BAIN MUSC 525
Post-Tonal Theory

Mathematics and the Twelve-Tone System

Return to: MUSC 525

For a mathematical perspective on twelve-tone composition, see:

Robert Morris, "Mathematics and the Twelve-Tone System: Past, Present, and Future,"
Perspectives of New Music 45/2 {JSTOR} (Morris 2007)

For a brief introduction to the topic, see the following pdf on the author's website:

https://ecmc.rochester.edu/rdm/pdflib/berlin.s.pdf

It is important to have a basic grounding in twelve-tone principles before attempting to read the music and mathematics research literature. My favorite introduction to the twelve-tone system may be found in Straus, Introduction to Post Tonal Theory (Straus 2016, Ch. 6). I also recommend reading the two "Composition with Twelve Tones" essays in Schoenberg's Style and Idea (Schoenberg 1984, 214-248).

I have also found the following sources to be helpful in learning twelve-tone theory: Babbitt's articles on twelve-tone music (see Babbitt 2003), Perle 1962 and 1977, Wuorinen 1979, Rahn 1980, Morris 1987, Morris 1991 a/b, Mead 1994, Straus 2001, Straus 2009, Pearsall 2012, and Jedrzejewki 2016. For more references, see the MUSC 525 Bibliography.

Video-based analyses:
I also recommend watching some of the analyses of twelve-tone compositions by Joseph N. Straus, et al. available online at:

1. Twelve-Tone and Serial Composition: The First 100 Years {JosephStraus.com}
2. Analyzing Modernist Music {YouTube Channel}
   
• Joseph N. Straus, The Art of Post-Tonal Analysis {YouTube Playlist} (Straus 2022)

In 2015, the Journal of Mathematics and Music (JM&M) dedicated a special issue to the mathematics of the twelve-tone system. After you read Morris 2007, you might find the following articles to be interesting reads:

1. Hook & Peck, Introduction to the Special Issue on Tone Rows and Tropes {JM&M; DOI} (Hook & Peck 2015)
2. Fripertinger & Lackner, Tone Rows and Tropes {JM&M; DOI} (Fripertinger & Lackner 2015)
3. Morris, Review of Hook & Peck's Tone Rows and Tropes {JM&M; DOI} (Morris 2015)
4. Mead, Remarks on Hook & Peck's Tone Rows and Tropes {JM&M; DOI} (Mead 2015)
   
   
   

Links

Journal of Mathematics and Music (JM&M) {TCL}

MusMat • Brazilian Journal of Music and Mathematics {MusMat}

Perspectives of New Music (PNM) {TCL}

References

Almada, Carlos, et al. eds. "Schoenberg: Music and Theory." Proceedings of the 6th International Conference MusMat (MusMat 2021). <MusMat>

Babbitt, Milton. 2003. The Collected Essays of Milton Babbitt, edited by Stephen Peles, Stephen Dembski, Andrew Mead, and Joseph N. Straus. Princeton, NJ: Princeton University Press. {GB}

Castine, Peter. 1994. Set Theory Objects: Abstractions for Computer-Aided Analysis and Composition of Serial and Atonal Music. Berlin: Lang. {DBLP}

Hook, Julian and Robert Peck. 2015. "Editorial: Introduction to the Special Issue on Tone Rows and Tropes." Journal of Mathematics and Music 9/2: 109-110. {JM&M; DOI}

Feitosa, Marco, et al. "An Intervallic Approach to Tone Rows: SerialGen, a Twelve-Tone Row Generator." MuseMat 2021. Available online at: <https://musmat.org/conferences/2021/12_Feitosa_et_al.pdf>.

Fripertinger, Harald & Peter Lackner. 2015. "Tone Rows and Tropes." Journal of Mathematics and Music 9/2: 111-172. {JM&M; DOI}

Jedrzejewski, Frank. 2006. Mathematical Theory of Music. IRCAM Musique/Sciences Series. Paris, Delatour. {GBd}

Lewin, David. 2007/1987. Generalized Musical Intervals and Transformations. New York: Oxford University Press. {GB; Full text: TCL}

Mead, Andrew. 2015. "Remarks on 'Tone Rows and Tropes' by Harald Fripertinger and Peter Lackner." Journal of Mathematics and Music 9/2: 173-178. {JM&M; DO}

Mead, Andrew. 1994. An Introduction to the Music of Milton Babbitt. Princeton, NJ: Princeton University Press. {GB}

Morris, Robert. 2007. "Mathematics and the Twelve-Tone System: Past, Present, and Future." Perspectives of New Music 45/2 (Summer, 2007), pp. 76-107 {JSTOR}

___________. 2015. "Review of 'Tone Rows and Tropes' by Harald Fripertinger and Peter Lackner." Journal of Mathematics and Music 9/2: 179-195. {JM&M; DOI}

____________. 1991a. Class Notes for Atonal Theory. Lebanon, NH: Frog Peak {GBd}

____________. 1991b. Class Notes for Advanced Atonal Theory. Lebanon, NH: Frog Peak. {GBd}

Morris, Robert. 1987. Composition with Pitch Classes: A Theory of Compositional Design. New Haven: Yale University Press. {Full text: JSTOR}

Pearsall, Edward. 2012. Twentieth-Century Music Theory and Practice. New York: Routledge. {GB}

Perle, George. 1997/1977. Twelve-Tone Tonality. Berkeley: University of California Press. {2/e: GB; 1/e: GB}

___________. 1991/1962. Serial Composition and Atonality, 6th ed. Berkeley: University of California Press. {GB}

Schoenberg, Arnold, Leonard Stein, ed, and Leo Black, trans. 1984/1950. Style and Idea: Selected Writings of Arnold Schoenberg. Berkeley: University of California Press. {GB}

Straus, Joseph N. 2022. The Art of Post-Tonal Analysis: Thirty-Three Graphic Music Analyses. New York: Oxford University Press. {GB; Full text: TCL; Oxford: Companion Website; Videos: YouTube Playlist}

_____________. 2016. Introduction to Post-Tonal Theory, 4th ed. New York: Norton. {GB}

_____________. 2009. Twelve-Tone Music in America. New York: Cambridge University Press. {GBd}

_____________. 2001. Stravinsky's Late Music. New York: Cambridge University Press. {GB}

Wuorinen, Charles. 1994/1979. Simple Composition. New York: Edition Peters.