Stockhausen Edition 5 CD Cover
No. 6: GRUPPEN (Groups)
for 3 Orchestras (including piano and electric guitar)
1955-57 [24'25"]

     GRUPPEN is probably Stockhausen's most famous work, mainly for its use of simultaneous 3 orchestras.  Happily, it also happens to be one of Stockhausen's most important creations, since it represents both a climax to Stockhausen's early work, as well as a prelude and fanfare to herald all of the compositions to come in the following 50 years.

     In 1955, Stockhausen received a commission from West German Radio (WDR) to compose a work for electronics and orchestra.  After 2 months in Switzerland he resurfaced with a new composition which didn't actually have any electronics in it, but was partly inspired by concepts he'd explored in electronic works such as GESANG DER JÜNGLINGE (which he was also working on at the same time).  Additionally, since this new work featured up to 3 different tempo layers at the same time, it required 3 separately-conducted orchestra groups.  This was because it was frankly impractical to have 1 conductor beat out 3 tempos at the same time (though this could be possible in a tape piece).  After realizing that he could place the 3 orchestra groups to the left, center and right of the audience, Stockhausen took advantage of the spatial possibilities of passing timbres from orchestra to orchestra in different dynamic shapes, speeds, densities and directions.

Time Structure
     The tempo, durations and pitch fields of GRUPPEN are all derived from a single sequence of 12 pitches (see 1st measure below).  This GRUPPEN tone row includes all 12 pitches of the chromatic scale and was designed to include all 11 possible intervals between 2 notes (minor 2nd, major 2nd, minor 3rd, etc...).  There are other reasons behind the choice of this pitch sequence (such as the use of a tritone as a transposition pivot and it's 2 hexachord retrograde nature) but perhaps the most interesting is one proposed by Sara Ann Overholt.  In her thesis on GRUPPEN (later published as a book) she describes finding Stockhausen's actual signature in this tone row.  The way Stockhausen embeds his name is not as simple as how J.S. Bach and Shostakovich had done this in some of their works, however.  It's a pretty complex process bordering on outright numerology but the first step is to assign KARLHEINZSTOCKHAUSEN to 2 and a half rows of 12 numbers with alphabet letters assigned to each of the 26 numbers.  Once the letters of Stockhausen's name is assigned a number, then the notes of the GRUPPEN tone row can be further teased out with some phrase rearrangement ("L-HAUSEN-KARLHEINZ-STOCK-R").  I've never read of Stockhausen mentioning that his name was in this tone row, but it's a very enticing theory. 

     The next step was to take this 12-note sequence and re-sequence it 11 times to come up with a total of 12 variations (including the original).  After adding 1 note to the end for "good luck", Stockhausen had a 145-note long sequence (actually the interval relationship from note 144 to 145 is the same as the relationship between note 1 and 2 and so, more than "good luck", it's more to do with a "snake eating its own tail" cyclic concept).  From this long, non-repeating sequence he derived 145 "groups", each with a different tempo, meter, duration, pitch range and timbral construction.

     For tempo, Stockhausen devised a "chromatic tempo scale" going from 60 to 120 (divided into 12 tempos, 1 for each chromatic note).  The tempo of each group was then chosen from the note matching the 12-step tempo-scale.  Fundamental durations were found by assigning values according to the octave register of the note in the GRUPPEN tone row (lower registers would get longer values such as whole notes, higher registers would get shorter ones such as 8th notes).  The pitch range of each group was derived from the proportionate sizes of intervals from one note to the next.  These assignments can be seen in the time-structure diagram from the Stockhausen Edition 5 GRUPPEN CD Booklet as well as by clicking below.
The full Time Structure can be seen in large format by clicking on the above graphic.
Special thanks to Stockhausen-Stiftung for usage of this graphic.

     A closer look at the first 2 GRUPPEN tone rows in the above time structure diagram is below.  "Proport." (in blue) shows the proportions derived from the changing pitches of the GRUPPEN pitch row.  Being proportions, these overlap, so that's why multiple orchestra groups are needed.  "Tempi" is derived from the chromatic tempo scale (12 speeds, 60 to 120 bpm) and matched to the pitches.  "Zahlwert" (values) is the basic duration based on the octave register of each note.  "Gruppen-nummer" is the Group number.  "Einschub" is a special "insert" group which is like an interlude outside of the large scale serial structure (more on that below).

Excerpt of the GRUPPEN Tempo Form Scheme showing the original GRUPPEN 12 note tone row and 1 re-sequenced variation.  These dictate tempo, duration and basic meter for the 24 groups shown, with 7 "Inserted groups" and a few combined groups bringing the total to 32 groups.

