International Academy of Electroacoustic Music / Bourges
1996: “Analysis in Electroacoustic Music”

Set Menu: Analysis (II)

The work Diagonal was conceived for the group "Multimúsica" (Gabriel Brncic, Claudio Zulian, and myself) and played on conventional instruments (viola, transverse flute and synthesizer) as well as other more exotic instruments (a number of diatonic and chromatic Pan flutes as well as percussion such as a metallophone and a set of tubular bells -both microtonal- this last being of my own fabrication).

The idea of the piece comes from the very word used for the title. It is a word that to me, in Spanish, appears charged with musicality -not to mention its derivates diagonalmente, diagonalización.

Its musical connotations are just as obvious as its Cartesian referents, i.e. moving from one point to another: two parameters (time and pitch, for example) may be represented in a two dimensional space, three parameters (time, pitch, intensity) may be represented in a three-dimensional space, four parameters (time, pitch, intensity, timbre/spectrum) in four dimensions and so on.

What attracted me in the beginning was the "linear" character of the system (after all, is not a diagonal a line?) i.e. an increase (or a decrease) along the "x" axis is accompanied by an increase (or decrease) that is always proportional along the "y" axis, the "z" axis, and so on.

The starting-point for the composition therefore consisted of drawing lines (diagonal lines, of course) on the computer screen using the software "MIDI Draw". Naturally, I could have used any other method. Then I applied the concept of "diagonality" to all parameters, which led me to conceive of a new instrumentation and the introduction of certain visual elements: the piece does not end with instrumental sounds but with the soft rustle of strips of paper bunting that the musicians throw to each other, thus tracing diagonal lines in the air.

There is no instrumental score (with written notes, that is) except for a chord of three notes that marks the start of each of the blocks. Each instrumentalist must begin the block with one of these three notes. Apart from this, there is a precise time diagram with some reference points, a few lines and here and there indications such as "play diagonally", "it's time for the grand diagonal", etc. These are mostly verbal instructions.

The pre-recorded electronic part -which overlaps the instrumental improvisation- is divided into 12 blocks and a Coda, contained within a tripartite structure. The first part A (4'18") covers Blocks I through VI, Part B (5'18") covers Blocks VII through XI, Part C (1'0") covers Block XII (Colofón) and the Coda. (fig. 11)

Also three in number are the generating parameters of the entire structure:

· Possibility / impossibility of drawing a diagonal
· Superposition / Juxtaposition
· Symmetry

Since a diagonal is a line, two points might appear sufficient. However, the very fact that it is a contact point between two opposing apices (opposite at least one angle) necessitates three points. The minimal condition for a polygon is three points, and although the notion of diagonal starts to become complex from the quadrilateral onwards, we may consider the diagonal as being one of the sides of a minimum polygon (the triangle). This is because diagonals of polygons mark out triangles, whereas those polyhedrons mark out a tetrahedron. To a diagonal, the polyhedron is paradise!

As a simple starter, let's take the numbers 1, 2, 3, which, when used to divide a unity, give us respectively 1, 1/2, 1/3. The unity in this case is the sum of the angles of a triangle, i.e. 180º. This first progression (Progression 1) brings about three possible cases:


Let's examine each of the three cases.

Case 1:
The triangle, and therefore the diagonal, are impossible. If the angle is 180º, this means that the sum of two sides is equal to the third. This impossible triangle is expressed by the division of the segment into 2 parts.


Case 2:
A right-angle triangle whose proportions we know, thanks to Pythagoras

Case 3:
An isosceles triangle.


The first case is impossible, the second and third case are possible. The first and third cases give me the two other basic structural criteria: juxtaposition / superposition and symmetry. Juxtaposition of two of the sides and their superposition on the third side (first case) on the one hand, and symmetry (third case) on the other.

As for the second case, it provides me with the Progression 2: 3, 4, 5. These three numbers correspond to the measurements of the sides and of the hypotenuse in the simplest and most elegant formulation of Pythagoras' theorem: a2+b2=c2. The sum of a+b+c equals 12 (what a coincidence, the same number as the twelve tone scale). Furthermore, the sum of the sides (3+4=7) will also be used as a measurement.

I really could not remain indifferent to the magic of the number 12. This led me to obtain no less than three series -I will explain the mechanism in a moment. The role of these three series is nevertheless very limited: they serve to generate twelve chords. The chords, which are the result of the superposition three-by-three of the juxtaposition of the three series, are used to indicate the start of each of the 12 blocks.


Series I is obtained from two measurements. Measurement 1 is generated by the superposition of the 12 possible interval distances:


as well as by the juxtaposition of the measurements 7 (sum of the sides) + 5 (hypotenuse)


Measurement 2 is obtained by the juxtaposition of the above-mentioned distances:



Let's us now generate another progression:

Progression 3 = Progression 2+ Progression 1


It will have five members ("1", "2", "3", "4", "5") such that:

"1" + "2"= "3"
"2" + "3"= "4"
"3" + "4"= "5"

Taking 12 as the measurement of "1", and 5 as the measurement of "2", we obtain the sub-segments:

the sum of which is exactly equal to Measurement 2.


Now we divide into each of the sub-segments:


which gives rise to Series 1:

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

The juxtaposition of the measurements 7, 5, and 12 determines the duration in seconds of the Coda (24)

and the size of the 12 remaining blocks is obtained by the accumulated addition of the members of Series I:

1 24+1= 25
2 25+7= 32
3 32+3= 35
4 35+6= 41
5 41+2= 43
6 43+4= 47
7 47+5= 52
8 52+11= 63 (1:03)
9 63+10= 73 (1:13)
10 73+8= 81 (1:21)
11 81+9= 90 (1:30)
12 90+12= 102 (1:42)


La división de la Magnitud 2 en tres subsegmentos iguales (triángulo equilátero), y la subsiguiente partición de estos en cuatro, da origen a la Serie II:

5,6,7,8,11,2,4,9,12,3,1,10 (ver fig. 4)

5+6+7+8= 26
11+2+4+9= 26
12+3+1+10= 26

26+26+26= 78


Drawings all possible diagonals between the apices of a grid formed by Series II gives as a result Series III (fig. 4). The apices indicate precisely the fractions 1, 1/2, 1/3 in a BASE SQUARE whose side measures 78 (Measurement 2).


Series III determines the distribution of the durations by block:

I 10 (1:13)
II 4 (0:47)
III 5 (0:52)
IV 7 (0:32)
V 11 (1:03)
VI 3 (0:35)
VII 8 (1:21)
VIII 12 (1:42)
IX 2 (0:43)
X 9 (1:30)
XI 1 (0:25)
XII 6 (0:41)

The three sections of the piece (A, B, C) are intertwined by juxtaposition whereas the blocks, inside each section, are intertwined by superposition (fade-out/fade-in between neighbouring blocks). The duration of the section, in comparison with the total duration, shows the same schema as the segments. In this case A (4:18) + C (1:00) = B (5:18), so that A+B+C= 10:36 (total duration of the piece).


By placing the apices at those points indicated by the measurements 5, 7, 12, 17 and 22 (as well as those hitherto included in the generation of Series III: 26, 39, 52) on the BASE SQUARE, we establish the 39 diagonals used in the piece.


The succession of the measurements of the respective hypotenuses admittedly does not produce a very linear graph, but that's a matter of poetical license.