The LICHT Super-Formula

Dramatic breakdown for SAMSTAG AUS LICHT based on Measures 14-16 of the LICHT super-formula
I feel that the sounds and intervals of the super formula are actually the characters, and
that the people who realise the sounds represent them. So the true actors in the entire work of
LICHT are actually the pitches and durations, the intensities and the colours, and the tone-forms
of the super-formula. It is difficult to explain to people that I, as a musician, see sounds like
people and experience them as such while composing.
 - Stockhausen, "FREITAG, World Theatre"

     This page describes the musical development of the 7-opera LICHT cycle from Stockhausen's LICHT "super-formula". A more general introduction to the LICHT cycle can be found here.

Going Towards the Light
     Stockhausen's 29-hour work LICHT ("Light") is basically a multi-disciplinary expansion of a 3-voice theme which he characterizes as a "super-formula".  This concept is the culmination of musical experiments which Stockhausen had been exploring since virtually the very beginning of his career.

     In his earliest works he explored ways to generate a variety of musical gestures by conducting  non-repeating serial operations on pitch, duration, dynamics and timbre (such as in KREUZSPIEL and PUNKTE).  This was followed by adding the concept of "points vs. groups" and tempo (KONTRAPUNKTE, GRUPPEN).  Then in MANTRA and INORI he added the idea of considering musical gestures themselves as elements to be quantified and put into a "row", so to speak.  These included gestures such as "a scale", "echos", "non-musical noises", "regular and irregular repetition", "clouds of notes around a central tone", etc... (see KLAVIERSTÜCKE V–X and MANTRA for more on this).  The behaviors were solidified into a "formula", which is a kind of serial row for musical "gestures" as opposed to notes (though it is basically note-driven). 

     While in Japan, working on a piece for a traditional Japanese Gagaku orchestra (DER) JAHRESLAUF, Stockhausen wondered to himself what aural factors gave the different kinds of ethnic world music traditions their identity.  He noticed that, for the most part, all of these global music genres still used common pitches and scales (for example, pentatonic, whole tone, diatonic, harmonic minor, etc...), but the element that gave each folk style a specific flavor was actually articulation and treatment of the pitches as they progressed from one note to the next.  Examples of these articulation types include microtonal glissandi, use of consonants or sibilant noises, tremoli, overtone vowels, etc...  All of these additional articulation gestures were used to further develop the formula technique used previously in MANTRA and INORI.

Stockhausen at Ryoanji, Kyoto Japan 1977
     Stockhausen also describes these individually-articulated pitches as "Tonformen".  The LICHT formula has 36 "nuclear" tones, and the full formula is basically the expansion of these 36 Tonformen nuclear tones into musical gestures which can each be somewhat characterized as having a "head", "body" and "tail".  This concept was also informed by his experience with electronic music, where each sound has to be described with an attack, body and decay. 

The LICHT Super-Formula
     Stockhausen hit upon the idea of layering the 3 formulas on top of each in order to get to the LICHT super-formula, which ultimately included just about every idea Stockhausen had used up until this time (with the notable exceptions being spatial music and intuitive music through texts or "plus-minus" symbols).  The 3 horizontal staves of the super-formula are named the MICHAEL layer, the EVE (EVA) layer and the LUCIFER (LUZIFER) layer, each of which is its own formula.  The super-formula was then divided vertically into 7 segments (2 to 4 measures long), one for each day of the week.  Each of these segments decides the overall form of each of the 7 operas in LICHT (each quarter note in the super-formula is expanded into 16 minutes of "real-time" to determine the durations of the acts and scenes - see top and bottom pictures of this page).

     Besides the 3 "character" formulas, each of these also have reduced versions which strip the formulas of their gestural elements and leave only pitches, duration and dynamics - essentially a melody-theme.  These are called "nuclear formulas" ("kernformel").  The nuclear formulas can also be further reduced to just their notes (without duration), which is basically a tone row in the pre-war Schoenberg-ian sense.  These are the "nuclear tones", also called "central tones" or "kernels".  MICHAEL has 17 nuclear tones (13 actually, but some are repeated), EVE has 12, and LUCIFER has 11.  Finally Stockhausen uses so-called head-motives derived from the first few notes of each of the 3 formula layers to act as character leitmotifs or transition signals.

