A series of these essays on the phenomenon called “Mathematical Music”, created by the scientist Sergey Filatov-Beckman, was prepared by Professor of the Moscow Conservatory, Doctor of Art History Marina Skrebkova-Filatova, a member of the Union of Moscow Composers Elena Sokolovski and friends of S. Filatov-Beckman, musicologists L. and S. Gartman (Germany).
Mathematical Music as a New Means of Developing Compositional Skills
About the Author
Filatov-Beckman Sergey Anatolyevich (1958-2020) was the Doctor of Pedagogical Sciences (PhD), Associate Professor of the Moscow Tchaikovsky State Conservatory and of the Russian State Specialized Academy of Arts. He was the author of more than two hundred scientific articles and four books, united by the term “computer-musical modeling”.
S. Filatov-Beckman was born on September 17, 1958 in Moscow, in a family of musicians. His great-grandmother was an outstanding pianist, professor of the Moscow Conservatory, Elena A. Beckman-Shcherbina; his great-grandfather Leonid K. Beckman was the author of the well-known New Year's song “A Christmas Tree Was Born in the Forest”.The grandfather – Sergey S. Skrebkov was professor of the Moscow Conservatory, his mother — Marina S. Skrebkova-Filatova was professor of the Moscow Conservatory.
S. Filatov-Beckman graduated from Moscow Lomonosov State University in 1981 in the specialty “Geophysics”. Since 1981, he had been working at the scientific research institutes in Russia and Germany. In 2001 he re-graduated from Moscow University in the specialty “Applied Mathematics and Informatics”. The topic of his PhD thesis was the application of the ideas of computer modeling to the field of the musical theory. He was the author of the phenomenon called “the mathematical music” which is synthesized by a computer and is a musical interpretation of the self-organization processes in the nonlinear dynamic systems (the author's term: “musical synergetics”). As a part of the thesis topic, the computer research was also carried out on the artistic features of the folk music (including Khoomei – Tuvan throat singing). He has repeatedly spoken at the scientific conferences in Russia (Moscow – Moscow University, Moscow Conservatory; St. Petersburg, Yaroslavl, Saratov, Tver, etc.), in Germany (Berlin, Frankfurt am Main, Hamburg), in Israel (Jerusalem, Beer Sheva, Ashkelon). He was a simultaneous translator from the German language.
Since 2001 he had been teaching mathematics, musical informatics, musical acoustics and physics for the musical sound engineers at the Russian State Specialized Academy of Arts. Since 2004, he had been working at the Moscow Conservatory, teaching the musical computer science for the students of the performing specialties.
S. Filatov-Beckman worked for many years at the intersection of mathematics and music and created a teaching methodology that in practice managed to combine such two areas as mathematics and music.
The range of the scientific interests of Filatov-Beckman was quite wide: mathematics and modern electronic and computer music, pedagogy and physics, philosophy, cybernetics, meteorology, acoustics, methodology. In his articles and books, he turns to the research of famous scientists – N. Garbuzov, S. Skrebkov, A. Losev and others.
Preface
The essays presented below reveal the content of the “mathematical music” phenomenon created by S. Filatov-Beckman, the methods of teaching it to the beginner composers and its concrete implementation in practice.
These essays present a story of the gradual formation of a methodology that involves the development of the creative abilities of young musicians who want to compose music. The scientist does not formulate any strict instructions for composition and does not reveal the secrets of any personal experiments in composition, since he is not a composer himself. He offers his methodology as one of the means designed to help novice composers develop their creative imagination to the greatest extent.
This technique was formed by S. Filatov-Beckman during numerous years of his pedagogical activity. The list of the students to whom the methodology was directed changed many times, and along with this factor, the details and nuances of the scientist’s work were refined.
Some aspects of the process that forms this technique are reflected in t he books and numerous articles of S. Filatov-Beckman, while many remained in the scientist’s archive. The task of S. Filatov-Beckman's relatives and friends was to publish important archival data, combining them with the already published material of his books and articles.
The essays presented to the readers do not constitute a finished work, since the scientist’s archive has not yet been thoroughly investigated, and in this regard, new materials will appear.
