Newton's applications would depend on the accuracy required. It certainly simplified the text, and provided a memorable emphasis to some important experiments and ideas. However, it implies a general theory about the relation of colour to music and, as such, it is fundamentally flawed. To equate a geometrical progression (musical notes) to a sine progression (values for refraction) is not possible, as they are not the same thing, and inconsistencies arise inevitably. The case established in Experiment 7 only applied in that one instance; as the angle of incident light changed, the visible spectrum would alter with it. At greater angles the spectrum lengthens, though at a decreasing rate, while the centre creeps away from the blue-green border towards yellow. The changes occur in the opposite way when the angle decreases. Moreover, the outer limits of the sines, red to violet, expand increasingly with greater angles and decrease more and more slowly as angles lessen. Their difference of 1 between overall sine values, insisted on by Newton, in fact changes by 10% if the light shifts by 7 or 8 degrees - a figure within Newton's experimental range.
   It is worth debunking the colour-music association somewhat, considering the portentous finality with which Newton pronounced it, and the way many since have treated it as some kind of occult law. Much of it is knowingly untrue: the one verity, if each coloured ray behaves consistently, is that the internal sines retain their musical pattern, despite changes in size to the colours they represent. And even that does not depend on his musical divisions, since any arbitrary set-up would have behaved just as well. Newton partially conceded this, in his Cambridge lecture notes on optics, by admitting that an equally-tempered scale would not produce a noticeable difference, and that spectral colours merge anyway:

"I could not, however, so precisely observe and define them without being compelled to admit that it could perhaps be constituted somewhat differently."

Illustration 3 : MUSICAL DIVISIONS OF THE PRISM from the "Hypothesis explaining the Properties of Light", 1675.

   Changes to the Western music system began to creep in during the late 15th century (due in large part to Arab influence), that brought a subtler understanding of music theory, and new instruments like the lute (al-'ud), with different tuning requirements. When Ramos de Pareja used non-Pythagorean note ratios, in "Musica Practica" of 1482, other theorists were outraged. Still, ancient Greek texts, translated during the Renaissance were showing the classical legacy to be more complex than previously imagined. And music practice continued apace, despite what theorists might say. By 1558, Gioseffo Zarlino was able to overturn the Pythagorean system, citing Claudius Ptolemy as his authority, in "Institutioni Harmoniche".
   Zarlino extended the narrow framework of numbers 1, 2, 3 and 4 (and their multiples, by which the Pythagorean ratios were formed), to included the numbers five and six, in his "numero Senario". In line with the injunctions in Ptolemy's "Harmonics" of the 2nd century, he also restricted the form of his new ratios to superparticulars: in these, the numerator of a fraction is one greater than the denominator. So, Zarlino's semitone, and major and minor tones - of 16 : 15, 9 : 8, and 10 : 9 respectively - were all superparticulars, and formed the basis for what was later known as just intonation. With these, ideal common chords of the keynote, third and fifth, could be constructed for singing sweet harmonies ( a common enough practice, in fact, since John Dunstable in the early 15th century). Confronted by this overhaul of music, the most comprehensive in a thousand years, theorists were kept busy trying to fit just ratios to the old Guidonian Hand - a task that Zarlino did not complete.
   Just intonation was not perfect - it failed to produce all the ideal fifths and thirds it aimed for - but it suited the needs of the 17th century better than the old Pythagorean ways. Nor was it the only theory around; equal temperament was a favourite with armchair musicians for its mathematical, if impractical, perfection. For keyboard players, the approximations of meantone tuning were convenient, providing for fluent transpositions between most of their fixed notes. At the same time, musical attitudes were changing in other ways: the ideas of keys and scales were emerging from the old system of modes. When Glareaus had posited the Aeolian and Ionian modes in 1547, starting on the notes A and C, theory was bowing to the inevitable. Musicians were already using them, along with accidental black notes they entailed. A certain democracy had arisen amongst the musical notes, and the revered Guidonian Hand was becoming obsolete.

Illustration 4 : THE GUIDONIAN HAND from Zarlino's "Institutioni Harmoniche" of 1558.

