Randomization : Variation and Ligeti’s Fascar movement from the Sonata for Viola : permutations and transposition : introducing rhythm variation through filtering (or erasing) pitch : Messiaen composes pitch and rhythm separately; further approaches to variation with rotation and L-Systems; Nigel Morgan’s Song from Array for solo violin.
One aim of this e-book is to be able to browse it rather than needing to read it sequentially. It aims to be an open-ended guide book and to stimulate ideas and possibilities. We probably learn best on an informal ‘need to do, need to learn’ basis: so it seems helpful to bring together not just an idea or concept that we might seek to explore, but also its rationale and illustrate its use in some kind of context. So let’s take one such, a preoccupation really, and regularly found as a process in computer-aided composition: randomization
Randomization as a creative and generative device has become embedded in contemporary musical thought. This has links with an awareness of the random ‘mark’ found in visual art of the Impressionists. Painters in this genre became very aware of how textural effects might be created by applying varying degree of randomness to a natural but often complex surface: the sky, the sea, a forest, grass in a field. ‘Randomness’ is everywhere we look, and to some extent is also present in what we hear. A dawn chorus is one such example that the composer Messiaen explored in Le Réveil Des Oiseaux. Birds do not sing in time, and their songs often contain seemingly random alterations. This is one reason for their fascination as natural composers. Edward Cowie’s Birdsong Bagatelles for string quartet is testament to that.
In music, variation, a recognisable change of state from a given subject, is an established way to produce good continuation. Composers ‘play’ with the business of varying material, and they often do it with great subtlety and invention. In a contemporary setting a composer such as Gyorgy Ligeti was a modern master. A good example of such invention can be found in the 3rd movement of his Sonata for solo viola (1992) where Ligeti uses a nine-note folk-like melody and creates sequences of ostinatos that gradually transform through variation. The movement is called Fascar, Romanian for ‘screw’ or to ‘wring’, and that’s what Ligeti achieves by gradually inserting different pitches into each five bar phrase until by the middle of the movement almost all the pitches originally present have been changed. For any composer seeking scenarios for computer transformations look no further than Ligeti! This sort approach is can easily be set up within a computer programme as a kind of interpolation. This observation owes much to the musicologist Benjamin Dwyer in Gyorgy Ligeti: In Foreign Lands and Strange Sounds by Louise Duchesneau, Ligeti’s assistant of over 20 years,
Ask yourself: what techniques do I use to vary a sequence of pitch? If we stick to a single parameter, a short pitch series in a fixed register, we soon reach a finite number of different orderings.
(setf motif ‘(c4 cs4 fs4 g4))
(setf m-all-1 (permute motif))
=> ((c4 cs4 fs4 g4) (c4 cs4 g4 fs4) (c4 fs4 cs4 g4)
(c4 fs4 g4 cs4)(c4 g4 cs4 fs4) (c4 g4 fs4 cs4)
(cs4 c4 fs4 g4) cs4 c4 g4 fs4) (cs4 fs4 c4 g4)
(cs4 fs4 g4 c4) (cs4 g4 c4 fs4) (cs4 g4 fs4 c4)
(fs4 c4 cs4 g4) (fs4 c4 g4 cs4) (fs4 cs4 c4 g4) (fs4 cs4 g4 c4)
(fs4 g4 c4 cs4) (fs4 g4 cs4 c4) (g4 c4 cs4 fs4) (g4 c4 fs4 cs4)
(g4 cs4 c4 fs4) (g4 cs4 fs4 c4) (g4 fs4 c4 cs4) (g4 fs4 cs4 c4))
This is all very well, but how might these permutations become a musical texture? And what about the context? This motif of pitches from the first pattern in the Slonimsky Thesaurus, generates in its four pitches a very distinct though ambiguous tonality. Imagine the pitches scored for log drums or marimba bars, then pick out (again at random) different reordered motifs to play in sequence.
(setf m-all-2 (rnd-unique ’6 m-all-1))
=> ((g4 cs4 c4 fs4) (fs4 g4 cs4 c4) (c4 g4 fs4 cs4)
(cs4 g4 c4 fs4)(g4 c4 fs4 cs4) (g4 c4 fs4 cs4)) (setf m-all-3 (rnd-sample 6 m-all-2 ))
=> ((fs4 cs4 g4 c4) (c4 fs4 cs4 g4) (fs4 c4 g4 cs4)
(fs4 cs4 c4 g4) (fs4 c4 g4 cs4) (fs4 g4 c4 cs4))
These little programs produce similar but different types of output: one has a unique sequence of patterns; the other allows for repeats.
