What are key tones in chemistry

Cellular and Molecular Music. About the gap between two tones

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1 Cellular and Molecular Music On the Gap between Two Tones Drafting a Topology of Music 2010/2016 Music and Molecules Using historical and experimental music examples, an attempt will be made in the following to create a topology of music. Usually notes are points and lines on the two-dimensional surface of paper. The notation of the tone sequences consists of characters on a surface. Nevertheless, these notes are interpreted as a temporal sequence, a temporal order. That is why music is considered the mother of all time-based arts. But music is also the mother of all technological arts. With the technical possibilities of today, the option opens up, so my thesis, of defining the music not as a temporal succession, but as a spatial side by side. I suggest that the classical interval theory should be followed by the theory of music that defines the gap between two tones as spatial neighborhood. So I am pursuing the idea of ​​a change from temporal to topological music. In doing so, I am not thinking of the reception of music in the room, of a sound dome, of a loudspeaker system distributed throughout the room. For me, spatial music does not happen in reception, but on the contrary during the production and generation of music. My proposal aims to use new geometric and mathematical methods to compose music in space instead of in time. The notes are defined as spatial points to which numbers or cells or molecules correspond. The relationship between the grades is compared to the relationship between the molecules. The physical properties of the molecules are correlated with the physical properties of the notes. So music becomes part of physics, chemistry and mathematics. There were already attempts in the 20th century, especially in the 1920s and 1950s, to change classical music theory. From twelve-tone music to serial music, the new music theories have devoted themselves to the question: What non-subjective rules do I use to get from one tone to the next? This is precisely the central question posed by composers. So far, the answer has been found in time-based composition rules. But on closer inspection we see that the composers were very early interested in a spatial arrangement of the tones, in spatial rules of composition. Arnold Schönberg already compared the chemical connections with the relationships between the tones in his theory of harmony (1911). For the material that is worked with consists of atoms and molecules for some and tones for others: “The material of music is tone; whereupon it works first, the ear. Sensory perception triggers associations and sets tone, ear and cellular and molecular music 383

2 Josef Lo schmidt, Molecular scheme from chemical studies, 1861, sheet 3 World of feelings in connection. Everything that is perceived as art in music depends on the interaction of these three factors. Nonetheless, if a chemical compound has properties other than the elements that compose it, and if the impression of art shows properties other than those that can be derived from each of its components, then one is justified in analyzing the overall appearance for some Purpose to use some properties of the basic components for consideration. The atomic weight and valence of the constituents also allow a conclusion about the molecular weight and valence of the compound. Perhaps it is untenable to try to derive everything that defines the physics of harmony from one of the components, for example from the tone alone. "'' Amazingly, when analyzing tone relationships, Schönberg refers to the analysis of chemical compounds and already uses the name" Molecular Weight «. This could already be the origin of a molecular theory of music. Furthermore, Schönberg describes the tone as a point in a kind of geometric series. That is why he speaks of “relatives” in the distribution of tones in the sense of “neighborhood”: “This explains how the series that was finally found is composed of the most important components of a keynote and its closest relatives. These closest relatives, who first make it a fixed point by keeping it in equilibrium through their forces acting in opposite directions. "2 Schoenberg speaks of" forces in equilibrium ". So he uses a thermodynamic vocabulary that was not unusual in Vienna at the time. The popular writings of Ludwig Boltzmann, the founder of thermodynamics and advocate of atomistics, which were still controversial at the time, were very well known in Vienna. Boltzmann Arnold Schönberg, Harmony, Universal Edition, Vienna, 1911; quoted from: 3 ed., Universal Edition, Vienna, 1922, lbid., MUSIK UND MEDIEN

3 Milan Grygar, score I project of the sound layers, Josef M atthias Hauer, tropical diagrams in morphological 1969, colored ballpoint pen, fixed pen, paper arrangement, here a) Polysymmetrical tropics; The numbering of the tropics corresponds to Hauer's tropical table of August 11, 1948, from: Vom Wesen des Musikalischen (1920), 1966, alongside James Clerk Maxwell and Josiah Willard Gibbs, is considered the founder of statistical mechanics. Maxwell's results, the determination of the distribution of the velocities of atoms in a gas as thermal equilibrium, were generalized by Boltzmann. We can see where Schoenberg got the phrase "forces in equilibrium" from. Schönberg's thesis of the fundamental tone in equilibrium is obviously based on the implicit vision that the clash of tones generates or changes the melodies and that the music arises from the dynamic energy of related and neighboring tones. Boltzmann studied gases in the non-equilibrium state, i. H. how the distribution of the "living forces", the kinetic energy, changes when the molecules collide. We owe the famous definition of entropy to Boltzmann: 5 = h log W. Where 5 is the entropy, h the Boltzmann constant, W the "thermodynamic probability" and "log" the natural logarithm. Like many writers and artists from the Vienna Secession and the Jung Wien group around Hermann Bahr, Schönberg was evidently relatively familiar with the popular writings of the most important physicists and philosophers of the time, such as Ernst Mach and Ludwig Boltzmann. Schönberg thus examines the relationships of tones in a similar way to how physicists examine the relationships between elementary particles and chemists examine the relationships between atoms and molecules. He wonders how a tone becomes a fixed point and which forces maintain it in a harmonious equilibrium. He asks how it is logically possible for a first tone to be followed by a second. What are the rules for doing this? How can you combine two tones at all? That is a question that comes from atomistics and molecular theory, e.g. B. a Josef Loschmidts, is derived. Loschmidt, who was Professor von Boltzmann, determined the size of the "air molecules" for the first time in 1865. In his work Chemical Studies. Constitution formulas of organic chemistry in graphic representation from 1861 he graphically represented 368 molecules through the spatial orientation of the atoms. His constitution formulas show the double and triple bonds of the atoms through the number of lines. CELLULAR AND MOLECULAR MUSIC 385

