2: APPLIED MATHEMATICS
It was commonly held that Pythagoras had investigated sounds made by different hammers striking an anvil, in the 6th. century BC. As a result, he became the first to fix music with immutable mathematical laws. This legend can be traced back to the musical writings of Nichomachus in the first century, and the Roman Boethius was to include the legend in his "Fundamentals of Music", some 400 years later. Followers of Pythagoras refined music's principals with the aid of a monochord, a single-stringed musical instrument. An harmonic series of notes was arrived at, that could be replicated by dividing the string into proportions of its length (similar to the way the neck of a guitar is divided by frets). The octave interval, basic to most tuning methods, was established by dividing a string length exactly in two. The full string and the half-string make sounds that uncannily resemble each other, though they are of markedly different pitches (like a bass and a soprano singing the same note). When sounded together, the different notes should not interfere with each other; if they were even slightly out of tune, interference could be heard as fluctuations called beats. Dividing the string by three or other odd numbers produced different notes, also considered harmonious. Using ratios between the numbers one, two, three and four, Pythagoreans divided the monochord into related pitches. From these, musical modes were elaborated that had moral as well as aesthetic force in Greek society.
Of course, little of this is true - the bit about Pythagoras, I mean. By the end of the 16th. century, the Galilei father and son had proven one main tenet of the tradition was false in fact - the note struck by a hammer would depend more on the weight and shape of the anvil than on the hammer itself. All manner of knowledge was ascribed to Pythagoras, and his name still adheres to fundamental music intervals. Those who invoke him are claiming, by proxy, to be fellow initiates in the universal mysteries available to the Pythagorean cult. Newton, for one, cited their authority in support of his own axioms; he speculated that the celebrated philosopher of antiquity had uncovered the secret to measuring the heavens, by experimenting with proportions of weights and string lengths.
In contradistinction, Marin Mersenne published the first systematic study of harmonics, as "Harmonie Universelle" in 1634. Here he established that the pitch of a bowed note was determined by the frequency at which the string vibrated. In turn, frequency depended on the properties of an individual string - its length and diameter as well as the tension applied to it. This could be expressed as an algebraic formula that we can recognise today: in the mid-17th. century, harmonics offered a true mathematico-experimental framework that might be applied to light and gravity, as much as to strings and pendulums.
Mersenne understood the vibration of a string to be even more complex; as well as the pure tone produced by a wave the length of the string, there are a number of partials that sound less clearly but contribute to the quality of the note. It was later confirmed that any string will allow for multiple waves to occur, chiefly those that fit neatly along its length. Their wavelengths are determined by dividing the length by whole numbers: as well as the octaves that result from repeated divisions by two, other important partials result from an odd number of waves fitting along the length of the string (notably the dominant, or fifth note of the scale, produced by division into thirds). The quality and quantity of partials contribute to the overall sound produced. A patient listener may be able to detect some of these overtones (piano tuners find they have to), though quieter partials are hard to hear and, generally, the ear cannot detect pitch differences towards either end of the auditory range.
Illustration 1 : THE HARMONICS OF 'PYTHAGORUS'.
When the unfettered string XY vibrates, it moves up and down in the shape of the sine curve 1. This is its fundamental tone, or first harmonic, the deepest note it can sound. When cut at A, the string and its fundamental wavelength are reduced by half. XA oscillates in the form of curve 2, twice as fast as the full string, XY, and its tone is pitched an octave higher. Halving again, the quarter-string XB vibrates in the shape of the sine curve 4, and the note rises another octave above that of XA. The unimpeded string, XY, is capable of vibrating in a combination of these curves, adding the second and fourth harmonics to its sound.
To some extent, these (and many other) quicker movements are present in the note produced by XY. Each harmonic can be heard by lightly touching the string at a node - those still-points of oscillation, at A or B. In musical practice, the string is pressed hard, to effectively shorten its length and produce an entirely new note with its own characteristic harmonics. It was important to Pythagoreans (as well as many later theorists) that any new note should be mathematically related to the harmonics of the original string.
