In strict imitations and canons, the risposta (consequent) may simply repeat the proposta (antecedent), but can also repeat it with some changes. In the generally accepted
theory of imitation and canon the following transformations are admitted: augmentationdiminution, inversion, retrograde motion, and combinations thereof. We call these transformations strict, since for each there is a rule according to which the risposta can be derived from the given proposta unambiguously. However, there can be other strict rules.
Hence, the methods of converting the proposta into the risposta are not limited to those listed above. What might these new methods be? In trying to answer this question generally and systematically, we use a mathematical concept of function. Function is a univocal correspondence between two sets. In our case function is a rule according to which the risposta can be derived from the proposta in a well-defined manner. Sets are melodies, the proposta and the risposta. The elements of a set are notes. A note in a melody is an object that has three dimensions: duration, pitch, and sequence number. At first we demonstrate conversions of durations into durations, pitches into pitches, and numbers into numbers. Always beginning with classical transformations (augmentation, inversion, retrograde motion) we then proceed to different methods. Then we consider “more exotic” conversions: durations into pitches, pitches into durations, etc. At the end of the article we attempt to demonstrate that some nonclassical transformations exist not only in our imagination, but also can be encountered in musical practice: e.g. Four Canons by Sergei Zagny — score, record.
We express our gratitude to Ivan S. Soshinsky for his assistance in preparing this article, and to Keith G. Hammond and Yuri V. Spitsyn for a great number of corrections and important improvements in the English text.
1. Ekimovskiy, Victor A. 1987. Olivier Messiaen. Zhizn’ i tvorchestvo [Olivier Messiaen. Life and Works]. Moscow: Sovetskiy kompozitor. (In Russian).
2. Johnson, Tom. 1996. Self-Similar Melodies. Paris: Editions 75.
3. Kholopov, Yuriy N. 1978. “Kanon. Genezis i rannie etapy razvitiya [Canon. Genesis and Early Stages of Development].” Teoreticheskie nablyudeniya nad istoriey muzyki [Theoretical Observations on Music History], 127–57. Moscow: Muzyka. (In Russian).
4. Kohoutek, Ctirad. 1976. Tekhnika kompozitsii v muzyke 20 veka [Composition Technique in the 20th Century Music], translated from Czech by Konstantin N. Ivanov, edited and commented by Yuriy N. Rags and Yuriy N. Kholopov. Мoscow: Muzyka. (In Russian).
5. Melik-Pashaeva, Karina L. 1987. Tvorchestvo O. Messiana [Works of O. Messiaen]. Moscow: Muzyka. (In Russian).
6. Tsenova, Valeriya S. 1985. “Muzykal’no-teoreticheskie problemy tvorchestva moskovskikh kompozitorov pervoy poloviny 1980-kh godov [Musicological Problems in Moscow Composers’ Music of the First Half of 1980s].” Diploma thesis, vol. 3, Tchaikovsky Moscow State Conservatory. (In Russian).
7. Zagny, Sergei A. 1988. “Ob izvestnykh i neizvestnykh sposobakh imitatsii [On Known and Unknown Ways Methods of Imitation].” In Kolichestvennye metody v muzykal’noy
fol’kloristike i muzykoznanii [Quantitative Methods in Study of Musical Folklore and Musicology], 259–82. Moscow: Sovetskiy kompozitor. (In Russian).
8. Zagny, Sergei A. 1988. “Preobrazovanie muzykal’nykh postroeniy [Transformations of Musical Structures].” Diploma thesis, Tchaikovsky Moscow State Conservatory. (In Russian).
9. Zagny, Sergei A. 1992. “Imitatsiya: traditsionnye i netraditsionnye preobrazovaniya melodii [Imitation: Traditional and Nontraditional Transformations of Melodies].” In Muzykal’noe
iskusstvo 20 veka: tvorcheskiy protsess, khudozhestvennye yavleniya, teoreticheskie kontseptsii [Music Art of 20th Century: Creative Process, Artistic Phenomena, Theoretical Conceptions], 75–95. Moscow: Moscow Conservatory. (In Russian).
10. Zagny, Sergei A. 1999. “Imitation: Traditional and Nontraditional Transformations of Melodies.” Perspectives of New Music 37, no. 2 (Summer): 163–87.