GRUPPEN

Stockhausen Edition 5 CD Cover
(©www.karlheinzstockhausen.org)

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.

Development
     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. 
(© www.karlheinzstockhausen.org)

     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.
(© www.karlheinzstockhausen.org)

     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.

CD Track	Beginning Group	Dominating Behavior	Duration                        3-Layer Sound Mapping       LEFT - CENTER - RIGHT
1	1	Bowed string masses w. gentle flute, percussion and harp accents move from orchestra to orchestra	1:00
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	1:01
6	16	Violin solo begins, supported by strings, vibraphone, and some percussion	0:56
7	20	Energetic recap of the previous group	0:13
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	0:57
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	0:26
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	0:47
15	53	Swelling mass with tremolo guitar/vibraph. in foreground	0:16
16	56	Tutti moving from left to right	0:16
17	62	Sound masses of strings, guitar & perc	0:35
18	70	Tremolo strings, then a drum solo (Orch.3) w pizz contrabass accents	0:30
19	72	Harp and contrabass duo, vibes & harp duo	0:35
20	74	Percussion (metals), guitar	0:13
21	75	2nd Guitar solo with drum crescendo	0:25
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	0:32
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	1:10
32	119	Swelling brass chords between all 3 orchestras with perc accents	0:17
33	120	Piano cadenza w quiet brass accents	0:33
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)	0:29
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	0:29
41	150	Sparse, descending arpeggio motif dominated by winds	0:40
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	1:09
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
LEFT - CENTER - RIGHT

     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.

Links
Stockhausen's Notes on GRUPPEN 
Stockhausen's Essay "...How Time Passes..." (PDF)
GRUPPEN Sound samples, tracks listings and CD ordering 
Buy the Score 
GRUPPEN Wiki
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)

LICHT-BILDER

SONNTAG AUS LICHT
Scene 1 LICHTER- WASSER	Scene 2 ENGEL-PROZESSIONEN	Scene 3 LICHT- BILDER	Scene 4 DUFTE-ZEICHEN	Scene 5 HOCH-ZEITEN	Farewell SONNTAGS- ABSCHIED

2011 - from and_vfx blog

No. 77: LICHT-BILDER (Light Pictures)
3rd Scene of  
SONNTAG AUS LICHT/SUNDAY from LIGHT
for tenor vocalist, trumpet with ring-modulation, basset-horn, and flute/alto flute with ring-modulation
(accompanied by video projection)
2002/2003 [~42']

Introduction
          LICHT-BILDER (Light Pictures) is the 3rd Scene of Stockhausen's dramatic music work SONNTAG AUS LICHT (SUNDAY from LIGHT), which was the last-composed entry of his 7-part, 29-hour opera cycle LICHT (Light).  LICHT is a work for acoustic and electronic operatic forces, divided into the 7 days of the week (one opera for each day).  This opera cycle revolves around 3 archetype characters, MICHAEL, EVE and LUCIFER, and over the 29 hours each of these characters are introduced, come into conflict, face temptation and finally come into union.  The music is almost entirely based on a "super-formula", which is a 3-layered melodic-thematic representation of the 3 characters.  These formula-themes are together and separately threaded throughout the opera's vocal and instrumental fabric.  Story-wise, actors and narrative can (and often do) change from scene to scene, and the libretto text is sometimes made up of non-traditional grammar (or even purely phonetic sounds).

     SONNTAG (Sunday) is the Day of Mystical Union, specifically between MICHAEL and EVE.  The scenes in SONNTAG do not have a dramatic plot connecting them, instead the actual theme of union between the 2 characters is achieved through musical, visual, spatial and even olfactory means.  LICHT-BILDER is the 3rd scene of SONNTAG AUS LICHT.

LIGHT-PICTURES
     LICHT-BILDER was originally named "Venerations of EVA-Maria, 7 Healings, 7 Pictures of the Week Days", but was eventually changed to "Light Pictures", and it's main text instead became more concerned with describing "the essences of 7 large spheres of life" (each linked to a day of the week) and praising God.  The 7 days are characterized in 7 day-sections, with Thursday also divided into 7 subsections and Sunday into 4.  The vocal text of each of these sections begins with a different "veneration of God".  The MICHAEL/EVE unification aspect is represented by intertwining fragments of the MICHAEL and EVE formulas (melodic layers) from the LICHT super-formula.

