YLIPAA ART

Sibelius Academy / Centre for Music and Technology

When discussing composition using computer, it usually implies paragimata of computer assisted composition and computer generation based on rigid mathematical formulae. What hasn’t been discussed so far…

is composition or performance made by computers, taking in consideration the computer’s ability – or inability – to calculate in time. This possibly nondeterministic character of computing can be of importance, leading possibly to variable, creativity-like results.

Turing machines, such as computers, are considered finite-state machines, which produce the same result with every run. By software, we can obtain some characteristics of nonlinearity, which seems to reveal variety using exactly the same algorithm. Computer generations based on deterministic methods usually require many runs to obtain aesthetically and intellectually pleasing results after tweaking the initial conditions, but software simulated, nondeterministic, dynamically complex applications’ musical results rely in pure luck, which implies even more randomness than using pseudorandom operations alone. This is a bold statement, but it gives a turing-obsessed machine a chance to have its unique, artistic moment in a universe.

The liberation – or destruction – of the tonal universe started at the beginning of the 20th century. Soon after that, the second school of Vienna sought importance in liberating all the other parameters of music. Stockhausen, Ligeti, Schaeffer and many others followed the concrete paved path and formed a new paradigm for music which culminated in the works of Xenakis, a Greek mathematician and an architect. After formalizing music as a multidimensional space-time construct, many formalists has followed his ideas and intentions. This may be seen as the inevitable historical development concurrent with that of the natural sciences, mathematics and arts, that is, human expression.

Soon came along the spectralists of the 1970’s. Amazed by the possibilities of computing, they discovered a beautifully diverse microcosmos inside a single tone. Through computer-aided visualization of the Fourier transform, observed overtones with differing gain factors and individual gain envelopes led to a new paradigm of music composition. The overtones had certainly been recognized by philosophers and physical scientists much long before modern age composers got their eyes towards on an oscilloscope. Nevertheless, the aesthetic application of technologically observed physical quantities became of interest among European and American composition schools.

Being influenced by all the above-mentioned concepts, as a modern composer, temptation to amalgamate these major abstract compositional concepts of the 20th century together with the computational approaches is far greater than this survey can address. Nevertheless, that is the intention of this article. I intend to describe the fundamental principles of my attitude, methods and applications towards composition with the aid of a defining tool of our time, a computer. I will base this survey of recent composition methods by computer on my composition called Prime Numbers. The fundamental idea of the composition is to use serial methods over those of spectralism, coordinated entirely by computer in a nondeterministic manner. This is the basis for Prime Numbers to emerge as a complete continuous computer generation.

SERIALISM AND SO ON

As it was spectralists’ duty to break the conventions of the history, let us first give serialists their turn. At the beginning of the 20th century, by eradicating the importance of hierarchy of pitch organization, for a while, other parameters of music came to the foreground. Some would say exactly the opposite, but by using a strict gamut for pitch space, dodecaphonic technique reduced the amount of decision-making when composing with pitch. Some even preached about the democratization of pitch space – and just the opposite happened. Freed from pitch-centeredness, composers could express their impressionistic ideas using all the other parameters, exploring the sounding world with a greater diversity. (Perle 1991, 2) (Loy 2006, 312)

Another cycle of control emerged soon after the First World War. Experiments in serial control of other musical parameters expanded composers’ organizational concepts well beyond dodecaphony. Soon, total serialization of musical parameters became a collective mantra, giving rise not only to a new kind of music, but also to new analysis techniques: serial devices were found being used by many notable composers of earlier time, such as Beethoven and Mozart (Keller 1955, ?). As dodecaphonic music tried to get rid of tonality, Darmstadt school of serialism rejected thematic implications of twelve tone rows (Felder 1977, 92). They were mainly interested in the combinations of certain basic operations imposed on the proportions of some numeric device or a set. This trend led towards the use of electronics, for it was more easier to deal with precise numeric abstractions, ratios and their sonification using machines.

