Numerologinių Technikų Inovacijos XX A. Antrosios Pusės Muzikos

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Numerologinių Technikų Inovacijos XX A. Antrosios Pusės Muzikos 3. Rima POvilionienė Lietuvos muzikologija, t. 9, 2008 Rima POVILIONIENË Rima Povilionienė Numerologinių technikų inovacijos XX a. antrosios pusės muzikos kompozicijoje Innovative Digital Technologies in the Musical Composition of the Second Half of the 20th Century Anotacija XX a. kompozicijoje labai išplečiamos matematikos raiškos muzikoje ribos – tai ne tik ankstesnių epochų numerologinės komponavimo praktikos re-evoliucija, bet ir tik šiam laikotarpiui būdingi matematiško muzikos kūrimo būdai. XX a. muzikos komponavimo procesų naujovė – ypač ekspansyvus šiuolaikinių matematikos teorijų, pavyzdžiui, kompleksinių matematinių modelių, tapusių kompozicijos for- maliosios gramatikos forma, taikymas – į muzikos garsų erdvę integruojamos kompiuterinės grafikos, kompiuteriu generuojamų algoritmų implikacijos. Straipsnyje pristatomi matematiški XX a. antrosios pusės muzikos kompozicijos pavyzdžiai; aiškinamasi, kaip konkrečios matematinės procedūros diegiamos į muzikos tekstus. Tai atskleidžiama šio laikotarpio muzikos kompozicijų analizėmis. Reikšminiai žodžiai: muzika ir matematika, šiuolaikinės matematikos elementai, muzikos komponavimo algoritmai, L-sistema, fraktalų, chaoso teorijos, John Adams, Charles Dodge, Tom Johnson, Šarūnas Nakas, Vytautas V. Jurgutis. Abstract In 20th composition the boundaries of mathematical expression in music were notably expanded. It is not only the re-evolution of nu- merological composition practice of the previous periods, but also the way of mathematical composition of this period. The innovation in the processes of 20th-century music composition is particularly expansive application of contemporary mathematical theories, for instance, the application of complex mathematical models, which have become a form of formal grammar of composition: integrated computer graphics, and the implications of computer-generated algorithms in the space of music sounds. The article presents instances of mathematical writing of musical compositions in the second half of the 20th century. It also attempts to find out in what forms concrete mathematical procedures are implemented in musical texts. This is disclosed by analysis of musical compositions of this period. Keywords: music and mathematics, elements of modern mathematics, music composition algorithms, L-system, theories of fractals and chaos, John Adams, Charles Dodge, Tom Johnson, Šarūnas Nakas, Vytautas V. Jurgutis. Įvadas komponavimo procesų naujovė – ir nepaprastai ekspansy- vus šiuolaikinių matematikos teorijų taikymas, pavyzdžiui, Matematikos įtaka muzikai – ilgaamžiškiausias tarp- kompleksinių matematinių modelių taikymas – integruo- dalykinis dialogas, koncentruotu pavidalu pasireiškęs jamos kompiuterinės grafikos, kompiuteriu generuojamų XX a. muzikos komponavimo praktikoje. Remdamiesi algoritmų implikacijos į muzikos garsų erdvę. pastebėjimais, teigtume, jog XX a. komponavimo procesas Pažymėtina, kad XX a. kompozitoriai labai individua- pagrįstas itin fomalizuotomis struktūrinėmis idiomomis, liai pasirenka, manipuliuoja ir diegia į savo kompozicijas kurių