     One important side-effect of using the proportional pitch relation to obtain tempos and durations was that groups would overlap, since one group always starts before the previous one has finished.  As mentioned earlier, this is because each note is expressed as overlapping ratios (above), so Group 2 starts after Group 1 reaches 10 duration units, while Group 1 continues on for 8 more.  Group 3 starts after Group 2 has reached 3 units, but Group 2 continues for 4 more, etc...  This was the moment that Stockhausen realized that it would be good to have 3 spatially separate orchestras in order to have different groups conducted at different tempos.  After this momentous leap, Stockhausen decided to explore the many interesting spatial possibilities available, so he added 3 "free" sections (this concept of "Einschub" (inserts) resurfaces time and time again, from MIKROPHONIE II to CARRÉ to MANTRA, etc...).  These 3 insert sections (Groups 16-22 (above), 71-77, 114-122) were composed outside of the GRUPPEN tone row rules and all 3 orchestras are in the same tempo (actually Group 7/8 is also kind of an "accelerando interlude" since all 3 orchestras are in the same tempo there as well).  These sections also feature some solo instruments as well as certain timbre groupings, most notably the brass in Group 119.  These new groups eventually brought the total to 174 groups.

     The pitch range of a group was based on the pitch interval from 1 time-structure note to the next (from the GRUPPEN tone row).  However the actual sequence is in retrograde (backwards) from the original sequence of the pitches - i.e., the pitch bandwidth interval for Group 123 is from the interval connected to Group 56 for example.  This pitch range only dictates the main pitches, there are many cases where melodic lines go beyond the prescribed pitch limit.  This is analogous to frequency filtering in electronics, where a parametric equalizer can accentuate and emphasize certain frequencies, but some outside frequency peaks can still bleed through.  Having said all that, this concept of the retrograde interval sequence dictating the pitch bandwidth is reported by Jonathan Harvey (as well as Gottfried Michael Koenig) and even he admits that there are so many exceptions to this scheme that it's pretty hard to actually hear any of this.
(score © Universal Edition)

"Formant" Structure
     The actual orchestration of each group is also partly based on the GRUPPEN tone row but in a different manner.  Stockhausen created a "duration-scale" by dividing a whole note duration into 12 decreasing sizes (1/1, 1/2, 1/3, 1/4, 1/5...1/12).  It turns out that the 12 durations Stockhausen came up with proportionately match the overtone series of a fundamental pitch.  Using the pitches of the 145 initial groups (derived from 12 versions of the original GRUPPEN signature melody-row) Stockhausen derived rhythmic patterns for each group.

     This technique of subdividing duration to get rhythmic patterns was used to create timbral material.  As Stockhausen had recently discovered in his experiences with electronic music, speeding up rhythmic pulses (phases) creates different timbral colors.  For example, a slow snare roll can be heard as individual hits, but a very fast drum roll becomes a solid sound (this effect is also well-represented in Structure X of KONTAKTE).  By changing the rhythm of the pulses in a cycle one gets different kinds of timbres.  After layering these tone row-derived rhythms on top of each other, complex overtone-based timbres on an orchestral level arise.  Group 7 (below) is an example which Stockhausen uses in his own published analysis of GRUPPEN ("...How Time Passes...") and it has up to 26 of these layers - which he calls "formants".

     However Stockhausen stepped way out of the laboratory when he decided to look out his window in Switzerland and make sketches of the mountain landscape.  These sketches became the literal "shape" of the orchestral forces at work.  In "...How Time Passes...", Stockhausen includes one of these sketches with the the height of each "mountain" measured in "formant-spectra" - which translates basically to complexity of timbre (or more broadly, orchestration).  This method of using the Swiss landscape to shape the ledger lines was used for Group 7 (as shown), but I doubt that it was feasible to make such landscape sketches for 145 groups (assuming he only had one window to look out of), so I assume many groups were orchestrated based on intuition and balance, but it's a fascinating method of breaking out of the mathematical tempo calculations.

Format sketch for 7 measures (Group 7).  "Bogen = "ties"
(from "...How Time Passes..." (Stockhausen))

Above is the sketch of Group 7 Stockhausen provides in his analysis.  Below is the slightly edited scored result (I took out some extra rest measures so that the image would be clearer).
(score © Universal Edition)
     This score representation is slightly misleading because the formant level number doesn't necessarily dictate the assignment of instruments on an orchestral score (also the 3 orchestras should be combined in one score, not stacked as above), but it just happens that the lower orchestral instruments in this case carry simpler formant rhythms better than higher, lighter instruments.  In any case it still shows somewhat what a Swiss landscape looks like in musical score!

