Stockhausen Studie 1

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Aug 3, 2024, 12:36:33 PM8/3/24

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In his treatise, Helmholtz explores Ohm's law using a number of experiments that are conceptually similar to Stockhausen's additive synthesis techniques in Studie I and Studie II. In the most remarkable of these experiments, Helmholtz actually builds analog sine-wave generators which he uses to additively synthesize complex timbres that imitate various musical sounds. The experiment comprises three parts. First, he uses an array of tuned resonators (now known as Helmholtz resonators) and theoretical analysis to separate the sinusoidal constituents from various musical sounds, like the several vowels made by the human voice when singing, flutes, reed-pipes, bowed strings, piano and so forth. He reports, for instance, that "with the aide of resonators, it is possible to recognize very high partials, up to the sixteenth, when one of the brighter vowels is sung by a powerful bass voice at a low pitch."7 He also quantifies this more precisely with a number of tables that show the relative amplitude of each sinusoidal constituent in various musical sounds.8

Stockhausen begins his electronic studies with a research question identical to Helmholtz's. In his notes from the period, Stockhausen contemplates: "The wave-constitution of instrumental notes and the most diverse noises are amenable to analysis with the aid of electro-acoustic apparatus: is it then possible to reverse the process and thus to synthesize wave-forms according analytic data? To do so one would ... have to take and combine simple waves into various forms..."12 This question is an exact description of parts one and three of Helmholtz's experiment (separated here by the colon), followed by a statement of Ohm's law (after the question mark). In Studie I, Stockhausen, like Helmholtz, attempts to answer this question by synthesizing complex timbres using the harmonic series as a starting point. In Stockhausen's tone combinations, however, although each pair of notes belongs to the same harmonic series, all of the constituents do not. For instance, the first mixture in the piece comprises sine waves whose frequencies are 1920, 800, 1000, 625, 1500, 1200 Hz13. 1920 and 800 are the 12th and 5th partials of a tone whose fundamental is 160Hz, and 800 and 1000 are the 4th and 5th partials of a tone whose fundamental is 200Hz, but all six sine-waves, taken together, do not share a common fundamental14. In Studie II, Stockhausen goes a step further and breaks all ties with the harmonic series. Instead, he uses combinations of 5 sine waves whose frequencies are related to one another by the 25th root of various powers of 515 (which forms a scale whose basic interval is about 111.451378 cents, or just bigger than a semitone, and which does not repeat at the octave). Furthermore, in Studie II, rather than combining the sine waves directly, he plays them in rapid succession into a reverberation chamber and uses the ensuing compound echo as his musical material. In choosing to abandon the harmonic series and diffuse the tones through a reverberation chamber, Stockhausen relinquishes the possibility of using additive synthesis to recreate traditional musical sounds (as Helmholtz did), but simultaneously opens the possibility of creating hitherto unheard timbres.

Amongst Stockhausen scholars, there seems to be a general consensus that Studies I and II are a technical and sthetic failure. One of Stockhausen's stated goals each piece was to create "an extreme homogeneity of the basic sounds" of the piece16. Robin Maconie, however, feels that both pieces fail to achieve this timbral homogeneity. From a technical perspective, Stockhausen's equipment was not accurate enough to allow him to control the phase relationships of the individual sine waves. Maconie claims that the lack of control of phase prevented Stockhausen from creating unified timbres in Studie I. He says that "Until the experiment had been tried ... it was easy for composers to hope that the differences in the phase relationship of partials would not greatly affect the tone quality." He says that Stockhausen "may have been puzzled, not to say disappointed, at having had only qualified success in fusing synthetic partials into coherent, unified tone-colors. Whether or not Meyer-Eppler was able to identify the source of the problem as phase-relationship is difficult to know for certain... It is possible that he did not realize how important a factor it was in timbre-synthesis..."

