Large-Scale Duration Organization in 'Hodoi tô Ergô'

This paper describes the underlying idea behind large-scale durations in a composition called ‘Hodoi tô Ergô’ (200102). We begin from a set of abstract numbers and gradually work towards the actual score. All the intermediate steps are documented along with an appropriate visualization of the data. This work (except for the final score) is realized inside a computer program called PWGL [1]. 1 Background There are not so many ways to divide one second for human instrumentalist, but many ways to split e.g. 367 seconds. Instrumental music is then essentially macroscopic and complex ratios of numbers can be realized in a large scale only. Let us take a dozen of marvelous chords (and marvelous relations between them) having equal durations; after too many repetitions it will become boring: the game of durations is neutralized. Something should happen in quantities (durations), since qualitative differences (harmony) aren’t enough. Even further, the pattern should function without different qualities in a form of pure quantities. Black and white figure should elevate to a dignity of quality. There is a tension between the clock-time and the way how different musical entities are perceived in time. I.e. different entities occupy the same duration in a different way. Despite this, in the ‘Hodoi tô Ergô’ I wanted to give independency for pure durations and in the beginning I didn’t know, what kind of musical entities would materialize in those abstract durations. This paper is organized as follows: first, the instrumental groups are defined and some general aspects discussed. Second, the durations in a large scale are generated. Third, these durations are arranged for the groups. Finally, five musical examples are shown from the score. 2 Instrumental Setup in Hodoi tô Ergô The ten players of ERGO ensemble were split into three groups: (S)oprano group (piccolo, flute, violin and vibes), (M)iddle group (piano, harp and percussion) and (B)ass group (bass clarinet, cello and marimba). One could perceive ‘Hodoi tô Ergô’ in three ways: first, as a 3-voiced ‘supermelody’, where one group is a substitute for a simple note. Second, as a stereophonic constellation where the groups are placed as follows: S-group on the left, M-group in the middle and B-group on the right. Finally, as a one 3-voiced multitimbral instrument, which is ‘triggered’ by the percussion player. He/she gives attacks for the groups: metal for the S-group, wood for the M-group and membrane for the B-group. We will focus on the middle section of the composition. It is speedy and dense music, contrasting to the static intro and coda. Let us see how one can split 367 seconds among the groups. 3 Durations In a Large Scale The middle section is one large polyrhythm, lasting 367 seconds in a tempo 1/4=72. The polyrhythm comes from the relation of different speeds of groups. The speed means here slow time interval when each group has its possibility to appear in the foreground (‘Hauptstimme’ in Schoenbergian sense). Time intervals (i.e. delta-times indicating when the next event will occur) are expressed here and onward as 1/16 notes: pulse/tic/raster like this, made it easier to maintain the qualitative difference between strong downbeat and unaccented beats in a traditional measure oriented notation. 3.1 S-group (41): B-group (43) I chose two arithmetical series: +41 (0,41,82,123...) for the S-group and +43 (0,43,86,129...) for the B-group. Next there was a union of these series in ascending order: (0,0,41,43,82,86,123,129...) and the delta-times were calculated (0,41,2,39,4,37,6...). Length of the series comes from the lowest common multiple of 41 and 43 (=1763=367 seconds in a tempo 72). We get a complete, symmetrical series of delta-times, which is closed after 43 repetitions of 41 or 41 of 43. The Figure 1 shows the visualization of these delta-times using 2D-editor [2]. So, the durational skeleton comes from the interplay of two slow pulses, 41*1/16 and 43*1/16 (ca. 8.5 and 9 seconds). The differences of delta-times are equalized until the first quarter and come to a head in the middle. These two linear series converge to the closure at the end (deltatimes=0). Figure 1. Delta-times from the union of two arithmetical series. 3.2 M-group (49) Let us modify the previous diagram by adding one arithmetical series more, +49 (10.2 seconds). This addition is for the M-group and starts a 1/16 note before the common start of Sand B-group: (-1,48,97,136...). Combining this with the union of +41 and +43 series and calculating the delta-times we get: (1,0,41,2,5,34,4,11,26,6,17...). This series contains all delta-times formed by three different periodical sequences (Fig. 2) The perfect symmetry of Figure 1 is now distorted resulting in a more complex set of delta-times: the total ‘lifespan’ of two groups (S and B) is now shared with a third one, M-group. The original shape is heavily distorted but clearly recognizable. Figure 2. The two linear series in Figure 1 are modified with another series (+49). 4 The Organization Of Durations After getting Figure 2, I analyzed the stream of durations for localizing the max/min and average values and especially their timings. Also finding out the regular patterns and their developments is rewarding since it gives ideas for the other aspects of music as well. My aim was to see beforehand the potentials of durations from this predetermined structure. 4.1 The grouping of durations Figure 3 shows ‘isorhytmical’ structure presented as durations. One can follow the ‘drama’ of pure differences. For instance, the areas where regular patterns break could be interesting. The whole structure is divided in 12 macroperiods. Criterion for this is explained in Section 4.3.