Page  00000205 A Brief Survey of Mapping in Algorithmic Composition Paul Doombusch Interactive Information Institute, RMIT University email: Abstract Mapping concerns the connection between structures, or gestures and audible results in a musical performance or composition. Algorithmic composition is sometimes the process of imagining a gesture or structure - perhaps physical or visual - and then applying a mapping process to turn that structure of the conceptual domain into sound which may display the original conception in some way. This article looks at mapping from the point of view of algorithmic composition, particularly where persistence is an issue, such that the structure (conceptual domain) is embodied and perceptible in the musical result. 1 Introduction Mapping as part of algorithmic composition has not been so thoroughly investigated as mapping in instrument design. This is possibly because it is usually an integrated part of the composition process and not seen as a discrete stage of practice. Mapping in algorithmic composition is different from mapping in instrument design because composition is a process of planning and instruments are for realtime music production. The use of mapping which is integral to an instrument has both differences and similarities to the mapping that is an inherent part of algorithmic composition. This paper investigates how composers use mapping in their practice and moves towards a taxonomy of the process. 2 Definition Of Terms Compositional structures and ideas can take many forms, but they are often abstract in some way, to a greater or lesser degree, from the music that is composed. Composers sometimes use visual ideas of shapes, mathematical functions, physical processes or phenomena and so on as ideas for creating music. Mapping is the process of taking the (possibly abstract) compositional structures and generating musical parameters. Gesture is an ill-defined term, which for the purposes of this paper means a musical concept; it is not a physical movement. A musical gesture is a planned change (randomness can be planned) in musical parameters as part of a piece of music. The parameters could be, for example; timbre, density, intensity, timing, pitches, and so on. A compositional gesture is the underlying conception, structure and planning of the musical gesture. "Conceptual domain" is a term that will be used to cover the entire conceptual area of compositional practice, which includes other organisation strategies as well as compositional gesture. 3 Mapping And Composition Algorithmic composition can be the practice of composing music through manipulating a number of compositional gestures to produce a musical piece. The manipulation is usually part of the compositional process, where a multitude of gestures are arranged and then through another process, possibly very direct, musical parameters are extracted. It is possible to do this from the micro scale of sound synthesis through to the macro scale of the structural model of the piece. Mapping is the process of extracting the musical parameters from the compositional gestures or structure, such that one set of data is mapped onto another. It should also be noted that algorithmic composition might use specific organisations of data that do not necessarily require "gestures"; for example, Markov chains, finite-state machines, fractals and space grammars. These constructs may also require significant mapping processes to output musical data. 205

Page  00000206 There may appear to be some similarities between the methods of mapping in algorithmic composition and the mappings used in the general practice of the sonification of data. This may sometimes be a valid observation with some pieces of music, and there is certainly some overlap between the two areas. However, as the intent of sonification is to illustrate a process or some data, it is primarily instructional. This type of mapping is not the focus of this paper, which is primarily about composition and mapping. The closest sonification comes to composition, what Larry Polansky calls "manifestation" (Polansky 2002), is where the intent is to creatively use a type of sonification of a formal or mathematical process to create a musical idea. For the purposes of this paper, that process will be included as a part of composition. 4 Historical Examples A famous example of the mapping process is the part of Pithoprakta (1955-56) as reproduced in Formalized Music pp17-21 (Xenakis 1991). Xenakis used the Brownian motion of gas particles, combined with Bernoulli's Law of Large Numbers, as his basic model for the cloud pizzicato glissandi section. After calculating, statistically, over 1000 velocities of gas particles as given instants of time (as the measurement of this was impossible), he then graphed them on an XY plane and directly mapped the straight lines of the velocities to glissandi for 46 string instruments. Xenakis divided his graph vertically into 15 pitched sections, each corresponding to a major third. This was then mapped to the ranges of the string instruments. The mapping was directly of pitch in the vertical direction. This is a particularly direct and concrete example of mapping. All intensities and durations are the same, but to ensure the sensation of a cloud of particles, Xenakis used a complex temporal arrangement of overlapping timing subdivisions that are factorially-unrelated. This is a complex mapping of linear time, designed to represent the instantaneous nature of the movement of the gas particles. Along with many other algorithmic composers, the mapping phase is implicit in much of Xenakis' work. He often used direct mapping as a result of the deliberate organisation of one set of data in such a way that is maps directly to musical