Introduction

The present study comprises an analysis of Klavierstuck I by Karlheinz Stockhausen. The piece comes under the category atonal as it does not (at least at first glance) make any reference to the tonal system of Western music. Unlike many other pieces which also fall into this category, the piece explicitly attempts to be "random", this is, not only does it avoid such systems as the tonal system, it attempts (at least in initial impression) not to utilise any other organisational systems (for example serialism). It will therefore be the aim of this study to establish how far this attempt is realised.

Analytical Method

Much of the data on which the following analytical observances are based was generated by computer. The pitch, rhythm and dynamic information from the score was encoded in a way suitable for the computer to present as a database-like interface to a simple programming language which was then used to process the data in any way the analyst wished in order to ascertain points of interest from the piece.

Hexachords

The pitch organisation of the piece is based on the alternating adjacent hexachords, C, C#, D, D#, E, F and F#, G, G#, A, A#, B. Each hexachord is described in the graph labelled 'Hexachords' in Appendix A. Out of the seventy two hexachords which make up the piece, twenty four can be termed 'anomalous' and are signified by dotted lines connecting their component pitches in the graph and have their anomalous pitches circled and their respective expected pitches written in hollow notation in vertical alignment.

The three anomalous hexachords in bars 7 to 8 (App. A, II) initialise a pattern which the remaining anomalous hexachords follow: in each case the anomalous pitch is the first of the hexachord and the expected pitch is the same as the the first (and therefore anomalous) pitch of the following hexachord. More specifically in the case of these three: not only is the anomalous pitch of the second hexachord (E) the same as the expected pitch of the first, it is also the same as the expected pitch of the third; the three are symmetrical.

The remaining anomalous hexachords all occur together (barring two correct ones in bars 42-43) from bars 35 to 50 (App. A, V-VII). Table 1 lists these hexachords (indexed by the hexachord's number and the bar number of its initial pitch), their actual initial pitch (the anomalous pitch) and their expected initial pitch.

The pattern set up by the first three hexachords (bars 7 to 8) is continued here: the expected pitch of an anomalous hexachord is the same as the anomalous pitch of the following hexachord. Furthermore, in this block of anomalous hexachords the pattern forms a cyclic unit: the expected pitch of the last hexachord in the block (#63) is the same as the actual pitch of the first hexachord in the block (#44).

Table 1 also shows the two correct hexachords (#51 and #52) which represent a break in the abovementioned pattern. Firstly, they cannot fit the pattern because they do not have an anomalous pitch or an expected pitch and secondly, the expected pitch of the hexachord immediately preceding the first correct one (#50, a G#) is not the same as the actual pitch of the hexachord immediately following the second correct one (#53, a G) though the two pitches are only a semi-tone apart; it is as if two pieces of the puzzle don't quite fit.

Figure 1 shows the actual and expected initial pitches of each of the anomalous hexachords from bars 35 to 50.

The abovementioned cyclicity is re-enforced by this figure: the final D#'s tie mark is to the initial D#. A similar unity can be discerned between the first seven hexachords (those which come before the break): the first and seventh each have D# as their initial pitch. However, because there are an odd number of hexachords here, the cyclic unity is not formed by identical actual and expected initial pitches, but by identical actual initial pitches only; thus the cycle is different from its encapsulating cycle.

No such pattern exists in the eleven hexachords preceding the break though it is interesting that both the initial seven and last eleven groups of hexachords end with the same pair of actual and expected initial pitches (in opposite roles) repeated twice (D# and A#).

Time and Barring

The third staff of the graph at Appendix A (labelled 'Divisions') shows how the piece can be divided into numerical sections based on its time signatures and barring. The six major divisions are show by the brackets underneath the staff while the thirty six minor divisions (six in each major division) are show by the brackets above the staff. The number of crotchets which each minor division encompasses is represented by the figure above the corresponding bracket. The number of crotchet beats of each group of six minor divisions is a combination of 1, 2, 3, 4, 5 and 6, with none of the six groups repeating the same combination:

Table 3 describes how each of the divisions was calculated.

The stared (*) divisions in Table 3 present some complications. Firstly, the division of size 1 in major division I is made up of three quavers which is a crotchet and a half, not a single crotchet. Secondly, the division of size 2 in the major division IV is made up of two crotchets and a semi-quaver where it should be just two crotchets. Apart from the these two problems this pattern is comprehensive and is very likely intentional on the part of the composer.

