We now mention the topics covered in Stockhausen's article, in the order in which they appear. Following this section, we will return to some of them in more depth.
Beginning with the statement "Music consists of order-relationships in time..." [p. 10] Stockhausen introduces his concept of time-intervals between maxima or minima of acoustical sound-pressure waves, denoting them as "phases" (a misuse of an already-established word, which raises immediate objections from scientists [Fokker1962], [Backus1962]). He mentions that pitch and duration are really two different localized aspects of a common underlying phenomenon - we perceive a sound-event as a pitch if it occurs at a rate faster than around 16 cycles per second, but we perceive it as a duration if it occurs at a slower rate. (we will discuss some problems with this concept in the next section)
He discusses the traditional notation for durations, of note heads either divided from or multiplied by a fundamental unit which is generally assigned a metronomical tempo. If a whole note is taken as the fundamental, for example, then half notes are obtained by dividing a whole note unit by 2, quarter notes divide the whole unit by 4, etc. If an eighth note is taken as the fundamental, then quarter notes are obtained by multiplying the eighth note unit duration by 2, half notes multiply the eighth unit by 4, etc.
Stockhausen compares this system against the traditional notation for pitches, finding it lacking. Pitches can be named, in the chromatic scale, which are not whole number multiples or divisors of a fundamental unit. He looks at different ways that a scale of differing durations might be established. In passing, he suggests that a keyboard might be devised which would cover a scale of equally-tempered increasing durations, instead of increasing pitch values (we return to this duration-keyboard in the final section, where we also consider a proposal by Henry Cowell for a new rhythmic instrument).
Harmonic or "modal" scales are more easily established, as several examples show (which we treat in depth in a later section of this paper), but he is searching for a chromatic scale of duration, in order to apply his serial composition principles to it (we showed in the first section that his serial system requires an exhaustive scale of equally-spaced proportions in order to treat a given musical parameter, like duration). In fact, he criticizes the use of a harmonic series of durations in serial music up until that time as "stylistically inappropriate" [p. 14] for use in serial composition.
Following a criticism of poly-rhythmicality (also stylistically inappropriate), he brings out the notion of group structure, in which individual relationships among notes are lost in the texture of the whole group. This Group-structure is one of the tools he used, starting in Gruppen, to advance his serial system beyond the limits it reached in pointillism, where the parameters of each note were treated as isolated individual values and the music had an overall static effect.
At this point we are reminded of the words of Iannis Xenakis, also critical of this polyphonal serial music, but criticizing from a different angle, as an advocate of applying stochastic principles to musical parameters, rather than serial principles:
"Linear polyphony destroys itself by its very complexity; what one hears is in reality nothing but a mass of notes in various registers. The enormous complexity prevents the audience from following the intertwining of the lines and has as its macroscopic effect an irrational and fortuituous dispersion of sounds over the whole extent of the sonic spectrum. There is consequently a contradiction between the polyphonic linear system and the heard result, which is surface or mass. " ([Xenakis1971], p. 5)
Stockhausen points out that there is an inherent contradiction between the inner structure of continuous musical tones, which are composed of overtones related harmonically (the "harmonic scale of perception" [p. 20]), and their organization in 12-note series, which have no harmonic or any other hierarichal structure (the "chromatic scale of perception" [p.20]). The introduction of this contradiction with the 12-tone compositional technique is, he says, what Schoenberg really meant by the "emancipation of the fundamental tone" [p. 20]. We shall have more to say about this issue too, in a later section of this paper.
He proposes that this contradiction should also be introduced into the sphere of rhythm -- that the internal structure of a rhythmic section, corresponding to a single complex tone in the sphere of pitch, should be considered separately from its placement among other rhythmic sections in a rhythmic series, corresponding to the placement of a single tone somewhere in a 12-tone row. This would allow the natural harmonic structure of related rhythmical sub-divisions to unfold, within an individual rhythmic unit, but would also allow his serial system to treat that entire harmonic structure as a single unit, to be placed before, after, or alongside other different units in accordance with the proportional series governing the macrostructure of the piece.
