Cosmology in sounds (on Leo Brouwer's "La Espiral Eterna") by Eduardo Fernandez

The idea that music is, or should be, a reflection of the cosmic order of things is very old, dating from at least the Middle Ages and the beginnings of polyphony. In the Renaissance there was still explicit recognition of this idea. With the evolution of the musical language, music became more and more the expression of an individual, the composer, displacing the aesthetic accent to the personal instead of the universal. In the second half of the 20 th century, a few works have appeared where the old ideal of music as a sign of the cosmos is taken up again; none of them is as convincing and accomplished, in my opinion, as Leo Brouwer's La Espiral Eterna , written in 1971.

In my experience, the Espiral is one of the very few works written in a really modern style that are able to captivate any audience, even those unfamiliar with this style (and the Espiral is unthinkable without a very wide repertoire of previous works; the most notable antecedents would be Continuum for harpsichord, by Ligeti, and the early works of Stockhausen and Penderecki). By "modern", I mean here only that this work could not possibly have been written before; by no means am I taking sides as to what is "modern" or "avant-garde". It is difficult to be on the "vanguard" if you don't know where "forward" lies. However that may be, audiences seem to sense intuitively that there is a deeper truth in the work that goes beyond the sound surface, however impressive and original it is; that listening to this work implies something more than merely following the musical discourse. In fact, so does any player who takes it up for studying. This article intends to try to find out why this happens, to explore the ways in which this work is conceived, and perhaps to serve as a companion to the study of the work.

I am well aware that programmatic interpretations are out of fashion. Nevertheless, a performer must have something on mind when tackling a new work, and in my view it is far better to have an image obtained through analysis and imagination than the beautiful blonde (or blond) in the second row. The analysis that follows may well be a delirium, not unlike that of finding universal numerical clues on the Gizeh pyramid. Anyway, I venture to put it forward, hoping that someone else will also find it interesting. Please refer to the score, Gitarren Archiv 423, B. Schott's Söhne, Mainz.

The basic idea of the work is manifest in the quotation that appears on the first page: "Por primera vez se reveló en los cielos la famosa estructura espiral empleada con derroche por la naturaleza en el mundo orgánico" (more or less: "For the first time was revealed in the heavens the famous spiral structure that Nature had employed with such abandon in the organic world"). The quotation is taken from a book on astrophysics, "The Structure of the Universe", by G.J. Whitrow. The message is clear: the same structure appears in heaven and in living creatures; the whole Universe is made up in the same way, following the same laws, forming the same structures at every level of being. Whitrow's phrase implies, in its original context, nothing more than a recognition of the same basic structure in galaxies and the organic world – not necessarily a revelation of the underlying principle of the Universe. Brouwer's imaginative step anticipated, in fact, scientifical discoveries still to come. We have to remember, this is several years before Carl Sagan made these ideas a commonplace of contemporary culture (and he did not appear in Cuban TV, as far as I know...)

The work is divided in four main sections, designated A,B,C, and D. Let's take them one by one.

1. THE GAS CLOUD

The Espiral begins with a ppp , almost inaudible; the object presented is not necessarily small, but only far away (yet). We are approaching it, so it seems a good idea to start not from ppp , but from silence. What object is this? We hear a group of three notes, D-E-D#, repeated continuosly at high speed (as fast as possible, in fact). It is not too farfetched to see in this group a spiral in embryo:

E

D#

D

To "see" the spiral shape, imagine a curved line connecting the three notes; the repetition, that makes us listen to the three notes in circulation, is an exact correspondence of a rapid rotation of the group. To the eyes of the imagination, we have here a spiral rotating very quickly before our ears, and it does not seem too far-fetched to take this as an analogy (in sound) of the first stage in the life of a star: a gas cloud. The pattern of "two steps forward, one back" is even reflected in the strings in which the notes are to be played: 3 rd , 1 st , 2 nd . This pattern is of course the basic cell of the whole work.