Summary

Numerical analysis of the piece reveals the important significance of the number six. The pitch organisation is based on hexachords of which there are seventy two in total, a number divisible by six giving twelve (twice six). There are twenty four anomalous hexachords giving a ratio of correct to anomalous hexachords 2:1 (which is parts of three, or, as 4:2, parts of six).

The numerical division scheme is also based on sixes. There are six major divisions each containing six minor divisions which are made up of the sizes one to six.

The presence and strictness of these two organisational systems would seem to go against the abovementioned premise that the piece attempts to be random. However, it is important to remember when considering music generally and atonal pieces more specifically the difference of perception brought about by difference of view point; these systems are very interesting and important to the analyst who is aware of their presence, but to the performer and listener they are of little importance as they can neither be realised by the former or perceived by the latter.

Secondly (and as will be explained in greater detail later on), the aural effect of a hexachordal pitch organisational system is one of randomness; it causes all the intervals to be atonal.

Analytical Method

While numerical patterns are very important in that they give an insight into the composer's methods and intentions, of equal importance to the analyst is how the piece is perceived aurally. Appendix B contains graphs detailing those elements of the piece which can be perceived aurally. Its second and third staffs show the dynamic and pitch contours respectively. The dynamic contour graph shows the progression from one prevailing dynamic in the piece to the next (i.e. although a mixture of dynamic markings are used in bar 20, the overall effect will be of fff because that is the loudest dynamic present). Similarly, the pitch contour graph picks out the most important notes (usually the highest where multiple notes occur simultaneously) in order to show the general shape of progression of tessitura. The pace graph shows how many chronologically distinct attacks occur per quaver.

Aural Sections

The brackets underneath the top staff of Appendix B show how the piece can be divided into fifteen sections on the basis of aurally significant elements. Herefollows a description and justification of each section division.

The first section ends at the end of bar 2. Although the dynamic contour graph here does not suggest a section division (the dynamics at the end of the first section and start of the second section are the same), both the pitch contour and the pace graph do. The division coincides with a point on the pace graph where the pace is 0, more specifically, the moment is silent and almost coincides with the beginning of a general movement upwards in pitch contour. The relevance of the change in pitch contour to the section division is justified by the silence which occurs after the first two notes of upward movement (end of bar 2): the effect is that the upward movement begins at the B in bar 3 rather than the lower Bb in bar 2.

The third section begins at the beginning of bar 5. In this case all three graphs point to an obviously aurally distinct character between bars 4 and 5: there is a sharp increase in dynamic from pp at the end of bar 4 to f at the beginning of bar 5; a general rise in pitch contour started at bar 3 ends at the end of bar 4 and a new upwards shape begins at bar 5; and there is a decrease in pace from bars 4 to 5 and sudden increase in pace in bar 6. This change of pace could also signify that the notes in bar 5 function more like an upbeat to bar 6 and consequently that the section begins in bar 6. However, the function of bar 5 as the beginning of a section gives a symmetry with the previous section: a moment of slow paced, loud notes followed by a bar's worth of quick paced notes.

Although the division between the third and fourth sections (from bar 6 to bar 7) occurs in the middle of a downward pitch contour, there is a sudden change of pace from demisemi-quavers to quavers. The division is also accentuated by a considerable change of dynamic from bar 6 to bar 7 (ff to pp). Although the dynamic of the notes immediately following the pp in bar 7 (C# onwards) is as loud as in the previous section, the slower pace and downward movement give the bar a different character to its preceding bar. Also, the typical speed of performance is such that that the quaver of pp lasts for a relatively long time.

The key aural signal to the division between sections four and five is the crotchet rest at the beginning of bar 9. Dynamically, the two sections are similar (both loud) and the pitch change is only a dip in the generally static surrounding tessitura. The pace change, however, is quite considerable: the fifth section only comprises four chronologically independent attacks whereas the last bar of the previous section is made up mainly of semi-quavers.

Bar 11 comprises the last attack of the fifth section and is made up of one nine-note chord whose notes last for varying lengths of time. The result is that by the end of the bar there is virtually no sound at all. This, coupled with the initial rests of bar 12 (the beginning of the sixth section), provides the main divisional phenomena between the two sections. Although the dynamics of the close of section five and the opening of section six are similar (fff and f respectively), the length of time between the two attacks makes the dynamic contrast between the sections significant as well. The first note of section six continues the level contour of the previous two sections. Its following movement, however, begins a new upward contour.

Section six is a long section made up of fast paced movement and erratic pitch and dynamic change which shows no sign of abating until bar 18. Although there is no gap in the sound (i.e. no rest) in bar 18, the moment where the pace is 0 and the following slower paced movement of bar 19 give a good indication of a change of character. Immediately following bar 18, both the dynamic and pitch contour continue their erratic movement.

