RESEARCH Rachel Wells Hall Department of Mathematics and Computer Science Saint Joseph’s University 5600 City Avenue Philadelphia, PA 19131 I study Mathematical Music Theory and Ethnomathematics.

The set of two- and one-note chords forms a thrice twisted Mobius strip, with the one-note chords on the boundary forming a trefoil knot. (Click for a larger version of this image.)

My research involves the application of mathematics to music theory – that is, the use of mathematics to describe and analyze musical structures such as rhythms, scales, chords, and melodies. I am currently writing a book that incorporates a historical survey of the field and my original research. My current projects concern defining distance in orbifolds representing chords and chord types, relating integer tilings to patterns in African drumming, and applying the theory of homometric sets to chord theory. In addition, I study Ethnomathematics (the study of mathematical thinking found outside of what has been traditionally defined as “mathematics”). Previously, I studied Operator Algebras and completed my doctoral thesis on Hecke C*-Algebras.

Publications

• Chirashree Bhattacharya and Rachel Wells Hall. Geometrical representations of North Indian thaats and raags. In Bridges: Mathematical Connections in Art, Music, and Science. R. Sarhangi, ed., Pecs, Hungary, 2010.
• Article (PDF)
• Rachel Wells Hall. Linear contextual transformations. Quaderni di matematica: Theory And Applications of Proximity, Nearness and Uniformity, Somashekhar Naimpally and Giuseppe Di Maio, eds., 23 (2009) 101–130.
• Article (PDF)
• Rachel Wells Hall. Review of Generalized Musical Intervals and Transformations and Musical Form and Transformation, by David Lewin, Oxford University Press, 2007. Journal of the American Musicological Society 62 (2009), no. 1, 205–222.
• Article (PDF)
• Rachel Wells Hall. Report on the 2008 Bridges art exhibition. Journal of Math and the Arts 2 (2008), no. 4, 197–204.
• Report (PDF)
• Rachel W. Hall. Neo-Riemannian geometry. In Bridges: Mathematical Connections in Art, Music, and Science. R. Sarhangi and C. Séquin, eds., Leeuwarden, Holland, 2008. (see “Linear contextual transformations” (2009) for the final version)
• Rachel Wells Hall. Geometrical music theory. Science 320 (2008) 328-329.
• Summary
• Full text (HTML)
• Article (PDF) This is the author's version of the work. It is posted here by permission of the AAAS for personal use, not for redistribution. The definitive version was published in SCIENCE, 320 (2008) http://www.sciencemag.org/cgi/content/short/320/5874/328
• Rachel W. Hall. Math for poets and drummers. Math Horizons 15 (2008) 10-11.
• Article (I have also written a longer version of this article for my book.)
• Rachel W. Hall. A course in multicultural mathematics. PRIMUS 17 (2007), no. 3, 209-227.
• Article (PDF)
• Rachel W. Hall. Review of The Math behind the Music, by Leon Harkleroad, Cambridge University Press, 2006. Journal of Mathematics and the Arts 1 (2007), no. 2, 143-145.
• Review (PDF)
• Rachel W. Hall and Paul Klingsberg. Asymmetric rhythms and tiling canons. Amer. Math. Monthly 113 (2006), no.10, 887-896.
• Article (PDF)
• Audio files
• New results (Summer 2005)
1. The number of asymmetric rhythm cycles modulo inversion
2. The number of r-note asymmetric cycles modulo inversion
• Rachel W. Hall. Playing musical tiles. In Bridges: Mathematical Connections in Art, Music, and Science. R. Sarhangi and J. Sharp, eds., London, 2006.
• Article (PDF)
• Joseph E. Flannick, Rachel W. Hall, and Robert Kelly. Detecting meter in recorded music. In Bridges: Mathematical Connections in Art, Music, and Science. R. Sarhangi and C. Sequin, eds., Banff, Canada, 2005.
• Article (PDF)
• Rachel W. Hall and Paul Klingsberg.  Asymmetric rhythms, tiling canons, and Burnside's lemma.  In Bridges: Mathematical Connections in Art, Music, and Science. R. Sarhangi and C. Sequin, eds., pp. 189-194.  Winfield, Kansas, 2004.
• Article (PDF)
• Audio files
• Rachel W. Hall and Kresimir Josic.  The mathematics of musical instruments. Amer. Math. Monthly 108 (2001), no. 4, 347--357.
• Article (PDF)
• Rachel W. Hall and Kresimir Josic.  Planetary motion and the duality of force laws.  SIAM Rev. 42 (2000), no. 1, 115-124
• Article (PDF)