We direct your attention to the fact that all the musical examples adduced in the essays have been collected by the scientist over many years.
The entire material is protected by copyright law.
The First Essay
In 2015 S. Filatov-Beckman’s book “Computer – musical modeling” was published. In 2017, due to some additions, a reprint of the book appeared. Now it carries the title “Special Pedagogy. Computer-musical modeling”. In this book such fields as mathematics, geophysics and music intersect — this intersection gives rise to the concept of the phenomenon of mathematical music created by the author. The book contains a description of not only this phenomenon, but also the first stage of the author's pedagogical methodology, which is aimed at developing composition skills based on this phenomenon. The author planned to describe the further stages of the development of his methodology in a new book. Our essays cite the material from the book published in 2017.
To clarify the essence of the mathematical music, let us turn to quoting some texts from the book. On page 117 we read: “The mathematical music is the result of mapping numerical sets into acoustic frequencies; from the viewpoint of the modern mathematics, this mapping can be interpreted as the result of the influence of some operator on the arrays of numbers. In essence, the role of such an operator is performed by the Cybercom converter program, which returns a music file in MIDI format”.
What number sets are meant? They are the result of numerous mathematical and geophysical experiments of the scientist, in which he displayed the processes occurring in the Earth's atmosphere. S. Filatov – Beckman did not limit himself to the mathematical results of research and converted, that is, translated these results into sound. This conversion is the basis for creating the phenomenon of the mathematical music.
The scientist reveals his idea of creating the phenomenon of the mathematical music: “We will consider and analyze a number of numerical experiments, the results of which were translated into monophonic sound lines and further processed in the form of a polyphonic fabric. The basis of these experiments is a problem from the field of theoretical geophysics (processes occurring in the earth's atmosphere). The numerical experiments are based on the sixth version of the author's musical-acoustic model MARC, which includes the so-called homogeneous equation of the heat influx in the Earth's atmosphere” (ibid.).
S. Filatov-Beckman conducted many experiments to obtain numerical geophysical data – the graphs, diagrams, calculations are adduced and described in detail in the book (chapter 5) and many dozens of published articles. Such experiments are the mathematical side of the mathematical music, and we will not touch on them in these essays.
What are the monophonic sound lines reproduced by means of the computer? They consist of numerous different elements, quite contrasting with each other. The scientist describes the methods of working with these lines, that is, he approaches the phenomenon of the mathematical music from the musical side itself.
Let’s recall that S. Filatov-Beckman is a geophysicist and mathematician by education. But he also had some musical knowledge: he graduated from the Gnessin music school in the piano class, later became an auditor of the music courses in Germany, while working there in his specialty. However, the scientist who did not consider his musical knowledge sufficient, often turned to the professional musicians for advice. When publishing his book, he received a number of valuable advices and professional help from them.
Conducting research at the intersection of the natural sciences and the musical art, the author asks the question: “What is the role and the tasks of the computer technology in modern music science?” – and further answers: “Music science can be understood as a complex of modern sciences about music. These are the theory and history of music, the analysis of performing arts, instrumental studies, musical acoustics and psychology, musical informatics, musical sound engineering and a number of other disciplines” (p. 71) and further explains that “music is a kind of the global information process. The idea of music as an information process makes it possible to apply the ideas and methods of modern information theory (cybernetics and informatics) to the musical science. These disciplines belong to the cycle of mathematical sciences, which emphasizes the natural organic connection between music and mathematics, which has its roots in ancient times” (ibid.).
S. Filatov-Beckman outlines the meaning of the computer-music technology of a mixed type, speaking about its vast possibilities: “The application domain of this technology is the computer musical creativity and the computer analysis of the elements of musical performance. If earlier the concept of musical creativity was associated mainly with composition and performance, now the computer helps to significantly expand the field of musical creativity, including the analysis of the performance of musical works. The computer analysis belongs to the modern direction of mathematical modeling – “soft modeling” – and it can be called computer-musical modeling” (ibid.). The analysis of the interpretations of musical works or their fragments by different performers is supplemented by the analysis of the performance of the same work on different instruments of the same type, for example, on different pianos or different guitars, and so on. In such cases, the computer captures changes in the sound of even the smallest details of the performance, which is very important for the performer. The author conducted similar experiments in the student classrooms and described them in his book (pp. 89, 95).
