2007 Chicago REU Lectures on Mathematical Music Theory

• REU Program.

• Abstract. This year we incorporate recent work of Dmitri Tymoczko and Thomas Noll. See also:
  
  Musical Actions of Dihedral Groups.
  Joint with Alissa Crans and Ramon Satyendra.
  American Mathematical Monthly, Volume 116, Number 6, June-July 2009, pp. 479-495. 17 pages.
  
  Generalized Contextual Groups.
  Joint with Ramon Satyendra.
  Music Theory Online, Volume 11, Number 3, September 2005. 29 pages.
  
  and the lecture notes Music and Mathematics.

• 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, 1926-1939 (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, 1920-1923 (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, 103--135,
  Grazer Math. Ber., 347, Karl-Franzens-Univ. Graz, Graz, 2005.
  Lecture Notes.
  Special Characteristic Morphisms and Subobjects.
  Summary.
  Problems pdf.
  Musical Example: Skrjabin (1872-1915), 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.

• Lecture 5: The Geometry of Musical Chords.
  July 20th, 2007.
  Tymoczko, Dmitri. The Geometry of Musical Chords.
  Science 313 (2006), no. 5783, 72--74.
  Lecture Notes.
  Musical Examples: See article and website.
  

• 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, 1822-1824 (Alberto Lizzio).
  
  
  

2006 Chicago REU Lectures on Mathematical Music Theory

• REU Program.

• Abstract. See the draft Music and Mathematics and the paper Generalized Contextual Groups.

• 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, 1926-1939 (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, 1920-1923 (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, 1822-1824 (Alberto Lizzio).
  
  
  

Student REU Projects on Mathematical Music Theory

• Padraic Bartlett, Triads and topos theory.
  

• Hannah Gordon, The neo-Riemannian group.
  

• John Sternberg, Conceptualizing music through mathematics and the generalized interval system.
  

• Milan Kidd, An introduction to the practical use of music-mathematics.
  

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 Well-Tempered 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 Well-Tempered 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