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Wolfram Summer SchoolFounded in 2003

16th Annual Wolfram Summer School, held at Bentley University June 24–July 13, 2018


Kristoffer Josefsson

Class of 2014


Kristoffer Josefsson has studied mathematics at Gothenburg University and spent several years doing research in discrete differential geometry at TU Berlin under a Marie Curie scholarship. Thereafter, he spent three years working as a specialist geometer for Foster + Partners in London. He is interested in applications of differential geometry in the arts in general and architecture in particular.

Project: Tonnetz Automata

I will create a mapping from a simple generative system (CA or Turing machine) to a Tonnetz in order to create chords and melodies. This way I will be able to evaluate the system from a musical point of view.

There are a couple of steps:

  1. Find a suitable Tonnetz. For example, the two-dimensional model above corresponds to almost-even chords of three notes. Call this space T. But we might look at examples with more notes per chord too.
  2. Fiber the space T with (S^1) or ℝ; call the fiber N. This is the space in which the melody gets generated. This melody will automatically move between chords according to the position on T.


Voice-leading Tonnetz is one out of three interpretations of Tonnetz:

  • Acoustic Tonnetz
  • Common-tone Tonnetz
  • Voice leading Tonnetz

Only the voice-leading Tonnetz has a (simple) geometric dual relationship: the note-based triangular net and the chord-based hexagonal net.

The mapping can be seen here:

There's also a connection to Neo-Riemannian theory and the groups generated by PRL moves. Note that D=L\[SmallCircle](R^-1).


The Generalized Tonnetz

Favorite Outer Totalistic r=1, k=2 2D Cellular Automaton

Rule 188595