# Noam Elkies

### Canonical Forms: The Mathematical Structure of Musical Canons

#### April 7, 2016

*Public Lecture***Time:** 4:30pm**Location:** Varis Lecture Hall, Granoff Music Center 155**Reception:** 5:30pm**Location: **Murnane Lobby, Granoff Music Center

**Abstract: **Musical canons, from simple rounds like Three Blind Mice to the compendium of canons Bach compiled in his Musical Offering, have a history almost as long as that of Western music itself, and continue to fascinate musical composers, performers and listeners. In a canon the same melody is played or sung in two or more parts at once; this melody must therefore make musical sense both as a tune and in harmony with a delayed or otherwise modified copy of itself. How does one go about constructing such a melody? This challenge has a mathematical flavor. It turns out that some kinds of canons are so easy to create that they can be improvised in real time, while other kinds are more demanding, and in some cases only a handful of examples are known. The talk will be illustrated with both abstract diagrams and specific musical examples, and may also digress into generalizations of canons (the forms known collectively as 'invertible counterpoint') and the reasons--besides showing off-- that so many composers incorporate canons into their music.

### Poisson summation and packing problems

#### April 8, 2016

*Colloquium***Time:** 4:00pm**Location:** Science and Technology Center 136**Reception:** 3:00pm**Location:** Science and Technology Center Atrium

**Abstract: **The Poisson summation formula asserts that the sum over the integers of "any" function \(f\) on \(\mathbb R\) equals the sum over integers of the Fourier transform of \(f\). We give some examples and applications, including evaluations of familiar sums (such as Euler's solution

\(1 + \frac 14 + \frac 19 + \frac{1}{16} + ... + \frac{1}{n^2} + ... = \frac{\pi^2}{6} \)

of the Basel problem) and less-familiar ones such as

\(\frac 12 + \frac 15 + \frac{1}{10} + \frac{1}{17} + ... + \frac{1}{n^2+1} + ... \)

We then generalize to functions on \(n\)-dimensional space \(\mathbb{R}^n\), and report on the recent application to the solution by Viazovska et al. of the problem of densest packing of \(\mathbb{R}^n\) by identical spheres in the "magic dimensions" \(n=8\) and \(n=24\).

The Norbert Wiener Lectures were initially funded by an anonymous gift to the Department of Mathematics. The 2016 lectures are co-sponsored by the Department of Music and Tufts STS. All talks are free and open to the public.