Periodicity blocks

Fokker's article "Unison vectors and periodicity blocks in the three-dimensional (3-5-7-) harmonic lattice of notes" is added to the HF-website:
http://www.xs4all.nl/~huygensf/doc/fokkerpb.html

Manuel Op de Coul coul@ezh.nl

Hi Paul,

Thanks for the ref. on periodicity blocks.

http://www.ixpres.com/interval/td/erlich/intropblock1.htm
<http://www.ixpres.com/interval/td/erlich/intropblock1.htm> .

Yes, I now know what they are, and the articles were very clear.

thanks!

Robert

(Apologies if this is double posted.)

Quick question: Is there anything in the definition of a periodicity
block that requires all intervals within the block to be larger than
the block's unison vectors?

I can't find such a requirement stated explicitly, but Erlich's
"Gentle Introduction" does seem to imply it -- a unison vector is
spoken of as "too small a distinction to keep on your instrument"; a
block containing a smaller interval would seem to be inconsistent with
that point of view.

On the other hand, way back in 1999 (the url:

/tuning/topicId_5524.html#5524?var=0&l=1

) Erlich wrote 'Bohlen's "diatonic" JI scale ... is a periodicity
block with UVs (1 2) = 245:243 and (4 -1) = 625:567'. Yet that block
contains intervals of (-2 0) = 27:25 = 133 cents, smaller than 625:567
= 169 cents.

Hence the question...

- Rich Holmes

That should be (1 2) and (4 -5) = 16807 : 16875

http://www.kees.cc/tuning/perbl.html
http://www.kees.cc/tuning/s357.html

--- In tuning@yahoogroups.com, "rsholmes" <rsholmes@...> wrote:
> ) Erlich wrote 'Bohlen's "diatonic" JI scale ... is a periodicity
> block with UVs (1 2) = 245:243 and (4 -1) = 625:567'. Yet that block
> contains intervals of (-2 0) = 27:25 = 133 cents, smaller than 625:567
> = 169 cents.
>
> Hence the question...
>
> - Rich Holmes
>

--- In tuning@yahoogroups.com, "Kees van Prooijen" <lists@...> wrote:
>
> That should be (1 2) and (4 -5) = 16807 : 16875

But that gives a 13-note scale, not one of Bohlen's 9-note scales as
Erlich was discussing.

> http://www.kees.cc/tuning/perbl.html

I see here the statement "My interpretation of good periodicity blocks
is ... they do not contain smaller intervals than the defining unison
intervals ...", so I know how you feel about the answer to my
question; is that the consensus opinion?

> http://www.kees.cc/tuning/s357.html
>
>
> --- In tuning@yahoogroups.com, "rsholmes" <rsholmes@> wrote:
> > ) Erlich wrote 'Bohlen's "diatonic" JI scale ... is a periodicity
> > block with UVs (1 2) = 245:243 and (4 -1) = 625:567'. Yet that block
> > contains intervals of (-2 0) = 27:25 = 133 cents, smaller than 625:567
> > = 169 cents.

- Rich Holmes

hi Rich,

--- In tuning@yahoogroups.com, "rsholmes" <rsholmes@...> wrote:
>
> (Apologies if this is double posted.)
>
> Quick question: Is there anything in the definition of a
> periodicity block that requires all intervals within the
> block to be larger than the block's unison vectors?
>
> I can't find such a requirement stated explicitly, but Erlich's
> "Gentle Introduction" does seem to imply it -- a unison vector
> is spoken of as "too small a distinction to keep on your
> instrument"; a block containing a smaller interval would seem
> to be inconsistent with that point of view.

As you pointed out, Paul Erlich did state that explicitly,
altho I don't know if Fokker's work actually states this (and
don't have time to study it again right now), but i certainly
read it this way.

(Fokker's work had been studied by a handful tuning theorists
up to 1999, and it was a discussion which Paul and i had
at his house in February of that year which resulted in his
"Gentle Introduction" being posted on my website and in turn
led, AFAIK, to the rest of the tuning cyber community getting
interested.)

The name "unison-vector" implicitly carries with it the idea
that the interval bounded by the two ratios which define it
is small enough to be considered a "unison", whereas all
the intervals defined by the set of ratios contained
within the periodicity-block are to be considered as
"degrees" of the tuning, or "steps" of the scale for
the smaller blocks.

This reasoning in turn implies that none of the intervals
within the block should be smaller than the unison-vectors,
which is the reverse-negative way of saying the same thing
you asked.

If you search the archives of this list sometime around
2000 or 2001 (i think), you'll turn up a long-running debate
about whether the word "unison" should be a part of this term,
the argument being that "unison" carries the very special
connotation of a "single sound" which only applies to
the actual unison ratio 1:1. I personally have no problem
with Fokker's term, because "unison" and "unison-vector"
are two different things (rather like "milk" and
"chocolate milk"). The "unison" refers specifically to
a heard sound, whereas "unison-vector" refers specifically
to a theoretical construct on a lattice-diagram, which is
an interval that is never actually heard.

-monz
http://tonalsoft.com
Tonescape microtonal music software

So would you say, then, that the "periodicity block" mentioned here:

/tuning/topicId_5524.html#5524?var=0&l=1

is in fact not a periodicity block, since it contains intervals
smaller than one of its unison vectors?

- Rich Holmes

rsholmes wrote:
> (Apologies if this is double posted.)
> > Quick question: Is there anything in the definition of a periodicity
> block that requires all intervals within the block to be larger than
> the block's unison vectors?

There are two papers on the Huygens-Fokker website dealing with periodicity blocks. One in English and one in French. So you can check them if you want it from the horse's mouth.

Fokker certainly doesn't mention such a constraint. He says unison vectors should be "unisonous" and chooses sensible examples. If you're using stranger vectors it's up to you whether you think they work as unisons or not. One thing, though, is that he does say that any number of unison vectors are also a unison vector. So for a meantone block you could add 10 syntonic commas to get a unison vector 215 cents wide. But it'd be perverse to do so.

I think the only thing a periodicity block has to be is periodic. It doesn't have to follow the parallelogram construction. So can't you always choose unison vectors that are sufficiently small?

> I can't find such a requirement stated explicitly, but Erlich's
> "Gentle Introduction" does seem to imply it -- a unison vector is
> spoken of as "too small a distinction to keep on your instrument"; a
> block containing a smaller interval would seem to be inconsistent with
> that point of view.

Yes, but we generally don't agree on the definitions. You can adopt any mathematically precise rules you like in an given context, as long as you're explicit about it. There hasn't been a classical text on this after Fokker that we can refer to for accepted definitions.

I prefer to keep the definitions simple. And, in cases like this, vague.

Graham