Rhythmic Resultants

A new and interesting idea of Schillinger is the rhythmical resultant[6].

Two different rhythms, when heard at the same time, create another which is the resultant of the two.

Figure: Some rhythmical Resultants according to the Schillinger System of Musical Composition[6]

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Ligeti, in many of his works have used the resultant rhythms techniques in very practical ways. In his Second String Quartet, for instance, the uneven subdivisions of the beat serves to create an asynchronous yet still ``calm'', animation.

In the fragment shown below^3.6 and in the following bars, we often get a stacking of subdivisions 4/5/6/7 of the beat.

That makes no single attack, except one occurring on the beat, of any part simultaneous with any other. This is a very effective and practical way to create an illusion of a faster playing speed. In such a beat, when players play simultaneously four, five, six and seven notes on a beat, the actual rhythm which is perceived is actually 34#34 minus 3 (for the ``on-beat'' simultaneous attack) equals 19 distinct notes in a beat.

Figure: String Quartett N.2, First Movement, bar: 43. Uneven subdivisions of the beat.

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If the metronome speed is one second a beat or faster, the resulting ``speed'' of the notes, 19 notes per second would almost equal the perception threshold which is of 20 discernible ``notes'' per second, or 50 millisecond for a musical event.

As stated earlier^3.7 experiments made in the electro-acoustic music studios showed that for durations below 50ms. musical elements, in our perception tend to dissolve into a continuum.

This technique, being a highly effective and ``economical'' one, because it surpasses performers and instruments physical limitations for the creation of an over-fast animation, is extensively used by Ligeti. Most notable areas of its use are for creating over-fast pizzicato sections which would be otherwise impossible to play, over-fast trills and huge wave-like motions.

In Atmosphères, at bars: 25-28, the tutti-trills are such an application. A gradually accelerating trill which starts at the First Violins and expands to the whole orchestra (except for the double-basses) is scored by using beat subdivisions ranging from 3 (eight-note triplets) to 15 (quindidecimal tuplets of 32.nd notes.

In the section which is the subject of this study^3.8 the rhythmical design of the parts is devised to prevent any synchronous attack and to avoid all kind of accentuation.

For this effect Ligeti did set in the up (Violins I) and bottom (Cellos) parts uneven tuplet beat-subdivisions. Five (quintuplets of eights) for Violins-I and three (triplets of quarters) for the Cellos. That will minimize the ``risk'' of having simultaneous attacks on the top and bottom parts to whose are perception is most sensitive.

The regularly subdivided inner parts (Violins II and Violas) uses the following rhythm palette:

Figure: Sample rhythms of the Violins II and Viola parts

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The outer parts, Violins I and Cellos whose are set in quintuplet and triplet subdivisions are using for the most part the same rhythms as those shown above. Because of their beat subdivisions as tuplets, despite the huge 48-voice polyphony there are actually only a very few simultaneous attacks. Almost all parts move ``independently''.

A similar rhythm palette is used during the ``raise'' which occurs at bars 45 through 48. From bar 48, just before the entrance of the double-basses, more quintuplets of 32nd. notes appear in the now even denser polyphony.

Double-basses are set in motion, yet not so fast due to the inertia of the instrument at the last two bars of the section.

The crescendo at bars: 52-53 is made stronger by the ``shrinking'' of the ``cloud''. As more individual instruments are joining on the same notes, near the end of bar:53 all instruments conglomerate in a very narrow range around middle-C.

The texture style created with the rhythms shown above, in the second Violins and Violas, is best described as an ``ornamental style''. Those rhythms combined with the neighboring intervals shown in the section: , page: give the ``cloud'' its characteristic ``inside-motion'' effect.

The crowded appearance of the notated rhythm is intended to get even more ``busy'' sounding at the performance for each instrumentalist will inevitably slide from the exact notated values. Syncopes and tuplets will make for them impossible to be on the exact rhythm.

On the other hand, the detailed scoring will also prevent the passage to ``deflate'' by having musicians playing in synchronization with their neighbors.