Re: Digest Number 293 &Algorithms

The algorithm I use to approximate ET degrees with ratios is Viggo
Brun's subtractive version of the Euclidean Algorithm. It us easier to
use and understand than Continued Fractions and converges slower as it
shows semi-convergents as well. It is closely related to the CF
algorithm as the number of times a given interval is subtracted is the
sames as the corresponding divisor term of the CF.

The Brun algorithm is easy to use when it is desired to approximate more
than two numbers simultaneously (e.g., find a temperament with good
3/2's, 5/4's, 7/4's, etc.). The corresponding ternary or higher-order
CF's tend to converge to rapidly, yielding only temperaments with very
large numbers of tones. J. M. Barbour developed a version which
converges slower, but Brun's algorithm is still easier to apply

Ervin Wilson has also used it to find constant-structures (MOS when when
only one interval and the interval of equivalent are being
approximated).

See Mandelbaum's dissertation and Brun's papers for details. These
should be listed in the Tuning bibligraphy.

--John