Persistant dispersal chain tunings

At one stage in the development of his metricity parameter Klarenz Barlo
needed a quasi-random chain of numbers to contrast with the highest
metricity value. He needed a chain so that it could be used to generate
ostinati that would be sustained through changes of metre, thereby
requiring as many members as there would be attacks in each member. The
series should also be quasi-random in that it maintained a equal
distribution throughout the measure. This meant that the first two terms
would each be distributed in the two halves of the available range, the
first three terms in each third of the range, the first four in each
fourth, etc.. Curiously, Barlowe needed a chain of eighteen members, but =
he
was unaware of contemporary research among Ramsey theorists who found tha=
t
the largest such chain could have only seventeen members. Not being able =
to
find a chain of more than twelve members at the time, Barlough was forced=

to utilize other means to determine metricity in his 'autobus' pieces. =


The composer and mathematician David Feldman has posted a complete list o=
f
these chains, each term of which is notated as a range between two
fractional values. These are available at http://www.math.unh.edu/~dvf

Although Barhloch orginally intended these series for generating metrical=

attacks, anyone working with frequency ratios will soon recognize somethi=
ng
vaguely familiar - i.e. ratios well-distributed within an octave. Taking
one of the longer chains at random, and then assigning a freshman sum rat=
io
to each term, one can construct collections of pitches which grow in
surprising ways while maintaining their distribution within the octave. I=
n
particular, I have worked with a tuning parameter whose extreme values ar=
e
derived on one hand from the harmonic series and on the other hand from
such a chain. While all of the materials used are just, the resultant
collections are often tonally ambiguous, but that's okay since I remain -=
-
as always -- ambiguous about tonality. =


Daniel Wolf =