     One of the main themes of LICHT-BILDER is "mirror reflections".  The interaction of the 4 players can be pictured as 2 people dancing, with their altered mirror reflections a few dance steps behind them.  In musical terms, the tenor vocalist's melodies are accompanied by the trumpet in a delayed and distorted imitation, and the trumpet is in turn immediately reflected by its own ring-modulation from the back of the auditorium (the delay time starts at 0 (unison), increases to a maximum delay and then returns to 0).  The bassett horn is accompanied by the delayed flute, which is also ring modulated (to be clear, none of these "delays" are through signal processing, they are actually scored). 

It might be interesting to consider that the whole idea of LIGHT PICTURES contains 

the retrograde character of photographs and also the perspective of pictures (delay scales).

 - Stockhausen

     The MICHAEL and EVE melodic formulas are chopped up and reordered backwards (retrograde), so these are also "reflection-melodies" (and the first time Stockhausen has used a retrograde form on the LICHT formula).  Finally, the melodic elements and vocal text are reflected in the movements of the soloists, which are also layered in forwards and backwards sequences.

Composition and Structure
     Like MANTRA and ZYKLUS, the architectural aspects of LICHT-BILDER are very interesting, but knowing them I think is not essential in order to enjoy the music (so it's just as well to just skip to my "Sound Impressions" at the end of this article).  But it does give alot of insight into how Stockhausen is able to grow an entire scene from just a minute's worth of melodic material.  In following the next paragraphs it's probably helpful to have my super-formula page handy...  (The following analysis is largely informed by Stockhausen's excellent 2005 Composition Course book on LICHT-BILDER).

Introductory Measures
     The layering of the stretched SONNTAG measures over the entire super-formula makes up the melodic material of the introduction (below on CD cover):

• The bassett horn plays an augmented form of 5 measures (m.7-11) from the LICHT super-formula's MICHAEL layer.  Here I label this the "LICHT-BILDER from SONNTAG" excerpt (LBS) because Stockhausen temporally stretched the 3 SONNTAG measures (m.17-19) over the entire super-formula and then selected the corresponding LICHT-BILDER measures from the super-formula.
• The flute plays the stretched MICHAEL layer (essentially 1 long pitch (F), but each measure here has different "approach" to the F).  I label this the "LICHT-BILDER from LICHT" excerpt (LBL) because Stockhausen compressed all 29 hours of LICHT on top of the super-formula and then selected the measures corresponding to the LICHT-BILDER scene from the super-formula.  This of course, is a very small excerpt - in fact, only 2 quarter notes.
• The tenor sings the augmented LBS  EVE formula layer, but at the end joins the bassett horn (MICHAEL's LBS).
• The trumpet plays the stretched LBL EVE formula, which hovers around G and G#.

CD Cover and introductory measures of LICHT-BILDER.
This fragment is also equivalent to the form scheme used to determine the
durations, transpositions and ring modulation tones of each day-sections.
(© www.karlheinzstockhausen.org)

     So just to reiterate, the very short LBL excerpt of the EVE and MICHAEL formulas is stretched out (through extreme augmentation) and layered on top of the LBS 5-measure excerpt.  The LBS excerpt comes from the middle of the super-formula because it's derived from an overlaying of only SONNTAG AUS LICHT (the 3 Sunday measures of the super-formula, m.17-19) stretched over the whole super-formula, and LICHT-BILDER is in the middle of SONNTAG.  The 2 quarter note long LBL excerpt (last 1/3 of m.17) comes from near the end of the full superformula because it's derived from an overlaying of all 29 hours of LICHT to get LICHT-BILDER (which is near the end of LICHT).  These 4 layers are all also slowed down to quarter-speed.

     The MICHAEL formula is also transposed down 3 steps, while the EVE layer is up 8 steps (this is because the MICHAEL LBL main note (F) is 3 steps up from the MICHAEL layer's initial note (D), and the EVE LBL main note (G) is 8 steps down from the EVE layer's initial note, C).  However, This is just another application of the super-formula to create contrast and variety and has nothing to do with "keys".