4 Thierry Delatour, 2013, molecular sound is generated with the help of a fundamental acoustic conversion method Schönberg obviously had the model of constitution formulas in mind when he wondered how ties and relationships of tones can be established and legitimized logically and rationally. The granular representation of the chemical compounds shows a remarkable resemblance to the graphic notation of the avant-garde music of the 1920s and 1960s. Let us compare, for example, the score of Milan Grygar's Score I Project of Sound Layers (1969) with Loschmidt's graphic representations of molecules. Without knowing his ancestors, namely Loschmidt and Schönberg, a French professor of physical chemistry, Thierry Delatour, developed the idea of ​​the constitution formulas further. H. he suggested treating the combination of tones analogously to chemical formulas. In the year 2000 he published his article »Molecular Music« in the Computer Music Journal. The Acoustic Conversion of Molecular Vibrational Spectra '.' In 2013 he published the extended text »Molecular Songs« in the volume Molecu / ar Aesthetics edited by Ljiljana Fruk and myself. 4 Delatour asked himself how microscopic, atomic vibrations in molecules can be transformed into audible sound vibrations. More precisely: How can scientific data, so-called »vibrational spectra«, be transformed into sounds and music? The translation of molecular vibrations into tones should be relevant both artistically as a musical composition and scientifically as additional information acquisition. The properties of molecular vibrations can be precisely determined with the help of spectroscopy: A molecular spectrum consists of a two-dimensional diagram with depths or heights below or above a baseline. The spectrum is the result of an interaction between electromagnetic waves (infrared, visible, ultraviolet) with matter (molecules, crystals, etc.). The X-axis, expressed in frequency or wavelength, is related to the energy of this interaction. The V-axis is related to the intensity of this interaction. With the specific band positions and intensities, each spectrum represents a certain chemical substance, to a certain extent the fingerprint of this substance. Molecules are made up of atoms held together by chemical bonds. These bonds are the result of random electron movements between atoms known as Thierry Delatour, nmolecular Music: The Acoustic Conversion of Molecular Vibrational Spectra << 1 in: Computer Music Journal, Vol. 24, No. 3, Fall 2ooo, lbid. 386 MUSIC AND MEDIA

5 Thierry Delatour, 2013, molecular sound waveforms homothetically to infrared interferograms of (a) liquid water, (b) ethanol, (c) benzene and (d) N-methylacetamide act as a type of dynamic cement. Molecular energies can be quantified at the microscopic level. There are therefore discrete energy levels from which frequencies or wave numbers (the inversion of the wavelengths) can be derived. Each molecule has singular vibration frequency values, i.e. a unique fingerprint. Molecular vibrational movements have numerical values ​​between 12 and 120 THz - an inaudible frequency. Therefore a method of acoustic conversion of the molecular vibration spectrum is required. So how are vibration spectra converted into acoustic signals? By recording such a spectrum and making its central frequencies visible on a computer screen. A high factor then transposes these frequencies into the acoustic range of 4,000 to 400 Hz. Then sine generators are tuned to these frequencies and play these frequencies at the same time. This is how the molecular sound is created. The intrinsic physical and chemical properties of molecules are used to create molecular music. The advantage of molecules as musical elements is that a molecule is a multidimensional musical oscillator with as many dimensions as the number of atoms in the molecule. This creates new types of resonance. The use of high technology and computers has deepened the musical composition from the macroscopic to the microscopic. The longing of Ferruccio Busoni and Edgard Varese for new instruments to generate new tones has been fulfilled by science. These new instruments are located below the human visible zone, where the inaudible tones normally arise. These new instruments require new mathematical composition methods. The result is new music that is not temporally defined as one after the other, but topologically as side by side, as neighborhood. Physics, chemistry and mathematics become the media of music. At the end of his life, Schönberg asks himself, as the sum of his musical experience in Style and Idea (1950), the seemingly simple question: How do I get from one note to the next? »If we now investigate want what it actually is: Relationship between ln biochemistry, for example, the (inter) molecular vibrations of cells are measured by means of vibration spectroscopy. In this way the music of the cells in the nürchester of the organism is made topologically visible in a spectrum. Cells and organisms could also become the basis for new music. CELLULAR AND MOLECULAR MUSIC 387

6 notes to each other, so first I ask the question: What is the possibility of having a second note follow a beginning first note based on? How is that logically possible? This question is more important than it looks at first glance; nevertheless, as far as I know, it has not yet been asked. Not yet, although one has already dealt with all possible and far-reaching problems, one has asked: Why can one connect two tones at all? My answer is: Such a stringing together of tones, if a connection is to be established through it, if a piece of music is also to emerge from it, is only possible because there is a relationship between the tones themselves. One can logically only connect what is related to one another: direct or indirect. In a piece of music, however, due to the lack of a musical relationship, I cannot combine a note with an eraser. In order to explain the relationships between tones, it is important to remember that each tone is a composite sound consisting of a most powerful fundamental tone and a series of overtones. One can say, and can test and prove this sentence to a large extent, that all musical events can be traced back to the overtone series, so that everything presents itself as the exploitation of simpler and more complicated relationships in this series. "6" So the question is what the The possibility of connecting the tones is based, answered: It is based on the fact that in the sounding tone and its closest relative, the connectedness of the tones and their resting together are repeatedly demonstrated, so that we do nothing other than imitate nature when we use this relationship . «7 Schönberg thus describes the possibility of connecting tones as a kind of relationship, referring to nature and natural science as a model. The renaissance music theorist Gioseffo Zarlino already understood music as an imitation of nature. 8 In order to clamp the tones together, just as chemical bonds hold the atoms together, Schönberg invented the idea of ​​the interval structure of the series: the series and its rules should act like a kind of dynamic cement, like the electron movements between the atoms, and legitimately connect the tones. " perhaps the most important influence of Schoenberg's method is not the 12-note idea in itself, but along with it the individual concepts of permutation, inversional symmetry and complementation, invariance under transformation, aggregate construction, closed systems, properties of adjacency as compositional determinants , transformations of musical surfaces, and so on. «9 But we also see that the methods of dodecafony, as they are described here, eg Symmetry, spatial parameters are. Anton Webern's Concerto for nine instruments, Op. 24 from 1934 consists of the simple basic form B B D and its variations: mirroring, inversion, inverse mirroring. The twelve-tone music by Schönberg and that by Josef Matthias Hauer are therefore already topological and not just temporal composition techniques. Temporal and spatial distances between the tones Composers ask themselves the question: How do I bridge the gap between one tone and the next? There is obviously a gap between the tones. 6 Arnold Schönberg, "Problems of Harmony" (1927), in: ders., Stil und Gedanke. Essays on music. Collected Writings 1, ed. v. Ivan Vojtěch, Fischer, Frankfurt am Main, 1976, p, here p. 220f. 7 Ibid., S Cf. Gioseffo Zarlino, Le institutioni harmoniche, Venice, George Perle and Paul Lansky, Serial Composition and Atonality, University of California Press, Los Angeles, 1981, S Musik und Medien

