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

3:  MIXING IT

Illustration 9 : "THE SCHOOL OF ATHENS", after Raphael's fresco in the Stanza della Signature, Vatican, 1510-12.

   Musicians and musical theorists had been aware of a relationship between proportion and harmony since antiquity. Western thought, generally, became saturated with notions of harmonic relationships connecting all things - the heavens, the earth below, the human body and soul. In "Timaeus", the only Platonic dialogue known to the West in the Dark Ages, the very structure of the heavens had been described as a cosmic harmony, with Pythagorean musical ratios offsetting the planets and stars from the earth in proper proportions. In the Myth of Er in "Republic X", Plato gave the heavenly bodies colours, as they appeared in the sky (Mars is 'reddish', for example), but each of their orbits was given a siren, singing an eight-fold harmony accompanied by the Fates, while the whole arrangement turned on a spindle:

"a line of light, straight as a column, extending right through the whole heaven and through the earth, in colour resembling the rainbow, only brighter and purer."

Illustration 10 : THE PIPES OF PAN,   from Athanasias Kircher's "Oedipus Aegyptiacus", 1652-4.

The seven pipe reeds held by the god Pan each inscribe a planetary orbit, symbolizing the ancient Pythagorean belief in a musical Harmony of the Spheres. The planets are arranged in the old Ptolemaic order, orbiting the earth. As a Jesuit, Kircher was forbidden by the Catholic hierarchy to represent the sun-centred system of Copernican astronomers; like many others in this situation, he privately favoured Tycho Brahe's modified scheme, in which the sun and moon still orbited the earth, but the planets were allowed to circle the sun. While Isaac Newton went even further than Copernicus, in reinventing the rules of the cosmos, he shared Kircher's taste for a musical metaphor: "And to what Agent did the Ancients attribute the gravity of their atoms and what did they mean by calling God an harmony and comparing him & matter (the corporeal part of the Universe) to the God Pan and his Pipe."

   It was this ancient legacy that Newton addressed in the Pipes of Pan, a proposed Scholium to the celebrated "Principia Mathematica" of 1687. Drawing on later commentaries, he rewrote the myth of Pythagoras, to claim the inverse-square law of gravitation had been known to the earliest sages. The same law had been discovered, Newton said, between a string's length and its tension for a given musical pitch, before it was applied to the heavens. Aristotle's crystalline spheres that carried the planets round the earth, and the geocentric cosmos of Claudius Ptolemy that had lasted for 1400 years, were symptoms of later Greek decline. To hide the truth from the vulgar, Pythagoras had fabricated the parable of the musical spheres, and Ptolemy had fallen for it. The time was ripe, apparently, for Newton to unveil the secret of gravity and reveal the cosmic harmony of spheres, governed by the octave of Apollo's seven-stringed lyre. The search for truly ancient sources, via the allegorical interpretation of Greek myth, was common practice in the 17th century, and Francis Bacon, in "The Wisdom of the Ancients" of 1619, had the seven reeds of Pan's pipes symbolize the influence of the seven planets on earthly affairs. He believed the popular correspondence to originate in Persian magic while Newton, in "System of the World", credited the ancient Egyptians and Chaldeans with getting it right.
   In fact, Newton's influences lay closer at hand: his academic environment was strongly influenced by the Cambridge Platonists, particularly Henry More, who encouraged the metaphysical overview. Even one of Newton's most famous remarks - "If I have seen further it is by standing on the shoulders of Giants" - can be traced to a Neoplatonic source. Writing to Hooke, Newton was paraphrasing Bernard of Chartres (as reported by John of Salisbury), who had likened his 12th century brethren to dwarves on giants' shoulders. Reprints of classical texts encouraged a reverence for antique thought: Homer, Herodotus, and Archimedes shared shelf space in Newton's library with Plato's "Republic", and Iamblichus's "Life of Pythagoras". The Orphic Hymns and Chaldean Oracles (which he translated) were there, alongside commentaries on the harmony of the spheres by Proclus and Macrobius. Of more practical use was Euclid's "Elements", a Greek work of the 3rd century BC. It provided the geometry for Isaac Barrow's up-to-date lectures on light, and he had his pupil Newton transcribe his notes. Euclid became the cornerstone of Newton's teaching methods and a starting-point for many of his mathematical adventures.