7 Days of the Week
     Stockhausen added up the first 7 numbers (from the 7 "days") of the Fibonacci sequence to get the value 53 (1 + 2 + 3 + 5 + 8 + 13 + 21).  He then divided the MICHAEL and EVE formulas into 53 connected fragments (the MICHAEL fragments are not always the same length as the EVE fragments).  Then he reordered these fragments in a backwards ("retrograde") sequence (fragment 53, 52, 51...3, 2, 1).

     On each day-section, the tenor and bassett horn take turns singing/playing these reordered fragments in a kind of overlapping dialogue, while their reflections (trumpet and flute) imitate them, at first in unison, but then falling farther and farther behind.  Eventually the reflections reach a maximum delay interval, after which they begin catching up to their partner.  There are roughly 4 cycles of this delay expansion and contraction in each formula traversal, and the cycles for the Tenor/Trumpet pair are not synchronous with the cycles for the Bassett Horn/Flute pair.  On some day-sections the Tenor/Trumpet pair plays the 53 EVE fragments and the Bassett Horn/Flute pair responds with the 53 MICHAEL fragments.  On other days the assignments are interchanged.  This "double delay-reflection" creates a 4-part polyphony which sometimes has elements of variation and canon structure.

Shown here are the last 4 measures of the MICHAEL and EVE formulas with Stockhausen's markings for LICHT-BILDER.
Green vertical dashes mark the 53 sections.  The numbers going from 0 up to 13 (or 20 for EVE) 
and then back down to 0 (reading right to left) are the delay values.
(from Stockhausen 2005 Composition Coursebook: Facsimile edition with sketches of LICHT - BILDER, 
Click to see the full LICHT super-formula with LICHT-BILDER markings)
(© www.karlheinzstockhausen.org)

     Each day-section also has a different register transposition for the MICHAEL and EVE layers, based on interval relationships between the initial pitches of the super-formula layers (in its original register, D and C) and 2 "guide scales".  The guide scale for the MICHAEL layer is the 3-step lower MICHAEL nuclear tone scale (16 notes, 1 for each section), and the guide scale for the EVE layer is the 13 main pitches from m. 7-11 in the MICHAEL formula (the 5-measure LBS excerpt, still 3 steps lower than the original).  EVE's guide scale is 13 notes because the 13th note is sustained for all 4 of Sunday's sections.  So, again, the transposition interval for each day-section comes from comparing each day's guide scale note with D (MICHAEL formula's first note) or C (EVE formula's first note). 

     For the Monday section, the Tenor/Trumpet pair starts with a pitch transposed (+m3rd) and augmented version of the 53rd EVE fragment.  The Bassett Horn/Flute pair responds with a similarly slowed and transposed (+m3rd) version of the 53rd MICHAEL fragment. The Tenor/Trumpet pair continue with the 52nd EVE fragment, the Bassett Horn/Flute the 52nd MICHAEL fragment, etc...  The Tuesday section has the same fragment sequences for the players, but a new transposition (EVE: +M6th/MICHAEL: -M2nd), the speed is faster and the delays are not the same exact intervals (though they still increase and then decrease).  The Wednesday section's main structural difference from the preceding patterns (besides tempo and transposition (EVE: +m2nd/MICHAEL: -m2nd) adjustments) is that the Bassett Horn/Flute pair starts, and the Tenor/Trumpet pair responds.  Thursday is divided into 7 subsections.  The Tenor/Trumpet and Bassett Horn/Flute couples take turns starting the subsections, but the duration of each is actually shorter than the original formulas, so some of the more "static" notes are skipped.  However each subsection has 11 dialogue pairs (11 "multi-fragments" each), and in each of the 7 subsections a different permutation of multi-fragments happens, each with their own register transpositions.
     Friday, Saturday and Sunday sections follow in similar fashion (each with their own tempo and pitch transposition modifications) and they are further described in the Stockhausen Courses 2005 Composition Class book (Facsimile edition with sketches of LICHT - BILDER)

Chloé L'Abbé (flute), Hubert Mayer (tenor), Fie Schouten (basset-horn), 2011
from and_vfx blog

Duos, Trios and Quartets
     Each of the day-sections also has duo, trio or quartet "inserts" where some voices briefly play in rhythmic unison. These unison parts were inserted in places where empty bars would result from the polyphonic arrangements of the fragments (for example during when the Bassett Horn/Flute pair is waiting while the Tenor/Trumpet pair play their fragment, or if the fragment has a rest).  The unison parts are derived from MICHAEL/EVE formula fragments. 