7 On the score, these distances are of a spatial nature; for performers and listeners, these distances are of a temporal nature. An implicit equation between spatial and temporal distances is hidden in the score. But it is clearly not these distances that give the music its temporal dimension, but rather the beat and the notes. The beat, a grouping of certain note values ​​with the same beat, is the basic temporal structure of music. The music is structured by the individual bars. The rhythm is created by the beats of the measure. The 4/4 time has four basic beats or beats, each worth a quarter note. The beat gives the music a metric structure. This metric structure of music as a tense can also be transferred to the tense of the moving image.Peter Kubelka viewed the individual film cadre as a note and lined up these film cadres according to musical principles (Arnulf Rainer,; Adebar, 1957). That is why Kubelka called his films »metric films«. A film can also be clocked. Kubelka, who is also a musician, follows the twelve-tone row scheme when constructing the cadre sequence. In the dodecafonia there are two basic transformations of a twelve-tone row. This results in the four modes of the twelve-tone row: 1. the basic or original row, 2. the cancer formation, which is caused by a vertical mirroring, i.e. when the original row is played backwards from its last tone, 3. the inversion, which arises when the intervals of the original series are replaced by their complementary intervals. Any interval that was directed upwards in the original sequence is directed downwards and vice versa. So it is a horizontal reflection. 4. the cancer formation of the reversal. For a long time, what Gotthold Ephraim Lessing said in his famous essay on Laocoon was true: poetry and music are the tenses of art, the arts of one another. Painting and sculpture are the spatial forms of art, the arts of juxtaposition. If it is true that painting uses completely different means or signs than poetry for its imitations; those namely figures and colors in space, but these articulated tones in time; if it is undisputed that the signs must have a comfortable relationship to what is designated: signs arranged next to one another can also only express objects that exist next to one another or their parts exist next to one another, but successive signs can only express objects that follow one another or their parts one after the other. «10 However, it is already evident in film, the art of the moving image, that although it is an image form, it is no longer an art of juxtaposition, but of one after the other, because cadre follows cadre as note follows note . The tense of the image begins with the film. This dissolution of the classical categories also allows the question: What is the actual event that is represented on a score? When we look at a score, we see the staff lines and the spaces between the notes. But what exactly is that between the notes? Is that space or time? Most composers and theorists still say "music is a tense". The terms rhythm, beat, measure, series and repetition emphasize the temporal aspect of the music. Perhaps the secret of music lies in the concept of the interval. Usually, interval is defined here as the distance between two tones. But the next definition already obscures the situation, a more explicit definition of the interval reads: The interval is the time interval that lies between two processes. The term »space« appears here and that is exactly the crux of the matter. In terms of sound, the 10 Gotthold Ephraim Lessing, "Laokoon or Beyond the Limits of Painting and Poetry" (1766), quoted here from: Günter Helmes and Werner Köster (ed.), Texts for Media Theory, Reclam, Stuttgart, 2002, P. 53. Cellular and Molecular Music 389

8 intervals defined in time, but in the notation they appear spatially. The juxtaposition of the notes on the paper is interpreted by the interpreter as one after the other. So we have to assume that the music is based on the distance between two tones, i.e. the intervals, but that the distance between two tones can be interpreted both spatially and temporally. The space between the tones can also be understood as a space in between, in which case the question arises, what do I do with the space. In 1901 the painter Adolf Hölzel published the essay "On Forms and Mass Distribution in Images" 11 in the magazine Ver Sacrum, the publication organ of the Vienna Secession, in which he demonstratively stated that he was not interested in the forms of objects and people, but rather only for the space between these shapes. He was one of the founders of abstract painting. Anton Webern transferred this idea to music. He was the first for whom the space between two notes was just as important as the two tones that delimit the space. In terms of time, the pause is worth just as much as the sound, because without a space there would be no bar. The spaces between the two notes are just as important as the notes themselves. This has been called the emancipation of the rest. John Cage's composition 4'33 "(1952), in which a composer sits motionless in front of his piano for exactly four minutes and 33 seconds, is nothing more than the application of Webern's theory, the celebration of the break. The main written work by Cage has the title Silence (1961). A book about music is called "Schweigen", after the silence in space or in the time between two tones. This book absolutizes Webern's idea of ​​the pause and Hölzel's idea of ​​the space Classical music theory, for example by Hegel, defines the distance between two tones as already described by Pythagoras and refers to the monochord: “With a monocord, where you can divide the string, the amount of vibrations is related to the parts in the same time certain length in inverse proportion; the third of the string makes three times more vibrations than the whole string. Small vibrations in high notes can be avoided because of their great speed ity no longer count; however, the numbers can be determined very precisely by analogy through the division of the string. [] The most interesting thing is the coincidence of what the ear finds harmony according to the numerical proportions. It is Pythagoras who first invented this coherence and was thereby induced to express relationships of thought in the manner of numbers. "12" The harmonic limit of this ascent is given by the ratio 1: 2, the fundamental and its octave; between these one must now also take the absolutely definite tones. The parts of the string by means of which one wants to produce such tones must be larger than half the string; for if they were smaller, the notes would be higher than the octave. In order to produce this uniformity, one must insert tones into the harmonic triad that have roughly the same relationship to one another as the fourth to the fifth; this is how all the notes come about, which form a whole interval, just like the progression of the fourth to the fifth. The space between the root and third is filled by the second when 8 of the 9 strings vibrate; this interval from the keynote to the second (from c to d) is the same as 11 Adolf Hölzel, "On Forms and Mass Distribution in the Picture", in: Ver Sacrum, No. 15, 1901, S Georg Wilhelm Friedrich Hegel, Enzyklopädie der philosophischen Wissenschaften im Floor plans: Part two. Die Naturphilosophie (1830), Suhrkamp, ​​Frankfurt am Main, 1986, p. 177f. 390 Music and Media

9 The monchord with the string divisions, as shown by Robert Fludd in De naturae simia seu technica macrocosmi historia (1618). that from the fourth to the fifth (from f to g) and that of the sixth to the seventh (a: h). «13 Hegel's philosophy of music is thus an interval theory of music. In his book Genesis of a Music (1949) Harry Partch writes “Long experience [...] convinces me that it is preferable to ignore partials as a source of musical materials. The ear is not impressed by partials as such. The faculty the prime faculty of the ear is the perception of small-number intervals, 2/1, 3/2, 4/3, etc., etc., and the ear cares not a whit whether these intervals are in or out of the overtone series. ”14 Hegel does not only speak of vibrations and numerical relationships, but also of lengths, i. H. of spatial parameters. So he allows spatial aspects of interval theory to resonate. Clearly, both temporal and spatial aspects can be recorded in numbers. The main thesis or main interest of my essay is to interpret Hegel's interval theory less temporally than spatially. The distance between two tones is not only a temporal distance, namely for the ear, but is also a spatial distance, namely on the score. According to this, music is part of mathematics, namely the mathematical discipline of topology. Topos means place. If music is topology, then it is a doctrine of space, not time. Topology, the mathematical theory of space, speaks of neighborhoods, specifically of spatial neighborhoods. In relation to the music, you can say f sharp, c sharp, g sharp, etc., but you can also say a, b, and c. B is the neighbor of A and C, C is the neighbor of B and what is the neighbor after C? The question of how do I get from one tone to the next turns into the question: How do I get from one neighbor to the next? Composition means nothing else than setting up rules to get from one note to the next. The most famous composition theory is harmony. After the results of these rules were no longer satisfactory because they excluded other audio experiences, e.g. Dissonances, other compositional teachings have subsequently been invented, such as serial music, serial music, twelve-tone music, the 13 Ibid., S Harry Partch, Genesis of a Music, University of Wisconsin Press, Madison, 1949, p. 87. Cellular and molecular music 391

