Ratio tonorum duodecim
Ratio tonorum duodecim, etiam dodecaphonia, serialismus duodecim tonorum, in musica est ratio compositionis ab Arnoldo Schoenberg compositore Austriaco (1874–1951) excogitata, quae facit ut omnes duodecim toni diagrammatis chromatici aeque saepe sonentur in compositione data cum vim ullius notae singularis prohibet per ordines tonorum adhibitos, certas duodecim classium tonorum series. Omnes toni duodecim ergo gravitatem plus minus parem sic accipiunt, et musica igitur in voculatione non est. Haec ars compositores medio saeculo XX movit.

Schoenberg ipse modum rationis descripsit "rationem componendi tonis duodecim qui solum inter se cognati sunt." Ratio late habetur genus serialismi.

Iosephus Matthias Hauer, civis et aequaevus Schoenbergianus, rationem similem excogitavit, hexachordas vel tropos inordinatos adhibens—sed ad rationem tonorum duodecim Schoenbergianam haud pertinentia. Alii compositores diagrammte chromatico rationaliter usi sunt, sed ratio Schoenbergiana in historia et aesthetica maximi momenti habetur.

Historia 

Ratio tonorum duodecim, ab Arnoldo Schoenberg compositore Austriaco anno 1921 et primum anno 1923 ad suos privatim descripta, proximis viginti annis paene solum adbibebatur a compositoribus Scholae Vindobonensis Secundae: Albanus Berg, Ioannes Eisler, Antonius Webern, et Schoenberg ipse. Haec ratio a rebus libere atonalibus annorum ab 1908 ad 1923 antecedebatur, quae, quamquam plus minus liberae, minutas intervallorum cellulas pro elementis solidante saepe habent, quae praeter expansionem ut cum ordine tonorum commutari possunt, et in qua notae singulares fungi possunt "elementa cardinalia, ad convenientes cellulae primae iterationes vel duarum vel plurium cellularum primarum nexum sinendum." Ratio tonorum duodecim etiam antecedebatur a "compositione seriali non dodecaphonica" libere in operibus Alexandri Skrjabin, Inguari Stravinsky, Belae Bartók, Caroli Ruggles, et aliorum adhibita. Oliver Neighbour affirmat Bartók fuisse "primus compositor qui gregem notarum duodecim pro fine structurali consulto adhibuit, anno 1908 in tertia ex quartis decimis bagatellis. "Schoenberg et Hauer necessarie in ordinem redegerunt et pro suis finibus dodecaphonicis definierunt obstinatum, technicam musicae modernae proprietatem undique circumfusum. Praeterea, Ioannes Covach strictam horum virorum distinctionem, a Perle et aliis auctoribus expressam, nimis exprimi putavit:
Distinctio scholarum Haueranae et Schoenbergianae saepe facta—musicam illius in hexachordis inordinatos condi cum musicam huius in serie ordinata condi—falsa est: cum Hauer res quae res troparum describi possunt, multum musicae tonorum duodecim Haueranorum seriem ordinatam adhibet.

Stricta autem scholae Vindobonensis secundae ordinatio, "per res utiles necessario temperata est: per interactionem inter congeries tonorum ordinatorum et non ordinatorum operabantur."

Haec ratio ante annos 1950 a Milton Babbitt, Luciano Berio, Petro Boulez, Ludovico Dallapiccola, Ernesto Krenek, Ricardo Malipiero, ac Schoenberg mortuo, Inguaro Stravinsky late adhibebatur, nonnulli ex quibus rationem extenderunt, res praeter tonos notarum (ut duratio et ratio impetus) imperandas, sic musicam serialem efficientes. Nonnulli etiam omnia elementa musica sub rationem serialem subiecerunt.

Carolus Wuorinen in colloquio interrogatorio anno 1962 affirmavit "plurimi Europaei," cum in America sint, "dicunt se rationem tonorum duodecim 'excessisse' et 'confecisse,'" sed "ratio tonorum duodecim diligenter investigata et in aedificium generale commutata est gravius quam ullum adhuc notum."