parameters. Another famous example of algorithmic composition and mapping is Charles Dodge's Earth 's Magnetic Field (1970). Here, Dodge uses data from the effects of the radiation of the sun on the magnetic field of Earth. A Bartels diagram showed fluctuations in the Earth's magnetic field for 1961 and this data formed the basis for the piece. Dodge mapped this data, the Kp index (a measure of the average magnetic activity of the earth) to pitches and rhythms. From the program notes of the recording of Earth 's Magnetic Field (Dodge, 1970), we may glean an insight into the mapping used: "The succession of notes in the music corresponds to the natural succession of the Kp indices for the year 1961. The musical interpretation consists of setting up a correlation between the level of the Kp reading and the pitch of the note (in a diatonic collection over four octaves), and compressing the 2,920 readings for the year into just over eight minutes of musical time." (Dodge, 1970) While the pitches appear to be a fairly direct mapping from the Kp index, some elements of the composition such as the timbres were chosen purely for aesthetic effect. An arrangement of the data that plotted the length of sections of the data against the maximum amplitude in the section was used to determine the speed and direction of the sound spatialisation and also the rhythms. The data was also sometimes read multiple times to generate the musical parameters. That, combined with the similarity of the fluctuations in the Kp index to 1/f noise data, contributes to the aspects of self-similarity in the piece. The two previous examples contrast different approaches to mapping. It may be linear and direct, but it may also be nonlinear and more complex. Both examples use the data as a structural component and the music achieves some structural unity for that. 5 How Composers Use Mapping There has not been the same formalisation of algorithmic composition and mapping as there has been for other musical composition. Additionally, there is little formal analysis of music in terms of mapping or compositional gesture. As there is no set method for defining the mappings or gestures or structures, each composer tends to use their own methods for their own reasons. Compositional structures and mappings are also used differently by different composers, thus making this a problematic area for analysis and study: For example, the structural model and mapping for a piece of music will have different applications for a 206

Page  00000207 composer who is focused on, say, spectral composition in contrast to another focused on some other form of algorithmic composition. It is clear from descriptions of how composers use mapping between the conceptual or abstract domain and the musical domain, that it is sometimes used in a similar combination of ways to how it is used in instrument design. That is; mapping one compositional parameter to many musical parameters, mapping many compositional parameters to one musical parameter and mapping many compositional parameters to many musical parameters. Additionally, there is another parallel between instrument designers and the composers, the mappings themselves may be linear mappings or nonlinear mappings (Hunt and Wanderley 2000, Hunt, Wanderley and Kirk 2000). However, in compositional mapping there is the additional possibility of repetitive nonlinear mappings and most importantly, each composer has their own combination of mapping techniques. 6 Questions Of Mapping To understand how composers use mapping, a number of composers were asked questions concerning their practice. The broad answers are summarised below. The composers were; Richard Barrett, Charles Dodge, Larry Polansky, Agostino Di Scipio, Rodney Waschka and Paul Doornbusch. The questions, asked were: 1. Do you have a consistent approach to algorithmic composition and mapping or does it vary and why? 2. When implementing a mapping strategy for (part of) a composition, do you organise this in a particular analytical way, decomposing the problem in a technical manner, or in a more creative and holistic way for a purely aesthetic result? 3. Is the mapping component of your compositions something that you think might be perceptible by a listener, or of interest to them and why? 4. Is the mapping component of algorithmic composition something that is pre-determined for you or is it part of a process of exploration? 5. Do you use individual mapping strategies for individual parameters or is there reuse of mapping strategies or a global system? (Ie are they monoparametric or multiparametric?) 6. What elements do you control algorithmically in a composition and can you comment on the function and importance of these? Is the mapping consistent within these elements or not? 7. Are the mapping schemas you use mostly linear mappings or nonlinear in some fashion, and why? 8. Can mapping be considered a composition technique in itself? There was deliberate overlap in the questions to elicit as much information as possible. These questions will be expanded upon later. 7 Results Of Responses The results of the composers' responses are summarised below, noting similarities and divergences. The answers supplied by the composers will be provided in another, larger, paper. In this way, it is possible to achieve the start of an overview of this area. I did not give the composers the choice of making multiple-choice decisions, such as the collation below presents the responses, this is an extrapolation from the responses, and sometimes (quite gross) generalisations have been made. Thus there is a possibility that I have misinterpreted a response by someone, this would skew the results below and I apologise if any mistake has occurred. The full responses will appear in a later paper. 