Then he proposes a system of equally-tempered "fundamental durations" ([p. 22]), analogous to the equal temperament of fundamental pitches. And finally he is ready for the application of the serial system to a set of fundamental durations, much as he applies it to a set of fundamental pitches (each with its own differing harmonic overtone series and resultant timbre). He works out an example which soon grows rather complicated, involving the variable overlap of different successive "groups of fundamental durations" [p. 24]. Adrian Fokker ([Fokker1962]) attempts to clarify this example, taken from the piece Gruppen, which we return to in detail later.
Continuing with his example, once he has applied the serial system to the set of fundamental durations, he discusses how one might use it again to choose the internal rhythm inside that duration, which is like choosing a timbre by constructing an overtone series on a fundamental pitch.
Furthermore, if different fundamental durations are present at the same time in a piece of music, they may require different, spatially separated groups of instruments, each with their own conductor, in order to be accurately performed (this is one of the main reasons why there are three orchestras in Gruppen and, later, four orchestras in Carre).
He mentions another contradiction, between "material and method, i.e. .... between instrumental music and serial music" [p. 29]. The composer who seeks to apply the strict serial method to musical durations finds that the written music which results is impossible for humans to play accurately! The choice then is to either "completely renounce instrumental music, and compose only electronic music" [p. 29], or else "seek a completely different path in composing for instruments, through a conception of musical time that is absolutely new". And he proceeds to explain how this might be done.
Not everyone would agree that the goal of completely notating events down to the smallest level of control is even worth pursuing in the first place:
".... What appears musically precise or specific is in fact not. The seeming exactitude of many aspects of musical notation makes many composers think that if only a reliable means can be found consistently to represent this 'exactitude' synthetically in sound, human misrepresentations of compositions will be no more. But when one comes to commit oneself electronically to a 'precise' event, never after its realization again to be inflected by interpretation, it develops that what one though was precisely specified in conception was really only generally outlined. It is gratuitous to speak of creating circumstances in which composers can realize every musical event 'exactly' as they want, since no composer ever *has* known exactly how he wants it, and since even the most exact vision of the event at the moment of realization will alter with time."
"Write a note for flute: no matter how well you know the note, a player can vary it within what you thought were minima. Write a short note: no matter how exact its length may seem in your mind, when you come to perform it, you will find a range of easily perceived variations that lies within acceptable limits. These are the lessons and resources of live performance. They point to the fact that even the most detailed and exact score is actually a set of generalities. They reveal the staggering size of the musical universe, the capacity to perceive which is one of man's rarest treasures. Even the smallest event in a musical fabric, an event of which there will be thousands of similar others in a single work, contains within it an infinity of variation and capacity for interpretation. This variation is not merely at the margins of the perceivable but contributes to the central range of meanings borne by the work." (Charles Wuorinen, in [Schwartz1973], p. 256)
Instead of rigidly precise durational values, human performances always involve some amount of variation. Stockhausen discusses what he calls "time-fields" [p. 30] with a precise durational value given at the center, and a certain bandwidth of leeway to either side, in which repeated performances may vary. That is, the field is a delimited area of possibility, in which the actual value of a given performance will take a certain value.
He suggests that the more precisely detailed that musical events are notated, the less precise their actual performance by musicians is. "...the more complicated the way in which a time-value was indicated, the less sure the performer was about when it should begin and end." ([p. 30])
He gives an example of four different ways to notate the same events (durations in the relationship of 4/3 to 2/3 of a larger fundamental duration) and predicts the relative accuracy of performance for the different notated versions. (in this brief example, the version he feels will be performed most accurately is written with larger note-heads, i.e. quarter notes, while the version he feels will be least accurate is written with sixteenth notes -- one wonders whether this leads to a simple generalization, namely that music written with quarter notes is easier to read than the same music rewritten in sixteenth notes? but such a simple conclusion would preclude the lengthy investigation which follows!) He proposes experiments, having instrumentalists perform differently-notated versions of the same complex passages, tape recording the performances, and analyzing the precise durational values found, to develop a scale of increasing and decreasing performance variability.
The goal, as always, is to find a parameter to which he can apply the serial system. So if he can devise an equally-spaced scale of performance variability, including all possible values, perhaps based on the way that a given group of musical events is notated differently, he can then choose an ordered series from the scale. And this would open the way to move serial music out of the realm of precisely specifying individual events, into the realm of specifying the overall limits of statistical collections of randomly varying events. Additionally, this points the way to new possibilities in instrumental music, which are not present in pure electronic music, in which there are no performers and hence no variability factors to work with.