The division between sections seven and eight is clearly marked by the 2/4 rest bar (bar 21) between them. Further to this, the chord in bar 20, similar to bar 11, comprises one initial attack and the gradual release of each note in turn; thus the silence between the two sections begins in bar 20. The following movement of section eight continues erratically both in pitch and dynamic and is in contrast to the held chord of bar 20.

Although there is no silence between bars 23 and 24, a sudden change in dynamic (from ff to pp) and slower pace creates considerable enough contrast between the two for the moment to be considered the divide between sections eight and nine. Also of interest is the inclusion of a pedal marking from bars 24 to 27 emphasising the contrast between the two sections.

The fff chord in bar 33 signifies the end of section nine. Although the immediately preceding notes are quiet, they only last for a short time and the character of the preceding phrase is loud; thus the chord functions as an end to the previous phrase not a beginning to the next. Also, there is no further movement after the chord until the pp G# four beats later.

The most obvious justification for the section divide between sections ten and eleven is the three and half crotchets rest in bars 37 and 38 and the 0 pace movement in the same two bars. The section divide occurs in the middle of a dynamically rising and falling figure (though, of course, there is silence in between), marks the end of an area of level pitch contour and marks the beginning of and area of generally rising pitch.

The quick paced, generally quiet notes of section eleven culminate in the fff chord of bar 42. This chord is accented as important not only by its sfffz marking but also by its preceding grace notes and crescendo marking. Section twelve contrasts the quick, quiet movement of section eleven with loud chords at a slower pace; deliberate in their execution.

The final two bars of section twelve mark the beginning of a short period of less erratic pitch contour: the contours here consist of smooth rising and falling figures. The divide between sections twelve and thirteen (bars 46 and 47) coincides with the divide between two such figures (see the falling figure from 46 to the first note of 47 and the rising figure beginning on the same note). The dynamics of bar 46 are generally loud (despite the final pp) while the dynamics of bar 47 are generally quieter. Another contrast between the two bars is that the notes of bar 47 construct a chord (i.e. are held down and sustained with a pedal marking) while bar 46's notes are all separate from each other.

Section thirteen continues at a fairly regular pace with periodic rising and falling in pitch contour and similar dynamic contour but with a longer period. There are two possible candidates for section divisions within the longer section 13. First is the sudden change in dynamic from f to pp from bar 47 to 48 coupled with fall and rise in pitch contour. However, the continuity of pace and sustained nature (see the pedal marking) of the preceding and following bars do not allow this moment to take on enough aural significance to be considered divisional. Second, the sudden change in dynamic from p to ff and short rest at bar 53 could signify a section divide. However, the rest only lasts for a dotted semi-quaver and the quietness of the preceding notes only for the same amount of time. The moment is too brief to be of significance and, more importantly, is followed immediately by a much more suitable candidate for the section division.

The nine-note chord in bar 54 marks the end of section thirteen. As in bars 11 and 20, the chord consists of one attack and each note ending at different points; thus perpetuating the divisional silence between sections thirteen and fourteen. The B and G which begin and span section fourteen do not quite fit any of the divisional patterns. The pitch contour of section fourteen is typified by upward movement starting on the following A in the left hand. Similarly, the dynamic contour of the section begins on the same note and moves upwards from pp. The B and G do not function as members of the section. However, they also do not function as members of the previous section as they have no common characteristics (such as quick pace).

Finally, the divide between sections fourteen and fifteen is signified by both the rest at the beginning of bar 57 and the change in dynamic from ff to mf. The pitch contour at this point is at a horizontally extended peak between the rise of the previous section and the fall of this section.

Symmetry

Figure 2 shows that all but four of the section divisions (fourteen in total, described above) coincide with one of the chords in the diagram. The section spanning bars 9 to 11 marks a turning point in the pattern of section divisions: whereas up to (and including) the chord at bar 9 the "active" side (that is, the side of the chord which has note activity on it, shaded in the figure) was the preceding side, from bar 11 onwards the active side becomes the following side; a kind of symmetry is centred around bars 9 to 11.

Pace

The significance of changes in pace has already been demonstrated in their influence on aural sectionalisation described above. The pace contour otherwise, however, shows little correlation with the other elements of the piece. For example, the pace of the loud section in bar 6 is quick while an equally loud section in bars 43 to 46 is of quite slow pace and there are no points where its contour follows that of the pitch.