With exposure to such examples the composing mind is quick to see the potential of bringing another function into play to produce more variations: transposition.
(setf m-all-4 (pitch-transpose ‘(0 1 -1 4 3 -1 0) m-all-3 ))
=> ((fs4 cs4 g4 c4) (cs4 g4 d4 gs4) (f4 b3 fs4 c4) (bb4 f4 e4 b4)
(a4 eb4 bb4 e4) (f4 fs4 b3 c4))
Let’s make one further expansion of the variation idea by introducing in the parameter of rhythm. Probably the easiest way to ‘make rhythm’ is to generate a sequence of identical note lengths and then erase (rather than delete) these lengths playfully.
(flatten (gen-filter-remove ‘(cs4 fs4 b3 e4)
(flatten m-all-4) ‘s nil)))
=>(-s c4 g4 -s – g4 gs4 d4 -s – f4 c4 bb4 b4 -s f4 -s bb4 a4 eb4
-s f4 c4)
This output list is in a notation called OMN that combines multiple parameters, here: pitch and note-length. Using the notation example, see if you can ‘decode’ the script.
Now we’ve taken out selected pitches and created a highly rhythmic variation. This is just one of many ways we could tackle this. Another would be to devise the rhythm we want as a separate parameter running alongside the stream of pitches. And then simply fill the note-lengths with the pitches. Some pitches would, of course, disappear.
(setf pitches ‘((fs4 cs4 g4 c4) (cs4 g4 d4 gs4) (f4 b3 fs4 c4)
(bb4 f4 e4 b4) (a4 eb4 bb4 e4) (f4 fs4 b3 c4))
(setf rhythms ‘((s s -s s -s s s s s s s s
s -s -s s s s s -s s s s s)))
; s = 1/16
=> ((s fs4 cs4 – c4 – cs4 g4 c4 fs4 cs4 g4 c4 fs4 – - c4 fs4 cs4
g4 – fs4 cs4 g4 c4))
What’s interesting here is that we’ve been able to merge together two parameter streams: of rhythms; of pitches. The result maintains the change in tonality outline of the motif (brought about by the transpositions), but it is now articulated rhythmically. Imagine the two phrases playing simultaneously on different instruments, say marimba and vibraphone.
This technique can be demonstrated in a complete composition, a Toccata for solo piano. This piece is based on the same Slonimsky pattern used in the examples above, but extended to include patterns in series – like this:
What we’ve encountered here is a very common form of musical invention. It is ably demonstrated by the observation of Yvonne Loriod that her husband Olivier Messiaen created much of his music in two separate places: sounding out at the piano; silently inventing at a table. Anything connected with pitch was auditioned, played over, even improvised upon. This was where pitch collections were brought together in chords. Very often such pitches were referenced to tonalities – sets of pitches arranged in a scalic way. But for rhythm, the invention happened separately and silently. It was only when the rhythm was in place the two parameters of pitch and note-length were superimposed on one another to make a composite whole.
By way of concluding this section, let’s return to further ideas of variation demonstrated in Liget’s Sonata for Viola described earlier in the text.
In my own Song, the penultimate movement of Array for solo violin, a more conventional variation ‘program’ is attempted. This makes use of the Lindenmayer system, a mathematical algorithm that in computer-based image-making is cable of producing stunning tree-like structures. Like Ligeti the influence on this music is one of folk-music: the piece should be played entirely on a single string with slides and folk-like effects and inflections. The six note ‘refrain’ is punctuated with L-System variations. Each refrain returns in a rotated form, its pitch and rhythm moving one ’step’ forwards.
L-systems are widely used in CAC systems and will be further explained and demonstrated in other compositions. One of the joys of this algorithm is that is can be explained and tried out on paper, but comes into its own in computation. The L-system remains one of the simplest yet most powerful of algorithms that demonstrate characteristics of ‘natural’ growth or evolution. To see a further example of L-Systems in a musical composition you can explore the web interpretation of the Author’s Treeness for viola and chamber organ. This includes a visual representation of the L-System in action as well as a concert recording of the music.
References, Links and Further Reading
Louise Duchesenau & Wolfgang Marx: Gyorgy Ligeti: Of Foreign Lands and Strange Sounds
Gyorgy Ligeti: Sonata for Solo Violin, Part 2 (III, IV, V)
Olivier Messiaen: Le Réveil Des Oiseaux
Nigel Morgan: Array for Solo Violin
Nigel Morgan: Toccata for Solo Piano
Nigel Morgan: Treeness for Viola and Chamber Organ