10 Minimal Music etc. The invocation of chance in aleatoric music, as well as the invocation of probability in stochastic music, are rules that define how the composer can bridge the gap, the abyss, between the tones. If we go back to the original instrument of music, the monochord, we recognize the problem of notation, the ambivalence of spatial proximity and temporal sequence. The monochord is a resonance box made of wood, a wooden body over which a string is stretched, in abstract terms: a line. There is a scale on the box: a line that divides this line in half. This line of wood is called a bridge. This bridge can be moved. By shifting it, you can divide the string further. The graduation can be used to precisely determine the division, especially the vibrating length of the string. This enables the intervals to be measured. Despite the name monochord, which means "single-string", there are also multi-string monochords that can be used to make the intervals sound simultaneously. But the ambivalence of spatial and temporal parameters already exists with the monochord. The division of a line into two halves and four quarters etc. is a spatial process. Division is a spatial process. The movable web creates spatial segments. A number proportion is used in the monochord. The vibrations or frequencies that arise in each case then generate the tones. Tones are therefore the representation of frequencies, i.e. a representation of how quickly the repetitions follow one another in a periodic process. The frequencies are the result of spatial divisions. These in turn are the representation of numerical or metric proportions. This division process can be interpreted as an interval, but one can just as easily say: A space is halved, thirds and eighth. If you transfer this idea to a piano, you can see that the keys have a spatial arrangement, namely that they are neighbors. Some are close neighbors, others are more distant neighbors. In the case of the piano, the sequence of keys or tones is defined as a spatial neighborhood. The rules of composition can therefore be set up as follows: The score is the instruction in which order this or that adjacent key is touched. It is therefore a matter of instructions that the pianist performs. The piano thus offers spatial and temporal parameters. The composer can determine in which order it will be played. The term sequence already predicts that a theory of the series can replace a traditional, harmonic composition. The composer determines the chronological order in which spatial neighbors become temporal neighbors. A chord that is created by touching several keys with several fingers at the same time and transforms a juxtaposition at a single point in time instead of one after the other, thus merges several spatial points at a single point in time. Instrumental music, like playing the piano, is always a bijection of space and time. Music as a tense, a purely temporal sequence, is only heard music or only sung music. Even music before the invention of the score was a tense. With the notation on a two-dimensional surface, the music paper, music becomes part of spatial art. Since the invention of instruments and the score, music has been the mutual transformation of spatial to temporal and of temporal to spatial parameters. One can even say that the spatial parameters dominate, both when composing on paper and when using the instruments. Let's take a clarinet as an example: The keys are a spatial arrangement, like points on a line. The score gives instructions as to the chronological order in which the spatially adjacent 392 music and media

11 Ulrich Rückriem, Teilungen-Partitions, 1970/1971, video produced in collaboration with Gerry Schum and Ursula Wevers for the video gallery schum flaps. By listening, people forget that the spatial coexistence still shimmers through in chronological succession. The sculptor and conceptual artist Ulrich Rückriem created Teilungen-Partitions (1971), a piece for Gerry Schum's video gallery. The English title already evokes the term "partitur". We see the division of a »line«, actually a stick made of wood, as a sculptural process. Rückriem takes this wooden stick and breaks it almost in half and this in turn in two halves. This creates half, quarter and eighth stretches. So he uses numerical proportions that he applies to a spatial route. Then he tries to make sixteenths out of the eighths, but he doesn't succeed. Rückriem believed that he had acted as a sculptor and created spatial proportions in the sense of Vitruvius. After all, he did nothing but cut up a stretch of space. But these partitions were not just proportions, they were also fifths, fourths, and thirds. So he not only acted as a sculptor, but also as a musician. Indeed, he gave a musical performance. In 2010, when I received the Austrian Cross of Honor for Science and Art, 1st Class in Vienna, I played my own piece instead of the usual musical accompaniment, which consisted of a continuation of Rückriem's ​​work. I drew a monochord on a blackboard, then halved, quartered, and eighthed the string of the monochord and next to it I wrote the integer ratios, i.e. the main intervals. Accordingly, I broke a provided wooden stick and declared the noise to be music. Music and mathematics The music theory goes back to the number theory of the Pythagoreans. The sentence: "Everything is number" is ascribed to Pythagoras. Important foundations of number theory and mathematics originate from him. These in turn served as the basis for his music theory. The Pythagorean number theory was based on the assumption that all phenomena of the cosmos can be explained as manifestations of integer number relationships, and this assumption also formed the basis of Pythagorean music theory. According to the ancient legend, Pythagoras discovered the melodious sound of hammers clinking together in a forge, the weights of which were in certain integer proportions. This observation formed the starting point for experiments and mathematical calculations, which form the basis for the theoretical description of CELLULAR AND MOLECULAR MUSIC 393

12 of musical intervals were made. With the knowledge gained in this way, Pythagoras founded music theory, which is based on twelve main intervals: The whole numbers 6, 8, 9 and 12 correspond to the fourths (number 9), based on the lowest note (number 12), the pure intervals. 5th (number 8) and octave (number 6). In musical notation, these four Pythagorean tones can be expressed, for example, with the tone sequence c 'f' g 'c ". Pythagoras is said to have also experimented with the monochord: The division of a string, similar to the division of a line (see back strap) takes place in Frame of integer ratios. These numerically divided strings generate the different tones or the scale. If a string vibrates, we get the fundamental tone, the tonic; if half of the string vibrates in a ratio of 1: 2, we get the octave. 2: 3 or 3: 4, those who vibrate to those who do not vibrate, is like the aforementioned ratio of wavelengths and frequencies. Whenever there is the same proportion between vibrating and resting string section, the same interval sounds and the theory of intervals was thus derived from spatial relationships, not from temporal relationships, that has been forgotten again and again Music a product of mathematics or is mathematics a product of music? I agree with the theories of Friedrich Kittler, who showed in two voluminous volumes what the relationship between mathematics and music is like. 15 The Pythagorean dream was that the world could be described by integer ratios. The result of this numerical sensitivity 16 is today the computer.The computer marks the beginning of a new age for music. The perfect mathematization of music becomes the model for a perfect mathematization of the world in the digital age. Computer-based music as music produced by a universal instrument, the computer, is purely numerical music, i.e. the completion of the Pythagorean dream. This dream filled the cosmos with the idea of ​​a harmony of spheres or music of the spheres, according to which tones arise during the movements of the heavenly bodies and the transparent balls (spheres) that support them, the height of which depends on their distances and speeds. This was based on the conviction that the cosmos is a whole that is optimally ordered through mathematical proportions and that therefore the same principles can be found in astronomy as in music. Johannes Kepler presented his model of a harmoniously ordered cosmos in Harmonice mundi ("World Harmony") in 1619 and tried to reformulate the idea of ​​spherical harmony within the framework of his knowledge of the movements of the planets at the time. The theory of the music of the spheres was thus a music theory of space. So music was meant to be a space-based art. With the mathematization of music it is paradoxically possible to go back to its origins and from there to work out a completely new conception of music, namely to deepen the already underlying implicit spatial theory of music and to make spatial neighborhoods instead of temporal intervals the dominant method of composition. Robert Fludd, a contemporary of Kepler, drew a cosmic monochord to show the relationship between the harmony of the spheres and the proportions of the string Friedrich Kittler, Musik und Mathematik I. Hellas 1: Aphrodite, Fink, Paderborn, 2006; ders., Music and Mathematics I. Hellas 2: Eros, Fink, Paderborn, cf. "Music as numerical sensibility" in this volume, S Cf. Gareth Loy, Musimathics. The Mathematical Foundations of Music, Vol. 1, The MIT Press, Cambridge / MA, 2006, S Music and Media

