©1982 by Rolf Gehlhaar

Step by Step...music for ears in motion
composition with computer-generated 'three-dimensional' sounds
produced at IRCAM, Paris 1981

The resolution of complex spectra through spatial phase-shifting: a report on research carried out at IRCAM 1979-1981

Introduction

Normally, the position of a listener in relation to the loudspeakers affects appreciably only the qualities of relative loudness and brightness of the sounds heard. For some years I asked myself whether it would be possible to find a method of creating a sound which, in direct opposition to this, would actually change in pitch according to the position and movement of the listener in the acoustical space; in other words, to create a static, unchanging complex sound, rather like an object standing in space, whose individual components would be audible only at certain well-defined points in an acoustical landscape. This landscape would be traversed by the listener, searching out those components. Furthermore, this process of of searching would be a process of interactive listening to a special kind of composition.

The results of several crude preliminary experiments - phase related sequences of pulses recorded on multi-channel tape loops accelerated many times their original speed, recorded with analog equipment and played back over 6 channels in the Electronic Studio of the WDR, Cologne in 1975 - indicated that the creation of such a sound might indeed be possible, but that digital signal processing, in particular very exact control over the phase relationships between the various tracks of pulses, and at least 8 output channels would be required.

This led me to IRCAM in Paris, where in the winter of 1979 I began again with preliminary experiments, this time employing the recently developed '4C' digital signal processor, designed by Pepino di Giugno and staff. The results were even more encouraging than before, although only four output channels were available. However, due to the limited availability of equipment and the lack of a suitable experimental space, thorough examination of the phenomenon had to be postponed to the following year, at which time I would have a sufficiently large space (13m x 13m), a dedicated 4C processor and PDP-11 (LSI-03) computer, 16 channels 16bit digital-analog converters (a very large number at that time), some equipment for acoustic analysis and 16 new loudspeakers of the same make and model. Furthermore, I would also benefit from the advice and assistance of Philippe Prevot, who developed all the software that I required for the operation of the 4C and for the generation and manipulation of the kinds of signals I required.

Signal Generation

Since I believed that the special sound I was looking for would arise out of the interference of sound waves in space - to be precise, out of the constructive and destructive interference of their spectra -, somewhat akin to the special kind of interference between wavefronts that results in a hologram, I began my search with simple pulse-type signals. The aim was to produce a pulse with an even as possible distribution of harmonic energy. Because of the signal format requirements of the 4C, these were best derived in a somewhat roundabout fashion from sine waves (see illustration below):

This quite primitive procedure made it possible, however, to generate the required signals to practically any specified length quickly and reliably, as it employed already existing software routines.

Experiments

Distributed in the manner as indicated in the illustration below:

the signal may be viewed as being a continuous one whose spatial origin only is shifted after the completion of a whole cycle. Theoretically, when heard at the exact centre of the circle of 16 loudspeakers, where it is being 'reconstructed' into its original form, this distribution of the signal should lead to the impression of a continuous sound with a frequency equal to 16 times the frequency of the period, i.e. the original 'harmonic' from which the signal was derived.

Indeed, this does occur. If we take for example a period rate of 6/second, then a frequency of 96Hz is heard in the centre. As the listener moves away from the centre, however, this continuous tone quickly disappears, to be replaced by a low, bright complex throbbing, not unlike the whirring of slowly approaching helicopter blades. As one continues to move away from the centre, the timbre of the sound undergoes a series of continuous changes towards the dark, until finally, upon approaching an individual loudpeaker, one hears a sequence of fairly bright impulses at a rate of 6/second. Another interesting feature of this sound is that if one moves in a circular path near the inner edge of circle of speakers, the apparent speed of the throbbing changes: it accelerates when one moves against the direction of the 'movement' of the pulse from speaker to speaker, and it decelerates when one moves with the direction of the pulses.

Using this type of signal, I experimented with many different patterns of distribution, particularly non-linear ones, letting the pulses 'hop' around the circle of speakers in the most varied ways possible, and going through a wide range of frequencies. I discovered little of greater interest than the effects mentined above, although signals which consisted of several whole cycles (derived from a sine wave in the same manner as described earlier, except that the 'harmonic used as the origin was a multiple of 16) led to much more complex and varied changes in timbre, especially at higher frequencies. I dubbed this phenomenon the sea gull-effect, due to the fact that I had the distinct impression of being pursued by high whirring flocks of chattering birds as I walked or ran, zig-zagging through the space.

The changes in timbre and sound quality were quite marked. As a result, my experimental space frequently became an entertaining attraction for several colleagues at IRCAM, who used this opportunity to roller-skate around and through the space, enjoying the interactive display of my 'sea gulls' and other strange whizzing sounds.

The effects mrntioned above - very distinct changes in timbre - were, however, not quite all what I was looking for, or at least, they were not distinct enough. I was convinced that the signal I was employing was more or less the correct one, but that some aspects of the way it was projected into the space were probably not quite right.

It seemed to me that one faulty aspect was probably its interaction with itself in the space as a result of the primarily cyclical, circular distribution: it was not leading to the kinds of interferences required. I therefore decided to examine different spatial distributions of the signal by systematising the great variety of possible configurations and examining the effects of each one. Another faulty aspect was probably the alteration or 'degeneration' of the signal caused by the loudspeaker, particularly as normal loudspeakers are not designed to project non-continuous pulse-type signals., especially infrasonic ones. This problem would be more difficult to address, as it would require an alteration of the signal in order to pre-compensate for the distortion. I would have to address this problem later, as there were no available algorithms for shaping the wavetables I was empoying and no one could give me any advice.