7.1 General remarks by composers: All composers use mapping in some ways. While composers may use mapping to move from the conceptual domain to musical parameters, mapping can work backwards for listeners, providing access to structures and concepts from the music. Composers tend to use complex ideas and structures for music. Focusing on mapping assumes that something preexists before the act of composition and not all composers may think or work like that. If mapping is a means of moving from the conceptual domain to the concrete domain, then it is important for all composition, but central to algorithmic composition. 7.2 Collated Responses: The tables below show the general response versus the number of composers who had that response. While the number of composers questioned means that this is a statistically minuscule sample, it does point to trends and variations. Note that this presentation of the results does not take into 207

Page  00000208 account the widely varying differences in approach by the various composers. These variations are of as much interest as the similarities presented here and they may be read in the other paper with the full responses. Question 1. General Response # I use a direct and consistent approach to mapping 1 Mixed; sometimes consistent, sometimes varied 1 I use a varied approach to mapping 4 Question 2. General Response # Mostly use mapping analytically 0 Mixed; mapping is sometimes used analytically 1 and sometimes creatively Mostly use mapping creatively 5 Question 3a. General Response # Mapping is perceptible by listeners 2 Mixed; mapping is sometimes perceptible to 4 listeners or perception is difficult Mapping is not perceptible by listeners 0 Question 3b. General Response # Mapping is interesting for listeners 4 Mixed; mapping is sometimes interesting for 1 listeners Mapping is not interesting for listeners T Question 4. General Response # Mapping is predetermined 0 Mixed; mapping is sometimes predetermined, 1 sometimes part of exploration Mapping is part of the exploration 5 Question 5. General Response # Use individual mapping strategies for individual 1 parameters (monoparametric) Both individual mappings strategies are used as 4 well as reusing some Reuse mapping strategies for multiple parameters T (multiparametric) Question 6a. General Response # I control as many elements as possible 4 I control most elements sometimes, fewer at other 1 times I control some elements only 0 0 Question 6b. General Response # The mapping is consistent for each element 5 The mapping is sometimes consistent for each 0 element The mapping is not consistent for each element 0 Question 7. An important outcome of this question was that composers tend to use linear mappings if the structure being mapped is complex and detailed. Conversely, nonlinear mappings seem to be used when the structure is either not so complex or has not so much detail. General Response # Mostly linear mappings are used (very complex 1 structures) A mixture of linear and nonlinear mappings are 3 used (depends on structure) Mostly nonlinear mappings are used (simpler 1 structures) Question 8. General Response # Mapping can be a compositional technique in 4 itself Mapping is part of compositional technique 2 Mapping cannot be considered a compositional 0 technique at all 8 Types Of Mapping From the composers' descriptions of how they use mapping, it seems that there is no set method for mapping data from the domain of the conceptual, gestural or structural to the musical domain. Linear and simple mappings are sometimes used by composers, but it is used significantly less often than more complex mappings. Simple mappings have been most clearly described as ratiometric (Polansky 2002). In this type of mapping, a doubling of the data to be mapped results in a "doubling" of the musical parameter. This could be, for example, an octave pitch displacement or a doubling of the loudness. Note that this second linear mapping might be perceptually linear to a particular person under particular conditions, but it would be mathematically nonlinear. Because ratiometric mappings are the easiest to perceive they are particularly useful when the composer has some data, gesture or concept that should be translated as directly as possible for the 208

Page  00000209 listener. The previous example of the pitch mappings fr-om Xenakis' Pithoprakta is an illustration of this. This particular example is musically successful because the data to be mapped is very complex from the outset. Simple mappings can be less musically successful if the data to be mapped is very simple. Complex mappings may be regularly nonlinear, such as an exponential law. For example, a square law will produce a fourfold increase in the musical parameter from a doubling of the data to be mapped. If this is perceptually based it can be perceived as a more extreme or less extreme mapping with changes in the data to be mapped. Other types of complex mappings are as varied as can be imagined. They can sometimes be related to a complex arrangement of the data to be mapped, or other potentially chaotic functions can be involved. This may obscure the original concept, compositional gesture, or data, or it may embellish it and give it another dimension. However, there is clearly a limit to how far such complexity in mapping can be taken before all sense of the original data is lost in the mapping. Polansky convincingly suggests that the cognitive weight of complex mappings degenerates rapidly and nonlinearly such that beyond a certain point everything is just "complex". It has been a repeated outcome from instrument design research that humans prefer complex mappings (Hunt and Kirk 2000). This would appear to translate somewhat into the domain of composition, given the previous situation