When I was composing GRUPPEN for three orchestras, I had a little room in Switzerland for three
 months, and there was a small window in front of my desk through which I could see the incredible 
shapes of the mountains on the other side of the valley.  
There are quite a few Groups in GRUPPEN which follow exactly the shape of these mountains: 
I became quite expert in drawing the outlines during that time.  

I would take such a shape and divide it vertically into musical measures of equal 
duration...fundamental durations - let's say they were whole notes.  Then I would add horizontal lines 
forming a grid, and subdivide each line going up from the fundamental into two, three, four units and 
so on, like overtones, but in the field of rhythm, until the shape was completely filled.  

So I was thinking in terms of shapes, of musical masses, 
and I could also make negative shapes, windows in these masses of sound.

 - Stockhausen, "Four Criteria of Electronic Music"

     Stockhausen here is very generous with his simplification of the orchestration technique used in GRUPPEN.  In his essay "...How Time Passes..." he goes into much greater detail of how the durational formant-spectra are composed from the serial GRUPPEN tone row proportions.  It's highly recommended reading, but requires a more than fair amount of concentration to follow...

     One writer, Ivor Morgan, analyzes GRUPPEN a bit differently (though eloquently), in that he sees it's fabric as a huge matrix of variations based on a 3-note arch figure (and it's upside-down inversion).  Stockhausen doesn't disavow this interpretation, but I think this melodic motif is a by-product of the GRUPPEN tone row's angular trajectory.

     The above describes the construction of GRUPPEN in extremely broad terms and leaves out things like repeated groups, combined groups, and other exceptions to the rules (also the first use of electric guitar as a solo instrument!), but as always, Stockhausen uses a concept like chromatic tempo assignment to first create a structural framework, and then applies intuition and musical taste (and "inserts"!) to make the final realization organic and balanced.

Spatial Mapping and Dominant Textures
     The aspect of GRUPPEN which is probably the most "audience-friendly" is the unique spatial experience of having 3 orchestras arranged in a horseshoe shape around the audience.  With some audio wizardry I was able to extract the center channel from the CD recording and separated the 3 orchestras into 3 tracks (arranged as columns: LEFT, CENTER, RIGHT) showing relative dynamic activity.  Basically, the wider the column, the louder the orchestra.  The right side numerical markings correspond to the CD track markers, with the beginning Group number in parentheses.  Non-serial based "inserts" are in red.

 Dominating Behavior  Duration                        3-Layer Sound Mapping 
1 1 Bowed string masses w. gentle flute, percussion and harp accents move from orchestra to orchestra

2 5 Low, angular string motif repeats, becoming transparent 0:25
3 7 Strings and perc swell, interrupted by 3 solo violin strokes, but continue on into a full tutti 0:28
4 8 Dissonant harp strums supported by brass hits create a "points" texture 0:23
5 9 Gong signals entry of rhythmic brass accents, changing to wind, vibraphone and perc. figures, eventually subsiding and ending on a brief clarinet trill

6 16 Violin solo begins, supported by strings, vibraphone, and some percussion

7 20 Energetic recap of the previous group

8 21 Cadenza for Orchestra 1 0:17
9 22 Orchestra 1 diminishes in energy with accents from the other orchestras (violin solo ends) 0:33
10 23 Accents (points) and groups featuring pizzicato strings, low brass, piano, and low perc 

11 37 Percussion, marimba, and vibraphone play tremolo/trill figures, leading to a brief harp solo 0:30
12 42 1st guitar solo with high strings and percussion

13 46 Clarinet motif signals return of low angular string motif from Group 5 0:18
14 49 Metal percussion signals pizz string point textures moving from left to right, supported by vibraphone, high strings and perc

15 53 Swelling mass with tremolo guitar/vibraph. in foreground 0:16
16 56 Tutti moving from left to right

17 62 Sound masses of strings, guitar & perc

18 70 Tremolo strings, then a drum solo (Orch.3) w pizz contrabass accents

19 72 Harp and contrabass duo, vibes & harp duo

20 74 Percussion (metals), guitar  0:13
21 75 2nd Guitar solo with drum crescendo

22 76 Clarinet, oboe, flute trio w perc. 0:14
23 77 Irregular drum rhythms with sustained metallic chords 0:28
24 78 Brass and percussion interlude   0:16
25 86 Groups of small masses followed by beginning of piano solo (Orch. 2) 0:33
26 92 Piano and high metals 0:12
27 95 Piano and high metals with brass/perc accents 0:25
28 102 Slower tempo, more brass elements
29 105 (-2 m.) Background string texture surfaces and joins the still proceeding brass/perc. texture 0:34
30 112 Brass/perc. groups become trombone dialogue (vocal cue) 0:27
31 116 Brass battle among all orchestras builds, piano accents join