   So steeped was the age in symbolism, classical and otherwise, that it pertinent to seek out any hidden meanings in Newton's mathematics. Contemporary scholars often remark how the very Law of Gravity is related to Newton's alchemical beliefs, as an occult force acting at a distance. (As to its true nature, the jury is still out.) In "Newton's Optical Writings", Dennis Sepper points up another cosmological significance, to the cube roots of squares - the formulation used in "Opticks" to scale down the octave, when measuring colours on plates. The same factor was known from Kepler's 3rd Law, deduced again in "Principia", to show the average distance of a planet from the sun is proportional to a cube root of the square of the planet's period of revolution. By this metaphor, each rainbow ring on the surface of the lenses could exemplify the orbit of a heavenly body.
   Pythagorean interpretations could also be applied to spacings of Newton's Rings - their areas and plate thicknesses that produce them - since these followed odd-number sequences. That would make the colours male, to those that worshipped numbers, while the dark spaces between them would be female, by following the even numbers. (The simple serendipity is spoilt when the same numbers are disguised, being doubled on the graph and halved when applied to thick plates.) In case this interpretation seems far-fetched, it may be noted that Newton consulted books on Cabbala, specializing in the mystical interpretation of numbers, letters, and symbols. Moreover, mathematics is notoriously prone to coincidences or, to put it another way, every common formulation accumulates its own mythology.
   The most significant example of numerology could apply to the musical notes themselves. They had a superparticular character, such as the major tone of 9 : 8, with numerators one larger than denominators (or n+1 : n where n is a whole number). The form was endorsed by Ramos and Zarlino, and used in music and theory from the end of the 15th century. In "Harmonics" of the 2nd century AD, Ptolemy had insisted on the use of superparticular musical intervals (with a few exceptions), including the semitones, tones, and thirds later incorporated into just intonation, as "those which are easily accepted by the ear". An English edition of "Harmonics" appeared in 1682, advocating the revival of the ethical virtues of Greek music. Its Appendix, comparing ancient and modern philosophies, was written by John Wallis, England's second mathematician, and Isaac Newton, her first, owned a copy. In "Opticks", superparticular ratios were applied to sine ratios too, so that the sines of refraction at either end of the spectrum were one apart. It is probable the author, as well as many of his readers, would have seen the connection between the overall sine values and just musical ratios: the beauty of this arrangement, as well as its mathematical convenience, compensated for any factual inaccuracy.

   A natural philosopher had greater scope for metaphor outside the rigours of mathematics. In the immaterial world of ideas, ancient traditions and classic forms helped to guide him, when probing the very soul of nature. One of these was the circle of music. Ptolemy had described how to make it from the Greater Perfect System of Greek music, saying "one could begin with the harmonic meanings of the single tones and bend the double octave into a circle". To keep its shape, the two end notes had to be fastened together, by "uniting the two tones". The musical circle was thus made ready for use. Ptolemy wrapped it round the night sky so the notes matched points of the zodiac, and enlisted the musical intervals as aids to interpreting a horoscope. (A similar procedure was used for medical diagnosis throughout the medieval period, but a human form was used instead of a musical one - from Aries at the head to Pisces at the feet, each house of the zodiac governed an area of the body and the cures for its ailments.)
   Johannes Kepler called Ptolemy's work ingenious nonsense, and did his best to improve upon it. He inscribed geometric figures within a musical circle, in "Mysterium Cosmographicum" of 1596. Using a monochord, he advised that "we must imagine that the string is not a straight line but a circle", so music, the regular solids, and the planets could be united in an elaborate cosmology. By 1619, in "Harmonices Mundi", he had scored little melodies with different characters for each planet, in accord with their subtlest motions. The music of the spheres reached its greatest refinement with Kepler, but it was Plato's creation myth in "Timaeus", from the 4th century BC, that provided the eternal prototype. In it, the Demiurge (or Craftsman) of the cosmos had first pounded together the three elements (of the same, the other, and the essence) like so much plasticine. From the mix, he cut off exact amounts corresponding to Pythagorean ratios for music - tones of 9 : 8 and 256 : 243 semitones. They were laid out in a broad strip, like a sheet of pastry, and cut lengthways into two thinner strips: one half was to become the fixed stars, the other the planets. Both strips, still containing musical divisions, were bent into circles of cosmic music:

"This entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point."