     Aside from these various simplified versions of the LICHT super-formula, these parts can all also be used in retrograde or inversion form (backwards or upside-down), though retrograde is only really used in LICHT-BILDER (and very briefly in LUCIFERs TRAUM).  All of this material is then stretched temporally, and transposed vertically in some cases.  Some of the musical material of LICHT may be based on a gestural element, and sometimes the material is an exact expression of the melodic content of a formula.  It is actually very rare, however, to hear the formulas used as thematic material in the "classical sense" - unlike say in a Tchaikovsky ballet, one usually doesn't end up leaving with a "tune" drilled into one's head.  In LICHT, the MICHAEL, EVE and LUCIFER formulas are used more as structural forces whose tonal characteristics exert a kind of planetary gravity over the surrounding musical ether. However, once a listener becomes more familiar with LICHT, the formulas do surface as "tunes" and, as Stockhausen implies in the quote at top, they become almost like "friends".

In other words, the instrumentation is actually an interpretation that I make on the 
spot. It even depends on the commission. I would never have thought of using a symphony 
band for this particular scene if it hadn’t been for this commission. But in the first scene, 
for example, where the basic idea was that Lucifer dreams a 13th Piano Piece, a piano 
is mandatory... 
 - Stockhausen, "SAMSTAG, World Theatre"

     In using this concept of a super-formula, Stockhausen gave himself a firm supporting structure and form on which to build musical materials.  As seen in the quote above from "SAMSTAG - World Theatre", the LICHT super-formula was written in 1977/78, but Stockhausen often didn't know the musical forces he would be using to express each day's opera of LICHT until close to the time he was about to actually begin working on a scene, and this method of development continued all the way up to LICHT's completion in 2003.  Many times the instrumentation of a scene was driven by commissions, or sometimes by the urge to write a work for family/friends. Also, because many elements are based on shapes and colors rather than pitch motifs, this gave Stockhausen plenty of room to incorporate virtually any kind of compositional timbres he wanted, both acoustic and electronic.

     The LICHT super-formula is shown below but it's instructive to note that this roughly 1-minute sequence of 19 measures almost never actually gets played simply as written.  The closest he gets to that is probably in the piece BASSETSU-TRIO from bar 160 to 189 (CD 55, track 15), or possibly the ROTARY Wind Quintet (CD 105, track 6, starting from 4:26). Both works are from the scene MICHAELION/MITTWOCH AUS LICHT. For the most part, the super-formula acts more like a basic pattern from which to form a much more complex quilt of lines and colors.
The LICHT Super-formula (English translation)
(The numbers in boxes refer to the tempo.)

Stockhausen calls the extra-musical gestures in the super-formula "accessories".  These include
  • Modulation - regular/irregular changes in timbre, rhythm, dynamic, etc...
  • Scale - even rising/falling notes connecting one nuclear tone with another
  • Variation - ornamental improvisation ("satellite tones") on 1 or more connected nuclear tones
  • Echo & Pre-echo - repetition with diminishing dynamic values, or an anticipatory note
  • Wind - usually "breathing" pitches with a very transparent timbre, but sometimes played as harmonics 
  • etc...
     The next sections describe how Stockhausen used elements from the LICHT super-formula to create macro and micro structures in the 7 LICHT operas.

The LICHT Super-formula segments for each day
Click to enlarge.
The 7 Days of the Week
     Each of the 7 operas takes a 2-to-4 measure segment of the LICHT super-formula and expands it to create a form structure which determines the durations of its Acts and Scenes.  Stockhausen's rule-of-thumb was that 1 quarter-note would be equal to 16 minutes in playing time (this rule was often stretched, of course). The following are the 7 days of LICHT with their respective measures in the super-formula (also marked in the above superformula itself by slightly thicker measure lines):

MONTAG    (Monday):    M 1-3
DIENSTAG    (Tuesday):    M 4, 5
MITTWOCH    (Wednesday):    M 6, 7
DONNERSTAG    (Thursday):    M 8-11
FREITAG    (Friday):    M 12, 13
SAMSTAG    (Saturday):    M 14-16
SONNTAG    (Sunday):    M 17-19

MICHAEL formula divided into day segments
and with Nuclear tones in diamonds.