32 119 Swelling brass chords between all 3 orchestras with perc accents 0:17
33 120 Piano cadenza w quiet brass accents

34 121 Low piano strikes w. quiet brass/perc. accents, becoming a 3-orchestra drum battle 0:33
35 122 Drums continue with brass accents and fanfares 0:34
36 122.5 Brass and drums climax (w piano)

37 123 Brass/drums coda with piano 0:14
38 127 Pizz strings, guitar, and perc. groups ignite dialogue with winds 0:22
39 134 Sound masses followed by wind cadenzas 0:19
40 142 Points and small groups

41 150 Sparse, descending arpeggio motif dominated by winds

42 153 Pizz strings and brass join previous descending motif, becomes more transparent with perc. but with dynamic spikes 0:48
43 156 Tremolo strings, wind and vibraphone figures, descending motif in winds, brief Group 119 recap

44 160 Motif continues with low winds/brass and metal percussion 0:28
45 165 Low brass and gongs, drum roll 0:17
46 168 Recap of brass and percussion climax 0:17
47 172 Tutti dialogue becoming sparser 0:24
48 174 "Coda": Tremolo strings with cowbell accent and final wind note 1:09

Group 119 Dynamic levels

     Group 119 (right) is the probably the most famous part of GRUPPEN, since the brass sound masses move very dramatically from orchestra to orchestra (R - C - L - R - C - L etc...).  This is preceded by a brass battle (Group 118) and followed by a piano solo (Group 120).

Sound Impressions
     In my opinion the 3 most important symphonic works of the 20th century would be Stravinsky's "Le Sacre du printemp" (for rhythmic innovation), Debussy's "Prélude à l'après-midi d'un faune" (for destroying "harmony"), and Stockhausen's GRUPPEN.  Stockhausen's achievement in creating a total serial construction basically completes the experiments begun by Schönberg, Webern and Messaien (and Stockhausen himself in KLAVIERSTÜCKE I).  Stockhausen had at this point liberated all aspects of musical parameter in late Romantic classical music from traditional designs.  More than this however, GRUPPEN opened up the vistas of spatial music ("sounds in space").  From GRUPPEN's 3 orchestral groups, Stockhausen would increase the number of orchestral groups to 4 in CARRÉ, 5 in MIXTUR and STERNKLANG, and ultimately make each member of an orchestra their own spatially independent unit in LICHTER-WASSER.  Also as important but less apparent, the creation of a chromatic tempo scale was a device which Stockhausen explored and used for the rest of his career.

     Still, the most famous aspect of GRUPPEN is the spatial element, and Group 119's waves of brass can be easily heard in modern film scores such as Don Davis' music for the Matrix films (though Davis actually uses two overlapping chords instead of one).  This use of space-motion in music (in 1957) would pre-date the quadrophonic sound systems of the 1970s and the octophonic (7.1) sound systems of today's high end home sound systems (as well as the NHK Ultra-High Definition 24-channel sound system).

     Personally, as a guitarist, this work also excites me in that it features an electric guitar as a solo instrument.  Very few concert works feature an actual electric guitar (actually none come to mind outside of an Yngwie J. Malmsteen project).  In fact, a case could be made that free improvisation guitarist Derek Bailey's entire musical output could be considered a variation of the electric guitar part of GRUPPEN.  So it's personally quite gratifying that one of the 3 essential works of 20th century art music features an electric.

Stockhausen's Notes on GRUPPEN 
Stockhausen's Essay "...How Time Passes..." (PDF)
GRUPPEN Sound samples, tracks listings and CD ordering 
Buy the Score 
Albrecht Moritz Review
Sonoloco Review
GRUPPEN at Worcester Polytechnic Institute
GRUPPEN video with score on Youtube
GRUPPEN, 2008 Spanish premiere (mono)  Pt 1  Pt 2  Pt 3
Compositional techniques in the music of Stockhausen (1951-1970) (Kelsall, 1975)
The Music of Stockhausen (Jonathan Harvey) (Amazon link)
On the Serial Shaping of Stockhausen's Gruppen für drei Orchester (Imke Misch)
Stockhausen's Musical Shapes: How a Master Composer Moves Sound (Sara Ann Overholt)
Karlheinz Stockhausen's Spatial Theories: Analyses of Gruppen and Oktophonie (Thesis, Overholt, 2006, see above)