Stockhausen distinguishes between performance variability which comes about merely by chance, due to the complex nature of the notation in a piece, and variability which has been precisely specified in its statistical parameters, i.e. as deliberately composed Variable form (even though the parameters like exact onset time, duration, precise pitch, etc. of the individual events themselves, by definition, even the number of events in a collection, are not.specified by the composer of the variable form).
He is still depending on the human performer to supply the variability (this is true in both the "chance" situation and in the precisely specified variability situation) rather than the use of stochastic processes ([Xenakis1971]) for the generation of event parameters. It is the limitation of a human performer, rather than an evaluation of a random variable in a Poisson probability distribution, which Stockhausen is interested in. In fact, the music written by Xenakis for real performers, notated from instantiations of stochastic variables, would fall under Stockhausen's first category, of music which is variable in performance due to the performer's difficulty in following the notation's precisely detailed instructions!
Stockhausen examines the possible ways of presenting a scale of, and serially choosing from, different field-sizes of variability, in the sphere of duration. In the case where several separated orchestras are each playing their own differen tempi, i.e. in Gruppen, the longer they play, the more likely it is that they will get out of step. Thus the variability field increases with the length of the segment of music. But no discrete steps can be isolated in this increasing process, so it is of no use to his serial system; the displacement, the amount that they are out-of-step with each other, may come gradually, continuously.
If the different orchestras are not playing fixed tempi. but are rather speeding up or slowing down individually, the situation is more complicated. Here, Stockhausen devises a discrete scale in which, for example, the number 0 refers to the situation where all the orchestras have constant tempi; 1 is the situation where all tempi are constant except for one which is varying, 2 is where two tempi are varying, and so on. This still allows him to apply the serial system to these higher-level, qualitative numbers - critics like Backus and Fokker wonder why he is so obsessed with this "mystical belief in numerology as the fundamental basis for music" ([Backus, p.20]). We will see later that Stockhausen continued to use this generalized, qualitative serial system, using it in the pre-compositional plans for Kontakte in denoting the movements of sounds among groups of loudspeakers.
Stockhausen discusses further the "statistical composition " [p. 32] of fields which contain events, but the events are not precisely specified, and are not even always separately perceptible. When the individual events in a mass-structure are separately perceptible, then the statistical quality falls away; the statistical quality depends on crowding in a short period of time ("Mass-structure means, then, merely the momentary opacity of a group" [p. 32]). We will compare Iannis Xenakis's statistical approach with this later.
Now he presents the `pointillistic' style, of precisely specifying all the parameters for each sound event , as merely a special case of the mass-structure in which all variability factors are zero, and the event is fixed compositionally as a point rather than a field (note that the performer still may not be totally accurate in the performance, so there will be some variability anyway, but it is of the kind he calls "chance" rather than of the composed kind).
Returning to his discussion of how different degrees of notational complexity, each representing the same sound-event to be performed, lead to differing amounts of variability, he wishes to devise a suitable notation for the variabilities (and of course, he will want an equally-spaced scale, in order to apply the serial system).
First, as an example of how NOT to do this notation, he discusses John Cage's graphic time notation [p. 33] in which time is represented by exact horizontal measurements on the printedpage (i.e., say, 1 centimeter of horizontal space equals exactly one second in time). The vertical axis represents pitch, in the usual musical staff. A sound-event composed to begin five seconds into the piece, of exactly one second in duration, at pitch A4 would be represented by a horizontal line beginning exactly five centimeters from the left edge, and exactly one centimeter in length at the A4 position on the G clef.
Stockhausen feels that this notation leads to much less certainty, in actual performance, of exactly where the duration begins and ends, even though the notation appears to be more precise than that using traditional note heads and stems. The reason is that the performer's eyes have to scan the horizontal line segment, measure its length, and translate that into a performance of a certain duration. No matter how long or short the line segment, the same interpretative procedure must be followed. Consequently, the variability factor is the same (according to Stockhausen [p. 33]) no matter whether the composed event is long or short - that is, there is imprecision in executing the onset moment of the event, and imprecision in executing the end moment of the event. On the other hand, with more traditional notation using note heads and stems, the performer's eye gets to count individual notes, which each have a known duration in relation to an underlying metrical time unit, or sub-divide the notes, again with an underlying metrical time unit available to assist in accuracy.