This object evolves in the whole of section A, and the parameters of its evolution are:

• the pitches on each group, or rather, the range of pitches on each group;
• the number of notes on each group, which determine a rythm of sorts: since all notes have the same duration (as short as possible), a group of 4 notes lasts longer than a group of 3 notes (this should be made clear in performance, avoiding the impression of, for example, a triplet on group 1 and four sixteenth-notes on group 2);
• most important, the differentials in each of the two previous parameters, that is, a)growth or lessening of the number of notes from one group to the next, and b)widening or narrowing of the range of pitches from one group to the next.

The 24 groups that appear in section A can be grouped themselves in seven cycles, as follows:

• Cycle 1: groups 1, 2, 3
• Cycle 2: groups 3, 4, 5, 6
• Cycle 3: groups 6, 7, 8, 9, 10, 11
• Cycle 4: groups 11, 12, 13
• Cycle 5: groups 13, 14, 15
• Cycle 6: groups 15, 16, 17, 18, 19, 20
• Cycle 7: groups 20, 21, 22, 23, 24

As we can see, every cycle begins and ends with a three-note group, and the number of groups on each cycle follows a spiral-like pattern as well: respectively, 3, 4, 6, 3 / 3, 6, 5. Or, substracting 3 : 0, 1, 2, 0 / 0, 3, 2. The reason to substract 3 is to find out the underlying structure, which as we shall see, is based on Fibonacci's series. This numerical series begins with 1, 1: each following term is the sum of the two preceding. So the third term is 2 (1+1), the fourth is 3 (2+1), the fifth is 5 (3+2). The series is then: 1, 1, 2, 3, 5, 8, 13, 21... This series is interesting for several reasons. First of all, it is often encountered in Nature (count, for instance, the number of leaves on each side of a branch). It gives the eye or ear the impression of a "natural" growing principle. Also, when the terms are big enough, the relationship between one term and the preceding one approaches the "golden measure" (approx. 0.618), which establishes the way of dividing a line so that the lesser part is to the bigger part what the biggest part is to the whole. This is routinely used in architecture, and has been used in music many times. Bártok made of it one of his main formal principles, and Brouwer uses it often.

If we consider the range of pitches, then the starting point is 2, and we have a different grouping of cycles:

1. 1-2-3;
2. 3-4-5-6;
3. 6-7-8-9-10-11;
4. 11-12-13-14-15-16-17-18-19-20-21-22 (the last two groups are below the minimum level of 2).

Please notice the growing number of groups on each cycle: 3, 4, 6, 12.

The dynamics follow a similar growing spiral-like pattern, but as it is more clear to the eye, I shall concentrate on the other parameters. It is important to notice that the dynamic evolution does not coincide with the groups, and to make this evident in performance.

The following table intends to give an overview of what happens in section A (leaving dynamics aside):

Cycle#/ Group #	Length in Notes	Difference in length	Range of pitches (in semitones)	Maximum of difference in range, per cycle	Effect
I. 1	3	0	2		Gas cloud
2	4	1	3	A. 1	Expansion upwards
II. 3	3	0	2		Contraction
4	4	1	3	B. 1	Expansion down
5	6	2	3		More activity
III. 6	3	0	2		Displacement upwards
7	4	1	3		Expansion upwards
8	8	5	4	C. 2	Expansion upwards
9	4	1	3		Displacement downwards
10	4	1	3		Displacement downwards
IV. 11	3	0	2		Contraction downwards
12	4	1	4		Expansion downwards
V. 13	3	0	4		Contraction
14	4	1	5		Expansion downwards
VI. 15	3	0	4		Contraction
16	7	4	3		Expansion downwards
17	7	4	4		Displacement downwards
18	4	1	4	D. 3	Expansion downwards
19	8	5	5		Expansion downwards
VII. 20	3	0	4		Contraction upwards
21	13	10	4		More activity
22	3	0	2		Contraction upwards
23	3	0	1		Contraction upwards
24	3	0	0		Contraction upwards

