
The monster prime graph
Here’s a nice, symmetric, labeled graph: The prime numbers labelling the vertices are exactly the prime divisors of the order of the largest sporadic group: the monster group $\mathbb{M}$. \[ \# \mathbb{M} = 2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71 \] Looking (for example) at the character table of the monster you can check that there is an edge between two…

Mamuth to Elephant (3)
Until now, we’ve looked at actions of groups (such as the $T/I$ or $PLR$group) or (transformation) monoids (such as Noll’s monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords. Elephantlovers recognise such settings as objects in the presheaf topos on…

Mamuth to Elephant (2)
Last time, we’ve viewed major and minor triads (chords) as inscribed triangles in a regular $12$gon. If we move clockwise along the $12$gon, starting from the endpoint of the longest edge (the root of the chord, here the $0$vertex) the edges skip $3,2$ and $4$ vertices (for a major chord, here on the left the…

From Mamuth to Elephant
Here, MaMuTh stands for Mathematical Music Theory which analyses the pitch, timing, and structure of works of music. The Elephant is the nickname for the ‘bible’ of topos theory, Sketches of an Elephant: A Topos Theory Compendium, a two (three?) volume book, written by Peter Johnstone. How can we get as quickly as possible from…