The sonic image of totally serial music has been described as ”punctual music”, emphasizing the ascetic, structural feeling the kind of music has. (Stockhausen 1998, 452) Described also as a celestial and galactic music, this music has an understandable analogue – mainly for historical reasons – in the concurrent cycles of celestial objects in a multiscalar system.

Where was it’s fascination, was also it’s treacherousness. Being hypnotized by their discovery, some die Reihe ”composers claimed that the sound programs which they presented to the public exhibited an achievement like that of a mathematical theorem.” (Flynt 1993) In his critique towards structure art in general, Henry Flynt defines the ”pieces” more as a combinatorial sculptures than music. In technical terms, ”The trouble taken by the composer with series and their permutations has been in vain; in the end it is the statistical distribution that determines the composition.” (Koenig 1970) On the other hand, I will state and stress, that the statistics were discovered because some phenomena were too complex to be reduced to the interactions of their parts – unless proven otherwise. Nevertheless, they are still of use as an analytical tool or as a model.

EMERGENCE OF THE SPECTRAL SCHOOL

The basis of spectral music is the physical structure inherent in a tone. This structure – investigated by the means of a sonogram – indicates parameter values, which are then assigned to the compositional space. Direct mappings typically include pitch, relative dynamics and event duration. Depending on the composition at hand, orchestration and variety of transformations are also techniques frequently used, influenced by the analysis. (Rose 1996) (Cornicello 2000) (Moscovich 1997) According to a recent source, the term ‘spectral music’ can be used to describe any music in which timbre is treated ”as an important element of structure or musical language.” (Reigle 2008, 1)

Fig. 1

The most prominent feature of spectral music is its pitch content, derived from the relative frequencies of overtones. (Fig. 1) Harmonic spectrum is defined as having integer relation between the overtones and the fundamental frequency, whereas inharmonic spectrum has fractional relation. Frequencies of the harmonic spectrum fuses well together and they are perceived as a single uniform sound object, giving arise to the concept of consonance, literally ”sounding well together”. Dissonance is considered in the case of inharmonic spectrum, where it is more easy to pick up discreet component frequencies. Hence, dissonant cycles align more seldom than consonant cycles. (Loy 2006) (Cowell 1930)

Overtone ratios give rise to the concept of interval. For historical reasons, each simple ratio has a unique name – and for historical reasons those names seem to apply only to pitch. More generally, ”interval” can be understood broadly as a measure of distance in any musical parameter. In the beginning of the 20th century, Henry Cowell investigated the application of harmonic intervals to the temporal domain, which will be dealt in the next chapter.

Despite the harmonic series being a physical phenomenon, some critics have been made about the claimed ”naturality” of spectral music processes. ”Is the relationship between sound structure and musical structure in early spectral music mimetic or metaphoric?,” asked Bayar in a recent spectral music conference publication. After investigating early spectral music pieces, he concludes composers to have ”failed to transform the acoustic properties of sound sources into communicable musical properties.” (Bayar 2008) It seems that another cycle of totalitaristic formalism is indeed required in order to approach ”naturality” asymptotically from a structuralist point of view. I’ve tried my best to formalize a rigid, self-contained structure based on the overtone ratios.

Despite the numerical basis of both dodecaphonic and spectral music, mapping of the integer series to the pitch space can be seen as the basis for their classification. Integers mapped to the root of square root of two result in semitonal chroma, whereas mapping to the multiples of the fundamental frequency result in uneven pitch spacing: the overtone series. Perle mentions the assumed justification of dodecaphonics: “The attempts of Schoenberg and others to derive the twelve notes from the ‘overtone series’ are so farfetched and self-contradictory that they hardly require discussion.” (Perle 1991) The history of tuning systems might reveal the verification to their argument, but more interesting is the fact, that a full cycle has been made from Pythagorean proportionality, thanks to the spectralism!