Preprints

• Rachel Wells Hall. Geometrical models for modulation in Arab music. To appear in Mathematical and Computational Musicology, Timour Kloche, ed., Springer, Berlin.
• Article (PDF)
• Rachel Wells Hall and Dmitri Tymoczko. Submajorization and the geometry of unordered collections. Preprint, 2010.
• Article (PDF)
• Rachel Wells Hall, The Sound of Numbers: A tour of mathematical music theory. Preprint, 2007.
• Book proposal (PDF)
• Rachel W. Hall and Clifton Callender (Department of Music, Florida State University). Homometric sets and Z-related chords. Paper presented at the Joint Mathematical Meetings, New Orleans, January 2007.
• Abstract (PDF)

Major presentations (see CV for more)

• “Multicultural mathematics: bringing the world into the mathematics classroom,” day-long workshop for teachers, Miami Dade College, February 2008.
• “Poverty and polyphony,” plenary talk, Bridges: Mathematical Connections in Art, Music, and Science, (Donostia, Spain, 2007), Society for Music Theory National Meeting (Baltimore, 2007), Joint Mathematical Meetings (New Orleans, 2007)
• Presentation (PPT) with embedded audio
• “Asymmetric rhythms and tiling canons,” invited address, MAA Eastern Pennsylvania and Delaware section meeting (2006), Vassar College (2007), Villanova University (2007), Shippensburg University (2007), IRCAM (Paris, 2005).
• Presentation (PPT) with embedded audio
• QuickTime movie of presentation (MOV) with audio
• Handout (PDF)
• “Math for Poets and Drummers” presentation, Haverford College (2005), Arcadia University (2005), Saint Joseph’s University (2004).
• Slides (PDF, with linked MIDI and .WAV audio)

C*-ALGEBRAS

I completed a thesis in the field of Operator Algebras under the direction of Nigel Higson. My thesis, entitled “Hecke C*-Algebras” concerns an analysis of Hecke algebras from the perspective of C*-algebra theory. A summary of the article “Hecke C*-algebras, unitary representations, and the geometry of trees,” which is based on my thesis research, is available here.

• Summary of ``Hecke C*-algebras, unitary representations, and the geometry of trees''
• Full text of my thesis ``Hecke C*-Algebras''

COURSE DEVELOPMENT

• Article (PDF) Multicultural mathematics is a course I designed for undergraduate liberal arts majors. This article details the course goals and topics covered. In addition, I provide sample projects, exercises, and an extensive course list. The course includes a unit on the mathematics of drumming.
• Course home page
• Abcdrums is a program for writing drum compositions created by Dr. Adlai Waksman (formerly of St. Joseph’s University). I used it in the Multicultural Mathematics course. The sound examples for “Asymmetric rhythms and tiling canons” were also created using Abcdrums.
• Abcdrums tutorial

UNDERGRADUATE RESEARCH

Two students, Bobby Kelly ‘02 and Joe Flannick ‘03, have written undergraduate theses under my direction.

• Bobby Kelly.  Mathematics of musical rhythm.  Undergraduate honors thesis, Saint Joseph’s University, 2002. Winner of MAA-EPADEL best student paper award, 2002.
• Thesis (PDF)
• Slides from Bobby’s final presentation (PDF)
• Joseph E. Flannick.  Rhythm detection in recorded music.  Undergraduate honors thesis, Saint Joseph’s University, 2003.
• Thesis (PDF)
• Audio files

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Rachel Wells Hall / rhall@sju.edu

Last modified 12/14/10 10:04 AM.

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