The manner in which a note is played, along with the sympathetic vibrations of surrounding strings and from the body of the musical instrument, result in a characteristic timbre. An instrument such as the violin will produce its richest sound when played in keys, such as A or D, that are aligned to the fixed tuning of its strings. A 'remote' key, such as F# major, produces less resonance; Paul Robertson of the Medici Quartet and the psychiatrist Peter Fenwick have noted that the combination of unfamiliar vibrations and difficult hand positions involved in the key of F# can put the violinist into a strange state, making a piece of music in that key feel strange. Different emotional states may also be associated with particular musical keys for historic reasons. Some keyless wind instruments of the Baroque period could only play in fixed keys and we have inherited that tradition of musical pitch to some degree, in spite of the greater flexibility of modern instruments. The bright edge of the brass was heard to best advantage in the keys of C or D, while keys of E or B flats were the natural provenance for the soft plaintiveness of the woodwinds.
Musical systems had been evolving for two millennia since Pythagoras, but they still contained potential discords. In medieval modal systems, whose main pillars were the octave, fourth and fifth, discrepancies were unavoidable: a cycle of perfect fifths or fourths never quite matches a corresponding cycle of octaves, differing by a small amount called the Pythagorean comma. (This was known to the Greeks, and Boethius gave calculations for the comma's value as well as that of other subtleties, such as the schisma.) In the 1500s, the Aeolian mode (based on A) and the Ionian (based on C) were introduced for liturgical use. These permitted greater flexibility in harmony singing and foreshadowed the A minor and C major keys comprising the white notes on keyboards. Still, to make pleasant music, singers had to adjust notes by small amounts as they went along - a process known as musica ficta. Clever composers, such as Adrian Willaert in the 16th century, could shift the tonal centre of a piece through many modes.
On instruments with fixed tuning, one answer to the dilemma was to make small adjustments to some pitches. A more homogeneous gamut of notes could be achieved by sacrificing their theoretical purity. As early as the 14th century, some organs had been built with multiple manuals and split keys to accommodate the vagaries of tuning. Clavichords and harpsichords, virginals and spinets, became popular with the growing bourgeoisie in the 15th century, providing fully-fleshed music to the homes of those with leisure enough to practice and enjoy music. But these instruments also had limitations: because each string's tension remained fixed throughout a performance, certain notes would require adjustment if the performer wished to change into another mode. Retuning the instrument was a tricky chore, and the ability to modulate within any piece of music was restricted.
Gradually, the ancient modes gave way to the key structure that dominates modern music. While the octave interval remained mathematically exact, only traces of musical modes remained in the internal structures of major and minor scales. In exchange, the musician gained the ability to play in any key, without retuning the instrument, and to thus modulate from key to key within the one piece. By the end of the 17th century, theorists like Andreas Werckmeister had devised methods to iron out the niceties of Pythagorean music, and Baroque composers commonly employed types of 'meantone' tuning. Johann Kuhnau's "Keyboard Practice", of 1689, contained seven partitas in ascending major keys (C, D, E, F, G, A, and B flat); a second volume of 1692 consisted of a similar sequence, in minor keys (c, d, e, f, g, a, and b). Using a meantone tuning, it was possible to play Kuhnau's major partitas in B flat, C, D, F, G, and A, and minor partitas in g, d, and a. Even so, the remaining works (E, b, c, e, and f) would have required retuning of the instrument.
The publication of J. S. Bach's "Well Tempered Clavier" of 1722, heralded the triumph of a tempered scale, in which all notes were more nicely spaced. Like his predecessor Kuhnau, Bach began his cyclic work at C. Moving up the scale semitone by semitone, he provided four pieces (preludes and fugues in both major and minor keys) for each of the notes - 48 works in all. As if this were not enough, Bach repeated the exercise with Book II in 1744. Each piece modulated from its native key to related ones and back again, while retaining a distinct flavour of its own (due to slight, residual unevenness in the tempering). Middle C and the white-note scale of C major gained pre-eminence, as the basis on which all other keys depend.
Illustration 2 : DESCARTES' MUSICAL SCALE and JOHN BULL'S "SPHERA MUNDI".
René Descartes' diagram of a tempered diatonic octave (left), first conceived in 1618, was published in "Compendium Musicae" of 1650. His pie chart, with some adjustments, served as a prototype for the colour music wheel Newton developed towards the end of the century. As a philosopher and mathematician, Descartes was concerned for an ideal but useful arrangement of just ratios, dividing a single octave that forms the circumference of a circle. (Note the schisma, for adjusting the tone on either side, from minor to major.) This theoretical pragmatism is of a different order to the utility required by practicing musicians.