• Monday - Bassett Horn & Flute Duos
• Tuesday - Bassett Horn, Trumpet & Tenor Trios
• Wednesday - Flute & Trumpet Duos
• Thursday - Bassett Horn & Tenor Duos
• Friday -  Bassett Horn, Flute and Tenor Trios
• Saturday - Trumpet & Tenor Duos
• Sunday -  Quartets

     In actuality there are many more unison duos than just these special inserts, since whenever the delay value is zero, the tenor and trumpet (or bassett horn and flute) automatically create a unison duo.  This happens 4 times (from 4 cycles) in each normal day-section.  Also most of the day-sections actually end in ensemble unison textures.

Durations
     Stockhausen had established that each quarter note in the LICHT super-formula (at tempo 60) is equal to 16 minutes in "real-time", so using this ratio the duration of LICHT-BILDER was initially calculated to be 32 minutes (the LBL excerpt is 2 quarter notes long).  The 32 minutes was divided into the 7 day-sections (proportional to 7 natural rhythmic divisions in the 5-measure LBS excerpt), and then the full MICHAEL and EVE formulas were speed-adjusted so that the 2 formulas would approximately fit each day-section.  The Thursday section is further divided up into 7 subsections with lengths proportional to a "MICHAEL rhythm" found in m.9 of MICHAEL's formula. 

Ring Modulation
     The trumpet and flute each get a different ring modulation "difference" tone in each day-section (ring modulation creates additional harmonies by adding and subtracting 2 different pitches - 1 is the acoustic pitch, and the other is played by a synthesizer).  The ring modulation difference tones for the flute are the notes of the MICHAEL nuclear tones (primary tone row of the MICHAEL formula).  The ring modulation tones for the trumpet are based on pitches from m.7-11 of the MICHAEL formula (the 5-measure LBS excerpt).  These are the same "guide scales" used to determine the formula transpositions.  In the Friday and Sunday day-sections the ring modulation tones from the MICHAEL and EVE formulas are exchanged, which increases dissonance in the reflected signals.

Music
     The final step, and the most important, was to make this elaborate construction "musical". This part is not as simply explained :) as the above architectural structure, since it is more about adjusting rhythms and articulations to fit dramatic flow and balance.  In one of his British lectures Stockhausen implies that after all the sketch work is done, the actual realization of the notes and durations is where the "real work" begins.  Many of LICHT-BILDER's small-scale elements were adjusted when Stockhausen composed its full form, and further modified during rehearsals where he was able to immediately make adjustments to make the work more aesthetically interesting.  The melodic material is also altered to fit the soloists' movements (described below) through things like glissandi, augmentation and adding octave transpositions.

Libretto
     The libretto sung by the tenor soloist is made up of non-grammatical words and phrases which praise God and correspond to the "essences of seven spheres of life".  The text for the beginning measures is "LIGHT PICTURES - 3rd Scene of SUNDAY FROM LIGHT".  The libretto continues as follows:

• Monday (texts concerning stones, hills, water): PRAISE GOD - pyramids - stairs of stone - planet - rotations - fountain - avalanche - volcano - meteor - crater - whirlpool...etc.
• Tuesday (texts concerning trees, plants, fruits): GLORIFY GOD - bellflower - common houseleek - rose bloom - daffodil - fir - golden chain - apple - snow-drop - nest spruce - lily of the valley - weeping willow...etc.
• Wednesday (texts concerning animals): ALL ANIMALS REJOICE IN GOD - oyster - lion - lark - crab crayfish cribbily - zebra - bee - snake...etc.
• Thursday is broken into 7 elements
•  Water: Give thanks to GOD, rain - wasserfall waterfall - tau dew - comet komet - fog - cloud...etc.
•  Earth: Give thanks to GOD, earth - plains - tundra - hillside - plateau - meteor...etc.
• Air: Give thanks to GOD, air - draft - autumn storm - air shaft - wind tunnel - air pocket - tornado - hurricane...etc.
• Ether: Give thanks to GOD, ether - crystal snow - space ship - moon - from east to west - prayer - rainbow - shooting star...etc.
• Eros: Give thanks to GOD, Eros - Amor - Susani - cypress - fir tree - cedar - palm - ray of love - love - silver water - lingam yoni - caress - kiss...etc.
• Fire: Give thanks to GOD, fire - sheet lightning - flame - blaze - glow - ashes - volcano...etc.
• Pealing of Bells: Give thanks ringing of bells - Ave Maria - bell rope - pealing - sexton - church bells...etc.

