Paul Rapoport's article

For those who want to know the formulas used to calculate the numbers
in the big file I've put together, or to know more about the theory of
equal temperaments in general I'd like to make mention of Paul
Rapoport's very interesting article
"The Structural Relationships of Fifths and Thirds in Equal
Temperaments", Journal of Music Theory vol. 37 no. 2, pp. 351-389,
Fall 1993.
It has three sections about respectively the perfect fifth, the
major third and the relationships between the two. It must be said
however that it's somewhat difficult material, but anyone with some
mathematical background should be able to follow it.
The intervals figurating in the article are:

p Pythagorean comma a perfect octave
k syntonic comma v perfect fifth, structurally just
d diesis t major third, structurally just
s schisma Theta major third, structurally Pythagorean
q diaschisma

I have found it practical to have an overview of the equations in
question while studying the article, which is this:

Theta = 4v - 2a
p = 12v - 7a
k = Theta - t
d = a - 3t
s = p - k
q = 2k - p
Schismatic major third = 5a - 8v.

Two errata that should be mentioned:
page 379, one line above the bottom line, replace
"Basic t is best t." with "Best t=5."
page 387, on the bottom line the second expression must be
19v-11a.

Manuel Op de Coul coul@ezh.nl

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