Topology

In classical Greek mathematics, three problems in particular influenced the course geometry subsequently took. The challenges facing ancient mathematicians included doubling the cube, trisecting an angle and squaring the circle. It was not until 1882 that the last and most famous problem was shown to be impossible, with Lindemann’s proof that pi is transcendental. The problem involves, essentially, constructing a square equal in area to a given circle.Dante’s Divine Comedy ends with a reference to this impossible task to illustrate how it is impossible for a time-based, finite human mind to imagine the eternal. It strikes me as parallel to my own concerns in composition with trying to represent timelessness inside the bounds of a time-based medium - a hopeless task, but one in which the attempt yields interesting results. Squaring the Circle is so named because it represents this aesthetic thread, which runs through most of my music.

The composition was begun during a reading of The Divine Comedy. Reflecting on the way that the timeless landscapes of Hell, Purgatory and Heaven are temporarily imbued with time by the presence and movement of the human figure of Dante, I imagined a melodic strand running through a static musical landscape. Applying a maxim of Brian Eno - “Go to an extreme, come part way back” - I went about composing music which subtly suggests this notion rather than embodying it in a clear, direct manner - the concepts informed the music, but the composition was primarily concerned with intuitive choices of pitches and rhythms.

Nevertheless, several aspects of slowing and stopping time are explored in the work. Resolution and cadence are avoided as devices which cause an excess of forward direction. Instead, vertical harmony and the suspension of instability are favoured for their static qualities.

One way time is distorted is in the use of inexorable processes (like those of Tom Johnson), in which expectation is stifled by impersonal system. This results in the listener “chunking” the material, which alters the sense of time passing. Process passages are essentially nonlinear, as there is little hierarchy. Motion results from unchanging principles, and is not perceived as progression.

Constantly present is the use of functional harmony – such as chord progressions Bach would have used to lead directly through a passage – compressed into single sonorities, as if time has been frozen and extended.

Dazzle Camouflage was composed in 2003 for Topology. The title refers to a peculiar type of camouflage used on merchant and military ships during the first and second world wars. It was the brainchild of British artist Norman Wilkinson, who noticed that ships seemed to be at their most conspicuous when painted black (the norm at that time.) Since there were an alarming number of ships being sunk by German submarines, a solution was urgently needed. Wilkinson’s only aim with his so-called dazzle designs was to confuse the submarine commander sufficiently so that an attack would miss the target, or would not take place at all. The principles of this camouflage were that bold lines and colours were to be painted all over the vessel (the port and starboard did not match) in a way that would not reveal the contours of the ship–rather, the designs were chosen to suggest a false perspective. The point which is easy to miss is that this was not supposed to hide the ships or to blend in with a background (as is the aim of most other camouflage), but to mislead a potential attacker. The visibility of the ship is of no significance. That the designs were efficient is well documented from ships’ logs, which often mention that an unknown vessel has been spotted heading on a certain course at a certain speed. What is remarkable is that these observations of course and speed (and even the type of vessel) were very often far from the truth. Dazzle designs continued to be used on ships well into the second world war, but became obsolete with the advent of radar. It is amusing to note that the admirals were very much in favour of dropping dazzle canouflage (in the USA it was known as razzle-dazzle) as they were uncomfortable with their destroyers being painted up in bright fanciful designs. In England especially, dazzle design caught the attention of artists and fashion designers. There was even a Great Dazzle Ball at the Chelsea Arts Club.
Dazzle Camouflage has 5 movements.