Notae

Bibliographia 

 Alegant, Brian. 2010. The Twelve-Tone Music of Luigi Dallapiccola. Eastman Studies in Music, 76. Rochester Novi Eboraci: University of Rochester Press. ISBN 978-1-58046-325-6.
 Babbitt, Milton. 1960. Twelve-Tone Invariants as Compositional Determinants. Musical Quarterly 46(2):246–259. Fasciculus praecipuus: Problems of Modern Music: The Princeton Seminar in Advanced Musical Studies. DOI 10.1093/mq/XLVI.2.246. JSTOR 740374 (situs venalis).
 Babbitt, Milton. 1961. Set Structure as a Compositional Determinant. Journal of Music Theory 5(1):72–94. JSTOR 842871 (situs venalis).
 Benson, Dave. 2007 Music: A Mathematical Offering. Cantabrigiae et Novi Eboraci: Cambridge University Press. ISBN 978-0-521-85387-3.
 Brett, Philip. Britten, Benjamin. Grove Music Online ed. L. Macy. https://web.archive.org/web/20080516041031/http://www.grovemusic.com/.
 Chase, Gilbert. 1987. America's Music: From the Pilgrims to the Present. Ed. 3a. Music in American Life. Urbanae: University of Illinois Press. ISBN 0-252-00454-X; ISBN 0-252-06275-2.
 Haimo, Ethan. 1990. Schoenberg's Serial Odyssey: The Evolution of his Twelve-Tone Method, 1914–1928. Oxoniae: Clarendon Press; Novi Eboraci: Oxford University Press. ISBN 0-19-315260-6.
 Hill, Richard S. 1936. Schoenberg's Tone-Rows and the Tonal System of the Future. Musical Quarterly 22(1):14–37. DOI 10.1093/mq/XXII.1.14. JSTOR 739013 (situs venalis).
 Lansky, Paul, George Perle, et Dave Headlam. 2001. Twelve-Note Composition. The New Grove Dictionary of Music and Musicians, ed. 2a, ed. Stanley Sadie et John Tyrrell. Londinii: Macmillan Publishers.
 Leeuw, Ton de. 2005. Music of the Twentieth Century: A Study of Its Elements and Structure. Ex Nederlandico a Stephano Taylor conversus. Amstelodami: Amsterdam University Press. ISBN 90-5356-765-8. Primus textus: Muziek van de twintigste eeuw: een onderzoek naar haar elementen en structuur (Traiecti: Oosthoek, 1964). Ed. 3a, Traiecti: Bohn, Scheltema & Holkema, 1977. ISBN 90-313-0244-9.
 Loy, D. Gareth. 2007. Musimathics: The Mathematical Foundations of Music. Vol. 1. Cantabrigiae Massachusettae et Londinii: MIT Press. ISBN 978-0-262-12282-5.
 Neighbour, Oliver. 1954. The Evolution of Twelve-Note Music. Proceedings of the Royal Musical Association 81(1):49–61. DOI 10.1093/jrma/81.1.49.
 Perle, George. 1977. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. Ed. 4a., retractata. Berkeleiae, Angelopoliti, et Londinii: University of California Press. ISBN 0-520-03395-7.
 Perle, George. 1991. Serial Composition and Atonality: An Introduction to the Music of Schoenberg, Berg, and Webern. Ed. 6a, retractata. Berkeleiae: University of California Press. ISBN 978-0-520-07430-9.
 Reti, Rudolph. 1958. Tonality, Atonality, Pantonality: A Study of Some Trends in Twentieth Century Music. Westport Connecticutae: Greenwood Press. ISBN 0-313-20478-0.
 Rufer, Josef. 1954. Composition with Twelve Notes Related Only to One Another, conv. Humphrey Searle. Novi Eboraci: Macmillan. Prima editio Theodisca, 1952.
 Schoenberg, Arnold. 1975. Style and Idea, ed. Leonard Stein cum translationibus Leonis Black. Berkeleiae et Angelopoli: University of California Press. ISBN 0-520-05294-3.
 207–208 Twelve-Tone Composition. 1923.
 214–245 Composition with Twelve Tones (1). 1941.
 245–249 Composition with Twelve Tones (2). Circa 1948.
 Solomon, Larry. 1973. New Symmetric Transformations. Perspectives of New Music 11(2):257–264. JSTOR 832323 (situs venalis).
 Spies, Claudio. 1965. Notes on Stravinsky's Abraham and Isaac. Perspectives of New Music 3(2):104–126. JSTOR 832508 (situs venalis).
 Whittall, Arnold. 2008. The Cambridge Introduction to Serialism. Cambridge Introductions to Music. Novi Eboraci: Cambridge University Press. ISBN 978-0-521-86341-4; ISBN 978-0-521-68200-8.

Bibliographia addita 
 Covach, John. 1992. The Zwölftonspiel of Josef Matthias Hauer. Journal of Music Theory 36(1):149–84. JSTOR 843913 (situs venalis).
 Covach, John. 2000. Schoenberg's "Poetics of Music," the Twelve-Tone Method, and the Musical Idea. In Schoenberg and Words: The Modernist Years, ed. Russell A. Berman et Charlotte M. Cross. Novi Eboraci: Garland. ISBN 0-8153-2830-3.
 Covach, John. 2002. Twelve-Tone Theory. In The Cambridge History of Western Music Theory, ed. Thomas Christensen, 603–627. Cantabrigiae: Cambridge University Press. ISBN 0-521-62371-5.
 Krenek, Ernst. 1953. Is the Twelve-Tone Technique on the Decline? The Musical Quarterly 39(4):513–527.
 Šedivý, Dominik. 2011. Serial Composition and Tonality: An Introduction to the Music of Hauer and Steinbauer, ed. Günther Friesinger, Helmut Neumann, et Dominik Šedivý. Vindobonae: Edition Mono. ISBN 3-902796-03-0.
 Sloan, Susan L. 1989. Archival Exhibit: Schoenberg’s Dodecaphonic Devices. Journal of the Arnold Schoenberg Institute 12(2):202–205.
 Starr, Daniel. 1978. Sets, Invariance and Partitions. Journal of Music Theory 22(1):1–42. JSTOR 843626 (situs venalis).
 Wuorinen, Charles. 1979. Simple Composition. Novi Eboraci: Longman. ISBN 0-582-28059-1. Reimpressus 1991, Novi Eboraci: C. F. Peters. ISBN 0-938856-06-5.

Nexus externi 

 Adams, Ricci. Generator matricum, www.musictheory.net
 Román, Dan. Ars tonorum duodecim, musike.cmpr.edu
 Solomon, Larry. Novae Transformationes: Praeter P, I, R, et RI, solomonsmusic.net
 Jedrzejewski, Franck. Bodi dodecaphonici et topologia verborum, recherche.ircam.fr 
 Javascript calculator tonorum duodecim et explicator ordinis tonorum, composertools.com
 Quadratum tonorum duodecim, www.andreaslehmann.com
 Repositorium datorum de ordinibus tonotrum et troporum, www.uni-graz.at

 
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