13 Music and Harmony Collateral damage to this theoretical deficit is the famous complaint since Theodor Adorno that modern music had cut the tape to the listener as if it had ever been the issue. Since Pythagoras, music theorists have thought of mathematics and not of the channel capacity of the human ear. However, Hegel and Schönberg of all people also referred to the human ear in their music theory. Classical music already suffered from the fact that it appealed to people's short-term memory by overemphasizing rhythm and melody. Recognizing a melody primarily means that the brain can process a sequence of tones, i.e. the listener can predict the next tone himself. So the listener no longer needs a composer. He does the composer's work himself. The question of how do I get from one note to the next is answered by himself, because he already knows the next note in advance. Rhythm, melody, and beat are nothing more than forms of redundancy and a trivial probability. Melodies are nothing more than simple probability calculations. The brain has its own rhythm. A living brain generates electrical signals on its own. The frequencies of these signals can be easily observed on a screen in an intensive care unit, for example. The moment the amplitudes of the signals flatten to a horizontal line, the brain is dead. We have to accept that the brain produces music that we do not hear, that is, its own, "proprius music". The miracle of music arises when signals penetrate the brain from outside and the "proprius frequencies" mix, add, subtract, amplify, etc. with the external frequencies. swing «brings. The superimposition of vibrations (one's own and those of others) creates the »swing«. This phenomenon is commonly referred to as the intoxication of music. The more improbable the superimposition of internal and external signals, the higher the compositional performance and the more difficult it is for the brain to predict the next note. But a good brain longs for signals that it cannot decipher, i.e. for music that cannot be predicted. At best, listening to music means that the brain cannot find a solution to the code that the music provides. It is therefore surprising when theorists and composers complain that the connection to the listener has been cut. This tape was cut again and again. And it is the conservative composers of today who are trying to re-establish this connection. The overemphasis on rhythm, melody and beat has led to masterpieces in classical music with the corresponding complexity. But basically this overemphasis has transferred the music of the 18th and 19th centuries into the program music of the 20th century, into the film music. Not only is classical music the most popular background music for many films, in order to create or reinforce the moods that the picture dictates. No, the Oscar-winning Hollywood composers imitate classical music themselves. Wagner is, so to speak, The Godfather, the godfather of Hollywood music. It is not for nothing that second-class concert composers have become first-class film music composers. It would be worth a chapter of its own to examine this transformation in more detail. 18 Classical music has simplified the question of how I get from one note to the next somewhat. When using the twelve main intervals between the keynote and 18 cf. Theodor W. Adorno and Hanns Eisler, Komposition für den Film, Europ. Verl.-Anst, Hamburg, 1996; Originally published in English under the title Composing for the Film, Oxford University Press, New York, Cellular and Molecular Music 395

For 14 octaves it stayed within the consonances, those are the intervals and chords that are perceived as resting in themselves and not "in need of resolution". Anything beyond that is considered dissonance. Such intervals are particularly dissonant, the frequencies of which have "complicated" numerical relationships, such as the major seventh (15: 8), the minor ninth (32:15) and the minor second (16:15). In his investigation of the twelve main intervals, Bach discovered the phenomenon of consonance and dissonance. From this he established the rule to only stay within the consonances. His solution was the Greek theory of harmony from whole numbers, built up within an arithmetic series. Bach built his theory of harmony on the geometric series, so he indirectly recognized that the piano is about keys or notes as a spatial neighborhood. The result of this theory of composition is the famous "well-tempered piano". Bach recognized intuitively that there is a categorical difference between instrument and notation. If I transfer the vibration field of a string to paper, I can also represent this using non-integer ratios. Bach saw the sequence of tones as a mathematical problem, formulated it as a topological relationship. So that the ratio of 1: 2 between tonic and octave is maintained when all neighboring tones have a constant vibration ratio, it is necessary that this is 1: 12 2. In a sense, Bach destroyed the Pythagorean dream. The well-tempered piano is the triumph of mathematics as music, because it uses not only the whole rational numbers, but also the irrational numbers. The twelve tones can be determined by the geometric proportions and the root can be drawn from them. So Bach almost proposes a topological solution, which Hauer and Schönberg continue. The twelve-tone music of Hauer and Schönberg developed from this equality of all tones by smoothing the arithmetic relationships through geometric models. Arnold Schönberg has advanced beyond consonances into the realm of dissonance. ”[T] he terms consonance and dissonance, which denote an opposite, are wrong. It depends only on the growing ability of the analyzing ear to familiarize itself with the distant overtones and thus to expand the concept of artistic euphony in such a way that there is room for the entire natural phenomenon. What is far away today may be close by tomorrow; all that matters is being able to approach. And the development of music has taken the path that it has incorporated more and more of the possibilities of harmony in tone into the realm of artistic means. «19 However, he kept the number 12 of the twelve main intervals. He concentrated on the twelve tones that arise from the interval theory between the tonic and the octave. Dodecafonia or twelve-tone music sets out a clear rule as to how I get from one tone to the next. According to Arnold Schönberg, this method “consists of the constant and exclusive use of a series of twelve different tones. This means, of course, that [] no tone is repeated within the series and that it uses all twelve tones of the chromatic scale, although in a different order. ”20 Schönberg reduced the problem of the gap between the tones to the question: How do I get from? one tone to the next within twelve tones? That is why his method is also called twelve-tone music. He established the rule that within a series 19 Arnold Schönberg, Harmonielehre, Universal-Edition, Vienna, 1911; quoted from: 3rd edition, Universal Edition, Vienna, 1922, S Arnold Schönberg, "Composing with twelve tones" (1935), in: ders., Stil und Gedanke, Fischer, Frankfurt am Main, 1976, S, here S music and media