Its only other significance is when it is considered by itself: it is a very smooth contour. The pace often remains constant for three or four quavers at a time and when it changes, it normally changes by step and therefore, by itself, plays no part in the randomness which the present study aims to identify.

Pitch Contour

The pitch contour graph at Appendix B shows that some interesting shapes and patterns occur in the pitch of the piece. Firstly, large scale rising and falling of contour occurs five times from bar 1 to bar 7 (App. B, I-II), another five times on bars 45 to 47 (App. B, VI-VII) and twice more in bars 55 to 59 (App. B, VIII). The relationship of these shapes to the aural sectionalisation of the piece has already been discussed above, but they also relate in some places to the dynamic contour of the piece. The dynamic contour of bars 5 to 7 roughly follows the shape of the pitch contour of the same span. Similar correspondence occurs at bars 45 to 47 and again at bars 55 to 59 including here the little inflexion of the contour at bar 57.

Another interesting pattern of pitch contour occurs in bars 13 to 19 (App. B, III). Here many of the changes of direction of the pitch contour occur after a movement for three notes in the previous direction. There is also a certain amount of correspondence between the dynamic and pitch contour here. Above I described this section as erratic and this is exactly the effect that the regular change of pitch direction gives. Also see the very large intervals here which enhance the erratic nature. Similar use of regular change of contour direction occurs at bars 24 to 32 (App. B, IV-V) and bars 34 and 35, though not for such an extended period.

Another interesting element of pitch basis in the piece is the considerable use it makes of major seventh intervals (and compound minor seconds, their inversions). They regularly occur at the chord points of the piece. For example, bar 9 has major sevenths between the notes in both the right hand and the left hand; both the right hand and left hand chords of bar 11 span major sevenths; the chord of bar 18 contains two major sevenths; the chord at bar 20 is made up of three major sevenths; the chord at bar 33 is made up of two major sevenths; the chord at bar 37 has major sevenths in both the right and left hands; the chord at bar 42 contains four major sevenths; the chord at bar 54 contains four major sevenths; the figure of the last two bars is made up of a series of major sevenths; and there are numerous other examples in the course of the piece.

Of course, the basis of the pitch system on two hexachords will inevitably give rise to chromatic intervals though it also allows for more tonal intervals and, indeed, other chromatic intervals; why chose major sevenths in so many cases? And why chose major sevenths for the chords which only use four notes from the hexachord? (e.g. bars 9 and 37). It could also be argued that the range of notes that can be played at a time (particularly as in bar 11 where all the fingers are occupied) would not span much more than a seventh; but why then do the sevenths occur in bars such as 9 and 37 where only four notes are used? Or at 20 and 42 where some of the notes are played first as grace notes? And similarly, why are sevenths favoured at points where the notes are separated in time (bars 60 and 61) if span of hands is a consideration?

The major seventh is a very "atonal" interval and its likely that the composer has favoured it over other intervals because of this; he wishes the piece the sound random and he achieves this through a process of carefully calculated randomness. The method of achieving erraticness described above (frequent change of pitch direction) is another example of calculated randomness. The piece is not truely random, but employs an aesthetic of randomness.

Finally, as well as there being a number of very atonal intervals, there are also numerous tonal intervals such as the C to E (major third) in bars 1 and 7, the F to A in bar 44 which are separated by an octave or more. Such intervals lose most of their tonal quality when such large gaps exist between them. Even the less tonal intervals such as fourths and fifths lose any sense of tonality when separated in this way (F# to B in bar 3, E to A in the middle of bar 7).

Dynamic Contour

The discernible relationships between the pitch contour and the dynamic contour are described above. These correspondences work on the large scale at which they were relevant for a discussion of pitch contour. However, once attention is paid to more detailed dynamic markings in the piece it soon becomes clear that the composer is keen to convey further a sense of randomness through their disposition. Looking, for example, at the first bar: the first three notes have the markings pp, fff and p and similarly, the C and D in the middle of bar 4 are both loud amongst quiet markings. These markings have the effect of breaking down any sense of beat (in tonal music strong beats and weak beats are often accentuated with dynamic markings) and thus add to the composer's random aesthetic though, again, are as a result of a calculated random effect, not true randomness.

The dynamic markings are also used to confuse any sense of progression in the piece. For example, the section break between bars two and three (described above) has the marking fff at both the end of the first section and the beginning of the next. Tonal progressions and phrase structures use dynamics to help shape phrases and would likely give a quiet to the end of shape such as bar 2's. The very loud note gives the feeling of unfinishedness (or "dominance" in tonal terms) and does not belong at the end of phrase. Similarly, the ff marking at the beginning of bar 41 interrupts the otherwise quiet phrase in which it occurs and anticipates its loud ending thus interrupting the progression towards the ending.