• Friday (texts concerning celestial bodies and star constellations): PRAISE GOD - Moon - Ursa Minor - Uranus - Phoenix - Phobos - Deimos - Sagittarius - Perseus...etc.
• Saturday (human saints): PRAISE GOD saints - Mary - Elizabeth - Stephen - John - Peter...etc.
• Saturday is broken into 4 sections: GOD in everything
•  1 (houses of God, noble trees and animals): God's house - minster - dome - church - cathedral - mosque - cypress - fir tree...etc.
• 2 (light, candle flame): candles - candlelight - eternal light - sunlight...etc.
• 3 (the invisible, transcendental): invisible - holy - in prayer - presence...God's Day.
• 4 (praying, Eva-Maria, LIGHT PICTURES - THOU): praying - glorifying - Mater - Ave - Sancta - Eva-Maria...GOD'S LIGHT PICTURES - THOU.

Live Performance

Marco Blaauw (trumpet), Hubert Mayer (tenor), Kathinka Pasveer (flute), Suzanne Stephens (basset-horn)
2004 Donaueschinger Music Festival World Premiere
(rehearsal photo Alain TAQUET)

     Each of the 4 soloists are amplified by speakers from above the stage.  The ring modulated signals from the trumpet and flute are projected from the rear of the auditorium, but at a lower volume than the stage speakers.  Up to 4 video screens (triangular in the premiere) are mounted above the soloists (though the content of the videos is left open).

     Like INORI and IN FREUNDSCHAFT (among many others), the soloists each have scored movements generally reflecting the melodic/rhythmic shapes being played or the text being sung. More specifically, each player has 53 movements per day-section (linked to the 53 formula fragments described earlier) with 9 movement types, with each section having a different type-distribution based on 7 permutations of the first 7 Fibonacci numbers.  These 53-movement sequences are then placed to fit with the melodic/thematic material (or the melodic material is adjusted to fit the movement).  The actual sequences change for each soloist and day-section based on different starting points and directions on a master sequence.  The movement types themselves are fairly open: go left, right, quick upward diagonal, rotate, etc...  The movement sequences for bassett horn and flute are in reverse of the sequences for tenor and trumpet (another reflection).

Score

Beginning of Monday section.  The first measure is the 53rd EVE fragment (see previous sketch).  The 2nd measure is the 53rd MICHAEL fragment.  The 3rd measure is the 52nd EVE fragment, etc... The fragments are adjusted to match the movement sequences (see arrow markings).   The staff indicated by the "~" is the ring-modulation "difference tone".
 (© www.karlheinzstockhausen.org)

Beginning of Tuesday section.  Same as above but note the changes of articulation to the fragments as well as the delayed reflections. 
 (© www.karlheinzstockhausen.org)

Sound Impressions
     There are several ways to listen to LICHT-BILDER, but I personally like to hear it primarily as a dialogue between the tenor vocal and the bassett horn, with the slightly ring-modulated trumpet and flute being the Greek chorus (not to diminish their roles of course!).  The tenor and bassett horn seem to "con-verse" using the melodies of EVE and MICHAEL as language phrases, with each soloist learning and then later practicing his partner's language (this concept of meeting through language exchange extends as far back as the first LICHT opera, DONNERSTAG AUS LICHT).  The trumpet and flute re-learn and re-translate from the tenor and bassett horn. Alternatively, a listener can concentrate on the interaction between just the pair members (tenor and trumpet, for example) and appreciate the variations of the delayed imitations.  One could listen just to the tenor/trumpet lines, then to just the bassett horn/flute lines, and then to the quartet in total as a double delay dialogue.  On the CD recording the voices are arranged from left to right: trumpet - tenor - flute - bassett horn (just as in the photographs).

     Also, after one gets to know the EVE and MICHAEL formulas pretty well from the previous operas, it's very fresh to hear them in this backwards reordered form.  Since the melodic fragments are still intact, recognizable EVE and MICHAEL melodic motifs still surface, but in places which are unexpected (if one is very familiar with them from the previous operas).

