# Wolfram Summer SchoolFounded in 2003

17th Annual Wolfram Summer School, held at Bentley University June 23–July 12, 2019

Alumni

# Kristoffer Josefsson

Class of 2014

## Bio

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.

### Background

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

• Acoustic Tonnetz
• Common-tone 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).

### References

The Generalized Tonnetz

Rule 188595