The Klavierstücke N.9 by Karlheinz Stockhausen, with its “characteristic repeated chords, obsessive trills and peculiar ending, is a highly characteristic and unmistakably idiomatic piece. Composed between 1955-1961, the piece exemplifies the composers intensive researches specially in the field of musical time.
The article “How time passes” by the composer published few years before this piece was unique and probably the first in 1957 to investigate the perception of musical time and to suggest new techniques for handling the musical time. “The temporal organization of musical elements” has never been studied, at least never so thoroughly. Musical rhythm was never processed in such a way.
The piece makes extensive use of the Fibonacci series in which each element is the sum of the two immediately preceding it. Those numbers are specially used in tempo and time (rhythm) relations. It also clearly sets in use a fundamental rhythmical concept probably best defined by Pierre Boulez: “temps strié – temps lisse“.
“Temps strié“: can be only incompletely translated as “pulsated time”. The original meaning of the term is wider than that, it includes but is not limited to pulsation. More adequately it means any rhythmical construct which can be measured, compared, referenced through a clearly defined basic unit or pulse. This is similar to a drawing on a quadrillated paper, while “Temps Lisse” is literally “smooth(ed) time”, when there is no clear reference beat or pulse.
Those opposing concepts are clearly exposed in the beginning of the piece; the starting bars (1-3) with the equally spaced “pulsated” chords followed by the “smoothed” chromatic line motif at bar 4.
Regarding pitch organization, this piece clearly demarks with serial techniques specially by making abundant use of repeated identical chords, creating pitch reference axis by prolongation. This feature may even be called to be the “motto” of the piece.
This paper will however concentrate on the ending part, the coda of the piece. Specifically starting from the third bar of the page 6 of the score.
The Pitch Space Organization of the Opening Measures
To grasp the material, pitch-sets and “themes”, for analyzing the Coda, one should have first some information on the main constituting elements of the piece. Those are clearly set in the opening bars.
The piece starts by a large diminuendo on a 140 times repeated chord, 139 eights and one dotted quarter, from ff followed by f to pppp at 160MM for a eight note.
As it will be seen soon, the time pulse of 160MM and therefore the accuracy in performance of that precise tempo indication is of a crucial importance for the structural integrity of the piece. The ratio 160BPM/120BPM (or 160BPM/60BPM), which is 2/3, will be used extensively throughout the piece.
To count the number of repeated chords in this analysis there was a decision to take in the first bar. Shall one count 140, which is the correct number of chords: 139 time eight notes and one dotted quarter note or shall we take the number 142 from the time signature (142/8) as a “fake” (virtual) number of chords? I think there is a subtle notational effect in here. The diminuendo is so long and goes so far pppp that I believe the last dotted-quarter chord is supposed to “sound” as if there were still three (more) eight-note chords yet imperceptible ones.
In actual performance, if played well, the effect is (or should be) as if there are still three more repeated chords at the end of the bar, but so softly played that they are almost unheard.
All repeated chords all through the piece always appear as a succession of equal rhythm values and never with a “stop” as notated in the first bar.
The importance of this point is that if we assume there is actually 3 very-very soft (suggested but not actually played, “unheard”) chords in the space of the last dotted-quarter, we get 142 for the number of attacks for the first bar and we shall base our numerical relationships analysis on that first number. That is the option I will take for the present paper.
However the stop with a dotted-quarter note seems structural if we consider the connection to what may be called the “secondtheme” (bar 3).
The “thematic chord” of the piece is a well-known and widely used aggregate. The PC-Set Prime Form: (0,1,6,7), Forte Code: 4-9 with Interval Vector: .
One cannot miss the connection of this PC-Set with the beginning of Alban Berg’s op.1 Piano Sonata, literally the same notes.
In the second bar we have 87 chords with the same dynamic span. The time-length relationship between those two bars are 142/87 = 1.63218… A number very close to the golden ratio (1.618..)
The “Second Theme”
The third bar presents a very interesting setting of a straight chromatic scale. From the point of view of pitch-space organization, there are here many points to note.
The line is a straight chromatic one with “stops” at pitches D, F, A, A and B. Those are PC-Classes: 2,6,9,10,11. The gaps between them are filled with grace notes.
Those lasting “main” notes form the PC-Set (0,2,3,4,5,8), Forte Code: 6-Z39, interval vector: .
This set, (023458), has no common tone with the set of the previous chord (0167), and their interval vectors  and , even though they have common values, present very different characteristics.
Specially on IC’s 2-3-4 “the chord theme”s interval set has all 0’s and while the “second chromatic theme”s set has all 3’s.
In this beginning the composer presents us with two completely contrasting ideas. From the point of view of pitch organization, the first idea of a repeated chord affirms a strong reference sonority and prolongation with PC-Set (0,1,6,7) while the second “theme” (or idea) creates a “smooth”, unmeasured-like sound-space by delicately underlining PC-Set (0,2,3,4,5,8) from within a chromatic scale.
Time Organization in the Opening Measures
In the presentation of the two main musical ideas, at the beginning of the piece, the contrast in time organization is even more accentuated than the pitch differences.
With 1 for an eight-note, the durations of the second theme notes form a series: [3-8-5-13-5-8] if we omit the first “3” we have a perfectly symmetrical Fibonacci series: 8 (+) 5 (=) 13 (-) 5 (=) 8.
On one hand the omission of the dotted-quarter note in the beginning of the third bar as shown above, may suggest that this note (chord), which is identical to the opening one, is a part of the preceding bar, thus it makes a connection with the last chord of the first bar, on the other hand the duration (3) of that chord also fits in the Fibonacci series above.