Voices

As well as the correlation between pitch and dynamic contour and the patterns in pitch contour described in the sections which above I called erratic, one other pattern can be drawn from these sections. The pitch contour of bars 24 to 28 is very erratic while the dynamic contour is slightly less so and the pace contour is quite smooth. Thus, although the notes are erratic in pitch, they occur at regular intervals and at a fairly uniform dynamic. The pitch contour shows that the octaves at which the pitches occur are the same in most cases (also notice in Appendix A, IV that all these notes are members of correct hexachords). These three phenomena coupled with the sustained nature of the phrase give a sense of continuity to the notes which is similar to voices in tonal music. Notice particularly the notes at the extremes of pitch here: the high G# in bars 24, 26 and 28; the low A in the same bars; and the low E in bars 24 and 27. Also, to an extent, the Db to C major seventh in bars 25 and 27.

Sound Experiments and Notation

The chords at bars 11, 20 and 54 all consist of an initial attack where all the notes come in at the same time and are then followed by the gradual cessation of each note at different times. This could be an attempt to notate the waveform that occurs when a loud sound happens and is followed by an echo. And considered backwards, these three bars would give the same effect as if a recorded sound was played backwards.

This observation is linked to the meticulous detail in which the piece is notated. The performance direction at the bottom of the score reads:

The tempo of each piece, determined by the smallest note values, is "As fast as possible". When the player has found this tempo and determined it metronomically, all the more complicated time-proportions under the brackets can be replaced by changes of tempo.

Many of these time proportions are very precise (such as 11:10 in bar 1, 11:12 in bar 6 and 13:12 in bar 7 and the passages which come under two nested time-proportions such as bars 1, 6 and 47). The actual difference in tempo in which they result is very minimal, especially to the listener, and it is interesting therefore that the composer has taken the time to put them in. Similarly, a lot of the dynamic markings can be considered with the same scrutiny and found to be difficult both to implement as a performer and to comprehend as a listener.

Finally, could it also be argued that the notation of pitch is meticulous? Every note has an accidental sign which implies an element of particularity but also, the enharmonics are sensible (i.e. there are no occurrences of notes like Fb or B# and enharmonics in large chords are designed to suit the fingers well) which implies that the pitch notation adopted is simply necessary. As mentioned above, the size of intervals is partly what gives the piece its atonal nature. Throughout almost the entire piece both the contour and pitch relation could be retained even if the intervals were reduced to a single octave. However, as the atonal nature of the piece is based on its large intervals, such an alteration would completely alter the nature of the piece. Thus the pitch notation adopted is not meticulous but necessary.

Summary

The results of aural analysis fall into two categories: those phenomena which impose some order on the otherwise chaotic collection of notes which this piece comprises; and those whose intentional effect is to give the impression of randomness.

Of the first category the sectionalisation system is the best example. The ability to split the piece into smaller sections aids comprehension for the performer, listener and analyst. The pace crosses categories: it is generally quite smooth and therefore easier to listen to. However, there are also section where the rhythm is very erratic. Both the pitch and dynamic contour fall firmly into the second category; they represent the most calculated random effects in the piece.

Relationship between Numerical and Aural Analyses

The two systems described above result in two different types of perception by the listener. The hexachords will make the piece sound random; they break down any sense of tonality by not allowing repeated notes and forcing each semi-tone to appear every twelve notes (something very unlikely to happen in tonal music) and thus are another example of the composer's aesthetic of randomness. The barring and timing system, however, is not at all perceptible to the listener and because their divisions occur only at barlines and because the aural division can occur at any point there is no correlation between the two divisional systems.

Conclusion

An aesthetic of randomness is what governs almost all the compositional decisions made in this piece. The composer has chosen his pitches using a system of hexachords and distributed them in far apart octaves in order that they may sound random. He has disposed dynamic markings throughout the piece at points where they best break up any sense of flow or continuity in order that they may sound random. Most of his rhythmic figures are such that no sense of pulse can be derived from them; they are calculated in order that they may sound random. Every detail of the score is in place in order that the music may sound random. The only sensation of order in the piece comes from the discontinuities in flow caused by the chord bars detailed above.

This is not random composition, it is carefully calculated composition which, through its erratic sound world, is intended to give the impression of a randomly derived piece.