Page  00000001 Large-Scale Duration Organization in 'Hodoi to Erg6' Kimmo Kuitunen Kottaraistie 5D 00730 Helsinki, Finland Mika Kuuskankare Sibelius Academy DocMus Mikael Laurson Sibelius Academy CMT Abstract This paper describes the underlying idea behind large-scale durations in a composition called 'Hodoi to Ergo' (2001 -02). 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 t6 Ergo' 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 to Erg6 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 t6 Ergo' 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

Page  00000002 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). 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. I (104125 344 11266 17 18 823 10 10292 12) II (29 6 8 27 14 2 25) III (18 4 19 20 10 1122 16 3 24) IV (1752115 13 15 13219 112939) V (343436) VI (54343 1228120) V1- (21 127 13 3 33 5 5) V- (363434) IV- (9 2 30 118 22 13 14 14 15 20 6 17) Ilf- (24 2 17 22 10 1120 18 5 18) -> (see Figures 5 and 6) IIf (25115277729) Ii (12 130 10 9 24 8 17 18 6 25 12 4 33 6 2 41) Total duration: 1+258 111 147 181 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 S- and 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. 107 108 77 181 147 111 258 Figure 3. The groupings of durations in a chronological order. 4.2 Duration profiles for the groups The next important step is to sort out the durations for every group. Then one can see the overall nature of duration profiles for the groups; e.g. M-group starts with the maximum presence in the foreground and then gradually disappears (see M-group in the Figure 4). One can express this abstract (only quantities, durations) interaction between three groups by different gradients of the slopes: the presence in the foreground for S-group increases +2 in relation to B-group and +8 in relation to Mgroup. The corresponding gradients for the other groups are: M-group decreases -6 in relation to B-group and -8 to Sgroup and finally B-group increases +6 in relation to Mgroup and decreases -2 in relation to S-group. So the 'lifespan' in the foreground becomes weaker or strengthen in a double linear manner. Figure 4. Profiles for the lifespan in the foreground of each group. From the left: S-group, M-group and B-group respectively. 4.3 12 macro-periods Thus each group alternates between two possible relations defined by the order of the groups in time. When the order of the groups changes, all the groups get another 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.

Page  00000003 gradient for the development of durations. These turning points became important moments for articulating the overall of the piece by cadenzas and changes in textures. There are 12 turning points, which are expressed as lists of durations in Section 4.1 (see also Fig. 5). These macroperiods divide into two parts, which are mirror symmetrical. However the symmetry is not perfect because of a tiny 1/16 note difference at the beginning (M-group didn't have the same starting point as S and B-group had; it started 1/16 note earlier than the other groups). Macroperiod III'" background and after some cycles they reach the last iteration (see dashed rectangle), which was composed first (composing 'backwards'). The last iteration in the background comes a counter-event for the foreground. Iterations become gradually more consistence and the target for them is the next foreground in the beginning of the next macro-period. So the players have to take care of this double gesture: playing in the foreground and also developing material for the next foreground when playing in the background. The function of background was not to accompany the foreground but to make some 'noise' or interference to the overall situation. Just an opposite function for the background was the analysis of the harmonic flow of the foreground. I call it the 'depth' parameter when foreground harmonies are supplemented towards the fundamental, resulting in more 'simple' harmonic spectrum. (4) 'cut off'-object. There is an instance of another periodical system. It is a violent tutti stroke at regular time intervals (every 160th 1/16 note starting 3/16 after the start of S and B-group). It is like a concrete musical object thrown into the musical action to refresh the situation. (5) foreground shared by all the groups. Sometimes I needed tutti power contrasting the main procedure, counterpoint of three layers. The division into the groups is destroyed and all the instruments share the same foreground. 6 For Future Figure 5. 12 macro-periods and the developments of life foreground for each group. 4.4 Macro-period IIfIi" in metrical n4 In the following we present the macro-perio traditional music notation using the score-editor [1] (Fig. 6). It was very useful to see t predetermined structure from different repres( when trying to understand and articulate it. span in the otation d IIInv in of PWGL he same 1s 6 fAd 46 la fo:i........................................................ M --------------55555555555[[5555:._ ..;.; ---------------55555555 -----------------------2555 SS |^ ^ ^^^^^^U 91iiii^ 16 '( entations, The durations were organized rather mechanically in 'Hodoi t6 Ergo', i.e. they were never manipulated by e.g.: continuous (accelerating/decelerating) functions. And there - - was not any fermatas, i.e. the structure of durations was never frozen. Also the conscious interaction between durations and different parameters was almost absent. f:"" i^ These predetermined structures of durations are not to be taken dogmatically, but prevent the despotism of [I"nn. 'composer's choice'. I.e. changes are made in relation to the original, 'given' structure. Figure 6. The rhythm of foreground in the macro-period I 5 Measures 167-178 Next, we present some of the aforementioned abstractions in a musical context. The Appendix shows the measures 167-178 of 'Hodoi t6 Ergo'. The exerpt is on the border of macro-period IIIiv and 11". Many paths (hodoi) are overlapping in a 'stretta' before the last macro-period (see the Appendix for the corresponding numbers in the score): (1) last three indexes in the foreground in IIInv'. We can see the foreground pattern moving from S-group to M-group and B-group. After this the order is changed: S->B->M. (2) foreground as a polyrhythm. Event in the foreground can be a polyrhythm itself. Long notes are given different pulses by alternating between normal and tremolo playing. (3) background iterations - polyrhythms inside the polyrhythm [3]. Short musical cells are iterated in the 7 Acknowledgements The work of Mikael Laurson has been supported by the Academy of Finland (SA 105557). References [1] Laurson M. and M. Kuuskankare. "PWGL: A novel Visual Language based on Common Lisp, CLOS and OpenGL", Proceedings of the International Computer Music Conference, Gothenburg, Sweden, 2002 [2] Laurson M. and M. Kuuskankare. "PWGL Editors: 2DEditor as a Case Study", Sound and Music Computing '04, Paris, France, 2004. [3] Kuitunen K. "Polyrhythms inside the polyrhythm", A lecture presented in Ergo Projects/Northlands: CanadaFinland exchange 2002, Toronto, Canada, 30.9.2002.

Page  00000004 Appendix (measures 167-178 from the score 'Hodoi to Ergo'): (1) S S, (2) ------------- molto f pmoltof ~ L-A Fl. 1 molto f p Fr 3 4: Pff Per. 1 00.. iO molto(f i p in molto f V1r. Z mop fpp M mflto f 3 - 5 3 3 A i A 'Rol Pf sz if molto f mp& Pfz I A A A A A Harp f (4) S a f molto f ~~fT I\ JI TI Pere. 11 P... f: ý f B f::-P. --6ý ý -.. -I -- -~-- - - - - - -I Pere. III S3 P-ro.f ==== = ff (3) molto f Bel. f Poco ff 5 wpm y ESTI Ve. 4.-3 Poco f m' - Fl. I P FP P P P P P P I ff f 5~m~~~f~mlI olo V I molto f fA Fl. 1.oO==7 ff f -fmolt f M5 Pere. I; ff v mI " I mtto f v - v I n~t V Harff f molto fmlto FRI A AA ýv vv O f. ff sfz-fmo o A AA d, v I I Har molt ffI sfz to~~m ~ ~ ~ q I ot