Multiplication (music)

[[File:Bartok - Music for Strings, Percussion and Celesta interval expansion.png|thumb|350px|right|Interval expansion in Bartók's ''Music for Strings, Percussion and Celesta'': first movement (mm. 1–5) and fourth movement (mm. 204–209).<ref name=MS>Schuijer, Michiel. ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts''. Eastman Studies in Music 60. University of Rochester Press, 2008.</ref>{{rp|79}}]] '''Multiplication''' is a mathematical practice that can be applied to music. The operation multiplies the numeric value of musical parameters like notes or rhythms to create new ones. Like transposition, inversion, and retrogression, multiplication generates new material from melodic sources. The practice is particularly important in the field of twelve-tone technique and set theory.

==Background== Ernst Krenek was the first to describe the technique during a series of lectures in 1936. The antecedents of pitch multiplication can be found in the music of both Béla Bartók and Alban Berg. Bartók described the process as an "extension in range", where chromatic intervals are augmented into diatonic ones. The process can be seen in the outer movements of his ''Music for Strings, Percussion and Celesta''.<ref name=MS/>{{rp|77–79}} In Bartók's ''String Quartet No. 3'', the opening chromatic tetrachord eventually expands from a series of semitones (C{{music|sharp}}–D–D{{music|sharp}}–E) into a series of fifths (C{{music|sharp}}–G{{music|sharp}}–D{{music|sharp}}–A{{music|sharp}}).<ref>Antokoletz, Elliott. "[https://archive.org/details/bartokcompanion0000unse/page/260/mode/1up Middle Period String Quartets]". In ''The Bartok Companion'', edited by Malcolm Gillies. London: Faber and Faber, 1993. 259–60.</ref>

[[File:Chromatic scale and M7 multiplication.png|alt=When multiplied by seven (mod 12), the pitches of a chromatic scale yield a cycle of fifths.|thumb|When multiplied by seven (mod 12), the pitches of a chromatic scale yield a cycle of fifths.<ref name=MS/>{{rp|80}}]]As twelve-tone technique developed, composers and music theorists sometimes reduced pitches to classes where every occurrence of a note can be considered the same, regardless of its octave. Those classes are often assigned numbers to assist in analysis, especially of tone rows.<ref>"Set theory", ''The Harvard Dictionary of Music''. Fourth Edition. Edited by Don Michael Randel. Harvard University Press, 2003. 776.</ref> In addition to transformations like transposition, inversion, regression, and retrograde inversion, a composer could apply multiplication to their tone rows as a developmental technique.<ref name=CW>Wuorinen, Charles. ''[https://archive.org/details/simplecompositio0000wuor/page/98/mode/1up Simple Composition]''. Edition Peters, 1979. 98–101.</ref> The process is transposition by multiplication instead of addition. Where transposition by a perfect fifth adds the interval to each note value, in multiplication, each note value is multiplied by a fifth.<ref name=Morris>Morris, Robert D. ''[https://doi.org/10.2307/j.ctt1xp3ss4 Composition With Pitch-Classes: A Theory of Compositional Design]''. Yale University Press, 1987.</ref>{{rp|41f, 65}}

Since octaves are disregarded in pitch classes, modular arithmetic is applied. For any given collection of twelve tones, only three multipliers would yield a new set of twelve unique tones: 5, 7, and 11. When the chromatic scale is multiplied by 7 (mod 12), the result is a cycle of fifths. This is the same transformation found in Bartók's string quartet. Music theorists denote multiplication by the letter 'M' and the factor number: M5, M7, M11.<ref name=MS/>{{rp|76–80}} M5 and M7 are inversions of each other.<ref>Rahn, John. ''[https://archive.org/details/basicatonaltheor0000rahn_z1e6 Basic Atonal Theory]''. Longman Music Series. New York and London: Longman, 1980. 54.</ref> M11 inverts the tone row.<ref name=CW/>{{rp|101}}

==Usage== Herbert Eimert used the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation) for the operations later theorists would call M5 and M7. He described the process as akin to the vertical mirror operation of inversion and the horizontal mirroring of retrogression. Fourth and fifth transformations slant the mirror to the angle of their respective intervals.<ref name=MS/>{{rp|80f}} These two operations appear in works by Milton Babbitt, Robert Morris, and Charles Wuorinen's ''The Politics of Harmony''.<ref>Morris, Robert D. "[https://www.jstor.org/stable/833652 Some Remarks on ''Odds and Ends'']". ''Perspectives of New Music'' 35, no. 2. Summer, 1997. 238, 242–43.</ref><ref>Hibbard, William. "[https://doi.org/10.2307/832299 Charles Wuorinen: The Politics of Harmony']." ''Perspectives of New Music'', vol. 7, no. 2, 1969. 157f.</ref> M5 also occurs in jazz.<ref>Morris, Robert D. "[https://doi.org/10.2307/746016 Review: "John Rahn, ''Basic Atonal Theory''"], ''Music Theory Spectrum'', vol. 4, 1982. 153f.</ref> Theorists like James K. Randall, Godfrey Winham, and Hubert S. Howe also used the concept of multiplication to analyze music that was not twelve-tone.<ref name=MS/>{{rp|81f}}

