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Generating modal sequences (a remote approach to minimal music).

1. Palindromes (by Means of Multiplications of Prime4 Numbers)

The Usage of the "sieve of Eratosthenes" (an ancient algorithm for finding prime numbers) in music is rich in suggestions and consequences; on the psychological-musical level it appeals to the sense of repetition. One only needs to mark the multiples of prime numbers in time to reveal complex sequences which offer both remarkable regularities and surprises.

1.1 Let us take two prime numbers and mark their multiples on the axis of time. Let them be A = 3 and B = 5 (Example 1).

We are faced with an infinite sequence having the period

Listening to this sequence in time (two percussion instruments can mark these periodicities) we shall clearly perceive the union of two periodicities 3 U

5 . We are in the presence of a phenomenon specific to early American minimalism: the loop technique. Bringing a new prime number into play will considerably increase the complexity of the phenomenon. Let us add for instance C = 7 (a third instrument could mark this new periodicity-see Example 2). This

3 U 5 U 7 generates a much more complex infinite sequence; its period will be much longer and consequently more difficult to perceive.

1.2 In order suggestively to illustrate the increased difficulty of perception upon adding a new prime number, we shall show this phenomenon graphically. Let us take a rectangle with sides A = 3 and B = 3 (see Example 3).

We can cover it with one stroke of the pencil by successive diagonals starting from one corner. If we start from a3 we get:

In fact, in order to cover the rectangle with one stroke of the pencil it is necessary that its sides should be prime to one another (admitting of no other common divider than 1; any two or more prime numbers are also prime to one another). We are presented with a palindrome (it reads the forwards and backwards); the sum total of its elements is 15 = 3 x 5.

It includes the periodicities 3 and 5 .

The palindrome resulting from the product 5 x 7 is

In order to apply this graphic method to the union of three periodicities we shall have to resort to a parallelepided. For 3, 5, and 7 the parallelepiped will be as shown in Example 4.

Starting from one corner, it will be much more difficult to imagine (to perceive the series of successive diagonals which will illustrate the union 3

U 5 U 7 . In order to unite four elements we shall need a four-dimensional body, and so forth. Conscious perception becomes increasingly difficult.

1. 3 The traditional musical graphic method seems the easiest for calculating these unions (Example 5). This makes us reflect once again on the brilliant accuracy of classical musical notation.(1)

As a musical exemplification I propose my work Soroc (1984), which is an alternation of rhythmic segments on indefinite sounds with segments of definite sounds lacking definite durations.

2. Loops of Loops (by Means of Successive Subtractions)

In my Cartea modurilor (Vieru 1980), the fifth and final chapter ("Modal Sequences," pp. 148-73), starting from the calculation of finite differences, describes a method of generating systems of modal sequences. In the following paragraphs I shall apply the method of successive differences to palindromic modal structures.

We will deal first with five-element structures. The sum total of the elements of a modal structure (modulo 12) will always be 12, the modal structure being conceived as an infinite periodic sequence. For instance:

... 4 1 2 1 4 ( 4 1 2 1 4 ) 4 1 2 1 4 ...

Successive differences will always remain infinite periodic sequences (see Example 6).

The period of each sequence will be a multiple of 12:

9 + 1 + 11 + 3 + 0 = 24

4 + 10 + 4 + 9 + 9 = 36

6 + 6 + 5 + 0 + 7 = 24

2.1.1. The difference will also admit of the element 0 (zero), which is excluded from the modal structure. We shall therefore draw up a catalog of all modulo 12 palindromic expressions of five elements. They will be of an a b c b a type, where c is always an even number (see Example 7).

2.1.2. The sequence of differences of a palindrome is an antipalindrome in which the elements symmetrically arranged around the axis are symmetrical within the algebraic group modulo 12:

4 1 2 1 4

9 1 11 3 0

The 9 1 11 3 0 antipalindrome

where 11 + 1 = 0 and 3 + 9 = 0. The demonstration is obvious: Two adjoining elements a and b will become b and a in the mirror. The differences a - b and b - a will therefore be symmetrical within the algebraic group. For the same reason the differences of an antipalindrome are also palindromes. Successive differences will therefore be a permanent alternation of palindromes and antipalindromes.

2.1.3. Any palindromic sequence of five elements is the synonym of a palindrome in the catalog. In Vieru 1980, 53-54, a device will be found for transforming an adjacent expression into the equivalent modal structure (see Example 8).

2.1.4. If the successive differences are effected starting from (4 1 2 1 4), for instance, after forty-eight subtractions we come to the original palindrome in its identical expression (see Example 9).

It is therefore a loop of palindromes (a loop of loops), an infinite periodic series. By replacing the palindromes with their catalog equivalents we get the following sequence:

c o w h d n g a f s o f h e x c d n g t r b e r

If we start from (5 1 0 1 5) we get the sequence in Example 10.

We come now to the loop of palindromes

w b d n g a f s o f b e x c d n g t r b c r c o

which is a circular permutation of the previous one. One comes to this sequence only when one sets out from the following elements: a (or) b, c, d, r, s, t, w, x in their canonical catalog form. However, if one sets out from e (or) f,g, b, n, o, one arrives at the following loop sequence of palindromes:

c s r a g n d b x o b r i b f t g n d c w e c

Elements u and v make up a loop themselves; so do 1, m, and [alpha].

2.1.5. The palindromes i, p, q, y, and z, made up of even numbers alone, make up a loop themselves. Surprisingly enough for our intuition, the loop is very long: it takes forty-eight subtractions to get back to the starting point (Example 11).