In 1621, John Bull formed a double round-canon into circles (right), to echo the cyclic form of the music and represent the heavenly and earthly spheres suggested by its title. The musical score contains a whole melody, rather than Descartes' single octave. (Its shape is little different to the circular music depicted in Dosso Dossi's "Allegory of Music", painted almost a hundred years earlier.) Usually, Bull's compositions went well beyond the technical constraints of Descartes' system - for his virtuoso performances, he even had organs custom-built with fully-extended chromatic keyboards.
At its theoretical extremes, harmonic science had little to do with real music practice, and scientists had to play catch-up. Isaac Newton attacked the problems of musical tuning like a mathematical puzzle. Arithmetic, geometrical and logarithmic methods, as well as a customised units of 'microtones', were applied to diatonic and chromatic scales, to tuning systems including just intonation and varying temperaments. Ultimately all such attempts produce compromises, so Newton finally separated the octave's notes with an array of just ratios arranged in a palindrome - a series that reads the same way backwards as forwards - to provide a pleasing mathematical symmetry. In this form, an antiquated Dorian mode was likened to the array of pure colours in Newton's "Opticks" of 1704. His analogy satisfied only the most general notions of harmonic science. The colour music wheel was somewhat contrived, with sizes of notes generalized and colour areas approximated (though Newton misleadingly presented his results as a genuine discovery, made from objective measurement). Accepted musical practice was already far in advance of the approach he took in dividing colour into ROYGBIV. Many other existing formats, of scales and modes, could have loosely matched the distribution of prominent colours in the spectrum.
While contemporaries could justly criticize the relevance of Newton's musical mathematics, this did not affect the validity of his observations on the nature of light. With marvellous skill he unravelled white light and reconstituted it, mixing coloured lights in unprecedented ways. The colour-music code was almost as relevant to this purpose as the recently-discovered sine law of refraction, allowing Newton to deck out Descartes' geometrical rainbow in its colours. "Opticks" started from simple experiments with a prism - "the usefullest Instrument Men have yet imploy'd about the Contemplation of Colours", according to Boyle. Thus, tentative observations begun in the 13th century, with quartz crystals and glass flasks, were brought to fulfilment. Newton couched his findings in a framework of geometrical optics, a tradition dating back to Alhazan in the 11th century, and thence through Ptolemy to Euclid. The exploration of coloured properties of light, only partially explored by others, entered the realm of modern science with Newton's "Opticks". Along with it, for better or worse, came an articulated colour-music code.
The practical possibilities of colour music were explored by Louis-Bertrand Castel, as early as 1725. He discovered the idea of colour music from Newton's "Opticks", which he had reviewed in its French edition. Castel proposed a 'harpsichord for eyes', a keyboard instrument which would displayed a patch of colour whenever a note was struck. At first, he aligned spectral colours with the white-note scale of C major. The usual order of ROY G BIV was reversed (as indeed it was for some prism experiments in Newton's book), so violet coincided with the lowest note, C, and red with the c an octave above. The musical scale of C was orthodox, representing the contemporary standard rather than Newton's old-fashioned Dorian mode on D. While admiring Newton as a mathematician, Castel felt his physics uninspired, writing that "there is no need to borrow ambiguous traits from Descartes or Newton to embellish the work of God". Castel put aside his original Newtonianism, to hail the earlier colour music theories of Athanasias Kircher instead. Encouraged by the composer Jean-Philippe Rameau, Castel also embraced a triadic theory of musical harmony. He emphasized the notes C, E and G, which make up the common chord of C major, by allotting them the painters' primary colours of blue, yellow and red. The primaries began with blue on C, at the bottom of the octave, since Castel considered it potentially the darkest of the three. There, the expressive capabilities of blue could best represent the musical ground-base, so important to the theories of Rameau. By 1734, Castel had begun to build his first 'ocular harpsichord', using twelve hues of colour aligned to a twelve-note chromatic scale. Each hue was varied further, by twelve degrees of tone from light to dark, so Castel could cycle his scheme through many octaves.
Illustration 3 : NEWTON'S FIRST COLOUR MUSIC (left) & CASTEL'S HARPSICHORD FOR THE EYES (right).