(Click to enlarge)
The MICHAEL Formula
     The MICHAEL formula has lots of ascending and descending figures and its characteristic note interval is the perfect 4th, which is also featured in the MICHAEL "head motif". Some of the main gestural features ("accessories") featured include ornamentation/improvisation (Measure 2) of the 3 notes in M1, colored pauses (as tongue clicks) (M3), 3 echoes (of M4 in M5), timbral variation (M7), scale runs (M9), rhythmic modulation (M12), pre-echo (M14) of the repeated note in M15, rhythmic pulses (M15), trilling and dynamic modulation (M18).  The signature melodic intervals of the MICHAEL formula are: ascending/descending perfect 4ths & 5ths, descending minor 6ths.

     It's worth pointing out that the "echoes" in the Tuesday segment "fade away". Additionally, the timbral changes found in measures 8-10 (Thursday segment) are done through the use of mutes (the "vowel" symbols actually represent a scale of 7 mute positions, from fully "closed" to fully "open"). The repeated notes in the Saturday segment each have different held dynamic attacks, while the final Sunday segment has different "pulsed" dynamic attacks. These are just some examples of the precise elements of the formula which make it more than just a melody theme.

     The formula is most easily heard in the beginning of MICHAELs REISE UM DIE ERDE in the ABSCHIED segment.  There, the trumpet plays the formula as a solo and the features of the musical gestures are clearly felt and heard. If Stockhausen Edition CD 30 isn't readily handy, this piece can also heard in a live rendition here.

LUCIFER formula divided into day segments
and with Nuclear tones in squares.

(Click to enlarge)
The LUCIFER Formula
     LUCIFER's formula includes more dissonances and has a more "martial" feel.  It has a head-motif of a major 7th (Measure 2).  The augmented 4th or tritone (named by 18th C. musicians as "the devil in music", appropriately enough) also features prominently.  Some of the main gestural features that are displayed here include a march-like regular pulsation (M1), colored tremolo (flutter tongue, M2), broken ascending scales (M3), irregular rhythm (M4, 13, 15, etc...), ornamental improvisation (M5), colored silence (M12), colored silence with counting (M7, 14, 19), timbre modulation (M8-10 (with phonetic vowels), M16), and tempo modulation (rhythmic deceleration) (M17).  One place that the LUCIFER formula can be heard relatively intact (and at a pretty brisk clip) is in MICHAELs REISE UM DIE ERDE in the trombone part of the 2nd STATION segment.  The signature melodic intervals of the LUCIFER formula are: ascending Major 7ths, ascending minor 6ths.

EVE formula divided into day segments
and with Nuclear tones circled.

(Click to enlarge)
The EVE Formula
     EVE's formula holds all 12 notes of the chromatic scale and has more "smooth" legato transitions between pitches as well as the simplest rhythms.  It's characteristic interval (used in the EVE head motif) is the "cheerful" major 3rd.  It has many of the same gestural features as the MICHAEL and LUCIFER formulas with the notable additions of broad pitch glissandi (M3), colored pitch bending (M8, 9), microtones (M15), rhythmic colored noise (breathing noises, M12), yodeling (M13) and different degrees of vibrato (M1).  One place that this formula can be heard relatively intact is in MICHAELs REISE UM DIE ERDE in the flute part of the 1st STATION segment.  The signature melodic intervals of the EVE formula are: ascending Major 3rds, ascending tritones, descending octaves.

     As mentioned above, all 3 ("elaborated") formulas in the form of the 3-layered super-formula can be heard (at reduced tempo with minor variations) in the piece BASSETSU-TRIO from bar 160 to 189 ("Viertes Formel-Trio", track 15 on Stockhausen Edition CD 55).

Formulas from Nuclear Tones and Nuclear Formulas
A graphical representation of the 16 (13 + 3) MICHAEL Nuclear Tones. This is a 12 tone pitch row with the 1st note (D) repeated as the last pitch.  Then 3 more notes (F - F# - F) were added as a transition between the 12th and 13th notes.
Here the MICHAEL Nuclear Tones have now been given note durations.  This is now called the MICHAEL Nuclear Formula.
This Nuclear Formula (like the full Formulas and the Nuclear Tones) are divided into 7 "day" fragments, to be used as motivic devices.
Each of the 16 notes in this Nuclear Formula was "elaborated" to create the full MICHAEL Formula (marked with diamonds in the MICHAEL Formula image earlier).
     Below is Stockhausen's colored sketch for the Nuclear Tones - sequenced pitches he wanted to use in the super-formula.  These sequences can be used to generate rhythmic figures and ostinato figures without being subject to the prescribed rhythms in the formulas and nuclear formulas. MICHAEL's nuclear tone sequence has 13 tones plus 3 extra repeated tones (the unconnected pitches in segment 5 below). EVE has 12 notes and LUCIFER has 11 notes. Each of these mini-segments can be used alone or in combination as motivic material.
LICHT Nuclear Tones. Glieder = limbs.

