Music and Mathematics

Winter 2002

John Rahn




Week 1: introduction and discussion; group theory and graph theory.

Week 2: discuss Chapters 1 and 7 of Lewin 1987

Week 3: discuss Chapters 8 and 9 of Lewin 1987

Week 4: discuss Lewin 1987 Chapter 10 and papers by Oren Kolman

Week 5: discuss Chapters 1 and 3 of Lewin 1993.

Week 6: discuss Carey and Clampitt, Self-Similar Pitch Structures

Week 7: discuss  Callender, Formalized Accelerando

Week 8: open for discussion and presentations

Week 9: open for discussion and presentations

Week 10: individual conferences and paper-writing




Annotated Select Bibliography



Birkhoff, Garrett, and MacLane, Saunders. A Survey of Modern Algebra. NY: MacMillan, 1977. (Or any good text on abstract algebra.)


Callender, Clifton. Formalized Accelerando: An Extension of Rhythmic Techniques in Nancarrow's Acceleration Canons. PNM 39/1 (Winter 2001): 188-210.


Carey, Norman, and Clampitt, David. Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogs. PNM 34/2 (Summer 1996): 62-87.


Gordon, Charles K. Jr. Introduction to Mathematical Structures. Belmont, CA: Dickenson Publishing Co., 1967.


Haralick, Robert. *The Language of Mathematics.: Unpublished paper, University of Washington College of Engineering, 1991.


Haralick, Robert. *A Consistent Labeling Theoretic Approach to Music.: In the proceedings of the *Music and Science: symposium, University of Washington, February 1991. CCISM, Music DN-10, University of Washington


Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. (See related articles by Lewin, Klumpenhourwer, and others, and subsequent analytical case book by Lewin (1993).)


Lewin, David. 1993. Musical form and transformation : 4 analytic essays. New Haven: Yale University Press.


Kolman, Oren. Generalized interval Systems: An Application of Logic. Unpublished paper.


Mazzola, Guerino. 1985. Gruppen und Kategorien in der Musik : Entwurf einer mathematischen Musiktheorie. Berlin : Heldermann.


Mazzola, Guerino, and Zahorka, Oliver. 1993. Geometry and Logic of Musical Performance. SNSF Report, University of Zurich.


Morris, Robert. Compositional Spaces and Other Territories. PNM 33 (1995): 328-59.


Morris, Robert. 1987. Composition with pitch-classes : a theory of compositional design. New Haven : Yale University Press.


Rahn, John. 1980. Basic Atonal Theory. NY: Schirmer Music Books. (The exercises and optional sections contain information on the group theory of Z12.)


Rahn, John.  1995. Some Remarks on Network Models for Music. In Musical Transformations and Intuitions: A Festschrift for David Lewin, ed. Raphael Atlas and Michael Cherlin. Pendragon Press.


Reiner, David. Enumeration in Music Theory. American Mathematical Monthly, January 1995: 51-54.


Rumelhart, D. E. and J. L. McClelland. Parallel Distributed Processing: Explorations in the Microstructure of Cognition. In two volumes. Cambridge, MA: MIT Press. (See the "Nets" bibliography on my web site for related work in Computer Music Journal, and by Gjerdingen and others.)


Suppes, Patrick.  1972. Axiomatic Set Theory. NY: Dover.


Vuza, Dan. Supplementary Sets and Regular Complementary Unending Canons. Serialized in four parts in PNM 29/2, 30/1, 30/2, and 31/1.



additional remarks:

 on “Neo-Riemannian Theory”


There has been a remarkable efflorescence of theory based on groups of tonal transformations, originating in Lewin’s book, propelled further by Brian Hyer,  Rick Cohn, and John Clough’s group at SUNY Buffalo (including Jack Douthett and David Clampitt), and energized by a conference at Buffalo in 1993. Much of this is collected in a special issue of JMT (Spring 1998, vol. 42/2), with a good introductory essay by Rick Cohn.


additional remarks:

 Scale Theory


John Clough also re-energized tonal  scale theory (somnolent since the Renaissance) in work since the 1980s, initially focussing on properties of the diatonic scale and variants of this. Jack Douthett, a mathematician, entered into a lengthy and fruitful collaboration with Clough, and Clough’s students David Clampitt and Norman Carey (now teaching at Eastman and Yale) wrote a brilliant paper in MTS which jolted this line of research further along. You will see plenty along these lines in the journals.