15 is not to come back to the same tone. So he actually proposed a graph-theoretic solution, in the sense of the traveling salesman who doesn't want to go through any place a second time before the journey is over. The row composition resulted from the twelve-tone row, from which the serial music later emerged. His contemporary and actual inventor of twelve-tone music, Josef Matthias Hauer, basically invented a more interesting variant of twelve-tone music. He had started to develop his own form of twelve-tone music from his principle of "building block technology". His Nomos op. 19 (1919) is considered to be the very first twelve-tone composition. At the end of 1921, Hauer discovered the 44 tropes (»constellation groups«, »twists«) and in 1926 the twelve-tone »continuum«. Compared to Arnold Schönberg's method, however, Hauer's theories received little attention. Hauer's twelve-tone compositions are like his terms, e.g. B. Tropics, constellation groups etc. already indicate almost topological terms. Hauer already composed and thought in spatial neighborhoods. Music and Mechanics As is so often the case in art, the creators of novel concepts lack the appropriate vocabulary. They try to develop something new within the framework of old reference systems and use historical examples for explanation and justification. You then speak of "the structure of crystals or growing plants" in order to legitimize musical or poetic construction models. Today we know that from Hauer to Pierre Boulez, the actual goal of the musicians was the development of genetic algorithms. It is astonishing how close Schoenberg and Co have come close in their will to formalize mathematical theorems. The results of metamathematics, from Kurt Gödel to Alan Turing, have shown that everything that can be formalized can also be calculated, and that everything that can be calculated can also be mechanized. That is the Church Turing thesis. Similarly, Schönberg recognized that his formalization of the composition through the twelve-tone row actually means a mechanization of the music. That is why he built a twelve-tone turntable and also used a twelve-tone row slider, which told him which tone is next according to his rules. This mechanization and formalization of music opened the door to predictability and thus provided the basis for computer-aided music. Usually the composer wants to keep control of all musical elements. One could call this process "subjective choice". However, the compositional control can be transferred from the person of the composer 1. to other composers, 2. to the interpreter and 3. to objective processes. Methods 1 and 2 are subjective choices to bridge the gap between two tones. Film music composers in Hollywood lead the way, but assistants work the details out and down. Here the composer gives up some of the control voluntarily. But even in the classical compositions with written notations, subjective, imprecise times and deviations by the interpreter are possible. Let's not talk about free jazz, where maximum subjective improvisation is desired. In the second half of the 20th century, composers such as Pousseur, Stockhausen, Brown, Cage, and Boulez voluntarily delegated control of the musical material: either to the performer or to objective procedures such as throwing the dice. The choice of which bridge to build or which path to take to overcome the gap between one note and the next should therefore be delegated to objective processes rather than subjective options. The rules of twelve-tone music, or cellular and molecular music 397

16 serial music represent such objective processes, but also chance operations, because these are independent of the subject, Mauritius Vogt Hufnägel bent into various shapes and thrown on the floor. He interpreted the way in which the nails fell and lay as a score or notation for the music. 21 In 1751, William Hayes squirted ink from a brush onto music paper and then used playing cards to add stems, lines, etc. to it. 22 Random operations have a long tradition in music. However, the computer offers new possibilities for non-subjective processes. 23 The techniques and options of musical dice games have been refined by the computer. According to Gary M. Potter, Wolfgang Amadeus Mozart, Joseph Haydn and Johann Sebastian Bach also used dice game techniques. 24 Mozart had a dice on the surface of which he wrote notes and then threw the dice. A cube has six sides, it is a geometric object and this defines the spatial proximity of the notes. So Mozart composed spatially or topologically with neighborhoods. His guidance. So many waltzes or grinders to compose with two dice [] without even knowing anything about the composition musically was not published until 1793 by Johann Julius Hummel. A random number is generated with two dice. This serves as a row index for a table in which the numbers of the individual measures are contained. The columns are to be selected according to the order of the throws. The bars numbered consecutively on a sheet of music are played in the order given by random numbers and the table. The table in which the spatial neighborhood of the tones is specified is a topological description. In 2007 Götz Dipper transferred this playful, random composition to the interactive sound installation Mozart-Würfel. 25 The aesthetics of classical music were formally so rigorous that it became a mechanical game with which music could be composed without much knowledge or practice. 26 Musical games flourished in the second half of the 18th century. 27 Diedrich Nikolaus Winkel, the inventor of the metronome, has Johann Philipp Kirnbergers 21 Mauritius Vogt, Conclave thesauri magnae artis musicae, in quo tractatur, Labaun, Vetero-Pragae, cf. Karl Gustav Fellerer, Sound and Structure in Western Music, Working Group for Research des Landes Nordrhein-Westfalen, Heft 141, Springer Fachmedien, Wiesbaden, 1967, S William Hayes, The Art of Composing Music by a Method Entirely New: Suited to the Meanest Capacity: Whereby All Difficulties are Removed, and a Person Who Has Made Never so Little Progress Before, May, with Some Small Application, Be Enabled to Excel, J. Lion, London, cf. Herbert Brün, "From Musical Ideas to Computers and Back," in: Harry Lincoln (ed.), The Computer and Music, Cornell University Press, Ithaca / NY, 1970, p; David Cope, Experiments in Musical Intelligence, A-R Editions, Middleton / WI, 1996; ders., Virtual Mozart (Experiments in Musical Intelligence), CD, Centaur Records (CRC 2452), Los Angeles, 1999; ders., Virtual Music. Computer Synthesis of Musical Style, The MIT Press, Cambridge / MA, Garry M. Potter, The Role of Chance in Contemporary Music, Diss., School of Music, Indiana University, Bloomington, 1971, unpublished; quoted from Loy 2006, S The developer Götz Dipper describes the functionality of the Mozart cube as follows: The installation Mozart cube implements the historical composition game in the form of a modern gaming machine in which the notes are placed on virtual "rolls". For each of the sixteen bars there is a role that can be set in motion by the visitors and played back individually or together. In contrast to the historical dice game, visitors can bypass chance here and put together their own versions in a targeted manner. This enables them to explore and understand how the game works. Compare online:, as well as "music as numerical sensitivity" in this volume, p. S, here especially p. 40f. 26 Johann Philipp Kirnberger, The always ready polonoise and minuet composer, Winter, Berlin, cf.Pierre Hoegi, A Tabular System Whereby Any Person without the Least Knowledge of Musick May Compose Ten Thousand Different Minuets in the Most Pleasing and Correct Manner, Welcker s Musick Shop, London, Musik und Medien