     Another interesting thing here is that this scene actually encapsulates the 7 days of a week in a very true way.  Every day-section basically traverses the same retrograded MICHAEL and EVE melodic formulas from beginning to end, but each is also very different in the details.  A couple days like Thursday and Friday are "special days".  This is a good parallel with real life on a daily basis -  get up, go to work, have lunch, dinner, nightlife, sleep, enjoy a weekend break...  But every day is still different in the details (at least, hopefully!).

     For a more narrative-based appreciation of LICHT-BILDER, Albrecht Moritz' rundown (see link below) is excellent and points out lots of cool moments.  I didn't explore the text too much here, but it is covered in the LICHT-BILDER CD booklet as well as in the Moritz rundown and Stockhausen's introduction to the premiere(below).

Links
LICHT-BILDER Sound samples, tracks listings and CD ordering 
Buy the Score
Stockhausen 2005 Composition Coursebook: Facsimile edition with sketches of LICHT - BILDER
Wiki Entry
Albrecht Moritz Rundown
Stockhausen's Introduction to LICHT-BILDER at the Rehearsals of the 2004 World Premiere
2004 Studio CCMIX Paris Premiere (excerpt)
2011 Musikfabric/Oper Köln SONNTAG Production
2011 Production Images and video from and_vfx
2011 Production Youtube clip
SONNTAG AUS LICHT in Rehearsal 1 (YouTube, German)
SONNTAG AUS LICHT in Rehearsal 2 (YouTube, English/German)
2011 SONNTAG AUS LICHT Production Dance Company Page

KLAVIERSTÜCKE I–IV

Graphic Analysis of KLAVIERSTÜCKE III based on pitch, durations and dynamics
(Dieter Schnebel, Die Reihe V4, 1958)

No. 2: KLAVIERSTÜCKE I–IV (Piano Pieces 1 - 4)
1952 [total time for all 4 ~8']

Development
     Like many children of his time, Stockhausen was forced to learn to play the piano as a child by his father.  In fact his grandmother sold a cow so that the family could have a piano.  Stockhausen, like most children, wasn't exactly keen to learn piano, but he eventually got good enough to the point where he was able to comfort wounded enemy soldiers in German hospitals during WWII with jazz standards.  Later in music conservatory he learned classical repertoire including Ravel, Schönberg, Bartok, Hindemith, etc...and actually did analyses of works by Bartok, Debussy and Webern.  When Schönberg's 12-tone technique came along (a method of creating non-traditional melodic variety), Stockhausen was fascinated, but it wasn't until he heard Messaien's initial experiments with serial organization of musical attributes ("organized chaos" of not just pitch, but also duration and dynamics) that he became galvanized into exploring this new frontier of abstract rhythm and melodic organization.

     (For an introduction of 12-tone music I invite you to read "How 12-Tone Music Works" from my Beethoven blog.)

     Stockhausen's first 4 solo piano pieces chart his early path from atonal, arrhythmic, non-thematic textures where each individual note/chord is treated as a motif ("point music") to works where notes with similar attributes are grouped together and treated as single objects.  Though these piano pieces are marked Opus 2, they actually precede Opus 1, KONTRA-PUNKTE, which would explore the next logical step of mixing points and groups in a kind of "shrinking orchestra" structure.

...the KLAVIERSTÜCKE I–IV were premiered at the 1954 Darmstadt Summer Courses 
by Marcelle Mercenier, and (were) transformed by the audience 
into an unusually vociferous whistling-concert. 
- Stockhausen

Avant-Garde Piano Music of the 50's (or: How I Learned to Stop Worrying and Love Serial Music)
     For most listeners, then and now, the piano music coming out of Europe (and America to a lessor extent) in the 1950s is pretty difficult to listen to.  After Beethoven and Brahms, people like Schönberg, Webern and Berg (the so-called 2nd Viennese School) began composing atonal works which used all 12 notes (instead of the usual 8) of a scale and tended to abandon the traditional concepts of melody-chord layers and sonata development form.  Many (most) of these kinds of works are still not accepted by the general music-loving public, even almost a hundred years later.  However the use of 12-tone "dodecaphonic" technique was basically a way for composers to "break out of the box" of late Romanticism, and these early masterpieces actually still had quite alot in common with late 19th century music.