Castel's endeavours captured Enlightenment imagination, to provide a talking point throughout Europe. In Germany, the composer Telemann praised his marvellous instrument, saying "this play of colours will please, for music is nothing but a pleasure". The mathematician Leonard Euler doubted that the fluttering, coloured cloths of Castel's harpsichord could afford enjoyment. He prefered an analogy with artwork, since "painting rather seems to be that to the eye which music is to the ear". Voltaire was skeptical, too: he dubbed Castel "the Don Quixote of mathematics", and suspected his performances would "shock, dazzle and exhaust the sight". In St Petersburg, the Imperial Academy of Sciences held a public meeting in 1742, to discuss the merits of colour music. Whether colour came from vibrations of the aether, as sound was carried by vibrating air, was debate by learned gentlemen. One question on the agenda asked: "Can colours, if arranged in a particular manner, provide a deaf person with the same type of enjoyment as we experience when our ears perceive a harmonious consonance of musical tones?" In the French encyclopaedia, Denis Diderot rendered homage to Castel, and took a special visitor to his workshop in 1751:
Illustration 4 : "MUSIC IS LIKE PAINTING." Francis Picabia, 1913-17.
Colour musicians often justified their work with references to science, spirituality and the grand order of nature, while the press shared their enthusiasm for a potentially revolutionary new art form. After World War I, some film-makers had turned their attention to colour music as an ideal subject for abstract animations. Sometimes, they might collaborate with an artist (Viking Eggerling with Hans Richter) or work with a colour organist (Oskar Fischinger with Alexander Laszlo). Their valuable innovations were often obscured by later advances in mainstream film - the advent of talkies and colour films - and public attention diverted to bowdlerized versions of colour music, such as Disney's "Fantasia".
Not until the rock shows of the 1960s and 70s did colour music regain a large audience, with psychedelic performances synthesizing light and sound. Pioneering work in electronics and computing enabled animators to participate in major films as well - John Whitney's contribution to the Stargate Corridor sequence in "2001: A Space Odyssey" being one example. The effort to co-ordinate colour and music on domestic computers has led to further software advances. In exploring possible interrelationship of colour and music, programmers have been obliged to analyse anew the formal elements, to map flexible links between any arrangements of pitch, colour, shape, movement and so on. Video makers and live performers have been able to take advantage of the broad theories supplied by traditional colour music. Even something of its persistent mysticism has been readily assimilated in the age of multimedia. Modern computing and animation experts can (and often do) claim a lineage that extends back through colour organists and animators of the early 20th century, to Castel's Ocular Harpsichord in the 18th century.
Illustration 5 : "LOUIE FULLER AT THE FOLIES-BERGÈRE", Jules Cheret, 1893.
Richard Wagner had put out the call for a Gesamtkunstwerk in 1850, a new kind of theatre that would synthesize music, verse and staging into a unified, total artwork. His Beyreuth playhouse introduced the wedge-shaped amphitheatre, hidden orchestra and darkened auditorium, to which audiences are now accustomed. With the introduction of arc lighting (and incandescent globes soon after), theatrical illusion was near complete. Loie Fuller, the Parisian dancer, put these effects to good use, timing her movements in response to atmospheric lighting. Wafting diaphanous veils under ever-changing coloured lights, she inspired Toulouse-Lautrec and D. W. Griffith and influenced Isadora Duncan and Martha Graham. Others were also effected by the Wagnerian spirit. Kandinsky wrote stage pieces from 1909 to exemplify the new values: "The Yellow Sound", "Black and White" and "Violet", employed the colours themselves, in motion to music, as the central characters. Sadly, his works proved too difficult to mount. Around the same time, Schoenberg composed "The Lucky Hand", to be accompanied by a range of colours according to no known theory:
Though Newton's "Opticks" went some way to provide a consensual model of the spectrum, his colour-music analogy of ROYGBIV was no more than a pretty conceit. In a sense, it is the prolongation of a medieval tradition, which joined music theory with other mathematical arts - specifically arithmetic, geometry and astronomy - in the quadrivium studied at universities. Since optics was essentially a geometric discipline, the inclusion of a musical component gave Newton's conclusions an extra stamp of academic authority. Descartes, Huygens, and many others of Newton's peers wrote on the subject, as would Euler and d'Alembert among the scientists that followed them. The association of music with colour was also acceptable, following a philosophical tradition that ranked sight and hearing at the top of a hierarchy of the five senses. Since Plato and Aristotle, colour and light, along with form, were considered the sole means by which we detect objects visually, while music was considered the most refined form (and the most readily analyzed) of auditory stimulus. While Castel might question the ROYGBIV colour-music code that Newton stipulated, he did not doubt that such a correspondence existed. Nor did Euler: though he felt his own theory of light superior, he adopted the code with little alteration. It suited Euler's wave theory of light (an hypothesis popular in some circles, as distinct from Newton's corpuscular theory), comparing sound travelling through air to movement of light through an aether. The belief in an overarching unity of colour and music continued, and even today parallels are being drawn, between sound and pure, spectral light.
