## Sunday, 24 May 2020

### COMPOSING MUSIC BY SIMULATION FROM MULTIPLE POINT PROCESSES

They are lots of articles in the literature about using stochastic processes e,g.Markov Chains to compose music. I don't know whether the following suggestion is original.

A log-Gaussian Cox Process driven by an Ornstein-Uhlenbeck Process.

This process was first proposed by Tom Leonard (Journal of the Royal Society, Series B, 1978) . It is a special case of the Doubly Stochastic Poisson Process described by my undergraduate mentor David Cox in the same journal in 1955

Suppose you wish  to compose a piece of music on the time interval (0,T) and assume that each note may assume one of M different characteristics, Then the times of occurrences of any particular characteristic in the tune may be represented by points in the interval (0,T), These could be specified by the composer  (see above diagram) . If repeated for each of the N characteristics, then the N sets of points (when superimposed with different labels on the same interval) defines the entire composition.

For any particular characteristic, it would instead be possible to simulate the points from a random point process on the interval (0,T). If a different point process is used to generate the points for each characteristic, then the entire composition can be simulated.

A log-Gaussian Cox Process can  be parameterised using a mean value function M(t) for t assuming values in the interval (0,T),  See, for example, the smoother curve in the above diagram.

If the Cox Process is driven by an Ornstein-Uhlenbeck process. then a variance V and a correlation parameter R also need to be specified.

It is left to the composer to come up with an imaginative choice of the continuous function M. He may then use a standard computer package to simulate the points of occurrence of that particular characteristic, for specified V and R (which could of course be fiddled with to improve the quality of the tune).

If the composer can devise a different choice of continuous function M for each of the N characteristics, then his entire composition can be simulated from the N thus defined Cox Processes, e.g.by keeping V and R fixed and specified across all simulations.

This suggests to me to suggest a whole neball park, where the ingenuity of the composer is still an essential ingredient.

## Friday, 1 May 2020

### SAMPLE SPACE--A NEW PUBLICATION BY THE DEPT OF STATISTICAL SCIENCE AT UCL

SAMPLE SPACE

FROM  PROFESSOR RICHARD CHANDLER