konstruktyvus valdymas neretai turi matematinę bei skirtinguose medžiaginiuose ir struktūriniuose lygme- prigimtį. Be to, XX a. kompozicijoje labai išplečiamos nyse naudoja matematines technologijas. Matematikos matematikos raiškos muzikoje ribos – tai ne tik ankstesnių įtaka muzikos komponavimo procesui nemaža dalimi lėmė epochų numerologinės komponavimo praktikos re-evoliu- originalius, XX a. muzikos kompoziciją signifikuojančius cija, bet ir tam tikros novacijos. XX a. kompoziciją galima numerologinio pobūdžio rezultatus. Tai I. Xenakio, K. Jo- įvardyti kaip kokybiškai naują matematikos ir muzikos neso muzikos kompozicijos, paremtos stochastiniais pro- sąlyčio etapą, kuriuo siekiama „subendravardiklinti“ cesais, Markovo grandžių logika; G. Ligeti, V. Jurgučio, muzikinių ir skaitmeninių struktūrų prielaidas. Tokiais Š. Nako, Ch. Dodge’o kompozicijų struktūrinės analogi- „bendraisiais vardikliais“ čia tampa matematiniai grupių jos su fraktalų teorija; chaoso teorijos įtaka G. L. Nelsono, teorijos dėsniai, Markovo grandžių logika, atitinkamai I. Xenakio kūriniams; algoritminės L-sistemos kreivės interpretuojama kaip melodinių procesų dėsnis; kon- tapo muzikos kūrinio grafiniu pirmavaizdžiu arba garsų krečiomis matematinėmis teorijomis pagrįstos muzikos audinio studijos rezultatu G. L. Nelsono, T. Johnsono, temperavimo praktikos (pavyzdžiui, Myhillo dėsnio H. Kyburzo opusuose. taikymas diatoninio garsaeilio intervalų darybai) (Journal Straipsnyje pateikiami matematizuoti XX a. muzikos of Mathematics and Music, 2007, p. 53). XX a. muzikos kompozicijų kūrimo prielaidos tyrimai ir aiškinamasi, 70 3. Rima POvilionienė numerologinių technikų inovacijos XX a. antrosios pusės muzikos kompozicijoje kokiomis formomis konkrečios matematinės procedūros atskirti du algoritminio muzikos komponavimo aspek- diegiamos į XX a. antrosios pusės kompozicinius tekstus, tus: pirma, kaip kompozitorius manipuliuoja pasirinktu analizuojamos muzikos kompozicijos. Aptariami šiuolaiki- algo rit mu, organizuodamas muzikos kompozicijos para- nės matematikos implikacijų pavyzdžiai XX a. muzikoje, metrus; antra, kokie matematinių algoritmų pavyzdžiai remiantis praktiniais pavyzdžiais ir literatūros šaltiniais, taikomi XX a. antrosios pusės muzikos kompozicijoje. skirstomi į keletą tipų, kaip antai: Kompozitoriaus dalyvavimą algoritminės muzikos • algoritminiai procesai – savitas XX a. muzikos kom- kūrimo procese galima apibūdinti dvejopai: pirma, jis ponavimo reiškinys; pasirenka tam tikrą algoritmą, kuriuo koordinuoja atskirus • matematizuota muzikos grafika. Čia dėmesys elementus – garsų aukščius, ritmą, dinamiką; antra, kom- krei piamas į grafines garsų aukščių struktūras, grafinius pozitorius nustato principą, kuriuo vadovaujantis minėti modelius, kurie taikomi kaip komponavimo algoritmas/ muzikos elementai sujungiami tam tikrais santykiais, t. y. pirmavaizdis ir kaip kūrinio struktūrinė interpretacija/ muzikos kūriniui algoritmai gali būti taikomi kaip kom- analizė. Prie matematizuotos muzikos grafikos tipo taip poziciniai kodai, lemiantys vieną ar kelis komponuojamo pat priskiriamas skaitmeninis garsų užrašymas ir skaitme- kūrinio parametrus. ninė notacija, L-sistemos grafika ir struktūrinės kalbotyros Remdamiesi įvairių tyrinėtojų pastebėjimais (Järveläi- įtaka; nen, 2000; Dodge, 1988; Burns, 1994) ir tuo, kokio po- • fraktalų teorijos įtakos reprezentacijos – tyrinėja- būdžio algoritminiai veiksmai taikomi muzikos kūrimui, mi fraktalų geometrijos analogai muzikos kompozicijų išskirtume pagrindinius algoritminius XX a. muzikos struktūrose (pseudofraktališkumas, fraktaliniai paveikslai, komponavimo būdus. Tai: fraktalinis principas kaip komponavimo priemonė); 1. De ter mi nuoti (nu sta ty ti, api brėž ti) algoritminiai • stochastinė muzika, jos praktika, Markovo grandys; procesai – patys įvairiausi konstruktyvūs veiksmai, atlie- • kitų šiuolaikinės matematikos teorijų sklaida kami laikantis konkrečių taisyklių ir atliekami paties (chaoso, tikimybių, transformacijų, grupių teorijų įtaka kompozitoriaus. Kai ku rie tokie algoritmai funk cio nuo ja muzikos komponavimo procesui). me chaniškai, juos leng va atsekti kom po zi ci joje. Kaip matyti, principinė XX a. antrosios pusės kompo- 2. Kompleksiški matematiniai algoritmai. Jų raiška zicijos naujovė yra gar si nių struk tū rų skait me nizacijoms yra sunkiai pastebima, nes dažniausiai susiduriama su naudojami mo de liai, ku rie at ėjo iš formaliosios gramati- sudėtingos kompiuterinės muzikos kompozicijomis, kurių kos, frak ta lų ge o met ri jos, chaoso ir kitų modernių XX a. generavimo procedūrų nustatymas ir analizė įmanoma tik matematikos te ori jų. Tai ir lėmė itin suaktyvėjusios in di- kompiuteriu ir remiantis nuosekliu paties kompozitoriaus vi dua lios kūrėjo kompozitoriaus intencijos, ins pi ra vusios procedūrų atkūrimu (aprašu). Algoritmo sąvoką XX a. ma te ma ti kos, fi zi kos, struktūrinės kalbotyros elementų antrosios pusės muzikoje tyrinėtojai daugiausia sieja implikacijas mu zi kos garsų erd vėje. su matematinių elementų implikacijomis4. Kristine Burns algoritmus muzikoje skirsto smulkiau (Burns, 1. Algoritmo sąvoka XX a. muzikoje 1994, p. 4–6): • stochastiniai procesai – atsitiktiniai, tikimybių XX a. muzikoje algoritminėmis dažniausiai vadinamos procesai. Stochastinių kompozicijų autoriai – I. Xenakis, kompiuteriu sukurtos muzikos kompozicijos; kū rė jo C. Ames, J. Teney; su Markovo procesais eksperimentavo ki ši masis čia kiek įmanoma minimizuojamas. Algoritmo 1 kompozitoriai L. Hilleris, L. Isaacsonas, K. Jones (H. Jär- sąvoką su muzikos sfera imta sieti tik XX a., ją pasiskolinus veläinen teigimu, Mar ko vo gran džių principai nepaprastai iš kompiuterijos ir informatikos. Kristine Burns teigia, jog tinka muzi kinėms me lo di joms generuoti. Järveläinen, algoritmo muzikoje apibrėžimui artimiausia algoritmo, 2000, p. 10); taikomo medicinos srityje nustatant klinikinę diagnozę, • chaotiniai procesai, apimantys fraktalines struktūras. interpretacija – „laipsniškas procesas, <...> kai atsakymas Kompozitoriai, į kūrybą implikavę matematinius chaoso į klausimą kelia kitą klausimą“ (Burns, 1994, p. 2). principus – Ch. Dodge’as, J. Harley, G. L. Nelsonas, A. Alpernas al go rit mi nį mu zi kos komponavimo procesą R. Waschka II ir kt.; vadina „for ma lių pro ce sų nau do ji mu kuriant mu zi ką, kai • gramatikos (angl. grammars) procesai – struktūrinės žmogus dalyvauja minimaliai“2 (Al pern, 1995, p. 1). Itin kalbotyros tyrinėjimų taikymas muzikai. Gramatinius aktyvų
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