   When Newton attached sol-fa syllables to the spectrum, in his 1675 "Hypothesis" and 1704 "Opticks", he was following the conventional path of music. Becoming so confident of his approach, he belatedly added a chapter on colour music to his original lecture notes from Cambridge. Newton was surely aware of more contemporary ideas: he had reviewed Francis North's "Philosophical Essay of Musick" in 1677, where the scales of C and A were used to define sharp and flat keys, along with their sol-fa names. But Newton preferred a conservative D scale, related to the first of the Gregorian Church modes, the Dorian, with no added accidentals. To find its sol-fa, he followed the Guidonian Hand. Starting on the note D, on the hard hexachord of G, he obtained the first sol and la; 'mutating' to the natural hexachord of C, gave him the following fa, sol, and la; and the final mi, fa, and sol came by mutating back to the G hexachord. (The only alternative, mutating between the C hexachord, and the soft hexachord of F, would have given a G scale with a B flat included: such a choice was contra-indicated by later remarks in "Opticks".) The omission of ut and re was normal, in a system brought to England from Geneva in 1596, that used only four of the syllables. Penelope Gouk, in "Music, Science and Natural Magic in 17th Century England", proves it (at least for an F scale) by quoting a popular ditty:

   In his undergraduate days, Newton had been a little more experimental with musical scales, playing with them as mathematical toys. Using the second G-to-G mode as his model, he had tried different methods for dividing it into twelve semitones. With ratios, he made a symmetrical array of half-notes (except for the central two between C and D). They would combined into the standard just intervals for the distances between white notes - a major tone of 9 : 8, a minor tone of 10 : 9, and a 16 : 15 semitone. In an alternative approach, Newton explored microtones, tiny equal fractions of an octave that could be combined to approximate any kind of interval. After trying a dozen divisions, he found 1/53 of an octave the most satisfactory. Adapting them to just intonation gave a semitone of 5 units, and major and minor tones of 9 and 8 units each. The same fraction was used for the basis for Mercator's tuning, later in the 17th century. Christiaan Huygens and others, however, preferred 31 microtones per octave, for use with meantone temperament. Intended for instruments with fixed tuning, the system required keyboards with movable sets of strings, or split keys to activate separate strings. With the few elaborate devices built, the gap between the mathematical ideal and its implementation became apparent. But Huygen's '31-tet' still has its adherents today, and the advent of electronic music has witnessed many modern composers developing their own microtonal methods, in the hope of escaping the imperfect rigidities of equal temperament. Newton shared their distaste, at least in theory, though his early notes reveal he had tried twelve equal divisions for the scale. Overall, Newton approached music as a mathematical exercise (albeit with a philosophical dimension) rather than as a creative endeavour.
   So what made Isaac Newton persevere with colour music? There was no hint of the subject for the first 125 pages of "Opticks", which dealt mainly with effects and properties of colour produced by prisms and lenses. With little warning, the spectrum was given its first musical makeover in Experiment 7, of Book 1, Part 2. In the Definition immediately before it, Newton changed gears; up till then he had been speaking "not philosophically and properly, but grossly", for the benefit of "vulgar People". Having been won over, the reader is given the telling insight that light rays are not coloured themselves, but merely "stir up a Sensation of this or that Colour". The senses as we experience them were considered illusory, creations of the mind or impressions on the sensorium. Objects that caused sensations, outside us in the physical world, had no sensual characteristics. The sound of a bell or a musical string, was no more than a trembling of the air; a thing had no colour, only a disposition to reflect or refract certain light rays. It was their motion - of trembling air or speeding light particles - that impacted on our sense organs, to create the form of sound or colour in the sensorium. Up to that point, we can only distinguish any order in light and sound separately, through physical experiments and mathematical analysis. Newton suspected the very different stimuli for sight and hearing were converted, by the physiology of the human body, into somewhat similar experiences.
   According to Query 23, at the end of Book 3, it was aether (or some like substance) that conveyed the motions in vibrating waves, along the auditory and optic nerves to the sensorium. So colour and sound, at least in the way they reached the brain, were much the same thing. Query 14 reinforced the similarity: since it was known from the laws of music, that harmony arose when two or more vibrations of the air were in due proportion, a similar process might cause colour harmony. Newton questioned whether proportional vibrations of the aether in the optic nerve might not cause pleasing colour sensations. He cited gold and indigo together as an example; in an earlier draft for "Opticks", orange and indigo, red and sky-blue, and yellow and violet, were considered harmonious pairs "for they are fifts", while adjacent colours were discordant, as being "but a note or tone above and below". It was in the speculative realms of an hypothesized aether and proportionate harmonies that Newton framed his analogy between colour and musical sound - not in the field of science as we now know it. Such was the proper procedure, so it seems, for a natural philosopher of the 17th century.

CONTINUED...