that linear mappings may be most appropriate for very complex data. In one way, the variety of sophisticated, creative and exploratory approaches to mapping, as embodied in the works of algorithmic composers, is a wonderful outcome and it ensures the variety of music from algorithmic composition. In another way, it means that mapping strategies are reinvented by every composer and young composers do not have easily accessible models to work from. If someone created a categorisation of compositional mapping strategies and a catalogue was built of mapping techniques, then other composers could refer to this and build on it in their own practice. This would be of particular use to student composers (Polansky 2002). In this way, mapping in algorithmic composition could be demystified and more complex, varied and musically appropriate practices could be developed by building on the work of others. The counter argument to this is that there is something valuable in the effort of developing sophisticated mappings for oneself. There is certainly interest in the effort required to play an instrument and how mapping relates to this (Ryan 1992). Something which has not been explored here, but which would be of interest to some composers, are cultural associations with mapping. For example, are there culturally invariant mappings with some parameters such as pitch or intensity with height? One example from instrument design that has caused problems is the original Theremin mapping of intensity with proximity to a horizontal antenna such that to make a louder sound the performer would bring their hand down. This was done for practical reasons so that without a performer there is no sound, but modem theremins allow this to be swapped because of the cognitive dissonance caused by the playing action. 9 Conclusions The foregoing indicates that composers who use algorithmic techniques certainly use mapping from one domain to another, often in very sophisticated ways. Also indicated is that there is no ideal solution or single method, but that it is a process of exploration. Very simple or obvious mappings are sometimes problematic in composition probably because of the oversimplif~ication, but they may be appropriate under some circumstances such as with very complex structures or data. More complex mapping strategies are more common and more musically useful, but very numerous and their application and musical usefulness appears to be an aesthetic judgement of the composer. There seems to be almost as many approaches to mapping as there are composers. It is the character of composition, being an expansive and creative activity, that if ever a theory provided a "standard", "optimised" or "ideal" mapping technique or repertoire, that composers would ignore it and go beyond that. Having said that, there would appear to be something to gain by creating a catalogue of historical and current mapping practices, even if only in a pedagogical sense (Polansky 2002) and this could be started by expanding on the questions presented above. The nature of composition and composers means that there will never be "solution" to the mapping problem in algorithmic composition, that it will remain a part of the exploration by every composer. 209

Page  00000210 10 Further Work The questions outlined previously could form the basis of a more comprehensive questionnaire to gather information on how composers use mapping. Since mapping is a fairly new area of interest in general, this could be a resource for student composers. However, much care must be taken in such an analysis because every composer has different techniques and these differences must be taken into account as well as the similarities. 11 Acknowledgments I would like to thank and gratefully acknowledge the contributions of the composers who made much of this work possible. They are; Richard Barrett, Charles Dodge, Larry Polansky, Agostino Di Scipio and Rodney Waschka. Also, much of the material in this paper is the result of conversations with Paul Berg, Larry Polansky and Marcelo Wanderly. References Wanderley, M. M. and Battier, M. eds. 2000. Trends in Gestural Control of Music. Paris: IRCAM Cadoz, C. and Wanderley, M. M. 1999. Gesture-Music. In Trends in Gestural Control of Music. Paris: IRCAM CDROM. Dodge, C. 1970. Earth 's Magnetic Field program notes. In Nonesuch LP H 71250 EARTH'S MAGNETIC FIELD: Realizations in Electronic Sound. Dodge, C. and Jerse, T. 1997. Computer Music second edition. New York: Schirmer Books Hunt, A., Wanderley, M. M. and Kirk, R. 2000. Towards a Model for Instrumental Mapping in Expert Musical Interaction. In Proceedings of the 2000 International Computer Music Conference. San Francisco: International Computer Music Association, pp. 209 - 212. Hunt, A. and Wanderley, M. M. (eds.) 2000. Mapping of Control Variables to Musical Variables. Interactive Systems and Instrument Design in Music Working Group. Website: ng.html Hunt, A. and Kirk, R. 2000. Trends in Gestural Control of Music. Paris: IRCAM Kramer, G. 1996. Mapping a single data stream to multiple auditory variables: A subjective approach to creating a compelling design. In Proceedings of International Conference on Auditory Display - ICAD'96. Matossian, N. 1986. Xenakis. London: Kahn & Averill Polansky, L. 2002. Manifestation and Sonification. Unpublished paper. Roads, C. 1996. The Computer Music Tutorial. Cambridge, Massachusettes: The MIT Press Ryan, J. 1992. Effort and Expression. In Proceedings of the 1992 International Computer Music Conference. San Francisco: International Computer Music Association, pp. 414 - 416. Thieberger, E. M. 1995. An Interview with Charles Dodge. Computer Music Journal 19(1): 11-25 Xenakis, I. 1991. Formalized Music Pendragon Revised Edition. New York: Pendragon Press 210