Music
2011 MAA MiniCourse on Geometry and Algebra in Music Theory
2009 Chicago REU Lectures on Mathematical Music Theory
This year we incorporate recent work of Thomas Noll, David Clampitt, and Manuel Domínguez on scale theory and algebraic combinatorics on words.
I also plan to discuss Vuza canons and tiling problems in music. These are topics researched by Dan Tudor Vuza,
Emmanuel Amiot,
Moreno Andretta,
Harald Fripertinger, and many others. For other topics, see also:
Musical Actions of Dihedral Groups.
Joint with Alissa Crans and Ramon Satyendra.
American Mathematical Monthly, Volume 116, Number 6, JuneJuly 2009, pp. 479495. 17 pages.
Generalized Contextual Groups.
Joint with Ramon Satyendra.
Music Theory Online, Volume 11, Number 3, September 2005. 29 pages.
and the beginner lecture notes Music and Mathematics. Here is what we did in the 2009 REU:

Lecture 1: Introduction to Mathematical Music Theory, Scale Theory.
July 6th, 2009, in Eckhart 133.
Slides pdf. (Some figures from Rahn, Carey, Clampitt, McCartin)
Problems pdf.
Musical Examples: Schumann, Album for the Young, Numbers 8 and 10, 1848 (Rico Gulda).
Bartok, Mikrokosmos, Volume IV, Number 101, Diminished Fifth, 19261939 (Jenö Jando).
Debussy, Preludes, Book 1, Number 2, Sails, 1910 (Paul Jacobs).

Lecture 2: Scale Theory and Algebraic Combinatorics on Words: Extension of Christoffel Duality to Christoffel Conjugates, and
the Characterization
of the Ionian Mode via Divider Incidence and Sturmian Standardicity.
July 7th, 2009, in Eckhart 133.
Problems pdf.
References: Noll, Thomas (2008): ''Sturmian Sequences and Morphisms: A MusicTheoretical Application", In: Mathematique et Musique. Societe Mathematique de France, Journee Annuelle.
Clampitt, David, Manuel Domínguez, and Thomas Noll (2009): "Plain and Twisted Adjoints of WellFormed Words", In: Mathematics and Computation in Music, Proceedings of the Second International Conference on Mathematics and Computation in Music, Springer. (Available online for free from a university computer!)

Lecture 3: Continuation of
Scale Theory and Algebraic Combinatorics on Words: Extension of Christoffel Duality to Christoffel Conjugates, and
the Characterization
of the Ionian Mode via Divider Incidence and Sturmian Standardicity.
July 8th, 2009, in Eckhart 133.
See Lecture 2 above for problems and references.

Lecture 4: Vuza Canons.
July 9th, 2009, in Eckhart 133.
Problems pdf.
References: Dan Tudor Vuza, Supplementary Sets and regular complementary unending canons, Perspectives of New Music, Numbers 29(2), 30(1), 30(2), and 31(1).
Journal of Mathematics and Music, Special Issue on Tiling Problems in Music, Vol. 3, No. 2 (available through our library).
Emmanuel Amiot’s Website on Rhythmic Canons.
Harald Fripertinger’s Website on Rhythmic Canons.
Harald Fripertinger’s Enumeration of Vuza Canons of Length 72 with Audio Samples.
Harald Fripertinger’s Enumeration of Vuza Canons of Length 108 with Audio Samples.
Rachel Hall’s Research Site, with lots of papers and audio examples.
Steve Edward’s Site Tiling Plane and Fancy.

Lecture 5: Group Actions in neoRiemannian Music Theory.
July 10th, 2009, in Eckhart 133.
Slides pdf. (This is a big file, it will take a few seconds to download.)
Slides from 2006 pdf and pdf.
Problems from 2006 pdf and pdf.
Musical Examples: Hindemith, Ludus Tonalis, Fugue in G, 1942 (Siglind Bruhn).
The Beatles, Abbey Road Album, Oh! Darling, 1969. Chords.
Beethoven, Ninth Symphony, Second Movement, 18221824 (Alberto Lizzio).
2007 Chicago REU Lectures on Mathematical Music Theory

Lecture 1: Introduction to Mathematical Music Theory, Scale Theory.
July 16th, 2007.
Slides pdf. (Some figures from Rahn, Carey, Clampitt, McCartin)
Problems pdf.
Musical Examples: Schumann, Album for the Young, Numbers 8 and 10, 1848 (Rico Gulda).
Bartok, Mikrokosmos, Volume IV, Number 101, Diminished Fifth, 19261939 (Jenö Jando).
Debussy, Preludes, Book 1, Number 2, Sails, 1910 (Paul Jacobs).

