The Topos of Triads

Paul Slevin (University of Glasgow)

Friday 23rd January, 2015 16:00-17:00 Maths 204


In my talk I will discuss Padraic Bartlett's paper "Triads and Topos Theory", an REU project carried out at the University of Chicago in 2007 which is based on the work of mathemusican Thomas Noll. We say that two pitches are equivalent if they differ by a whole number of octaves (i.e. that the frequency of one pitch is an integer power of 2 times the other), and if we restrict to our modern system of musical notes, there are twelve equivalence classes of pitches. The cycle of fifths gives us a natural bijection from these "pitch classes" to the group $\mathbb Z/12$, and so we can try to translate musical ideas over to the mathematical setting - for example, an interval is an order pair and a triad is a three element subset. In fact, the monoid of affine transformations that preserve the C major triad ({0,1,4}) defines a monad on the category of sets, and it turns out that the Eilenberg-Moore category for this monad admits the structure of a topos with a subobject classifier which is fairly easy to describe. Whenever you have a topos, you have a notion of topology, and somehow in the above example this topology corresponds to the emotional topology of music. At the end of the talk we will apply some of the ideas to a specific piece of classical music.

You do not need to know anything about music to understand this talk, but it would help to know the extreme basics of category theory. If you don't, then at least you will get to hear some music.

Add to your calendar

Download event information as iCalendar file (only this event)