Illustration 6 : "MUSIC." Luigi Russolo, 1911.
Though he joined the Futurists as a militant painter, Russolo soon turned his attention to music. He created a range of novel instruments, called intonarumori, and published "The Art of Noises" in 1916. His concerts were attended by the musical and artistic elite of Europe, as well as the press and rowdy Dada protesters. Russolo's concert career was curtailed by talking pictures in the 1930s, and he turned to studying folk music and Eastern mysticism (unlike many other Futurists, he avoided joining the Fascisti). By trying to break down the distinction between musical sound and everyday noise, in exploiting the secondary vibrations of his instruments, Russolo prefigured the concrete music of the 1950s. His legacy remains in computer music, where his notation system is still used.
Russolo's painting might suggest a belief in correspondences of colour to music. The clearest clue is provided in a manifesto on "The Painting of Sounds, Noises and Smells", by his comrade Carlo Carrà: - "...rrrrrrreds that shouuuuuuut, greeeeeeeeeeeens that screeeeeeam, yellows, as violent as can be." Other Futurists, the brothers Ginna and Corra, committed themselves to a spectral colour-music code inspired by their Theosophical beliefs. Bruno Corra's "Abstract Cinema-Chromatic Music" provides an intriguing account of the techniques the brothers used, employing the code first on a colour piano then translating the effects to film in 1910-12.
Music has evolved over millennia, through many compromises, to a relatively 'impure' state, and its contrived formulae are not echoed in the natural phenomena of the spectrum. The idea of a colour-music code lost much of its importance for scientists in the 19th century. Many, like Thomas Young and Hermann von Helmholtz, wrote separate works on both music and colour. While denying any link between the two phenomena, they still felt the need to address the claims of colour music. The spectrum of light was found to contain no equivalent of the musical octave, let alone the intervals within it. A comparable light octave might be envisaged by doubling the lowest red frequency, but doing so takes one immediately beyond the range of visible light. The spectrum, from red to violet, can barely span three-quarters of one such 'colour octave', let alone encompass the cycles of octaves used in music. Nor can any unifying principle be meaningfully divined from the separate vibrations of light and sound: disparities between them are clear and fundamental. To connect pitch and colour by a relationship of frequencies would require a formula so convoluted as to be ridiculous - not that this has stopped many from trying.
The ROYGBIV sequence retained its appeal for more general audiences; the classic status of Newton's "Opticks" inspired many variants, with spectral colours assigned to notes of a scale. The idea was adapted to specific purposes - as an aid in teaching music, as a theosophical demonstration of an occult system, and so on. But oft-times it was invented anew, by those searching for a relation between the realms of the ear and the eye. Variations in colours and in notes will almost automatically invite comparison; seekers can arrive at a colour-music code they genuinely believe to be new, but may in fact be known of old. There are still those that marvel to discover that frequencies of certain notes, when doubled forty times over, can fall within the measurable range of visible light (a factor Young had noted, but discounted as meaningless). In some quarters, the size of the spectrum has been exaggerated beyond the range of average vision, to create a colour octave. Some have claimed to see extra colour, faintly, particularly at the red end of the spectrum. Others have consciously squeezed the musical octave to fit the spectrum, though Helmholtz rightly pointed out that such a process made arithmetic nonsense of the original musical proportions. Naive or cynical, such conjuring tricks with dimensions have taken hold in New Age movements, bolstered up by references to popular and exotic mystical beliefs. With all the variations, the most pervasive influence on colour-music codes today remains the prototype Newton supplied in his "Opticks" of 1704.