Cage has done away with all notions of proportionality in time-relationships, since there are no notes to be grouped or sub-divided. Therefore, Stockhausen feels, Cage has made all time proportions less distinct, and the result is a continual disorientation in time. Rather than a feeling of timelessness (remember, Stockhausen's investigation is trying to determine how to get away from the precise specification of time, to allow a variability that is nevertheless precisely notated in its extent, which might indeed produce a sensation of timelessness in the listener, but he is still very interested in preserving the sense of relative proportions among different time intervals, which is required by his serial system), Stockhausen feels that this leads to a sense that time is bound to one plane (literally, the horizontal plane on the paper) and therefore is equally present at each moment. And of course this is no use to him and the serial system since it cannot be ordered in degrees of increasing or decreasing performance variability.
Stockhausen returns again to his search for a possible notation of variability in time. He wishes to devise a scale of variable-field sizes which can be serially treated, yet which contain continuous time inside them, yielding qualitative rather than quantitative control. He looks for examples of variable time in traditional notation [p. 34].
He discusses the `grace-note'. In a group of grace notes, to be played `as fast as possible', the individual notes are usually not identical in duration or evenly spaced in onset time, and this is an example of what he is looking for. The speed with which the individual notes can be played (he uses the piano keyboard as his example) depends on their location on the keyboard, relative to each other. Thus the movements of the pianist's arm, which are different for each grace note in the group, is one factor which determines the individual variability of each note (but the traditional composer does not notate the arm movements! instead, they notate the individual grace notes). The reaction time of different performers is another factor, and so is the resonant character of the room in which the performance is given, because the notes must be played slower to remain clearly distinct in a more reverberant hall. But these latter factors are constant across the whole group of grace notes, and thus play no part in Stockhausen's desired notation, while the arm movements are different for each note, and thus are a potential source of information to be notated by him.
He speculates that a series of variable field-sizes would correspond to a series of actions taking various lengths of time, much as the pianist's arm takes various lengths of time to reach the different members of a group of grace notes, and therefore causes the grace notes to occur at varying lengths of time. He might specify a variable number of preparatory movements before each note can be struck, for example (and a serial series of differing proportions among the movement-lengths could be established). The duration of `rests' [p. 34] depend for their onset on how a sound decays to silence, and for their conclusion on how soon the performer is prepared to play the next note. This preparation can be mental if the musical notation is esoteric, or it can be practical depending on the physical properties of the instrument (i.e., the hammer which strikes a piano string takes a finite amount of time to recover before it is ready to strike the note again, and a string player needs a certain time to re-position the bow for the next attack).
So he is interested in a way of making an equally-spaced scale out of the variable action-times needed for producing different sound-events. He mentions John Cage again, who is not interested in constructing such a scale, but who has given elaborate instructions, on occasion, which require appreciable amounts of time to carry out a musical action. The example given is "engage the right-hand pedal, attack the note staccato and immediately allow the pedal to spring just so far back that the note goes on sounding softly as an echo" ([p. 35]) and the presumption is that Stockhausen is interested in using such an instruction as perhaps one of the larger values in a proportional scale of action-times. (Klavierstucke XI is where he implemented such ideas)
The next situation he examines (from Zeitmasse) involves a woodwind player, who must pause to breathe now and then. This gives a natural grouping to the notes that are played in one breath, and Stockhausen considers the relationship of the variability of the overall breath-group with the variability of each individual note within the breath-group.
The duration of the breath actually depends on the register, density, and loudness of the notes to be played in it (ignoring the physiological factors of the musician). "The lower and louder the notes, and the fewer sustained notes there are to play, the shorter the duration of the breath is." [p. 35] Supposing that the tempo indication "as slow as possible" is given, the individual note-durations must be distributed over as long a breath-duration as possible. So the length of the breath, overall, determines how much time is available for distributing among the individual note-durations, but the breath-duration depends on the register, density, and intensity of the individual notes. The composer can specify that the individual notes are each to take a fixed proportion of the overall breath length, which can vary, or they can allow variability of the proportion, of the overall breath length, that each individual note can take (again, with the overall breath length also allowed to vary). In this second case, then, there are two levels of variability, the breath length and the individual note length, which are interdependent.