"...distinguish the Circumference into seven Parts...proportional to the seven Musical Tones or Intervals of the eight Sounds, Sol, la, fa, sol, la, mi, fa, sol, contained in an eight, that is, proportional to the Number 1/9, 1/16, 1/10, 1/9, 1/16, 1/16, 1/9."

   The two men seemed to have had a strange, symbiotic relationship; Newton owned two of Salmon's books, including a manuscript copy of "Division of the Monochord" in the author's own hand. Their names are linked as far back as 1672, when Newton's "New Theory about Light and Colours" was read to the Royal Society. The Philosophical Transactions also record an account given at the same meeting, of Salmon's "An Essay to the Advancement of Musick". The "Essay" had raised the ire of traditional theorists, including John Playford, a prominent music publisher. (Salmon immediately responded with "Vindication of an Essay", which contained his music circle.) Exception was taken to the use of letters to replace the sol-fa, a practice Salmon advocated and which we use today. The technique seems closer to a mathematician's method and would have appealed to Newton, who named notes alphabetically, from o to u, in his early writings "Of Musick", in 1665.
   Shortly after Newton assumed presidency of the Royal Society in 1703, Salmon put in an appearance, the first musician to present himself since Berchenshaw in 1676. Under the title "The Theory of Musick reduced to Arithmetical and Geometrical Proportions" (which alone would have appealed to Newton), Salmon mounted a performance of a Corelli sonata for violins and viols. Over two meetings, he demonstrated his new, pure tunings for viols and lutes - a rather impractical just intonation for fretted instruments, more suited to equal temperament. Newton owned one of his books on the subject, and Salmon would have elicited general sympathy with his impassioned plea, for a return to the ethical effects of ancient Greek music. As for musical circles, they would come as no surprise to his learned audience. Several were included in Rene Descartes' "Compendium Musicae", written about 1618 but published posthumously, and translated into English in 1653 by Lord William Brouncker, a past president of the Royal Society.

Speculative music theorists required a precise justification to join one end of music to another, and usually found it in the cycle of octaves. Successively lower octaves were formed by continually doubling the length of a musical string, so a mathematical relationship connected two notes an octave apart. Another rationale was provided by overtones, faintly heard, when a single note is struck. The most prominent of them is an octave above the note's main pitch, though Marin Mersenne was able to detect many more. Newton was aware of overtones, duly noting in the early 1660s in "Questiones quaedum Philosophiae" that "Hence a sound & its eight are never seperate...Hence 8ths seeme to bee unisons". Whether a resemblance in notes an octave apart implied unity - sufficient to pin a musical circle together - was vociferously debated.
   Mersenne was the first to explore the mathematical relationships of sounds methodically, establishing a theory of musical consonances in "Harmonie Universelle" of 1636. But he could not even allow that a unison (created when two instruments play identical notes simultaneously) formed the one note: rather, a special kind of consonance, or harmony, was created. Isaac Newton added another dimension, as discussed in Jamie C. Kassler's "The Beginnings of the Modern Philosophy of Music in England". Commenting on two identical pitches heard from different sources, Newton decided that the pulses of sound in the air did not synchronize by the time they reached either ear. Instead, "unisons are rather a harmony of two like tones then a single tone made more loud and full by the addition". So how could they decide that two notes, as far apart as an octave, could unite to form a musical circle? It would take musical common sense to overcome the scientific hair-splitting. Thomas Mace, a singing master at Newton's Trinity College, provided a practical example in "Musick's Monument, or a Remembrancer of the Best Practical Musick" of 1676. The work was a manual on psalm singing and lute playing - a publication which Newton, Isaac Barrow, and others had subscribed to as early as 1665 - where Mace had this to say about the identicalness of octaves:

"You must know, that an Eighth, and a Unison, (in Musick's Nature) is the self same Thing in Effect; as I shall here demonstrate, by an Example. For, let a Man, and a Woman (or a Boy) sing any Song together, (Note, for Note) And the Woman, or Boy, will as Naturally (and cannot but) sing an Eighth, above the Man, as if they were both the same; which will not do in any other Chorde whatever besides."