TEMPORAL ANALOGY OF THE SPECTRUM

Largely unknown – and largely underrated – art theorist Joseph Schillinger states that “the types of series upon which a certain continuity is based determines the potential forms of development and growth of such continuity.” Then he represents possible number series, by which groupings (factorial continuity) and divisions (fractional continuity) of the basic pulse stream could be accomplished: natural integer series, arithmetical progression series, geometric progression series, summation series, series of natural differences and series of prime numbers. (Schillinger 1976, 85- 107) Cowell also describes two methods for deriving other layers from the basic pulse stream: respectively by grouping and dividing.

The structures can be categorized based on the method of their generation. (Ylipää 2012, 6) Grouping structure (Fig. 2) is arrived at by dividing a fundamental frequency of the basic pulse stream by the integer value of the index of the partial in the harmonic series. In this way, the interpretive layers are always slower than the basic pulse stream. Dividing structure (Fig. 3) is achieved by similar method, but instead of division, multiplication is used. These structures were also described by Boulez, but his description of the methods used for generating the structures are confusing (Boulez 1971, 52).

Fig. 2

Fig. 3

These two structures have a unique character. Grouping structure shares a common, basic pulse layer and sounds quantized. (Cowell 1930) Because of the constant time unit common to all the layers, music theorist Harald Krebs finds the structure to have a direct implication in metre. He describes several metrical concepts, such as interpretive layers, metrical consonance, metrical dissonance and hypermeter (Krebs 1999). This chromatic rhythmic structure is found also in much of the total serialist’s work. For example Stockhausen became anxious to avoid the ”uniform pulsation, which arises inevitably from superimposing of lines in chromatic durations.” (Griffiths, ?)

Dividing structure is dominated by stacked polyrhythms, each fitting inside a single beat of a basic pulse stream. It is striking that the construct resembles the modes of a vibrating string. According to Cowell, these two are completely analogous. ”The vibration lengths may thus be thought of as making a sort of pattern, in which the units start at the same instant, separate, and reassemble at a point a fixed distance away; and this they continue to do as long as the tones are sounded together.” (Cowell 1930, 47)

As has been shown, rhythmically complex constructs result when applying simple integer ratios to the generation of rhythm. In the history of music, intervals have been mainly considered as a vertical phenomenon. The reason for this might be that our perception of pitch differs from the temporal aspects of a sound: pitch is perceived as a single object. Nevertheless, in the case of a division method, a construct similar to the nodes of a vibrating string results. When this construct is sped up far enough, we will perceive it as a single pitch. Assuming the scalability in both directions, therefore, it is reasonable to present the construct as the sound’s equivalent in the temporal domain. As Cowell phrases it, ”a parallel can be drawn between the ratio of rhythmical beats and the ratio of musical tones by virtue of the common mathematical basis of both musical time and musical tone.” (Cowell 1930, 50-51)

The development of the compositional devices through history can be seen as the increased tolerance towards dissonant features in music (Cowell 1930). At the same time the systematic search for equal-temperament influenced current musical aesthetics, which culminated in the destruction of the hierarchical tonal system in the form of dodecaphony. The contrast of spectralism to the concurrent serialistic techniques is obvious: the former implies hierarchical structures with a sense of gravity, whereas the latter highlighted chromatic, equal division of the parameter space. One might ask, naturally, how might music sound like, which is designed using approaches of both equal and hierarchical organization of both space and time?

TOWARDS INVARIANT STRUCTURE

The isomorphism between horizontal and vertical dimension led to a construct presented by American spectralist James Tenney. In his purely mathematical piece Spectral Canon for Conlon Nancarrow, Tenney projects isochronous pulse streams over the first 24 harmonics, each harmonic pulsating at a different speed. Two fundamental rhythmic structures are eminent (Fig. 4 & 5): the first one builds up from the beginning, and the second one ends the piece. When describing the composition, Larry Polanski sees them being ”more as a fact of nature than as a composed piece” (Polansky 1983, 225).

When constructing a formal world within the software, one is overwhelmed by all the possible dimensions and connections between its parts. It is tempting to use plethora of random or statistical operations in order to control every single parameter in order to obtain variety. What is usually missed, is the unity.