The book by S. Filatov-Beckman, in addition to the scientific explanation of the phenomenon of mathematical music, contains some practical recommendations on the technology for creating examples of mathematical music. These recommendations are directly related to the author's many years of pedagogical activity. In the future, similar examples of mathematical music served as the basis for the creation of various musical works, and professional musicians took part in this process.
The scientist outlines several stages in the creation of samples of mathematical music. The book details the first step of this process. The author who created the sound line through conversion writes the following: “The sound line generated by the computer as a result of converting numerical data into acoustic frequencies is divided into an arbitrary number of fragments (this is done on the basis of the Cubase virtual studio). Fragments are freely combined with each other, creating a polyphonic musical fabric; it can cover the sound range up to 8-9 octaves” (p. 118).
Explaining the methods of processing a sound line and obtaining a polyphonic fabric therefrom, the author uses the technique of combinatorics. Thanks to this, you can get “...the increase and decrease of the number of voices; the creation of fabric according to the principle of the imitation-polyphonic and the vertical-cluster presentation; the combination of polyphony with monophonic sections, the sequential appearance of these sections, and so on. The results obtained can be colored with a variety of timbre colors, as well as supplemented by a number of processes and effects contained in the menu of music editor programs” (ibid.).
S. Filatov-Beckman demonstrated a sufficient number of the examples of the mathematical music in the book. One-voice lines, created by the computer on the basis of the solution of numerical equations and reflecting a number of geophysical processes in a specific musical language, have become the foundation for the transformation into various types of polyphonic fabric. These experiments aroused great interest among the readers of the book and the listeners at the scientific conferences, where the mathematical music was played repeatedly. S. Filatov-Beckman also mentions that his students were actively working on such examples, which awakened their creative imagination. This creative impulse became one of the goals of the computer-music technology of the scientist, who had been carrying out pedagogical work for many years. At the same time, mathematical music has been used more than once as an applied genre in the performances at the suggestion of the theater directors.
The scientist outlined a plan to continue his work in the form of a new book, revealing more deeply the musical aspects of his theory, but did not have time to implement this plan. His sudden death made it impossible to complete and formalize his research. He left a large number of individual texts, theses, abstracts, developments, which, however, allowed the friends and fellow musicians of S. Filatov-Beckman to bring his work to a coherent literary form. This essay includes many theses from his scientific heritage, as well as materials from his published books and articles. This and subsequent essays are a summary of the ideas of the scientist, who considered the work on the creation of mathematical music as the basis for his doctoral dissertation.
Let's try, using the methodological developments of S. Filatov-Beckman, to reveal in detail the process of obtaining a polyphonic fabric (and then – much more complex musical compositions, which will be adduced below) from different monophonic sound computer lines. Let us turn to an example from the book, which the author designated as 61ТТА (hereinafter, the numbering of the given musical examples contains the designations of numerical experiments, on the basis of which these examples were obtained: 61ТТА, 61ТТА8, 61ТТА12, and so on).
This audio line, displayed by the computer, has a fairly large length. Let's pay attention to the following: the line is absolutely monorhythmic, and the groups of 4 sixteenths and the conditional division into measures are introduced by the author only for the work convenience. But the pitch parameter of the line is very active, covering a range of several octaves and representing a variety of types of “the melodic profile”. There are, for example, movements of sounds at different intervals, jumps and stepwise “intonations”, long “rehearsals” on one sound or, on the contrary, breaks of the registers in the monophony. Please note that some of the examples are given both in musical notation and in sound notation, and the other part is only in musical notation. We give an example of one of the sound lines (line No. 1) in its entirety, since only a part of this line is published in the book.
What does this sound line reflect, what geophysical process has the computer translated into sound? “Experiment 61, based on the version 6 of the model, covers an integration time of 24 hours in 600-second increments”, the researcher wrote. And he explains the essence of this geophysical process: “For such a short period of time, the air temperature does not undergo any significant changes. Only in the high layer of the atmosphere (45-50 km) there is a process of the rapid temperature changes associated with a sharp decrease of the air density at such heights” (p. 118). The results of the discussed experiment were converted into a monophonic audio line.
