Music and Mathematics

Winter 2002

John Rahn

**Syllabus**

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of Mathematics.: Unpublished paper, University of Washington College of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 Unpublished paper.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.