75, rue de la Roquette

75011 Paris

tom@johnson.org

*Abstract: Self-replicating melodic loops
have already been discussed in the last section of the author's book *Self-Replicating
Melodies*, and in a more mathematical article by David Feldman, and employed
in several compositions by Johnson and other composers. This presentation is
an introduction to the idea, along with computer output of orbit structures,
and some other more recent observations.*

In New York in the early 70s a number of composers
of my generation were reacting against the American academic musical tradition,
best known in Europe in the examples of Elliott Carter and Milton Babbitt. We
wanted something simpler, something that spoke more directly to a wider audience.
Of course, our reaction was to become known as American minimalism, and the
best known segment of it was the repetitive music of Terry Riley and Philip
Glass and others, but there were other important forms: the long drone tones
of La Monte Young, the meditative sounds of Pauline Oliveros, the sustained
microtonal textures of Phill Niblock, the acoustical experiments of Alvin Lucier,
and of course, pieces on limited scales, such as John Adams' *Shaker Loops*
and my own *Four-Note Opera* (1972). As the '70s went on, however, I found
myself increasingly attracted to the more rationally organized forms of minimalism,
such as the logical melodies of Frederic Rzewski. I had been playing piano duets
with Philip Corner, another colleague who liked to count notes and calculate
logical sequences, and it was really in playing piano with him that I began
to appreciate melodic loops, sometimes playing them in unison in different tempos,
like this:

a b c a b c a b c a b c a b c a b c a b c...

a b c a b c...

This is a trivial case, since any loop of *n*
notes can be in unison with itself when played *(n + 1)* times slower or
faster. I wanted to go further, and find more interesting ways of making loops
within loops, and one day I decided to consult my friend David Feldman with
this question: How could I write a melodic loop of, say, 15 notes, in such a
way that one will hear the same melody if one listens to every note, or to every
second note? Feldman is himself a composer, but he is also a professional mathematician,
and he was able to answer the question a couple of days later. He explained
that I could write whatever notes I desired, as long as are were not more than
five notes in the scale, and provided that they fall together on the loop in
this way.

(0)

(1 2 4 8)

(3 6 9 12)

(5 10)

(7 11 13 14)

This means that note *0* can be anything,
but that notes *1, 2, 4,* and *8* of the loop must be identical, notes
*3, 6, 9,* and *12* must be identical, and so on. With this information
I wrote the first "self-replicating melody," which was to become *Rational
Melody No. 15* (1981). Sometimes we find extraordinary things and just don't
see any way of going further with them until much later, and it was more than
10 years later that I began to realize how many things this principle would
lead to.

I won't pretend to be able to really explain how
these orbit structures come about, as this is a job for a mathematician well
versed in group theory, but for musicians, let me demonstrate how one might
compose a self-replicating melody without really knowing about such things.
Let's say that we want to compose a seven-note loop that makes a copy of itself
at *2 : 1*. Since we must be able to hear the same melody at the two different
tempos simultaneously, we have to see how the seven notes of the fast loop will
fall against the seven notes of the slow loop:

0 1 2 3 4 5 6 0 1 2 3 4 5 6 0...

0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0...

Note *1* of the slow melody occurs at the
same time as note *2* of the fast melody, and note *2* of the slow
melody occurs at the same time as note *4* of the slow melody, so it is
clear that notes *1, 2,* and* 4* must all be the same. Notes *3,
5, *and *6* also must be identical, since they sometimes occur simultaneously,
so now we can deduce that there are three orbits in this form:

(0)

(1 2 4)

(3 5 6)

In other words, what I call "the solution
loop" must be a melody on a three-note scale that has the form *0 1 1
2 1 2 2*. Here you can see how the two different speeds fall in unison if
a seven-note loop employs three pitches in this way :

0 1 1 2 1 2 2 0 1 1 2 1 2 2...

0 1 1 2 1 2 2 0 1 1 2 1 2 2 0 1 1 2 1 2 2 0 1 1 2 1 2...

Naturally, if the loop at half time is in unison with the loop at the original tempo, the same loop at quarter time must be in unison with the loop at half time, and we can continue with voices 8 times slower than the original, 16 times slower, and so on, and they will all be in unison.

There is no way to construct an eight-note loop
in such a way that it will self-replicate at *2 : 1*, because *8*
is divisible by *2*. An eight-note loop can make a copy of itself at *3
: 1*, however, or *5 : 1*, or *7 : 1* in these ways:

3 : 1: (0) (1 3) (2 6) (4) (5 7)

5 : 1: (0) (1 5) (2) (3 7) (4) (6)

7 : 1: (0) (1 7) (2 6) (3 5) (4)

There are five different orbits at *3 : 1*
and *7 : 1*, and six different orbits at *5 : 1*, so we can have many
different notes in the scale. There is another orbit structure that I find more
interesting however.

