fibs and beyond

Hi everybody...

I got lost a while back during list site drift and, due to several
offbeat projects consuming my time, didn't get around to resubscribing
until now. It's good to be back, and it's good to see that most of
the e-faces are still around.

Paul Erlich has brought up the Fibonacci sequence again. To be
honest, while I've always been fascinated with it myself, I didn't
really know what to make of it musically. That is, until recently.
Many years ago I was heavily into weaving on a four-harness loom.
When a friend of mine sent me a paper several years ago on tiling the
plane with the usual diatonic and other scale forms, it hit me that
there was a deep relationship: math-weaving-music. The following is
only one possible direction this relationship might take.

The list may have discussed this before, in which case I apologize
for not doing my homework, but what really got me excited (after
slapping myself in the head and shouting "well, DUH!") was
surf-stumbling into the following web site at the computer science
department at Arizona State:

http://www.cs.arizona.edu/patterns/weaving/Sequences/sequences.html

All of a sudden I was compelled to ask, what happens when you apply a
modulus to an infinite sequence? In the case of Fibonacci, the result
(which can be seen visually at the above site which shows
treadle/harness sequences for looms) is extremely interesting and
provocative for compositional theory, whether you're considering
pitch-class sets, point or duration-class sets, or macro-design
proportions.

Again, hoping I'm not repeating what you already know, here's a
couple of examples. For reference, here's the classic Fibonacci
again:

1,1,2,3,5,8,13,21,34,55,89,... ad infinitum. Now, "ad infinitum"
isn't very helpful in a piece of music we hope will end in our
lifetime; plus, the larger the terms get, the more difficult they are
to "grasp" conceptually. So here's what happens when we begin to
short circuit the series by modding it out:

mod 2:
1,1,2,1,1,2,...
Pretty simple (like 2 sixteenths and an eighth repeated over and
over); as you can see, Fib mod 2 has a period of 3. Now things get
interesting.

mod 3 (period = 8):
1,1,2,3,2,2,1,3,1,1,2,3,2,2,1,3,...
For the next couple I'll just give the basic repeating sequence.

mod 4 (period = 6):
1,1,2,3,1,4,...

mod 5 (period = 20):
1,1,2,3,5,3,3,1,4,5,4,4,3,2,5,2,2,4,1,5,...

I'll let you work out the next few if you're interested - mod 6 has
period 24, mod 7 has period 16, mod 8 has period 12, etc. (BTW, the
above site incorrectly states that Fib mod 10 has no discernible
period; in fact, its period is 56). One more comment. The Fib mod 5
(and some other modded-out Fibs) can be stacked to show an additional
internal symmetry:

1,1,2,3,5,
3,3,1,4,5,
4,4,3,2,5,
2,2,4,1,5,
...

So the rows reflect Fib mod 5, and the columns reflect the
permutation (1342)(5) (i.e., each vertical term t = 3(t - 1) mod 5)

The Tribonacci sequence Paul has brought up, if we cap it at mod 5,
has a period of 31:

1,1,1,3,5,4,2,1,2,5,3,5,3,1,4,3,3,5,1,4,5,5,4,4,3,1,3,2,1,1,4,...

At mod 3, it has a period of 13:

1,1,1,3,2,3,2,1,3,3,1,1,2,...

The Arizona site (built by Ralph Griswold) has other interesting
"weaving" sequences, e.g.,

Pell
Perrin
Hofstadter's chaotic sequence
Triangular (pentagonal, septagonal, undecagonal) numbers [really
cool]
Versum [surprise in the midst of chaos - keep scrolling to see it]
Connell and Lower & Upper Wythoff sequences [irregular regularity]
Irrationals (e, pi, etc.) [not as irrational as you might think when
modded out and squinted at]

It also gives you a gate to the Online Encyclopedia of Integer
Sequences where you can look up any sequence ever discovered (at least
that's the claim).

Like I said, I didn't check all the past tuning issues I've missed,
so I hope I'm not repeating a discussion that already exhausted the
subject.

Steve

Stephen Soderberg
Music Division
Library of Congress

"Stephen Guy Soderberg" <SSOD@LOC.GOV> wrote:

> I got lost a while back during list site drift and, due to several
> offbeat projects consuming my time, didn't get around to resubscribing
> until now. It's good to be back, and it's good to see that most of
> the e-faces are still around.
>
> Paul Erlich has brought up the Fibonacci sequence again. To be
> honest, while I've always been fascinated with it myself, I didn't
> really know what to make of it musically. That is, until recently.
> Many years ago I was heavily into weaving on a four-harness loom.
> When a friend of mine sent me a paper several years ago on tiling the
> plane with the usual diatonic and other scale forms, it hit me that
> there was a deep relationship: math-weaving-music.

Here is a tool for visually inspecting a sequence for periodicity:
http://home.netcom.com/~eugenek/download.html It may be of some use to
you. It could also be of some use to algorithmic music experimenters.
One of Tim Thompson's KeyKit toys at his site accepts GIFs as input.
I've input some GIFs of recurrence plots from some of the other sites
that mention recurrence analysis into it and gotten some reasonable
results. If nothing else, recurrence analysis produces pretty pictures!
--
Bye for now

Allan Myhara
Winnipeg, Manitoba, Canada

Stephen Soderberg's claim that the period for the Fibonacci seeries mod 10
is 56 is clearly wrong, since the period mod 10 must be divisible by the
periods mod 2 (3) and mod 5 (20). A quick check shows that it is indeed
60 shown as follows:

1 1 2 3 5 8 3 1 4 5 9 4 3 7 0 7 7 4 1 5
6 1 7 8 5 3 8 1 9 0 9 9 8 7 5 2 7 9 6 5
1 6 7 3 0 3 3 6 9 5 4 9 3 2 5 7 2 9 1 0

David Bowen

David Bowen (Digest 965.20) is exactly right. The period for the Fib
mod 10 is 60 (not 56). Reason (but not excuse) for my error: I didn't
bother to think it through but simply counted the sequence terms as
presented on the Arizona site up to the reappearance of 1,1. I don't
feel too bad though, since, as I said, the guy who put this site up is
Ralph Griswold (early developer of SNOBOL and former head of
Programming Research and Development at Bell Labs - retired now and
into weaving and number theory). Evidently geniuses don't take a lot
of trouble balancing checkbooks (or are possibly checkbook impaired).
But I don't have the excuse :-)

Chagrinned,

Steve

Stephen Soderberg
Music Division
Library of Congress