Barlow's indigestibilty function

> Georg, I wonder if you care to comment on Barlow' harmonic complexity
> functions. We've had quite a lot of discussion on this list about
> consonance/dissonance measures and much has been said that would be directly
> relevant to his models, but no one has presented an explanation of how they
> are derived.
>
> -Paul

Paul-

Clarence Barlow basically wrote two important publications concerning the
derivation of his theory of tonality.
A 124-page article ("Bus Journey to Parametron") published in 1981 in the
Cologne Feedback Paper series, and a shorter article ("Two essays on
Theory") which was printed in one of the 1987 issues of the Computer Music
Journal.
In his very amusing Bus Journey, Barlow makes the assumption that the
"harmonicity" of an interval is related to the divisibility of the numbers
involved in the frequency ratio.
(He observed that this might be related with our mental ability to divide
circles such as a pizza in a given number of segments, equal in size. When
asked to give an order of how easily one can cut the circle in 2 to 9
segment, his friends gave orders which were surprisingly consistent.
Interestingly, the famous cognitive psychologist Roger Shepard conducted
research on mental complexity of numbers with similar results in the
mid-70s.

Hence, Barlow's (scientifically unproven, but irrefutable) assumption that
an analogous mechanism also determines the
harmonicity/consonance/concordance of an interval, and the "metrical
affinity" of simultaneous meters. He looked for existing theories and found
that Euler's "gradus" function gave a pretty good approximation of the
divisibility problem but with a fundamental shortfall.

Barlow: "Those epitomes of indivisibility, the prime numbers, have functions
larger than those of all of the smaller numbers, and the larger the primes,
the larger the function. This fitted well into the requirements; but no
difference was made between the functions of 1 and 2, or 3, 4 and 6, or 7
and 9, etc. I had to keep lookingŠ After some extensive if not exhaustive
experimentation, I finally came up with following seemingly workable formula
[http://www.mhs-muenster.de/Dozenten/Hajdu/Articles/LowEnergy.pdf , page 2
(1)]Š (Arranging the numbers 2 to 9 for example in order of increasing
indigestibility, one gets [2-4-3-8-6-9-5-7], which I would say compares
pretty well with the guess hazarded above".

From here, it was only a small step to the derivation of the "harmonicity"
formula [http://www.mhs-muenster.de/Dozenten/Hajdu/Articles/LowEnergy.pdf,
page 2 (2)] which also yields the "polarity" of an interval (a striking
effect when comparing a fourth with a fifth, or minor third with a major
third).

I hope this helps answering your question.

BTW, Barlow's pioneering paper can be ordered from
Frog Peak in the US (http://www.frogpeak.org/) or directly from Feedback
Verlag in Cologne (http://genterstr.hypermart.net/feedback.html).

Georg

Thanks Georg. Now I wonder if
you'd care to comment on your
application of Barlow's function
to your paper. You seem to be
after a notion of "consonance",
and bring up some recent
psychoacoustic concepts in
defining it, yet rather than
investigating the mathematics of
these phenomena themselves,
you bring in Barlow's function to
stand for consonance. Can you
elaborate on your reasoning
behind this move?