One of the very first new configurations I examined was a signal derived from the 8th harmonic, with the circle of speakers divided into two opposed semicircular groups of 8 channels, nos. 1-8 and nos. 9-16, as indicated in the illustration:

This arrangement immediately led to the first convincing results: in a corridor approximately 5m in length and 1m in width, running through the centre of the circle, bisecting the opposed groups of speakers, one could localise a number of continuous tones, all belonging to the harmonic series of the same fundamental.The most prominent ones were the 4th through the 12th harmonics of a fundamental frequency equal to the pulse frequency, with the 8th harmonic 'standing', as expected, near the centre of the circle. The same phenomenon could also be observed when only one group of the 8 speakers was operating, although at a greatly diminished degree of resolution and with a somewhat different spatial localisation.

An 'iso-harmonic' map, of such a single-sided distribution is shown in the following illustration, the lines indicating the paths along which a specific harmonic could be localised as a continuous tone:

Note that the 7th harmonic occupies a path crossing through the centre, the 8th harmonic being deplaced slightly. The paths are quite clearly defined; straying a short distance to either side results in an immediate loss of the continuous tone. In the areas where the paths lie very close to one another, one tone blends quickly into its neighbour as one crosses over from one path to another. Rocking back and forth between two adjacent paths, one hears an alternation between the two tones associated with the paths.

A spectral analysis of the signal - as it leaves the loudspeaker - reveals that each pulse generates a harmonic spectrum, the number of harmonics generated being inversely proportional to the pulse length, the distribution of energy being related to the steepness of the attacking and decaying flanks of the pulse, as illustrated in the following comparisons (all spectral analyses of the signal were made as it leaves the loudspeaker, with a sampling microphone directly in front of it) :

Similarly, a series of spectrograms plotted at various positions in the space where one hears distinct 'standing' harmonics reveal that the energies of those harmonics lying immediately to either side - sometimes to both sides - of the one which is heard as a continuous tone, are greatly reduced, sometimes even completely suppressed.

All the above data points to the conclusion that the staionary resolution of harmonics is due to a complex standing wave system created by the particular discontinuity and spatial distribution of the signal. The term standing wave is generally used to designate a field of sound waves having a fixed distribution is space as a result of the interference of progressive sound waves of the same frequency and kind. When such sound waves are of a simple and continuous nature, then the field is characterised by the existence of nodes or partial nodes and antinodes that are fixed in space.

In this case, because the waves have complex spectra, the interferences lead to stationary nodes and antinodes in the harmonics of the signal as well.The discontinuity of the signal is responsible for the fact that a particular component of the spectrum becomes audible as a continuous tone at those points in space where that part of the signal which is responsible for it is consistently reinforced through constructive interference (antinode); conversely, those components of the spectrum which are responsible for other harmonics are, at these same points, to some degree consistently suppressed.

The specific location or series of locations in the space at which a given harmonic is resolved into a continuous tone is a function of the wavelength of the signal: long wavelengths lead to a wide spread (greater distance between the isoharmonic lines), shorter wavelengths to a closer alignment.

Conclusions

The results of the above experiments, as well as related work with continuous complex sounds phase-shifted in a like manner (leading to a spatial resolution of the harmonic structure of the sound into rhtythmic components, each harmonic pulsating with a speed proportional to the degree of phase-shift, frequency of its fundamental and its harmonic number) were made public in a series of 'performances' which took place at IRCAM in January 1980. As information concerning the types of signals employed, their distribution, etc. may be found in the detailed documentation and score (Rolf Gehlhaar: PAS a PAS....music for ears in motion, Feedback Studio Verlag, Cologne), I shall touch upon it here only in so far as it is relevant to a general understanding.

Firstly, the space in which the signal is propagated should be as dry as possible (< 0.5 sec reverb.) and free of any interfering reflective bodies; the outdoors seems ideal. It was found that the ideal diameter of the circle of loudspeakers was 11m in a space measuring 14m x 14m, with the loudspeakers approximately 1.5m recessed from the walls. Within such a circle, 20-25 persons were a tolerable number; above this, due to reflection and interference caused by the bodies of the audience, the phenomenon was increasinlg seriously impaired. Moreover, a greater number of persons also impaired the physical freedom of the individual to explore the phenomenon, a very important part of the experience.

Secondly, as the appreciation of this phenomenon requires a certain amount of time for acclimatisation to it, it is recommended that very long duarations are used, giving the listeners ample time to explore the space. It must be pointed out that the phenomenon is a fairly delicate one, the success of which is easily disturbed by reflective bodies. Although any listeners within the circle unavoidably interfere, a number limited to the above maximum has very little disturbing efffect.

Further research in this area might include the following:
1) Experimentation with a wider variety of pulse type signals, particularly ones whose harmonic structure and energy dstribution may be specified by window functions or Walsh functions;
2) comparative research into the most suitable loudspeaker types, perhaps even a fundamental redesign of a complete loudspeaker system;
3) work outdoors.

Return to: Biographical Notes