Illustration 12 : ST PAUL'S CATHEDRAL, LONDON, 1675-1710, by Sir Christopher Wren.

   Robert Boyle had described how painters and dyers did it properly, in Experiment 12, Part 3, of "Experiments and Considerations Touching Colours". He noticed the craftsmen used only five simple or primary colours - white, black, red, blue, and yellow - and could match all other colours with these (in hue, if not in splendour). Modern painters' primaries of red, yellow and blue are instantly recognizable in the range, and Boyle duly listed their secondary mixtures of orange tawny, green and purple. The range of greys got from black and white mixes was remarked upon, thus isolating the tonal scale from colours. He included carnation, made from red and white, perhaps to indicate that each colour could also vary from light to dark. Newton knew of Boyle's primaries for painters, and noted an identical set in William Perry's rules, read to a Royal Society meeting in 1662. The physician Francis Glisson was to compile them into a coherent colour system in 1677, cross-referencing the three primary colours and their mixes to grey-scale values.
   Newton had uncovered an infinitely larger range of primaries in sunlight, which excluded black and white. The colour disc itself was purpose built to predict and analyse their mixtures, by summing the effects of each component - represented by the small circles, p to x, located on the circumference where colours were purest. Multiplying each amount by its distance to the centre and summing the results, found the centre of gravity of the resultant mix at z. The closer z got to the centre, the whiter it became. At the central white point, all pure colours (or at least four or five of them) were present in correct amounts, while black simply indicated an absence of light. The Scholastics and others had thought the reverse - colours were discrete combinations of black and white, modifications not components of light. In one sense, black and white were both alien to Newton's colour theories. He spent some effort early in his lectures attacking Aristotle, the founder of the rival view, which he refuted in Queries 27 and 28 of "Opticks", along with the wave theory of light.
   But Newton still had a problem: how were the violet and red ends of the spectrum to meet, so the colour circle could take shape? Overlapping the red and violet lights from two separate prisms (as he had earlier in "Opticks") created a violet "more bright and more fiery" than the spectral violet. It was allocated an ill-defined region around the keynote D on his colour music disc. Boyle also produced his "lovely Purple" the same way, and purple was the name Newton had generally given it, from 1675 in his "Hypothesis". Then, and later in "Opticks", he defined the most refrangible end of the spectrum as violet, after the colour of the flower, where previously he usually called it purple. The new purple was compound and heterogeneous, rather than pure and homogeneous. It was given a mere mention in the text to the disc, and no purple was named in the instructions of how to draw it. Purple had no place, by rights, among the simple colours of a normal spectral array; there the notes D were an octave apart, assigned to extremes of red or violet, depending on their pitch. Newton had essentially the same problem with colour as the makers of musical circles had with octaves - two like elements that should match, but didn't quite. He relied on their visual similarity to gloss over the difference, and leaned on the colour music analogy to carry it through. With a comment in his lecture notes, intended at one time as a closing remark for the first Book of "Opticks", he described:

"...the affinity existing between the outermost purple and red, the extremities of the colours, such as is found between the ends of the octave (which can in a way be considered as unisons)."