He proceeds to discuss the overall structure of a piece (Klavierstucke XI), which can also be composed in variable form. In his example, the duration of each group of notes is not fixed in advance, and neither is the order in which the groups are to be performed. At the time of performance, at the conclusion of one group of notes (which has been played with a certain variation in duration), the duration of the next group is determined, based on the some ratio taken with respect to the duration of the previous group.
The overall form of this piece, and the field of variability that contains it as a whole, "will become clearer, naturally, if it can be compared with that of other pieces in a cycle, or, above all, when it is played several times in succession." [p. 36] Thus this piece actually doesn't fully exist in any single rendition; the form can only be appreciated by experiencing several versions and noting the common tendencies shared by all.
Stockhausen explains why he doesn't consider this kind of piece to be an invitation to improvisation on the part of the performer. The fields of variability, with their fixed degrees of freedom, are "no occasion to invent something in addition to the composed structure". [p. 37] Instead, we experience time-proportions which can only arise through the performer's physical actions, where previously, time-proportions were usually notated in absoute time as counted on the clock or as counted against metrical units.
No longer can one look at the time-notation in the score, and check the `rightness' of a given realization against it. "In a field-composition, the parts of the score in which actions are notated give no information at all about the measurement of time-proportions -- the latter come into existence only at the moment when they are realised in sound, when they are played. In this case, the `rightness' of a realisation is checked against itself; tested, that is, in order to find out whether the action-times in the moment of playing stand in an organic relationship to the sound-times to be produced...." [p. 37]
He suggests that one could even interpret this situation as one where the performer, rather than "mechanically quantifying durations that conflict with the regularity of metronomic time" [p. 37], is actually precisely measuring and interpreting sensory information. That is, in fulfilling instructions like "play as fast as possible", the performer relies precisely on their own feeling of whether the notes are as fast as possible yet, while still sounding clearly and distinctly enough, rather than relying on some vague quantification of a variable duration on the clock. And the variable duration is then a precise consequence of the sensory information.
Finally, the notion of degrees of freedom can be applied to the act of composition itself -- the composer chooses from a range, with one extreme being total pre-determination of all durational quantities, and the other extreme being "chance" determinations, with the ability to select an intermediate value (presumably, in application of his usual serial principles....) before proceeding with the pre-compositional sketches that then determine the composition itself.
Having completed his study of variable field-composition in the sphere of durations, he now presents some important consequences in the sphere of pitches, i.e. in composing works for variable-pitch using this system of fields with degrees of freedom.
Since pitch would have to be continuously variable, only instruments capable of non-fixed scales could be used (no pianos, for example). The case of fixed-pitches would be a special case, where the field-size was zero, leaving no room for variation. Just as in the case of durations, the particular pitches could result from specific physical actions with variable distance required to accomplish them on the instrument, or they could also be determined by breaking up a group into parts and then instantiating specific pitches within the parts (this was done for durations in "Zietmasse", involving the number and length of notes possible to be played in one single breath of a wind instrument).
Then, too, the case where the pitch remained constant over the duration of a note would be the exception rather than the rule, for in the general case, the pitch would be the result of individual "phases", single cycles, which would not be exactly periodic, but would also be varying within specified limits. Thus a new instrument would have to be designed, allowing continuous variation between definite pitch and noise, and continuous variation of the fundamental frequency, as well as harmonic spectrum (we consider this again in the section where we compare Henry Cowell's new musical instrument, the Rhythmicon).
His final comments concern yet another contradiction, this time "between, on the one hand, a material that has become useless -- instruments that have become useless -- and, on the other, our compositional conception." Instead of dwelling on this contradiction, he has apparently chosen to reconcile himself with a new concept of musical time.
It seems to us that he is actually admitting defeat, or at least compromising, by modifying his compositional conception of total serial organization. Instead of the lofty goal of total control over all musical parameters, he has admitted that this is impossible, or at least leads to uninteresting results and is therefore useless. Instead, he now wishes to control only higher-level qualitative parameters with his system, in order to resolve the contradiction and continue composing for real instruments.
And it is clear tht he does not really regard traditional instruments as useless - otherwise why would he have continued composing for them? In fact, as ([Heikinheimo1972]) points out, once he completed the tape part for Kontakte, he never again composed a piece of pure studio electronic music; in all future electronic works, he included real instruments among the sound sources.
"Useless", indeed!
Next section (Specific Examples, part 5)
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