Lecture 2: Set Theory.
July 17th, 2007.
Slides pdf.
Problems pdf.
Musical Examples: Schoenberg, Five Piano Pieces, Opus 23, Number 1, 19201923 (Paul Jacobs).
Schoenberg, Six Little Piano Pieces, Opus 19, Number 6, 1911 (Paul Jacobs).
Wagner, Tristan and Isolde, Prelude, 1859 (Daniel Barenboim).

Lectures 3, 4: The Topos of Triads.
July 18th, July 19th 2007.
Noll, Thomas. The Topos of Triads. Colloquium on Mathematical Music Theory, 103135,
Grazer Math. Ber., 347, KarlFranzensUniv. Graz, Graz, 2005.
Lecture Notes.
Special Characteristic Morphisms and Subobjects.
Summary.
Problems pdf.
Musical Example: Skrjabin (18721915), Opus 65, Number 3(Nikita Magaloff).
For much more on topos theory and music, see:
Mazzola, Guerino. The topos of music. Geometric logic of concepts, theory, and performance.
In collaboration with Stefan Göller and Stefan Müller. Birkhäuser Verlag, Basel, 2002. 1335 pp.

Lectures 6, 7: Transformational Theory
July 24th, 26th, 2007.
Slides pdf and pdf.
Problems pdf and pdf.
Musical Examples: Hindemith, Ludus Tonalis, Fugue in G, 1942 (Siglind Bruhn).
The Beatles, Abbey Road Album, Oh! Darling, 1969. Chords.
Beethoven, Ninth Symphony, Second Movement, 18221824 (Alberto Lizzio).
2006 Chicago REU Lectures on Mathematical Music Theory

Lecture 1: Introduction to Mathematical Music Theory, Scale Theory.
July 5th, 2006.
Slides pdf. (Some figures from Rahn, Carey, Clampitt, McCartin)
Problems pdf.
Musical Examples: Schumann, Album for the Young, Numbers 8 and 10, 1848 (Rico Gulda).
Bartok, Mikrokosmos, Volume IV, Number 101, Diminished Fifth, 19261939 (Jenö Jando).
Debussy, Preludes, Book 1, Number 2, Sails, 1910 (Paul Jacobs).

Lecture 2: Set Theory.
July 7th, 2006.
Slides pdf.
Problems pdf.
Musical Examples: Schoenberg, Five Piano Pieces, Opus 23, Number 1, 19201923 (Paul Jacobs).
Schoenberg, Six Little Piano Pieces, Opus 19, Number 6, 1911 (Paul Jacobs).
Wagner, Tristan and Isolde, Prelude, 1859 (Daniel Barenboim).

Lectures 3, 4: Transformational Theory
July 11th, 13th, 2007.
Slides pdf and pdf.
Problems pdf and pdf.
Musical Examples: Hindemith, Ludus Tonalis, Fugue in G, 1942 (Siglind Bruhn).
The Beatles, Abbey Road Album, Oh! Darling, 1969. Chords.
Beethoven, Ninth Symphony, Second Movement, 18221824 (Alberto Lizzio).
Student REU Projects on Mathematical Music Theory
I mentored the following students on their projects during the 2006, 2007, and 2009 REU Programs.
One student, inspired by my REU lectures, wrote his paper entirely independently:
Undergraduate Lectures on Transformational Theory at Michigan
The links below were created from a series of lectures I gave in Math 107 on applications of group theory to music theory as part of my VIGRE project at the University of Michigan. It was inspired by the work of music theorist David Lewin. These ideas
are further developed in a joint paper with Ramon Satyendra, which has appeared in Music Theory Online.
I also gave a talk on these topics in the Undergraduate Math Club at the University of Michigan.

Start Here! The examples below pertain to this text: Music and Mathematics ps, pdf

Handouts and Musical Examples we did in class:
Bach's Fugue in d minor from the WellTempered Clavier
Wagner's Tristan Prelude, taken from John Rahn, Basic Atonal Theory. New York: Schirmer Books, 1980.
Hindemith's Fugue in G major from Ludus Tonalis
Handout for Lecture 2
Beethoven's Symphony 9, Movement 2

In class we listened to:
Bach's Fugue in d minor from the WellTempered Clavier, performed by Glenn Gould
Wagner's Tristan Prelude, performed by the Berlin Philharmonic under Daniel Barenboim
Hindemith's Fugue in G major from Ludus Tonalis, performed by Siglind Bruhn
The Beatles' "Oh! Darling"
Beethoven's Symphony 9, Movement 2, performed by Great Festival Orchestra under Alberto Lizzio
Acknowledgement
Some of the material on this website is based upon work supported by the National Science Foundation under Grant DMS0501208.