Efforts for unity in serial and spectral composition techniques are obvious, thus different. The tone row is the basis for serial constructions: other musical elements receive their values based on the transformations and transpositions of the row, thus constructing consequently the form. (Boulez 1971, 45) Spectrum of a tone forms the basis in spectralism: ”The shape of a work, then, might be generated by the protracted evolution of a single sound, thereby making it possible for spectral processes to influence the musical elements of harmony, melody, rhythm, orchestration, and form.” (Cornicello 2000, 2-3) Former method relies upon imagination of the tone row and its transformations, whereas the latter is informed by the measurement of physical quantities. To what extent is the basis followed rigidly is left for the composer’s judgment.

Fig. 4

Fig. 5

In the case of my composition Prime Numbers, the invariance of the musical structure is based on the primal series (hence, I use the term primal, for not to confuse with the dodecaphonic term prime), which is constant throughout the piece. Ten first prime numbers – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 – were chosen for their property not to align so often. This series is projected onto the ten members of each parameter: it designates the pitch structure, the relative tempi of pulse layers, velocity of each partial, duration of note event, every ADSR envelope component, and the transformation times between consecutive parameter values. The process of transformation can be heard directly in the sine wave part as a glissandi as the program interpolates between the given two frequencies.

When the overall form is concerned, the same relative intervals based on the primal series are projected onto the metaparameters – namely, two values controlling the density of a particular parameter – with a slight randomness in their allocation, which is necessary for avoiding the cyclic form. A unique random number series within each metaparameter designation cycle and the ratios based on the primal series are the essence of the structural pacing. This method creates plateaus of slow movements in contrast with fast structural events, which are the inevitable result of the hierarchic nature of the harmonic series. Observing the MIDI representation of the composition reveals the nature of the pacing as well as the recursion of each musical situation within a parameter space. (Fig. 8)

When calculating the member values of a particular parameter, few set operations are required for aesthetical reasons. Within a parameter, the member set can be formed by multiplying or dividing a metaparameter value with the primal series. The result is either grouping or dividing structure. Furthermore, the set can also be reversed. These operations enable parameter spaces to interconnect more sophistically. For example reversing the velocity set (rightmost image) in relation to the pitch set (the left one), the lowest pitch became also the loudest, and so on. (Fig. 6) By experimenting with set operations a pleasing combination can be obtained for the structure for a composition. (Fig. 7)

Fig. 6

CONNECTING FOR DYNAMICALLY COMPLEX BEHAVIOUR

Based on the notion of self-similarity, unity in plurality – and vice versa, each parameter is allowed to exhibit variety. On the notion of unity, each member within a musical parameter can have exactly the same value, or the values based on the primal series, or anything between. These two are the extremes of the possible values for any member in a particular parameter. Anything between is possible (within the limits of 32-bit floating point values), but worlds beyond that do not exist; these are the internal constraints of the system. For example, when concerning pitch, the spectrum at its maximum refers to just intonation, slowly compressing the ambitus of the pitch structure nearer to unity, where subharmonic beating occurs, going all the way down to a common frequency. Boulez and Cowell theoretize about possible scenarios of combinations of parameter spaces. In Boulez’s classification on the density of generation, my implementation comprises the first two situations: fixed, mobile and non-evolutionary, leaving the third category mobile and evolutionary out. (Boulez 1971, 53-54) The result of the parameter space treatment can be exemplified as follows: ”Crescendo and diminuendo in dynamics is the same in many ways as sliding tone in pitch; and undoubtedly a definite relation could be found between a certain curve of changing pitch; and a similar curve in a gradual reducing or increasing of loudness or softness.” (Cowell 1930, 83) Boulez continues: “Compare, for example, a succession of diverse timbres upon the same pitch and, conversely, a succession of diverse pitches linked by a single timbre, that is to say, interchange the two organiations so as to reverse their specific characters: uniquity and multiplicity.” (Boulez 1971, 39)

Complex behaviour emerges when all the parameters are involved in the forming of a structure. Temporal envelope values are interconnected to fit to the length of a note event, which is in turn in relation to the interonset interval. Interpolation between any consecutive values can happen almost