     Stockhausen however, (inspired in part by Messaien) not only broke out of the box but threw it down the stairs.  As mentioned earlier, additional music elements such as duration, volume, and timbre were also "shuffled" (purposefully), along with the trend of democratizing the 12 scale notes.  However, when all of these elements were evened out with equal levels of seeming randomness (especially without repetition or traditional form structures), they created a texture which was closer to an image of a cloud of insects, and became described as "point music".  Stockhausen on the other hand was interested in more than just pitch democratization.  In a 1992 lecture about his  piano music, Stockhausen lists differentiation of dynamics (attack), expansion of (pitch) intervals and extremely long (or short) durations of sound (or silence) as things which interested him from the start.

Listening
     Stockhausen has published a very thorough musical analysis of how KLAVIERSTÜCK I is organized, but he's also in later lectures (including the above-mentioned 1992 one) given some more "audience-friendly" tips on how to appreciate this complex music (link at bottom).  One of the structural signposts to listen for is the silences.  During a silence the listener can take a breath and reflect on the phrase that just came before it.  The same thing could be said of long held tones.  The things to listen for include tone colors, especially when notes in a chord stop or start at different times (during a loud chord strike, some notes may be held and some dampened).  Shapes of melodies, dynamic changes from note to note, extreme register leaps - these can all be considered identifying motifs.

Of the many approaches to this music, one might pay particular attention to how and when pauses 
are composed, how different in length and degree of silence one feels them to be, depending on 
whether loud or soft sounds, dense or transparent note groups are heard before and after the pauses.

- Stockhausen, from a 1957 radio broadcast (Toop translation)

     Another excellent practice which Stockhausen recommends is to close one's eyes while listening and pretend to play the piano along with the music (like playing "air-guitar", but with piano).  I have to admit, it drives me crazy in live concerts when people try to "conduct" with the orchestra, but at home it's actually a great way to stay with the music.  Stockhausen recommends doing this at-home piano-miming many times ("hundreds of times"!) for a piece until it's actually almost "learned".  It may seem silly, but Piano Piece 3 is only 30 seconds long, so it's worth trying for half a minute.  Try to listen for melodic shapes, the frequency of single notes vs. chords, changing speeds, and the differences in loudness for each note. 

     Having said all of the above, I'll still try to summarize a few general concepts about each piece without getting too much into, in my opinion, largely unmissed analytical minutiae. In these 4 pieces Stockhausen basically pits "points" against "groups" and develops drama from their evolving relationship.  Each piece however is to be played as fast as possible.

KLAVIERSTÜCK I
(Piano: David Tudor)

     This piano piece was the last to be written and is inferred by Stockhausen to be the conclusion of Opus 2's journey from "points" to "groups".  It was written in just 2 days and probably flowed quite intuitively from his pen.  In fact, only when analyzing it after the fact did Stockhausen notice the symmetry he'd unconsciously incorporated into it.  The groupings are based on things like register, dynamic curve, melodic direction, tempo, chord elements etc...  Though it also has the chordal and contrapuntal elements of II and IV, it's notable for having several pauses and sustained chords which make it easier to identify these groups/structures.  Stockhausen considers these "breaths" as a structural step forward from previous more "homogenous" serial compositions.  To me, it also feels like it's speeding up and slowing down throughout and is generally the most "violent" of the 4 piano pieces.

     Interestingly, some people (such as author/composer Jonathan Harvey) disagree with Stockhausen's published analysis of the piece, and label the groups differently.  Harvey also describes it as being organized in 2 alternating sets of pitches (C to F and F# to B), with some expanded chords (such as bar 11's 9-note chord with 5 different dynamic markings - above).

KLAVIERSTÜCK II
    KLAVIERSTÜCKE II (2nd to be composed) is either built on 2 trichords (D, Eb, F and C, C#, D) according to Harvey, or a rotation of 2 pentachords according to Herman Sabbe.  In any case, the texture is more harmonic (chordal) than KLAVIERSTÜCK III and IV's more pointillistic designs, and features more 2 and 3-note tone clusters, as well as use of the sustain pedal to create large resonating harmonies.  Compared to IV, this has a more wistful feeling, and doesn't have as many jarring dynamic leaps.