17 realized a formal game of dice in a mechanical form, the so-called Componium, one of the first machines to compose automatic music. Rational, non-subjective methods and rules of composition led to mechanical composition machines. 28 Graphical notations and graphical user interface The beginnings of this spatialization of music, both in composition and in notation, are already recognizable around 1950, probably under the influence of an anticipated future of music as acousmatics. In the 1950s and 1960s music tried to find a basis other than interval theory through serial composition methods, stochastics and probability theory. The individual choices between possible solutions with which the gap between the tones could and should be bridged seemed exhausted. Instead of subjective ones, one looked for objective choices. The change in the image of the notation is particularly relevant in this context, namely leaving the five lines and the notes as points. The question of how to get from one note to another was previously understood as the question of how to get from one point in time to another. But now one has gone over to looking for the way from one point in space to another. The points in space become continuous stretches and lines or large spots or thicker or thinner lines. The lines could curve and be smeared over the whole page and the points could also be arranged as mass distributions over the whole page. The composers drew lines criss-cross across the page, added dots and spots as desired, and left the interpreter free to interpret. Under the slogan »emancipation of the interpreter« he became an equal partner of the composer. The meaning of graphic notation, however, was not understood by music theory. The graphic notation was the decisive hint and the decisive turn from music as a tense to music as a spatial form. As the word "graphic notation" suggests, this is about drawings, not notes. This is about graphics, not scores. It's about point and line to surface, as the title of a book by Wassily Kandinsky (1926) is, about visual art and not about musical notation. The triumphant advance of graphic notation lasted for about a decade, but unfortunately it was canceled for two reasons: On the one hand, because music theory did not recognize this spatial turn and understood graphic notation as a contemporary whimsy of the music avant-garde. On the other hand, the musicians themselves did not have the necessary mathematical knowledge to be able to develop graphical notation into graph theory. The musicians found the new home of graphic notation in graph theory. The actual establishment of graphical notation began with the development of the graphical user interface in conjunction with the computer. The tones could thus be generated directly by the composer, without notation and interpreter. 28 Herbert Gerigk says: "The gimmick was in the air in the second half of the 18th century, always clearly marked as such." B. by »kirnberger, who we must first consider as the father of the subsequently swelling literature on cube music []. Every game is ultimately a reflection of true thoughts. The rationalistic age is seriously considering the possibility of mechanical composition in more than one place. «Ders.,» Würfelmusik «, in: Zeitschrift für Musikwissenschaft, Vol. 16, No. 7/8, 1934, S, here S Cellular and molecular Music 399

18 Anestis Logothetis, Odyssey, 1963, two scores The term graph was introduced in 1878 by the mathematician James Joseph Sylvester, based on graphic notations of chemical structures. 29 The first applications were chemical constitution formulas. ' 0 The first textbook on graph theory appeared. Graph theory defines a large number of basic terms: graph, neighborhood, path, cycle, circle, etc. An important application of graph theory is the search for the shortest route between two or more places. The locations are defined as nodes and the connections between locations as edges. In general, a graph is a set of nodes and edges. An edge is a set of exactly two nodes. If the set of nodes is finite, one speaks of a finite graph. If the edges are given by pairs of nodes instead of sets, we speak of directed graphs. Since the solution of graph theoretical problems (e.g. Euler circle problem, postman problem, Hamilton circle problem, traveling salesman problem) is often based on algorithms, graph theory is of great importance in complexity theory. In graph theory, the musicians would have found the new home of graphic notation. It would have been necessary to incorporate graph theory into music theory, then today we would have a music theory that would be adequate for the computer. In the musical interpretations of the graph schemes, the locations would be notes and the edges would be vectors etc. 29 James Joseph Sylvester, Chemistry and Algebra <<, in: Nature, Jg, Arthur Cayley, >> On the Mathematical Theory of Isomers <<, in: Philosophical Magazine, vol. 47, no. 314, 1874, Caylay is also known for coining the term nßaum "for mathematical graphs, cf. ders." On the theory of the analytical forms called trees ", in: Philosophical Magazine, Vol. 13 (Fourth Series), No. Bs, 1857, oenes König, theory of finite and infinite graphs. Combinatorial topology of the Strechenhomplexe, Chelsea, New York, MUSIC AND MEDIA

19 Music and Graph Theory There is one branch of mathematics that has inherited geometry: topology. One speaks of topological structures. Of course, these go far beyond the visual space of geometry. The famous mathematician Leonhard Euler introduced topology in 1736 with the Königsberg bridge problem. Four bridges over the river Pregel, which flows through the city of Königsberg, connect the two banks with the island A and two bridges with the island B and one the two islands A and B. The question was whether one can find a tour of from any point of view where you only cross each of the bridges once. Euler proved that such a path does not exist, since an uneven number of bridges lead to all four bank areas or islands. In abstract terms, Euler's question is: Can you find a tour where you cross each of the seven bridges exactly once from any point of view (A, B, C, D) and then return to the starting point? These four positions form the four nodes of a graph, which we distinguish by the numbers 1, 2, 3 and 4. We connect two nodes when a bridge connects the positions. This creates the following graph. Due to their arrangement, the edges form different graphs. A graph is called connected if there is at least one path between every two nodes, such as B. in the graph above. Then there are open and closed edges. Let us assume that the edges of a graph model a system of streets and imagine a postman, then we know that he wants to walk each route only once, if possible. This is called the traveling salesman problem or round trip problem. A traveling salesman who has 35 cities to visit clearly wants to avoid taking the same route several times. He will therefore think about how he can visit 35 cities without having to walk twice. This topological problem is part of graph theory: Is there a cycle that uses all edges exactly once? We observe the similarity between the prohibition of the repetition of a note in the twelve-tone row and the avoidance of the repetition of a path in the topology. Cellular and Molecular Music 401

20 I don't have to solve the question of how to get from one note to the next using harmony, I can also solve it using graph theory. Electrical networks can also be assigned to graphs. The possible currents in the network can be determined with the aid of the graph. The following nodes appear: the voltage sources, the grounding points and the network points at which a current branch can take place. The figure on the left shows an electrical network, the figure on the right shows its directional graph. The currents in the individual branches of the network obey Kirchhoff's rules, which are expressed by linear equations. Gustav R. Kirchhoff showed in 1847 that all currents in an electrical network can be determined as soon as one knows the currents in the branches of an exciting tree. 32 "Tree" is a graph theoretic term. Trees appear as data structures in the optimization of communication networks, in grammars for computer languages, etc. We know that electrical networks, i.e. graphs, can be realized physically. So it stands to reason that computers through which currents flow should not compose according to old analog rules, but according to Kirchhoff's and other rules and methods of graph theory. The round trip problem applies today not only to the security personnel of an industrial plant or salesmen, but also to robot arms when processing a workpiece or for the positions of drill holes on a circuit board (computer chip). Graphs also play a role in chemistry. A molecule is characterized by its structural formula, which indicates how many atoms of which type a molecule consists of. H 2 O consists of two atoms of hydrogen and one atom of oxygen. The binding possibilities can be represented by an undirected graph, which we call a structure graph. 32 Gustav R. Kirchhoff, "On the resolution of equations to which one is led in the investigation of the linear distribution of galvanic currents", in: Annalen der Physik, vol. 148, no. 72, 1847, S, here S music and media

