RE: A gentle introduction to Fokker periodicity blocks, part 1

Rick, you are right that there is nothing that doesn't predate Fokker (a
20th century physicist) in the periodicity block concept when applied to the
circle of fifths. The point of part 1 was so for readers to gain familiarity
with the terminology and concepts in the context of some easy examples. Part
2 will deal with 2-dimensional, 5-limit JI, which is where Fokker's original
contributions (though independently discovered by Kees van Prooijen and
probably others) can be said to begin.

>After a first reading, it seems to just be another way of saying
>that you can have a closed circle of pitches with some specified relation,

>and that's equivalent to any other closed circle on the same specified
>relation, by transposition through some interval.

>Is that right?

More specifically, I was trying to point out that the entire infinitude of
3-limit JI (Pythagorean tuning) can be collapsed to a finite set by choosing
just one small interval as your unison vector. Then, no matter how far out
you go along the chain, any note you pick is equivalent to one and only one
of the finite number of notes in your periodicity block. By way of a
familiar geometrical model, the chain of fifths becomes a finite circle of
fifths. This model sounds like what you were referring to. A peek into part
2: in 5-limit JI we need two unison vectors at a time, which would turn the
infinite sheet of fifths and thirds into a finite torus (donut shape). But
these closed geometrical figures are more appropriate when one is discussing
temperament, which may arise from the desire to "hide" the unison vector(s)
by distributing them equally (or unequally) among the consonant intervals.
Fokker himself gave his periodicity blocks in JI, using a (slightly
arbitrary, but of definite quantity) set of JI pitches to which no new pitch
could be added without having a unison vector between the new pitch and one
of the existing pitches.

Hope that clears things up,

Paul