Further, the scientist reveals the connections between the geophysical processes and their sound reflections: the sound line “...contains several sections consisting of the ostinato repeating sounds in the middle register: the line sections rise slowly, step by step, covering the range of the tritone”. And further: “The ostinato repetition of sounds is explained by slight changes in the temperature at each of the levels of the atmosphere” (ibid.), and “the gradual rise of sounds is due to the transition to the layers which are closer and closer to the Earth’s surface, where an increase and then a decrease of the temperature are observed”.
The author pays special attention to the ultra-high sounds (“do” of the fifth octave): “The more contrastingly sounds the sharp ejection of several sounds up 4 octaves with a return to the middle register”. And he explains this effect by the fact that the cause of the “ejection” is a noticeable dynamics in the upper layers of the atmosphere (45-50 km).
Let us recall that the scientist outlines several stages, while imparting practical recommendations on the methodology for creating the examples of the mathematical music. The first step of the first stage of work on a sound line is its arbitrary division into monophonic fragments. These fragments, taken from different parts of the line, can contain both several sounds and be quite extended. When dividing a sound line into fragments, the following conditions must be fulfilled: you cannot change one sound to another in the fragments (for example, to use the sound “do” instead of the sound “re”), as well as you cannot change the sequence of sounds received by the computer. Here are a few examples given in the book of S. A. Filatov-Beckman and obtained as a result of the arbitrary division of the line №1 into fragments:
The next step in creating a polyphonic canvas requires connecting monophonic fragments to each other vertically and horizontally.
As already mentioned above, the division of the audio line into fragments is completely free, therefore, the choice of the horizontal sequence of these fragments is also free. In this case, it is also possible to vertically transfer the line fragments from register to register up or down, but only by an interval of one or several octaves, since the sounds of the computer line fragments do not change in this case.
Thus, at the first stage of the technique, S. Filatov–Beckman allows some modifications of the sound line fragments. Further, the fragment modification technique generates very important and interesting results. These results will lead to the fact that on the basis of the computer-sound series it will be possible to create full-fledged musical works of various genres, which will be demonstrated in subsequent essays. The creation of such works is the goal of the author's methodology aimed at developing the creative potential of young composers. We will devote the second essay to the problem of modifying the fragments of the computer lines.
So, when the line fragments are layered on the top of each other vertically, a polyphonic fabric appears, containing from 2-3 to 7-10 voices or more. It is the search for the most interesting and richly colored polyphonic fabric that leads to the creation of the primary monorhythmic “musical compositions” and awakens the creative imagination of young musicians. It is this work that is the content of the first stage, which the author managed to reflect in the book (the subsequent stages, as mentioned above, are contained in archival records and will be presented in these essays).
In the following examples, the author demonstrates the results of such work. We adduce these examples and will analyze them in sufficient detail.
The initial fragment of the example No. 3 – the monophony on the sound “b-flat” – very quickly connects vertically with another fragment on the sound “d-flat” (the introduction of each new “voice” is marked with an accent), then with the next fragment (on the sound “e”); an ultra-high sound “с” of the fifth octave is superimposed and repeated on the resulting three-voice, and then more and more new fragments enter and layer, creating the cluster sounds that slowly move up. “Since the incoming “voices” are similar in the sound composition and the ostinato-rhythmic repetition, the nature of the presentation is associated with an imitative-polyphonic technique”, the author sums up. This is especially facilitated by the sequence of the introduction of “the voices”.
In the following example No. 4, the same sound computer line is divided into fragments differently, and their layering on each other presents a different picture. The first sounds (“b-flat”) and monophonic sounding coincide with the original sound line and with the previous example, and then the differences begin. These differences are especially noticeable, when five- six-sound clusters appear, gradually descending chromatically. The imitative – polyphonic technique (alternate entry of voices) is used to a lesser extent, and the movement itself from the ascending in the previous example changes direction to the slowly descending. Let’s adduce the second half of this example.