Fig. 7

instantaneously (> 80 ms) or during several seconds (all happening in near real time). This creates the impression of naturality: the accumulation of continuous interaction. (Senge 1990, 71-72) “Spectral composers do not make use of functional harmonic progressions the way tonal composers do; instead, the harmonies are often metamorphosed from one to the next.” Cornicello (2000) More importantly, real time interpolation is partly affecting the complex dynamical behaviour of the whole structure. Considering the computer’s ability to calculate intermediate points in real time as fast as possible, some ruptures are likely to happen due to the stress to the processor. This might be seen as a programming error, or as a personal feature of a particular computing system capable of variable computing performance. It stresses the uniqueness of each computational system and acts as an external constrain in complex dynamical systems (Loy 2006, 306).

The composition is a unisono of a piano and a tape part, so the complex behaviour is more detailed in the spectromorphological properties of synthesized sounds. The MIDI representation shows only the quaternary nature of sound: pitch, interonset interval, duration and velocity. (Fig. 8) Complex systems are characterized partly by a nonlinear feedback path. In this system, a practical approach obtaining complexity is the feedback of note events to the large-scale structure: every onset of a note is set to denote a value for a metaparameter in turn. First, values of the primal series are being scaled to fit inside the world limited by the user. These scaled values are being rotated in an ascending order from metaparameter to another in a randomized way: according to the

Fig. 8

dodecaphonic theory, each metaparameter must have been assigned a value before another randomized cycle begins. If the order of allocation is similar with every run, it will manifest a cyclical form, which is something I try to avoid for aesthetical reasons.

AESTHETIC CONSIDERATIONS

When combining just intonation of the sinusoids with equal-temperedness of the piano, interesting sonic clashes result. By placing the speakers near the resonant body of a player piano, sympathetic vibrations with the piano strings enhance the duality of these two different tuning systems, which is of theoretical and aesthetical interest.

Decision for using only sinusoidal oscillators as the sound source is justified by the attempt to draw attention to the structure itself, which is also formed by simple but slower oscillations. Other reason is the direct mapping of the parameter space to the piano. By playing in unison, the sinusoids are extending the sound envelope of a piano, creating textural and structural variety. In some cases, the glissandos of the sinusoids are predicting the upcoming movement, helping a listener to orient to becoming changes.

Ordinary listener’s ability to adapt to the composition is made comfortable because of the tonal-like functions embedded in the harmonic series. Harmonicity functions as a tonic and inharmonicity as other tonal functions. Primal, prime series acts as a thematic anchor point and the compression of the overtone structure functions as a development section or transformation feature. Cornicello confirms, that “The harmonic motion in spectral music has the potential to replicate the tensionrelease paradigm that has been the basis for Western music for centuries.” (Cornicello 2000, 3)

In some places, the repetitive nature of the pulse streams as the main rhythmical feature may become naive and irritating, but at the same time it is essential to reveal the structure behind the premeditated abstraction. On the other hand, some listeners have mentioned a notion of humour, acuteness and meditative feeling, mainly because of the temporal characteristics of the generation. The absence of common metrical unit creates the impression of natural flow, which may blur the distinction between a composition and a performance. Is the system the composition itself, from which computer performed a small part, or did the computer compose a piece based on the internal constraints which was perfected when designing the system? Is it necessary to make a distinction between the two?

Technically, too many interconnections are eminent in the piece to make a complete theory based on the initial abstraction that I came up with. In this piece, each parameter is individually controlled and no cross-dependencies have been made, which could be of interest in future studies.

RESULTS

Perceptual qualities of musical cognition could be used as a model for scaling the extremes of the parameters, but this is out of the scope of this article. Artistically pleasing results can be archieved by tuning the system by ear, implying good taste and reactivity to minor details as well as theirs effect to the whole structure. Depending on the primal series, the progression of the movement is highly dependent on its intervallic structure, influencing first on the spacing of the note events and consequently on the progression of the form. The final judgement for me was mainly influenced by the development of the form. A rigid form can compensate some minor disruptions in the scale of the note events.