Finally, j and k also make up a small loop of sixteen elements (Example 12).

2.2.1. Palindromic modal structures of six elements (modulo 12) evince certain specific aspects. We shall call the structures of an a b c c b a type (where a + b + c = 6) strong palindromes; their inventory is limited (Example 13).

2.2.2. Yet there is also another type of palindrome of an a b c b a d type, which we shall call weak palindromes. They have two axes: d a b c b a d and c b a d a b c. In order to draw up their inventory one shall take into account the condition c + d = even number. These palindromes are weak - but numerous (Example 14). To be found among them is 222222, which is strong and weak alike.

2.2.3. The differences of strong palindromes lead to small loops of weak palindromes (Example 15).

2.3. The subtractions of weak palindromes lead to loops of weak palindromes (see Examples 16 and 17).

The chemistry of the palindromic structures derived from successive subtractions is suggestive in the area of musical rhythm; synonymous palindromes offer remarkable possibilities for varied expansions of temporal values, while the interior connection of the loops imparts a sense to this reservoir of sounds and durations. A loop of loops became the canvas of my Versete for violin and piano.

3. Growing! (by Means of Successive Additions)

I shall now describe a suggestive system of sequences created by successive additions. We apply these additions to the modal structure (2 1 2 1 2 1 2 1), known as Messiaen's Mode II (aux transpositions limities). The successive subtractions soon end up in a reproducible sequence (Example 18). Dan Vuza, who studied these sequences in Vuza 1985b, speaks about reducible sequences and reproducible sequences, the latter replacing the term "irreducible sequences" which I myself use.

While subtractions are wholly determined by the starting sequence, each successive superposition by addition depends on the term from which the addition begins. In this case we shall agree to start each higher tier (marked by a "+" while " - " designates a lower tier) with the same term with which the symmetrical lower level begins. The +1 tier will therefore start with 11, very much like -1; the +2 tier will begin with 2; all the others will start with 8.

3.1. Each higher tier will be an infinite periodic sequence; the period will have a cardinal number which is a multiple of 8. The following interesting regularity can be noticed: the first two levels have the cardinal number 8; the next four tiers (2[sup.2]) have the cardinal 16 (2[sup.4]); the next eight tiers (2[sub.3]) have the cardinal 64 (2[sub.6]), and so on. As can be seen, this increase of the cardinal has a "genetic" character.

3.2. Each periodic sequence will preserve the same modal structure; its elements will belong to the mode {1, 2, 4, 5, 7, 8, 10, 11}, which is the level +1 as a matter of fact.

3.3. But the most interesting "genetic" regularity is the transition of a sequence of sequences of a certain cardinal number to the sequences having a double cardinal. For lack of space I shall only exemplify with tiers +15 up to +30, having sixty-four elements each (Example 19). As it progresses, the pair 4, 8 appears to proliferate. On the level +15 these elements cover only fifteen of the sixty-four elements. Next, the pair 8, 4 tends to occupy ever more space on the upper levels; for instance, on the level +22 the first line is completely taken, very much like eight of the sixteen columns; on the level +27 only fourteen out of sixty-four elements arc not 8 or 4; on the level +28 only ten "foreign" elements are left; on the level +29 only eight; when we are inclined to hope that they will totally disappear, their number increases on the level +30, while the level +31 offers us a period of 128 elements.

Next, the same "genetic" tendencies are resumed: the pair 8, 4 proliferates, starting from thirty-two elements at +31 to reach 128 with +61; at +62 the number of foreign elements grows dramatically, and at +63 the length of the period has doubled to 256,elements (Example 20).

In keeping with the same genetics, the next sixty-four levels will have 256 elements each, all belonging to the same model (Messiaen's Mode II); the pair 8, 4 seems to expel the other elements, only to sustain the same failure on the sixty-third tier, after which the number of elements in the period will again double to 512, and so on.

In conclusion, I shall illustrate these digital latencies by a rough musicalization, not resorting to other compositional metarules (Example 21). My String Quartet No. 8 (1991) also includes this fractal process.

Next, the same "genetic" tendencies are resumed: the pair, 8, 4, proliferates, starting from thirty-two elements at + 31 to reach 128 with +61; at +62 the number of foreign elements grows dramatically, and at + 63 the length of the period has doubled to 256 elements (Example 20).

In keeping with the same genetics, the next sixty-fours levels will have 256 elements each, all belonging to the same mode (Messiaen's Mode II); the pair 8, 4 seems to expel the other elements, only to sustain the same failure on the sixty-third tier, after which the number of elements in the period will again double to 512, and so on.

NOTES

(1.) For a more complete understanding of rhythm unions see Vuza 1985 and 1986.

REFERENCES

Vieru, Anatol. 1980. Carrea modurilor I (The Book of Modes 1). Bucharest: Editura Muzicala

Vuza, Dan Tudor. 1984. "Proprietis des suites periodiques utilisees dans la pratique modale." Muzica 34, no. 2 (February): 44-48.

_____. 1985. "Sur le rhythme piriodique." Revue Roumaine de Linguistique - Cahiers de Linguistique Theorique et Appliquie 22, no. 1: 73-103. (Reprinted in Musikometrika I, edited by M. G. Boroda, 83-126. Bochum: Studienverlag Dr. N. Brockmeyer, 1988.)

_____.1986. "Les Structures modales, instrument d'etude des modes et des rythmes." Revue Raumaine de Linguistique - Cahiers de Linguistique Theorique et Appliquee 23, no. 1:55-68.l
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Author:Vieru, Anatol
Publication:Perspectives of New Music
Date:Jun 22, 1992
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