## Exponential versus Myhill event distributions

This applet implements an exponential distribution and a Myhill distribution. It uses an 'entry delay approach' to generate event densities, that is, each event is preceeded by a random delay which was generated by one of these two distributions. Red events are placed using an exponential distribution. Black events are placed using a Myhill distribution. Use the slider to slowly increase the Ratio value, and notice that the Myhill distribution gradually approaches the exponential distribution.

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Musical Application: Using the Myhill distribution to deconstruct Schubert

Myhill distribution and additional comments from "A Catalog of Statistical Distributions" by Charles Ames. Leonardo Music Journal, Vol. 1 No. 1, 1991

The Myhill distribution can be thought of as a more flexible exponential distribution; one whose variance can be controlled. This control is reflected in a ratio whose value causes the Myhill distribution to more or less approximate the exponential distribution. When the ratio is high (above 128), the two distributions are indistinguishable. When ratio is low, the Myhill transform generates increasingly regular, periodic patterns, as opposed to the clustering and leaps of the exponential transform.

EXPONENTIAL DISTRIBUTION
This gives you lots of short entry delays with an occasional long one. A nice balance, but with no control over standard deviation from the mean event density. These comments were taken from Ames, LMJ.
``` f(u,m) = -(ln(1-u)) / m
where m = mean event density
u is a driver value (ie. random float between 0..1)
```

MYHILL DISTRIBUTION
This distribution maintains the balance of long to short durations that makes the logarithmic distribution appealing, additionally giving you control over mean AND variance. Also eliminates exceedingly short and exceedingly long values. These comments were taken from Ames, LMJ.
``` f = -ln[(U1-U2)u + U2] / m
where U2 = the (1-R)th root of R and U1 = U2^R
R is a ratio relating the maximum and minimum samples.
m is mean event density
u is a driver value (ie. random float between 0..1)
```
Ames on Ratio values:
Ratio represents approximation to pure exponential. It is the ratio relating the maximum and minimum allowed samples. Thus a ratio of 5.0 with a mean of 7.0 would yield a range from 2.8 to 14.0, since 14/2.8=5 and is centered on 7
Ratios over 128 makes Myhill behave indistinguishably from exponential.
Ratios from 4 to 16 produce approximation ranging from 50% to 75%
Ratios less than 2 are 'drunkenly regular'.
Closer and closer to unity ratios generate more and more regular event patterns.

Java programming by Nick Didkovsky, didkovn@mail.rockefeller.edu
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