Duration and tempo relations for the first 3 bars of the composition can be summarized as follows: In the “Duration in 8th.” column the symmetries: 139 – 3 / 87 – 3 and 8 – 5 – 13 – 5 – 8 are clearly visible. Furthermore if as explained in section: we count 142 attacks for the first bar and compare this to the length of the second bar; 142 by 87 we get the ratio: 1.632183908045… an irrational number interestingly close to the Golden Ratio
This first part of the composition is now continued by an interesting series of the same chord and silences: The row created by the repetitions of the chord and the silences in between is worth noting: [13-2-21-8-1-3-8-1-5-13-2-5-3]. This is actually a shuffled Fibonacci series: [1-2-3-5-8-13-21].
With these constituting elements we can now get to the point of this paper which is the Coda of the piece.
Overview of the Ending Section
From the notation we see three kind of notes in this section:
- Note-type A: Normal-size notes within grace-note groups
- Note-type B: Grace-notes groups
- Note-type C: Notes gradually added into pedal resonance, creating an aggregate (chord), in the low range of the instrument. This can be thought like a pedal chord [E-B-A] which is to be sustained until the end of the piece. This aggregate, the PC-Set Prime Form: (0,1,6), Forte Code: 3-5 with interval vector:  is a subset of the first chord.
The perceptibility of this last “chord” is created by the dynamics mf ff and f the range contrast (low to extreme low compared with the other notes in the section) and the resounding effect of the sustain pedal in that range.
Pitches as Groups of Grace-Notes
In this section the ending of the piece is examined bar by bar. The division in bars in this section of the piece is structurally significative because it presents groups of grace-notes with varying time-intervals (silences) between them.
The speed of the groups of notes and the evolution of the “waiting” times between each group is the subject of the next section.
Bar 1 of the Coda (measure 3, page 6 The first group of notes-bar starting the Coda section shows a clear analogy to the second “theme” of the piece.
The analogy is obvious when comparing the PC-Set of this bar (all notes taken as equal in weight) which is: (0-1-2-4-5-8), Forte-Code 6-15, interval vector:  with the “long” lasting “main” notes of the “second theme”: the PC-Set 0,2,3,4,5,8, Forte Code: 6-Z39, Interval Vector: .
Bar 2 of the Coda (measure 4, page 6 The second group, set in a bar of 5/8 duration forms the PC-Set: (0-1-2-4-8), Forte-Code 5-13, interval vector: , this is a subset of the previous bar.
Bar 3 of the Coda (measure 5, page 6 This third bar is one of the longest of the section: 21/8, note that the notes are to played as fast as possible but gradually slowing down, and one must wait for the remaining time of the bar throughout the section.
The second group of notes is the full-size notes, to be played mf,f of ff:
Considering the grace-notes only, pitch-classes C and F are most repeated ones, three times each. There are closely followed by C, D and B. Those PC-Classes form the set: Prime Form (0,1,2,3,7), Forte Code 5-5, Interval Vector: .
This set is related to the one of the first repeated chord: (0,1,6,7) and have related ones in many structurally characteristic places throughout the piece.
However, the “weighted” pitch-classes stated above, C-F-C-D-B are most clearly heard just before this Coda section in two different ways.
First it is stated as a chord, then slightly modified, as PC-Set (0,1,2,6,7) Forte Code 5-7, interval Vector: .
Bar 4 of the Coda (measure 6, page 6) This bar is one of 13/8 length which is the previous step from the last in the Fibonacci series used: 1-2-3-5-8-13-21.
In analyzing first the most weighted notes C,D,G, notated as full-size note-heads one can not miss the analogy with the initial chord of the piece C,F,G,C. The sets for the first chord of the piece and this passage being respectively (0,1,6,7) and (0,2,6) present analogy even though their interval vectors are somehow different:  .
The pitches notated as grace-notes display interesting analogies with previously heard elements too.
All pitches form the PC-Set (0,1,2,3,4,5,7,8,10) with Forte Code: 9-7 and the interval Vector: . But the way those pitches are segmented is very interesting from the point of view of the structural integrity of the piece.
All the notes of this bar can be segmented in several ways with or without the full-size notes of the left hand and the resulting sets can be examined and compared to the first chord of the piece.
Pitches in the remaining bars of the “Coda“
The remaining bars of the “Coda”, from the point of view of pitch organization can be similarly analyzed. One crucial point in here should be the separation of the grace-notes sized notes with the full-sized ones.
Those “parts”(grace-notes and full-size notes) can (and probably should) be examined both as different “voices” for the PC-Sets emerging by connecting even distant full-size note-heads but also as a kind of “PC-Set Polyphony” to display how two different sets are sounding simultaneously because they are clearly set apart by dynamics and texture.
For example connecting notes notated at the same or close dynamic levels (i.e. f and ff reveals structural points worth noting.
The sets are all centered around PC6 thus emphasizing the tritone and building around with minor-major seconds. This creates a continuity by referring to the sound of the first chord which has been set through repetition and prolongation and a central reference sonority in the piece.
Time Organization in the Coda
The instructions for performing this “Coda” are very peculiar: the “small” notes are to be played as fast as possible but one must wait for the remaining time of the bar before proceeding to the next but at the same time towards the end the speed of those “small” notes must decrease, they must go slower and slower yet the tempo of the beat duration which affects the elapsing time of each bar should not change.
This view can be sectioned as sub-groups in several ways. There are some centers of symmetries around some values.
This is a real masterpiece of the XXth. century, creating new idioms for the piano, it creates amazing sound effects while having an incredible structural integrity which reveals itself in analysis layer by layer.