Based on the same sound line, the author offers two more examples. One of them almost immediately begins in two voices, with a large break in the registers; then the voices quickly accumulate in the upper register, the ten-sound clusters appear; then the clusters lose sounds from ten to seven-four, and the sounds are interrupted. Obviously, this example contains a completely different division into fragments and their connections, than the previous examples (we adduce the beginning and end of the example).
In the following example based on the same line, starting as the previous example, new effects arise: polyphonic clusters (up to 15-17 sounds) include extremely low registers and huge gaps in the polyphonic fabric. “These clusters are characterized by a very wide pitch range (8-9 octaves). This example shows sharp contrasts between the sonorities containing wide “empty” register breaks and extremely dense clusters with a rapid expansion of the texture space”, the author comments on this option (p. 119).
In the classroom, S. Filatov-Beckman explained to the students how the clusters arise from the monophonic sound line obtained by him as a result of the experiments. These clusters are created from the original sounds of several previous fragments. We adduce an example-scheme, where only the initial sounds from seven different fragments are written out, gradually leading from the monophony to the seven-voice cluster. Here is the scheme:
Another way is to “verticalize” the monophonic fragment. The cluster in this case consists of as many sounds as the fragment contains – such a cluster can consist of 15 – 16 sounds. We give an example:
In addition to working on obtaining clusters from a monophonic line, S. Filatov-Beckman also proposed to the students to find such examples of a polyphonic fabric, where the ultra-low registers are used in combination with the ultra-high ones (Example No. 9 a) or the opposition of the monophony to the polyphonic clusters (Example No. 9 b). These case studies are taken from the researcher's archive:
The demonstrated examples and the examples analogous to them are stored in the archive of the author, who devoted a lot of time to working on the computer sound line. There are dozens of such examples.
The number of the polyphony variants created on the basis of only sound line №1, which were demonstrated above, is very significant (several different sound lines are proposed in the book, but a much larger number of them is stored in the archive). But not all of them are equally informative. When publishing the book, the scientist carefully chose those examples in which he, together with the students, achieved the best results and showed the textural-register possibilities of the variants.
As already mentioned, all the previously given musical examples were derived from the same sound line No. 1. We present for review two examples of the polyphony, created on the basis of other lines. They sufficiently contrast with each other in registers, as well as in the interval moves. These examples are included into the book.
Thus, the result of the first stage of working with the line is the search for different forms of the polyphony. So far, at the level of the first stage, all these examples of the polyphony are monorhythmic.
At the same time, the question arises: is the internal connection between the monophonic sound line obtained on the basis of the geophysical experiments and displayed by means of a musical computer program, and various variants of the polyphonic fabric, not lost? The scientist answers this question: “In some cases, the combination of the fragments of a monophonic sound line veils, and sometimes even completely hides the features of a numerical experiment that clearly appear in the original one-voice line”. However, as the author writes, “the latent influence of the original line can be traced even in the case of the volumetric cluster structures: the degree of variability of these structures strongly depends on the hidden textural features of the monophonic fragments” (p. 119). And further on the scientist writes that these examples “reflect the pitch, register, spatial aspects of the musical fabric, and also highlight some textural features, forming elements of the imitation-polyphonic, choral-cluster storehouses, the polyphony of layers, etc.” (p. 126).
Let us pose the main question of the new phenomenon called “the mathematical music”: what new quality, in addition to the transformation of a monophonic sound line into polyphony, do numerous examples acquire? This quality is as follows.
The most important difference between a computer sound line and the examples of polyphony is that this line, voiced by the computer, has nothing to do with the musical art, since it reflects a natural geophysical process. The examples of the polyphonic fabric, obtained on the basis of the fragments of one or another line, were created by a team of people directly connected with the musical art and showed creative imagination.
The author considered that it is possible to achieve more important and interesting results in the work on the monophonic monorhythmic sound line. The scientist writes about his creative plans at the level of the second stage: “In the future, it is planned to set the tasks of working with rhythm, articulation and loudness dynamics. In combination with a variety of timbres and acoustic effects, mathematical music should contribute to an even greater extent to the development of the creative potential of the students” (p. 126).
Computer Sound Lines
We also offer for analysis a number of the computer sound lines obtained by S. Filatov-Beckman. Some of them are reflected in the author's book, and some are taken from the scientist's archive.