21 The left figure shows the positions of drill holes on a printed circuit board (computer chip). The illustration on the right shows the shortest route that connects the boreholes with one another. This order is obtained as a solution to the associated round trip problem. It indicates how a drill should operate in an optimal way. In this figure we can see the structure graph of dioxin. Such graphs have symmetries, automorphisms and permutations. The graph theory seeks solutions for spatial neighborhoods. The application of graph theory for electronic networks for the production of music would therefore be obvious. So graph theory is a more up-to-date mathematical method to answer the question of how do I get from one note to the next. Music and Algebra There are other mathematical models such as the Clifford algebra, which could be used as spatial composition techniques. The Clifford algebra has several origins, on the one hand in the linear expansion theory (1844) by Hermann Graßmann. The aim of Graßmann's algebra was to generalize the concept of vector to higher dimensions in order to describe oriented surface elements, volumes, etc. So it is a kind of multi-dimensional geometry. Another source is the algebra of quaternions, discovered by Sir William Rowan Hamilton in 1843. The quaternions form a four-dimensional generalization of the complex numbers. Quaternions are suitable for describing rotations in three and four dimensions. William Kingdon Clifford discovered the Clifford algebra named after him, which he himself called "geometric algebra". 33 Clifford wanted to use it to unite Graßmann's algebra and Hamilton's quaternions in a common mathematical framework. In mathematics there are number systems with complex units, more precisely the complex numbers, the quaternions and octaves. One, three or seven elements can be fixed in each of these, which span the number space as a real vector space that can be multi-dimensional. The geometric algebra is based on a vector space that is provided with a metric. A geometric vector is a directed line that is characterized by a contribution or a direction. 33 William Kingdon Clifford, Mathematical Fragments, Being Facsimiles of His Unfinished Papers Relating to the Theory of Graphs, Macmillan, London, 1881; ders., Mathematical Papers, Macmillan, London, Cellular and Molecular Music 403

22 A vector is thus an oriented element of a one-dimensional space. But in geometry we need not only vectors (lines) but also other objects such as surfaces, volumes, etc. These can be understood as a generalization of the vector concept or as multi-vectors. The geometric algebra is thus a Clifford algebra over a vector space with a square shape. The vector space is real two-dimensional, the algebra real four-dimensional. The Clifford algebra is the initial object of a category, which is characterized by the fact that there is a morphism for every other object in this category. This gives rise to algebra morphisms. The Clifford algebra spans a vector space that goes far beyond the natural numbers. From this one could conclude that the rules that were previously applied to numbers and operations with numbers that were only written on a two-dimensional surface no longer apply if they are presented in multi-dimensional spaces. To put it simply: a normal number system is written down on a piece of paper, on a two-dimensional surface. The number space that is metaphorically spoken of is in reality no space at all in normal arithmetic operations, but just a surface. Therefore, the normal mathematical number systems are not sufficient for physical processes with higher dimensions or with n-structures. Math problems, e.g. B. the calculation of n-dimensional quantum states can only be solved as algebra. In this respect, the Clifford algebra enables the currently perfect description of quantum mechanics. Further research inspires the hope that we will develop a number space that is multidimensional to n-dimensional and that in this number space the previous notation systems such as digits, notes and letters will break away from the two-dimensional surface. The computer will make it possible to operate in a multidimensional number space and thereby create new constellations, connections, paths and bridges between notes, letters and numbers. The application of Clifford's algebra to music by means of the computer would be the first example of this hope, the beginning of a civilizational revolution: the transition from two-dimensional notation systems to multi-dimensional notation and composition systems. Roman Haubenstock-Ramati defined the musical form as "a temporal-spatial conception of musical events" in the form of New Music. 34 »Form is not invented by the composer, but replaced by a technically analyzable structure« 35 Movement, variation, repetition in the sense of Alexander Calder's mobiles became central parameters for this generation of composers. The »open form«, the catchphrase Umberto Ecos (Opera aperta, 1962) was practiced in all dimensions. Roman Haubenstock-Ramati describes his work Jeux for 6 percussionists (1960) as a vector field: “In Jeux 6, for example, the structure of the work is arranged horizontally in six and vertically in ten rows. In the case of the Jeux, however, not only a double: horizontal and vertical X-ray examination of the structure is possible, but a six-fold, pointing in six different directions (initial arrows). The whole form is closed, i.e. after completion of the first cycle the second cycle begins at the same point, etc. In this way the same overall structure (always the same) provides six different possibilities of ordering the same in ever new contexts, new perspectives ( always different). «36 34 Roman Haubenstock-Ramati, ot, in: Form in der Neue Musik, published in the series Darmstadt Contributions to New Music, ed. by Ernst Thomas, No. 10, 1966, S, here S Ibid. 36 Ibid., S Music and Media

23 Roman Haubenstock-Ramati, Vektorfeld zu Jeux for 6 percussionists, 1960 Xenakis will intuitively approach Clifford algebra without mentioning it. His music will not only operate in a two-dimensional vector space as in Haubenstock-Ramati, but in a vector space with a square shape. He will work with quaternions, which describe notations as rotations of words in three and four dimensions. The representatives of graphic notation from Roman Haubenstock-Ramati to Earle Brown sensed that they were on the trail of a new method of composition, namely the change from time-based to space-based methods of music composition. In the same booklet on Forms in New Music, for which Mauricio Kagel and György Ligeti also wrote, Earle Brown comments on the influence that Calder and Pollock have on his work, as does Karl Gerstner. He admits that his compositions do not correspond to the general idea of ​​how a composer works. He doesn't mind being called "a designer of programs" 37. Before actually composing with computer programs, the avant-garde composers had already intuitively understood their composition methods as programming.Brown refers explicitly to Karl Gerstner's book Program drafting 38: “Designing a Program is similar to the procedures I have described above and somewhat unlike the traditional concept of composing in that (as a negative attitude would say,) the final step of Definitive arrangement is left out [] This is a process of inclusion and expansion of the concept of a work of art rather than one / of deterministic contraction and exclusivity. One does not diminish the amount of meaningful control within a work but seeks to create the work as an entity, a quasi-organism, and to program a life for it within which it comes to find its shape []. «39 Brown already sees the future of music as the programming of life-like »quasi-organisms«, so to speak the generation of music by cellular automata, such as by Conway's Game of Life. But most of all it is astonishing how little Brown uses the term time and how often he uses the term space when he talks about his revolutionary compositions from 1952: “One of the first things that I ever wrote about form is the following, from notebooks, October and November, 1952; under the word, Synergy: to have elements exist in space space as an infinitude of directions from an infinitude of points in space to work (compositionally and in performance) to 37 Earle Brown, ot, in: Thomas 1966, S, hier S Karl Gerstner, Programs draft, Teufen, Niggli, Earle Brown, ot, in: Thomas 1966, S, here p. 63. Cellular and molecular music 405

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