   Descartes' scale included a schisma, between the string lengths 486 and 480. Known as the syntonic comma, it had a ratio of 81 : 80 and made up the difference between a major tone (9 : 8) and a minor one (10 : 9). It created a moveable note and could be added to the semitone and minor tone on either side, to make a minor third (6 : 5) in just intonation. Singing downwards from 405, the note at 486 would be used; 480 was preferred in the ascending direction, clockwise. Neither Salmon nor Newton included the nicety in their musical circles, where it would appear at D. Otherwise, Salmon's ratios were the same as Descartes', while Newton's symmetrical array distributed them differently. In other diagrams, Descartes showed how he came to this arrangement, by trying to fit just ratios to the Guidonean hexachords, a problem left unsolved by the musical theorist, Zarlino. In private notes, Newton showed an awareness of Descartes' book (or at least Brouncker's version of it), elaborating on his diagrams but moving away from hexachord analysis towards the idea of a musical scale.
   The scale was of little importance to Descartes when he penned "Compendium Musicae", so his ideal arrangement left the notes unnamed. Brouncker's version borrowed letters from another diagram, but the main difference comes from repositioning string-lengths around the circumference; proportions are altered, so as to swing the schisma clockwise. Descartes' numbers are reinterpreted as linear measurements around the circle - the half-way mark, for instance, between high and low E at 288 and 576, is A at 432, placed opposite here but not so in Descartes' original. Descartes intended to undo the effect of music's geometrical progression, by showing equal ratios at equal sizes. Brouncker's arithmetical 'corrections' highlight an important difference in method, though neither diagram is particularly accurate. (For instance, Descartes' minor tones differ in size, and Brouncker's 405 for B flat is 14 degrees too far clockwise.) Salmon and Newton would later emulate Descartes' approach, but depict the scale more accurately.

   To create even distributions in his colour music wheel, Newton supplied a list of fractions. Each was created separately, as if all musical intervals (and the colours they held) were located in the same position. Red was first, positioned between the string-lengths of 1 and 8/9 (where violet had been in the previous prism experiment). The space it took up was 1/9 of a string length, so it was given this fraction for the circle. Since green and violet occupied the same sized musical interval as red, they were also allocated 1/9 of the circle each. But blue and yellow took up smaller spaces, of 10 : 9 minor tones; if either were moved to the red end, they would span from 1 to 9/10 on the string's length - making their portions of the whole 1/10 each. Likewise, the 16 : 15 semitones of indigo and orange take segments of 1/16 apiece, calculated from the red end. These fractions Newton supplied were not perfect. A smaller ratio butted up to the red end of the string would coincide with red at that end, but not at its other. To correct this, by aligning geometric centres of three kinds of ratios, would entail complex sums that Newton decided to forego. Commenting on the colour music wheel, and its use as much as its construction, he concluded "This Rule I conceive accurate enough for practice, though not mathematically accurate".    He did have a choice: in 1614, John Napier had invented logarithms, a computational method seemingly purpose built for converting the geometrical scale into its circular form. When each number was considered as a power of 10, indices could be extracted that could be added instead of multiplied - or divided by, say, 12 to get the 12th root. All that was needed was a set of tables to translate numbers to logs and back again. These were supplied by Napier's friend Henry Briggs, and Brouncker used them extensively in his Animadversions on Descartes. The first Englishman to use logarithms on music, Brouncker divided the octave into 12, 15, and 17 equal parts, comparing their equal temperaments to the intervals of just intonation, in tuning tables for the lute. The "artificial numbers" were also applied to Descartes' own figures. Though there are no logs in the text of "Compendium Musicae", it seems perverse that the English mathematician should convert Descartes' diagram from its logarithmic form back to a geometrically-biased format.
   Of the four musical circles, Newton's colour music disc is the most accurately drawn - within one percent of what we might expect if logs had been used (though the error grew to 3% by the 4th edition of "Opticks"). Unlike Brouncker, Newton did not clutter his writing with logarithmic asides; instead, the list of fractions related directly to musical numbers used elsewhere in "Opticks". But musicians found the logarithmic method very useful, as it gave a way to directly compare sizes of intervals without clumsy calculations for ratios and string-lengths. But no common method was devised until 1884, when Alexander Ellis appended his cents table to the second edition of Helmholtz's "On the Sensations of Tone". The octave was divided into 1200 cents, so that an equal-tempered semitone had a cent value of 100; any interval could be expressed in cents, and the system is widely used today. In his private notes of around 1665, Newton constructed a very similar scale of twelve equal divisions - each with a value of one million, so that the octave amounted to twelve million - and compared his other ratios to these.

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Niels Hutchison, 2004. nielshutchison@yahoo.com