Thou Shalt!/Thou Shalt Not!, as its title implies, is a back and forth struggle. If you can imagine a superkeyboard on which each key plays, not a note, but a different texture with its own complex sonorities and rhythmic pattern - and then a winding permutational melody played on that keyboard – you might capture the feel of Thou Shalt!/Thou Shalt Not! As a result, the piece isn’t about notes, melodies, or even form, but about the clash and collision of sound images, all of them steely and sharp-edged. Piano, violin, viola, electric guitar, and sometimes bass clarinet act together, playing reiterative figures in 9/8 and 6/8 meter. They are continually interrupted, however, by the marimba and drums, who fight back with their own time-frame of quarter-notes, usually in groups of four (F-A-A-F), sometimes five or six. The bass clarinet often sits out as an observer to the conflict, growling its low E of disapproval over and over.The percussion’s intransigence eventually spills into the ensemble, for, halfway through, the bass clarinet begins barking buzzy multiphonics, while the guitar hits thick with half-step dissonances; these impose yet a third conflicting time-frame. The cadenza finally granted to the percussion makes it only more insistent, and, over the low E, it ends the piece by itself after the ensemble surrenders. In a Romantic conception of music, opposing elements are eventually synthesized into a new, more inclusive harmony. But Thou Shalt!/Thou Shalt Not! is essentially anti-romantic; its refusal to find a common rhythmic ground portrays the failure, and by extension the impossibility, of musical dialectic.
- Kyle Gann

Taking its title from Dutch artist M. C. Escher’s engraving, this contrapuntal miniature uses as an analogy the visual illusion playfully explored by Escher: the back-and-forth shift which occurs as the image conjured in the brain seems to point inwards, then outward, then inwards, continually flipping (but never in both states simultaneously). A musical analogy is the shift in perception between onbeat and offbeat; this is the preoccupation of Convex and Concave. Babbage’s Chop Chop makes a similar use of this opposition, presenting phrases in binary pairs – first on the beat, then off.
© Robert Davidson

Purcell Manoeuvres was written for the Michael Nyman Band in 1980 when the piece was selected for a workshop performance by the SPNM (Society for the Promotion of New Music) in UK. Since then different groups have performed it in several revised versions, including my own group Regular Music who recorded it for Rough Trade Records in 1986.
The material for the piece is taken from Henry Purcell’s Trio Sonata No. VI in G minor. In each of the sections (of which there were originally five though only Nos. 1, 2 and 4 are currently performed), the material from the original is subject to varying processes of pitch and rhythm change.Purcell Manoeuvres No. 2 is closest to the original in that virtually all pitches occur at the same moment as they would have occurred in the original. However, due to the process used the resulting sound is more like a distillation, or a suspension of these pitches in time and bears little obvious or immediate resemblance to the Purcell. The pitches are selected according to a simple but strict numerical system whereby the pitch on every sixth beat of the bass line, every seventh beat of the upper melodic line and every fifth beat of the lower melodic line are sustained and overlaid in a series of slow moving interwoven lines. Every pitch heard in the resulting polyphony would have occurred at the same point in the original, but all the original melodic patterns are missing. The original is there somewhere but it’s as if suspended or in a dreamscape.

My treatment of existing material is partly a desire to revisit the tonal material of our musical past and re-present it in a different way, but also by the contradictory notion that, after the development of serialism and extremes of the experimental era, a return to tonality has to deal with the possibility that there are no really new things which can be done with it. The Russian composer Vladimir Martinov recently stated ‘we now live in an era beyond composition’. Any return to western harmonic structures has to acknowledge that to an extent we are simply rearranging the furniture of the past.

Purcell Manoeuvres is also a good example of another feature of my work - the idea of ENSEMBLE. I’ve always been interested in watching a group of players and how they combine to make a whole rather than the individual expression of soloists. In my own composition I like the focus to be on the interplay between a group of musicians rather than on the virtuosity of one. Hence there are very few melodies featuring one particular instrument, pace is dictated by interlocking rhythms, somewhat raucous unisons and the wonderful energy and dynamic communication found between members of a group. When you get rid of the melodies the interplay of the accompaniment becomes the focus – the piece could therefore be said to be more about background than foreground, as much about the ‘nuts and bolts’ of music and the dynamic of the live performance as the finished product.

- Jeremy Peyton Jones

Most of your recent music has its starting point in mathematical phenomena. Is this now your main musical concern?Yes, I consider this my main emphasis particularly since around 1979-80. The Rational Melodies come from that time, and all are clear perceptible logical sequences. Since then things have gotten more complicated and the logic is sometimes less perceptible, but I’ve maintained a basic commitment to rationalism, to objectivity, to music that knows what it is doing.