(0 6) (1 5) (2 4) (3 7)

Here there are only four orbits, but they are
arranged in such a way that, if we set the starting times right, they permit
our melodic loop to make copies of itself at all of the above ratios, and in
fact*, 9 : 1, 11 : 1* (11 modulo 8 = 3) and every other odd numbered ratio
as well.

0 1 2 3 2 1 0 3 0 1 2 3 2 1 0 3 0 1 2 3 2 1 0 3 0 1 2 3 2...

0 1 2 3 2 1 0 3...

0 1 2 3 2 1...

0 1 2 3...

As I was studying all this, quite a few months
after completing *Self-Similar Melodies*, I found it necessary to write
a computer program that would compute the orbit structures of loops I might
want to use someday. Good mathematicians told me that my 58 pages of computer
output, which considered all the possible self-replicating loops of 36 notes
or less, is completely trivial, because they can figure out the orbit structures
for any given situation about as easily as we can punch out multiplication on
our calculators. But for me the exercise was not at all a waste of time, and
I often refer to my 58 pages of orbit structures when I am looking for one that
has particular characteristics. Let us look at all the possible self-replicating
loops of 13 notes, for example:

Loop of 13, ratio of 2

(0)

(1 2 4 8 3 6 12 11 9 5 10 7)

Loop of 13, ratio of 3

(0)

(1 3 9)

(2 6 5)

(4 12 10)

(7 8 11)

Loop of 13, ratio of 4

(0)

(1 4 3 12 9 10)

(2 8 6 11 5 7)

Loop of 13, ratio of 5

(0)

(1 5 12 8)

(2 10 11 3)

(4 7 9 6)

Loop of 13, ratio of 6

(0)

(1 6 10 8 9 2 12 7 3 5 4 11)

Loop of 13, ratio of 7

(0)

(1 7 10 5 9 11 12 6 3 8 4 2)

Loop of 13, ratio of 8

(0)

(1 8 12 5)

(2 3 11 10)

(4 6 9 7)

Loop of 13, ratio of 9

(0)

(1 9 3)

(2 5 6)

(4 10 12)

(7 11 8)

Loop of 13, ratio of 10

(0)

(1 10 9 12 3 4)

(2 7 5 11 6 8)

Loop of 13, ratio of 11

(0)

(1 11 4 5 3 7 12 2 9 8 10 6)

Loop of 13, ratio of 12

(0)

(1 12)

(2 11)

(3 10)

(4 9)

(5 8)

(6 7)

Since *13* is a prime number, all ratios
from *2* to *12* are possible, and four of these produce only two
orbits, which do not permit very interesting two-note melodies. It is not surprising
that the orbits are the same whether the ratio is *3 : 1* or *9 : 1*,
since *9* is three times three, but note that the results are completely
different when the ratio is 2 : 1 or 4 : 1 or 8 : 1. A number of other observations
might be made, but let me leave you to do your own research here and conclude
with something else. You may find this information useful, and probably won't,
but it is a good example of how computer lists can be useful, even when mathematicians
tell us that the information is trivial.

Naturally scales of only three or four notes often seem too limited, and one question that interested me ever since the beginning of this investigation was how to permit the complete chromatic. What is the smallest loop possible that permits 12 orbits, the complete chromatic scale? I posed this question to very good mathematicians like Feldman, and my friend Jean-Paul Allouche, and they couldn't tell me. It is not hard to find loops with 25 or 30 different notes, that can have 12 or more orbits, but it is difficult to say with any certainty that one has found the shortest one. Only after making my 58-page list was it clear, much the surprise of everyone, that the answer to my question was a loop much shorter than anyone expected, a loop only 16 notes long, with this orbit structure.

Loop of 16, ratio of 9

(0)

(1 9)

(2)

(3 11)

(4)

(5 13)

(6)

(7 15)

(8)

(10)

(12)

(14)

Unfortunately, the loop does its thing only at
a ratio of *9 : 1*, which is musically a little difficult to hear and work
with, but it is information that can perhaps be useful for composers who like
difficult things. It is something that a minimalist like myself can turn over
to maximal composers, like Elliott Carter and Milton Babbitt.

__Bibliography:__

Allouche, J.P., Johnson, Tom. "Finite Automata
and Morphisms in Assisted Musical Composition," in *Journal of New Music
Research,* Vol. 24 (1995), pp. 97-108).

Feldman, David, review of *Self-Similar Melodies,*
in Leonardo Music Journal, Vol. 8, 1999, pp. 80 -84.

Johnson, Tom (1996), *Self-Similar Melodies*
(291 pp), Editions 75, 75 rue de la Roquette, 75011 Paris. The discussion particularly
devoted to self-replicating melodies is PP. 233 - 277. The *Rational Melodies*,
*Loops for Orchestra* (to be premiered in the Steirischer Herbst in September),
and other scores of Johnson employing self-replicating melodies are also available
from Editions 75.

Johnson, Tom, "Automatic Music" in *Actes
des Journées d'informatique musicales 98,* pp. F1 - 1 to F1 - 5.