Pierre Boulez advanced the concept beyond simple multiplication by a single factor. He would often multiply one group of notes by another, creating a much more intricate complex of resulting pitches. As with all serial music, he utilized multiplication on additional parameters like rhythm and timbre.<ref>Boulez, Pierre. ''[https://archive.org/details/boulezonmusictod0000boul/page/n6/mode/1up Boulez on Music Today]''. Translated by Susan Bradshaw and Richard Rodney Bennett. Cambridge, Massachusetts: Harvard University Press, 1971. 39-40, 79-80.</ref><ref>Boulez, Pierre. ''[https://archive.org/details/notesofapprentic00pier/page/167/mode/1up Notes of an Apprenticeship]''. Texts collected and presented by Paule Thévenin. Translated by Herbert Weinstock. A. A. Knopf, 1968. 167f.</ref><ref name=LK/>{{rp|3–5}} Boulez' 1955 masterpiece ''Le Marteau sans maître'' demonstrates the technique, which is also found in his ''Third Piano Sonata'', ''Structures II'', ''Pli selon pli'', and several other works.<ref name=SH>Heinemann, Stephen. "[https://doi.org/10.2307/746157 Pitch-Class Set Multiplication in Theory and Practice]." ''Music Theory Spectrum'', vol. 20, no. 1, 1998. 72f.</ref><ref name=LK>Koblyakov, Lev. ''Pierre Boulez: A World of Harmony''. Chur: Harwood Academic Publishers, 1990.</ref>{{rp|32}}

In addition to Bartók, many other composers employed the concepts of multiplication while using different names for the technique. Howard Hanson called it "projection".<ref name=Cohn>Cohn, Richard. "[https://doi.org/10.2307/745790 Inversional Symmetry and Transpositional Combination in Bartók]." ''Music Theory Spectrum'', vol. 10, 1988.</ref>{{rp|23}} Nicolas Slonimsky liked the names interpolation, infrapolation, and ultrapolation.<ref name=SH/> Adriaan Fokker devised a tuning system where chords could be constructed through multiplication.<ref>Barbera, André. "[https://doi.org/10.2307/843798 Reviews]", ''Journal of Music Theory'', vol. 33, no. 2, 1989. 395.</ref>

Joseph Schillinger used the opening rhythm of "Pennies From Heaven" to demonstrate how squaring the durations generates new material. He also expanded multiplication into geometric space, citing the precedent in visual art. Schillinger had a parlor trick of multiplying by two all of the intervals in a Johann Sebastian Bach fugue and performing the results.<ref>Schillinger, Joseph. [https://archive.org/details/schillingersyste0001unse/mode/1up ''The Schillinger System of Musical Composition'', Volumes I] & [https://archive.org/details/bwb_C0-AVR-259/mode/1up II]. New York: Carl Fischer, 1941. I:79, 185, 211.</ref><ref>Slonimsky, Nicholas. ''Slonimsky's Book of Musical Anecdotes''. Taylor & Francis, 2014. 211</ref>

==References== {{reflist}}

==Further reading== *Eimert, Herbert. ''Lehrbuch der Zwölftontechnik''. Wiesbaden: Breitkopf & Härtel, 1950. *Hanson, Howard. 1960. ''[https://archive.org/details/harmonicmaterial1960hans/page/n4/mode/1up Harmonic Materials of Modern Music]''. New York: Appleton-Century-Crofts. *Heinemann, Stephen. ''Pitch-Class Set Multiplication in Boulez's Le Marteau sans maître''. D.M.A. diss., University of Washington, 1993. *Howe, Hubert S. 1965. "Some Combinational Properties of Pitch Structures." ''Perspectives of New Music'' 4, no. 1 (Fall-Winter): 45–61. *Krenek, Ernst. 1937. ''Über neue Musik: Sechs Vorlesungen zur Einführung in die theoretischen Grundlagen''. Vienna: Ringbuchhandlung. *Losada, Catherine C. 2014. "Complex Multiplication, Structure, and Process: Harmony and Form in Boulez’s Structures II". ''Music Theory Spectrum'' 36, no. 1 (Spring): 86–120. *Morris, Robert D. 1977. "On the Generation of Multiple-Order-Function Twelve-Tone Rows". ''Journal of Music Theory'' 21, no. 2 (Autumn): 238–262. *Morris, Robert D. 1982–83. "Combinatoriality without the Aggregate". ''Perspectives of New Music'' 21, nos. 1 & 2 (Autumn-Winter/Spring-Summer): 432–486. *Morris, Robert D. 1990. "Pitch-Class Complementation and Its Generalizations". ''Journal of Music Theory'' 34, no. 2 (Autumn): 175–245. *Slonimsky, Nicolas. 1947. ''Thesaurus of Scales and Melodic Patterns''. New York: Charles Scribner Sons. *Starr, Daniel V. 1978. "Sets, Invariance, and Partitions." ''Journal of Music Theory'' 22, no. 1:1–42. *Winham, Godfrey. 1970. "Composition with Arrays". ''Perspectives of New Music'' 9, no. 1 (Fall-Winter): 43–67.

Category:Musical techniques Category:Mathematics of music