Scale theory and Neo-Riemannian theory together have transformed the music theory of tonality into a modern mathematical study.


additional remarks:

 Similarity measures


The spate of articles on measuring similarity among pc sets originated in three articles published simultaneously in PNM 18 (1980), by Morris, Rahn, and Lewin. Morris generalized and extended a measure from Forte based on interval vectors, and interval-based similarity measures have remained a major thread in the subsequent discussion (including Erickson’s). Rahn generalized and extended Morris’s ideas to measures of subset-type content is sizes greater than 2 (interval). Lewin reformulated some of Rahn’s notions with different mathematics. Later, Mark Hoover closed some loopholes in Rahn’s ATMEMB measure, and Marcus Castren (Finland) wrote a good monograph surveying the similarity literature to his time and proposing a measure which improves Rahn. The most recent entry is Quinn (PNM 39/2). Some articles on formal voice-leading also lead from the interval-based similarity thinking.


additional remarks:

 on structure


Suppes, Patrick.  1972. Axiomatic Set Theory,  an undergraduate textbook, is a good place to start; Gordon, Charles.  1967. Introduction to Mathematical Structures is even more elementary.


Further work in this line needs elementary logic (first-order predicate calculus); see any good text, such as Benson Mates, Elementary Logic (Oxford). Slightly more advanced is a text such as Geoffrey Hunter, Metalogic: An introduction to the Metatheory of Standard First-Order Logic (UC Press). From Frege to Goedel: A Source book in Mathematical logic, 1879-1931 (ed. Jean van Heijenoort) is an excellent compilation of mind-boggling breakthroughs, including Goedel's most famous contributions (just the original research papers, translated into english when necessary). You can end up with Alonzo Church, Introduction to Mathematical Logic, Vol. 1 (Princeton, 1956). Vol 2 was never published because its scope reaches beyond current research.


At some point in this series you can enjoy Foundations of Set Theory, by Fraenkel, Bar-Hillel, and Levy (North Holland, 1973). Axiomatic Set Theory by Bernays and Fraenkel (North Holland, 1968) is a good practical construct to supplement Suppes.


The interesting structure of almost-not-structure is the subject of point-set topology, which could prove quite fruitful for music theory; e.g. Topological Spaces, by Eduard Cech (Interscience/Wiley 1966), though this is a somewhat quirky treatment of it; this connects with Haralick's approach.


For more specialized, algebraic structures (group theory, fields, category theory, etc.) see any good text on algebra, such as Birkhoff and MacLane, Algebra (introductory); Serge Lang, Algebra (advanced).


David J. Griffiths, Introduction to Quantum Mechanics, is a user-friendly guide to this curious corner, but you will need math through partial differential equations in complex fields as well as probability theory, group theory, etc.



additional remarks:

 on models


A classic collection of papers from 1960 called Logic, Methodology and Philosophy of Science (ed. Nagel, Suppes, and Tarski) snapshots some ferment in model theory, which you should follow up by reading Tarski (various works, use the library software). There is plenty of recent work in this area which is too profuse and diverse to detail. See issues of the journal Synth\se from the 60s-80s for one. Various authors have explored alternate universe theory (at times called discourse theory) as an extension to model theory. Much work has a computer-science connection, as in UW Professor Robert Haralick's highly abstract relational models for robot vision, applicable to music as in "A Consistent Labelling Theoretic Approach to Music" and its references (see Network bibliography).


For a very simple, non-mathematical, musical application see Rahn, John (1979) "Aspects of Musical Explanation." A simple, small model of a formalized music theory for tonal music is in "Logic, Set Theory, Music Theory."  College Music Symposium 19, no. 1 (Spring): 114-27. See also my "Some Remarks on Network Models for Music" for a sketch of a synthesis of relations, grammars, neural nets, and Lewin nets.


additional remarks:

 on explanation


The locus classicus on "scientific" explanation is Karl Popper, The Logic of Scientific Discovery (Harper, 1965). See also the short, undergraduate text Philosophy of Natural Science, by Carl Hempel (Prentice-Hall, 1966). The sociology of it is pioneered in the misnamed The Structure of Scientific Revolutions by Thomas Kuhn. Milton Babbitt pioneered the "scientific" model of explanation for music (various articles). Ben Boretz

in Meta-Variations, The Logic of What?, and various articles, and others such as myself ("How do You DU", "Aspects of Musical Explanation, " "Notes on Methodology in Music Theory") examined Babbitt's assertion skeptically or refigured it. More recently, Brown and Dempster misread all this, in the course of asserting again a cruder version of a quasi-Popperian model for explanation for music. (The whole exchange is in Journal of Music Theory Vol 33/1, 1989). 


There are plenty of alternatives to the Popperian model for explanation, especially for an art. For a somewhat faded snapshot of some, see my "New Research Paradigms,"  Spectrum 12/1 (Spring): 84-94, its bibliography; see also the general bibliography above.