The compositional method for creating the Prime Numbers was pretty straightforward – if the iterative process of programming over the years is not taken into account. All that was necessary, was the designation of the fixed primal number series, in this case the ten first prime numbers, then setting the limits of the parameter space, setting up recording equipment and hitting play. When it seemed that the piece ended, I stopped the process and there it was. The result was evaluated and possibly accepted as a piece of art. What is ironic, is that it is a product of partly aesthetic sensitivity, partly some notion of a fit abstract structure and partly of pure luck. In the case of Prime Numbers, the generation felt instantly as a whole, with its variations, unities, themes and climaxes. As an artist, it is my duty and privilege to follow the beauty, create worlds with intellect, and to recreate and mediate an experience, let it be abstract, concrete, intellectual or emotional. Art defies the motivation to proof anything: it can merely reveal something, which is tightly and merely in connection with the things one ingest. Art theorist Joseph Shillinger stated, that theories of art should be generative instead of analytical. (Schillinger 1976) There might be a personal bias to the statement, but it should not be bypassed with a shrug. Both approaches are fertile, but for a different reasons.

There is a great distinction between music based on axioms and music based on the ”simulation” of natural phenomena. The former implies formulation based on an analytical insight of previously observed instances, whereas the latter relies on the current insight of the perceived state of zeitgeist: the interplay between the current state of the arts and the sciences. Conscious or not, an artist is supposed to reflect the duality of the both, even though there is the risk of inventing the wheel again. Henry Flynt states, that ”There is no objective cognitive accomplishment in being a Bach initiate.” when referring to the efforts of structure art. (Flynt, 1993) As a counterargument, the current state of science and art can still make a difference if it is assumed that life evolves. At least it is healthy practice to revisit universal topics every now and then.

During the recorded history of humanity many scholars have stated that music imitates the nature. Information theory underlines, that ”… we must have a way to correlate past and future choices to the present before random choice techniques are of use in those musical styles that manipulate listener expectation.” (Loy 2006, 343) I would state, that simple integer ratios can have a cascading effect in a dynamically complex system, revealing the satisfying nature of the simplicity itself. What if nature uses simple integer ratios to effectively organize, store and retrieve data based on the necessity of temporal change, in which we as human beings are confined? In my opinion this resembles the famous problem of computation first postulated in the 1950’s, which conjectures the processes in polynomial time being equal to the processes in nondeterministic polynomial time, better known as the P=NP problem. At the moment, it is the most important open question in theoretical computer science, which in the end might be left without a final answer. Basis for the last argument is proved by Kurt Gödel in his incompleteness theorem.

”Methodological criticism, information theory, psychoacoustics, complexity theory, and other approaches discussed in this chapter are making important contributions to theoretical aesthetics that finally allow the dialogue about the nature of art to move beyond its fixation with Pythagorean proportionality. Perhaps the truest proportions in music are those that relate expectation, interest, entropy, and redundancy; perhaps the truest study of music structure requires understanding the nonlinearities of our perceptual and nervous systems as well as the self-organizing principles of nature.” (Loy 2006, 407)

Am I the last dinosaur of Pythagorean approach, even though combining ancient theories of aesthetics with the modern ones? Should one discard completely Pythagorean rational approach if previous endeavours have been doomed by the public to been in vain, or unsatisfying? Loy’s comprehensive book Musimathics Vol. I concludes to the chapter ”Calculating Beauty” in a statement, that an artists’ intentions allow us to ”apprehend the deeper significance of their art.” (Loy 2006, 407) Moreover, what is the effect of personality attributes to the decisions individuals make in order to map a particular subject from the scientific realm to the musical domain – and in the end, for sharing with others? How much of this process is subconscious? By the use of generative methods, it is indeed possible to reduce the amount of subconscious decisions made by an individual because of the models used, if the choice of the premises are not taken into account. Overall, rigid abstract structures have the potential to bring something new into the consciousness of human beings, being ”natural” or not.

BIBLIOGRAPHY AND REFERENCES

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