A few things are apparent from the table:

• if we look at the length of each group, although each cycle expands and contracts, the expansion is always greater than in the preceding cycle: 1, 2, 5, 4+4 = 8 (plus a 5), 10. (the longest group has 13 notes, # 21). The general tendency is, then, towards expansion. These numbers suggest that the Fibonacci series is in operation.
• The last cycle is particularly interesting: it presents the group with the most notes 13, in group 21), which translates as a great increase in activity, followed by a contraction .
• The maximum of difference per cycle in pitch range (a complicated way of saying how big this range gets on each cycle) is the first 4 terms in the Fibonacci series: 1, 1, 2, 3. Also an expansion, which not only overlaps with the cycling in terms of length, but one in which as the cycle gets bigger, the pitch range also gets bigger.

All this together, which of course is not meant to be perceived directly by the listener, conveys subliminally the feeling of an underlying order, in which the material obeys laws, however complex, and it is this feeling of order, proportion and activity directed towards a goal (however unpredictable at the moment) what the listener hears in this section.

What does this point to? My interpretation would be that the cloud is rotating more and more rapidly, as gravity pulls it together. The last note, which can be played either ppp or as a ff Bártok pizzicato, represents in my view the first "solids" generated in the process.

Another possible view is to consider the section in terms of attack and resonance. In this view, all of section A, excepting the last note, would be a "resonance", and the last note would be an "attack".

Once the first solid has been generated (big or small, ppp or ff , it doesn't really matter which) then the view of the process changes. We need to watch things from closer, and this is what happens in section B.

1. THE STAR BEING BORN
Section B begins reversing the resonance/attack pattern that comprised the whole of section A. The first three groups consist of a composite attack (noise+sound) plus one resonance each. The pitches of the attacks, E flat – F – C sharp are recognizable as the initial cell after an expansion (a whole tone instead of a semitone), and a retrograde. The spiral pattern is now:
F
E flat
C sharp

Seeing things bigger means that we are closer to the object. As we get closer, it becomes apparent that the same overall structure (attack/resonance) is encountered everywhere inside the object as well. In fact, B1 and B2 consist exclusively on this pattern.
The first three groups of B1 have contracting lengths: 10-6-5 (notice that the resonance is modulated by accelerandi/rallentandi, also a form of "spiraling" tempo. The last group is a transition to the"fast" section that follows.
B2 begins with three pitches that are, to no one's surprise by now, based on a slightly irregular expansion of the initial basic cell: C sharp – G sharp – F. C sharp was of course the last pitch played as an attack in the previous section, B1. From there, 3 ½ tones up is G sharp, and four tones down is F. The spiral is now much bigger (notice the octave expansion), with this shape:
G sharp
C sharp
F
Towards the end of B2, the attacks begin to multiply, first two, then four, then eight. Let us look closely at the last group, that closes B2: G#-F#-D-E-C-B-A#-D#. This is made up of three different versions of the basic cell:
Version 1: G# - F#..................................................................G natural (present on the first group of B3)
(inversion of original cell)
Version 2: D – E - ...........................D# (original cell at original state)
Version 3: C – B – A#
(permutation of original cell)
Something significant is happening in the gas cloud: the basic pattern is generating ever more complex versions of itself. In this context, I propose to see B3 as another change in scale, starting with the sound space divided in semitones and moving progressively towards a microtonal space. Of course, all of B3 consists in resonance, modulated by the change in sound color from normal to pizzicato. The spiraling pattern of ascent is clear in the score. The three notes at the end can be seen as a "flattened" version of the original cell.