KLAVIERSTÜCK III
    The first to be written (and probably the most analyzed), KLAVIERSTÜCK III is either built on a pentachord (A, B, D, Ab, Bb) (Harvey), or 3 tetrachords (D-F, F-G#, G#-B) (Robin Maconie).  I just know that this one has no left-hand "bombs" in the bass register.  Also I think it's Stockhausen's shortest composition at 32 seconds long.  If Opus 2 were a classical sonata, this would be the scherzo.

KLAVIERSTÜCK IV
    Composed third, Harvey claims that KLAVIERSTÜCK IV is based on 3 notes (Db, C, F).  It obviously has more notes than that so I'm not quite sure how that works.  But it's generally accepted that this is the "contrapuntal one", and the 2 voices are identified by dynamic attack (in other words, one line is always loud, and the other line is generally quiet).  As a listener, I find it a very interesting exercise to follow attack intensity instead of the usual melodic line.  Another way to look at it is 2 couples having a conversation at the same time, with one couple being loud, and the other couple speaking softly (this analogy attempts to characterize both dynamics and note register for each line).  Robin Maconie also notes that each note in succession is followed by or preceded by a rest, which probably gives it that "point" feeling.  In any case my impression is that it still swings more than the typical pointillistic serial piano pieces of the era.

Live Performance
     Watching a live performance can be very instructive since seeing the pianist's fingers leaping from place to place on the keyboard gives a pretty good idea of the point/group dialogue. 
 
KLAVIERSTÜCKE I–IV (Vanessa Benelli Mosell at Stockhausen Courses, Kurten)

Sound Impressions
     These  piano pieces have been analyzed quite a bit.  In fact, an analysis of the different kinds of analyses (11 including the author's own) of KLAVIERSTÜCK III has actually been written.  However, I think there's sometimes a temptation for academics to miss the forest for the trees.  For most listeners the best advice is the one Stockhausen gave in a lecture about MANTRA - you don't need to know astrophysics in order to appreciate the stars at night.  Personally, I like to imagine these pieces as Stockhausen improvising on the piano.  That's not to say that the notes are strewn about cavalierly, but more that these pieces were executed with emotional weight (along with thoroughly premeditated objectives).  This 1st set of 4 piano pieces definitely hits me as being more raucous that his later, much longer piano pieces, which can be appreciated from other, more architectural angles.

KLAVIERSTÜCKE I–IV
KLAVIERSTÜCK V
KLAVIERSTÜCK VI
KLAVIERSTÜCK VII
KLAVIERSTÜCK VIII
KLAVIERSTÜCK IX
KLAVIERSTÜCK X
KLAVIERSTÜCK XI
KLAVIERSTÜCK XII (EXAMEN)
KLAVIERSTÜCK XIII (LUZIFERs TRAUM)
KLAVIERSTÜCK XIV (GEBURTSTAGS-FORMEL)
KLAVIERSTÜCK XV (SYNTHI-FOU)
KLAVIERSTÜCK XVI (w. Sound Scene 12)
KLAVIERSTÜCK XVII (KOMET)
KLAVIERSTÜCK XVIII (MITTWOCH-FORMEL)
KLAVIERSTÜCK XIX (SONNTAGS-ABSCHIED)

Links and Sources
Stockhausen on the KLAVIERSTÜCKE (1955,1957)
Stockhausen's Own Analysis of KLAVIERSTÜCK I (1955, in French and English) 
Stockhausen on Piano Music (1992)
KLAVIERSTÜCKE I-XIV (Ellen Corver) Sound samples, tracks listings and CD ordering 
Buy the Score 
KLAVIERSTÜCKE Wiki
Sonoloco Review
Vanessa Benelli Mosell Rehearses Klavierstück IV with Stockhausen (Youtube, Italian)
KLAVIERSTÜCKE I-XI by Aloys Kontarsky (flac)
KLAVIERSTÜCK II Analysis (Schlüter, German PDF)
KLAVIERSTÜCK I Analysis (Takashi Matsudaira, Japanese)
Compositional techniques in the music of Stockhausen (1951-1970) (Kelsall, 1975)
The Music of Stockhausen (Jonathan Harvey) Amazon link
The Works of Karlheinz Stockhausen (Maconie 1976/1990) Amazon link