     The Nuclear Formulas were created by giving durations to the Nuclear Tones, essentially making them into short melodies.  Below are the 3 Nuclear Formulas ("kernformel") of MICHAEL, EVE and LUCIFER, in this picture layered on top of each other, though they are often used just by themselves.  Compared to their full Formula incarnations, it's clear that these are highly simplified versions of the LICHT Formulas before all of the "dialect accessories" were added in.  This makes them much easier to use as motifs and themes in a classical sense, especially when used for tutti groups of instruments such as chorus or brass.  Oftentimes these are broken up into fragments so that characters can "speak" with each other musically.
The 3 LICHT Nuclear formulas.  Each is divided into 5, 7 and 6 "limbs" (fragments used as motifs).
(original Nuclear formula transcription from "Into the Middleground: Formula Syntax in Stockhausen's LICHT" - Jerome Kohl)

     All 3 nuclear formulas can also be heard in the piece BASSETSU-TRIO from bar 117 ("Erstes Formel-Trio", track 12 on Stockhausen Edition CD 55).

     The first few notes of the 3 LICHT nuclear formulas are also used as the head motifs (below) of each character. These are somewhat like the "signatures" of each of the 3 characters of LICHT.
Head motifs of MICHAEL (perfect 4th), EVE (major 3rd) and LUCIFER (major 7th).
     With all of these musical formulas and their reversed/inverted/fragmented variations, Stockhausen had enough musical building blocks to work on LICHT for the following 26 years.

From Formulas to LICHT
     Stockhausen used the formulas, nuclear tones and head motifs in many different ways to generate the musical drama of LICHT, but probably the most obvious use was still generally based on pitch and duration (as opposed to the gestural elements).  A typical scenario could be to take a day's segment (say the SAMSTAG segment of measures 14 to 16 and use those 3 layers on top of the full LUCIFER formula layer to create a 4 layer polyphony, or to use the LUCIFER head-motif as a recurring figure during a scene.  In DONNERSTAG's MICHAELs REISE, the trumpet and the bassett horn have duets where they "trade licks" with each other, musically teaching the other fragments of their respective formulas.  In the HELIKOPTER STREICHQUARTETT the string players play the full super-formula with tremolo glissandi, but each string player switches from layer to layer.  In KATHINKA's GESANG each note of the LUCIFER formula gets it's own workout.  Some scenes use many interpolations of formulas, but with a single formula influencing the general tonal character of a scene.

     The only part of LICHT which is not fully based on the super-formula is Act 1 of DIENSTAG AUS LICHT, JAHRESLAUF, since it is an adapted staged version of DER JAHRESLAUF, which was completed before the super-formula was developed.  However, working on DER JAHRESLAUF inspired Stockhausen to begin work on LICHT, and the version used in LICHT includes additional vocal parts which use the LICHT super-formula.

     More detailed explanations of how the LICHT super-formula is used to germinate the individual scenes can be found in the posts describing the individual parts of the operas. A good place to start is the article on MICHAELs REISE.

"...structures can be composed into each element which accord with
the overall structure of the work to be composed.
The elementary micro-structures and macro-structures of a composition are derived
from the unique, total conception of the work..."
- Stockhausen writing about GESANG DER JÜNGLINGE in "Actualia", 1955/rev.1991

Dramatic breakdown for FREITAG AUS LICHT based on M12 & 13 of the LICHT super-formula.

LICHT Forces
The links below lead to brief synopses and breakdowns of the instrumental forces utilized in each work. 

Wiki Entry on LICHT
Into the Middleground: Formula Syntax in Stockhausen's Licht (Jerome Kohl) 
The Evolution of Macro- and Micro-Time Relations in Stockhausen's Recent Music (Jerome Kohl)
Time and Light (Jerome Kohl)  
Malcolm Ball's interview with Stockhausen on LICHT (1997)
Bernard Pulham's Site on LICHT
Generative processes in Stockhausen's Lichter - Wasser (Bianchi, PDF Thesis)
An Analysis of Karlheinz Stockhausen's Traum-Formel Pt 1 (Vickery, PDF Thesis)
The Saxophone Works of Karlheinz Stockhausen (Bunt, PDF thesis)