I propose that Maconie may be putting too much weight on the role of phase in additive synthesis. Helmholtz was also interested in the question of phase, and firmly opposes Maconie's view. He says: "I have thus experimented upon numerous combinations of tone with varied differences of phase, and I have never experienced the slightest difference in the quality of tone."17 He then declares that the timbre of a complex tone is in no way dependent upon the phase of its sinusoidal constituents. Although this is not, strictly speaking, true in all cases, modern acousticians agree that "for most complex sounds, the amplitudes of the harmonics have more influence than their phases in determining what we hear..."18. In any event, a simple experiment will determine whether this principle holds true in Studie I. Here I have used a program written in C to cause a digital computer to create two audio files. Each contains a 3 second mixture of 6 sine-waves of equal amplitude. The frequencies of the sine waves are 1920, 800, 1000, 625, 1500, and 1200 Hz, in accordance with Stockhausen's usage in Studie I. In the first file, all of the sinusoids are in phase, and begin at 0 radians at the beginning of the file. In the second file, the phases of the sine waves were randomly chosen to begin at approximately 0.843991, 4.206231, 5.705825, 3.048656, 1.869836, and 5.057321 radians, respectively. Here is a plot of the beginning of the waveform (sound-pressure on the ordinate plotted as a function of time on the abscissa) of each of the generated audio files:

This clearly shows a unique waveform in each file (as would be expected because of the phase difference). Nonetheless, the "in phase" and "out of phase" versions of the note mixture are aurally indistinguishable. From this it may be concluded that, if Studie I lacks timbral homogeneity, the reason is probably not phase. One may further raise the question of where, specifically, Maconie hears differences in timbre...

Seppho Heikinheimo also feels that the studies are a failure, and focuses on sthetic issues rather than technical ones. He feels that both compositions are monotonous and lack interest. He says that Stockhausen's method of composing Studie I, "in which everything is of equal importance gives rise to a host of new problems and even dangers. It easily results in the emergence of an even, monotonous greyness from which the listener finally cannot distinguish anything at all."19. He goes on to say that "...the way the tones are combined can only seem formal and monotonous ... each intuitively differentiated period in the composition resembles all the others..."20. He also feels that Studie II suffers the same defect. He says: "The same critical evaluation of Studie I holds true for Studie II, and the same criticism of of monotony is undoubtedly justified."

Although I consider both works to be a great success, I will, for the sake of completeness, point out a few more technical shortcomings in Studie II which Stockhausen scholars seem to have overlooked. Stockhausen was not only interested in additive synthesis, but also in a related technique now called "subtractive synthesis", which is defined as the use of filters to selectively remove sinusoids of unwanted frequencies from broadband noise. In a discussion of Studie II, he describes his interest in this technique: "An alternative fundamental method of producing electronic sounds is based, not on the addition of sine-wave frequencies... but instead on the separation of 'white noise' into 'colored noise'. Here electric filters are needed to split up the 'white noise' into bands of noise of any given breadth and density... " He then continues by saying that his use of a reverberation chamber in Studie II was an attempt to simulate sounds produced by subtractive synthesis, by adding broadband noise back into the audio signal. He says "In Studie II -- in the absence of sufficiently varied filtering systems, -- a special procedure was adopted ... [which] permitted the incorporation into the composition of the noise spectrum."21 A Fourier spectrogram (amplitude is plotted on the ordinate as a function of frequency on the abscissa) of the opening note mixture of the piece (mixture number 67), however, reveals that there is no broadband noise, aside from some low frequency tape-hiss under approximately 1000 Hz.

Only the original 5 sine waves are visible in this spectrogram. It would therefore be a mistake repeat Stockhausen and say, for instance, that "the use of noise solely as a musical element is the most important feature of Studie II",22 because there is neither noise nor anything else resembling subtractive synthesis.

The spectrogram also reveals another shortcoming. Stockhausen appears to have wanted each sine wave to be equal in intensity. He specifies that "the five frequencies are recorded at 0 dB", and that the reverberation chamber should have a "regular frequency response"23. This spectrogram, however, reveals a sharp roll-off. In fact, the highest sine-wave is barely visible (or audible). It is unknown whether this is the result of poor frequency response of the equipment (the reverberation chamber, microphone, tape machine, the tape itself...), degradation of the tape over time, side effects of modern mastering, or otherwise. In any event, the spectrogram plainly shows an undeniable gap between what Stockhausen set out to achieve and what he did achieve, both in terms of noise spectra and intensity.