2. BELOW THE MOLECULAR LEVEL


At first glance, section C seems to suggest to the player that "anything goes". This is, of course, not quite the case. The patterns to be played with both hands on the fingerboard are, of course, the basic cell: 3-1-2 with left hand, i-a-m with right hand. This comes out very clearly if, as Brouwer suggests, the left thumb is held over the strings, damping them (in the position cellists use for playing ultrahigh passages), so as to dampen the resonance produced behind the point of impact of the fingers. What will come out of these actions is an irregular overlapping pattern generated by the basic cell, mostly at tritone or fourth amplitude, expanding and contracting irregularly as the result of the speed cycles overlapping. This is definitely, in my view, a section where the player should have the score before his/her eyes when playing it; nothing is so difficult to generate as irregularity, and one should be reacting to the score every time as if it was the first.
At atomic level, all interactions are regulated by the laws of quantum mechanics: these laws are probabilistic in nature, and I think there is a correspondence between the object and its musical sign here: it would not make much sense to try to write precisely actions that must be indetermined, by their very nature.
The ending of section C suggests going still deeper down, into some kind of mystery; in fact, it is nothing less than the appearance of life forms, as the next stage of evolution. This happens before our eyes in section D.
3. LIFE APPEARS

Section D is divided in four. In D1, we have again the basic cell, expanded along three octaves and permutated (F#-G-A#). What is absolutely new and almost shocking is the way in which this cell appears. There is a rythm imposed on it; a recognizable, danceable, pleasant, Cuban rythm. This forms a sort of landscape against which the next section develops. I would say that this "landscape" corresponds to the appearence of plants and animal life, setting the stage for the next step.

In D2, the rythm becomes fast, and superimposed to this resonance are sharp, cutting attacks. The intervals used are, suggestively, sixths and thirds, the most "human" of intervals. Is it too farfetched to think that this symbolizes human life, permutating endlessly over the planet's landscape? Probably only a Cuban could think of human history as a dance, but isn't it a nice idea?... The rallentando and diminuendo at the end is announcing another change of scale, which comes about in section D3.

D3 takes the cell to the maximun possible expansion in the guitar: from lowest to highest note. Every group is played twice, so let's analyze what happens with them. The following table helps to see the process:

Group #	Nº pitches	Nº notes (= duration of group)	Effect
1	5 (E-D-C#-C-B), ascending and descending	8	Maximum expansion, quasi-chromatic scale
2	4 (E-F-F#-G), ascending and descending	6	Somewhat contracted, accelerating (less duration).Chromatic scale.
3	5 (E-D-C-B flat-A flat), ascending (but descending pitches)	5	Somewhat expanded, accelerating, whole-tone scale. Ascending movement only, seems to increase speed.
4	4 (E-D#-D-C#), ascending, with descending pitches	4	Contracted, accelerating, chromatic scale.
5	3 (G-A flat- D), descending	3	Contracted, accelerating, distorted form of basic cell.
6, 7	"Infinite", chaotic but based on basic cell (especially group 6, if seen as 3+4 pitches)	26	Explosion.

It does not take much imagination to see this as the final process in the life of a star: the contraction and explosion.

This generates a new complex solid, or a double attack (F#-G played as a Bártok pizzicato), followed by a resonance using the pitches of the basic cell in scale form. This pattern appears three times, and it is interesting to notice the pitches used in the resonances:

E-F-G flat

D-D#-E

C-C#-D

The spiral shape, this time composed of little spirals, is again in evidence. The duration of the patterns is also in spiral shape: 12-9-"infinite" (25 seconds).

The ending of the work, D4, suggests another gas cloud, for the moment chaotic, but as it loses itself in the distance of interstellar space (diminuendo to ppp and 6 seconds silence at the end), it is not hard to foresee what will happen: the whole process is about to begin again; in fact, it seems to go on behind the final silence. "In my end is my beginning..."

One last fanciful digression. The first three pitches of the work, and the last three, seem to suggest an open-ended spiral:

X-Y-Z?

D-D#-E

C-C#-D

No doubt the next step, which we won't be able to hear, will use the pitches E-F-F# ..!