Device to vary the sine waves which control the ring modulation.
(© Universal Edition)
No. 16: MIXTUR
for Ring Modulated Orchestra Groups (1964)
No. 16 1/2): MIXTUR
for Ring Modulated Instrumental Groups (1967)
No. 16 2/3: MIXTUR 2003
for Ring Modulated Instrumental Groups (35 players)
[ca. 27']

     MIXTUR (Mixture) is Stockhausen's 1st work to use sine-wave-controlled ring modulators to distort and enhance acoustic musical instruments.  In this piece he applies this effect to 4 groups of instrument types: woodwinds, brass, bowed strings and plucked strings. Soon he would also apply this technique to voice ensemble (MIKROPHONIE II) and piano duo (MANTRA).  It was used much later in LICHT in DONNERSTAG's MICHAEL's REISE (applied to piano in the Bali Station) and in SONNTAG's LICHT-BILDER (applied to flute and trumpet).

Ring Modulation
     Basically, ring modulation is a distortion  of the microphone input of an instrument so that additional harmonics, sub-tones and overtones surface.  These new sounds can deviate dramatically from the original pitch.  The amount of deviation is controlled by a sine wave tone (indicated in the score and controlled by another player).  What the audience hears is a mix of the original orchestral instruments and the new pitches produced from the ring modulation.

     To be more specific, the sum and difference (addition & subtraction) of 2 frequencies (one is the microphone input, the other is a sine-wave) are produced, creating both a lower and a higher pitch.  For example, if the pitch A (440 Hz) is RM'ed (ring-modulated) with a sine-wave tone of 200 Hz, the output will be 640 Hz and 240 Hz (the sum (440 plus 200) and difference (440 minus 200) frequencies).  In general, if the sine-wave tone is from 1 to the matching frequency of the main input (440 in the case of A), there will be a pair of increasingly higher and increasing lower tones.  If the sine-wave is from the main tone and up, both new tones will be increasingly higher.  Stockhausen used very high sine-waves (for example, 12000 Hz) in TELEMUSIK so that both of the output tones were extremely high pitched (an A 440 would become 12440 and 11560).  In MIXTUR, the sine-waves generally don't go much higher than 1200 Hz or so.  When the chosen control sine-waves create frequencies which are in the low to middle range, frequency "beating" often occurs from the dissonance between the 3 notes (original plus the sum and difference tones).  When the beating is very fast it creates textures similar to a Doctor Who "Dalek" voice.

     One very interesting effect is when the original sound is a steady held note, but the sine wave is varied.  This creates a kind of swooping theremin-like effect.  Additionally, the ring-modulated sounds are actually much more than 2 additional pitches, since each original tone also contains its own natural overtones and harmonics.  So what happens is the effect is multiplied exponentially, depending on how loud the original sound is and how complex its timbre.  In some cases even the sum and difference tones get ring-modulated.  In order to control these complex sounds, Stockhausen would eventually employ a wide array of equalization filters (lo-pass, high-pass, octave filters) to mold or chop off unwanted high or low frequencies.
Sine Wave generator for 1965 performace (WDR Studio)
Sine Wave generator for 1977 performace (WDR Studio)
Stockhausen Edition CD 8
    MIXTUR is in 20 sections, each an independent "Moment" with no real sequentially thematic connection with the previous or next Moment.  The Moments each have a somewhat descriptive title and for a given performance can be played in sequence from Moment 1 to Moment 20, or alternatively in backwards sequence (there are also several Moments which can be switched/exchanged with each other for a performance - see below).  Stockhausen preferred that the forward sequence be played, and then the backwards sequence, with an intermission in between.  The orchestra is organized into 5 instrument type groups (percussion, woodwinds, brass, bowed strings, plucked strings), and in the table below, the 5 right hand columns indicate which of the groups play for each Moment.  Sometimes a plucked (pizzicato) string group will play bowed (arco) material, and vice-versa.  These are indicated by parentheses.
     Finally, in Moments 4 and 11 there are 2 "inserts", which could be considered a kind of improvisation or "flashback".  Jerome Kohl's 1981 thesis "Serial and Nonserial Techniques in the Music of Karlheinz Stockhausen from 1962-1968", observes that these inserts are the first time Stockhausen uses a "quote" from an external music source in his work (preceding TELEMUSIK).  Stockhausen, for his part, describes the quote of "La Marseillaise" as being "played backwards" (in a 2001 interview with Julia Spinoza (FAZ)).
Moment Name Dur. Description Instrument Groups
1 Mixture 0:34 7 accented string plucks, short wind phrases, high harmonic string figures, muted brass swells and a climactic cymbal strike ("unknown timbre glissandi") PERCUSSION
2 Percussion 1:05 Ringing cymbal and gong strikes, also gong "scraping" with plastic/cardboard boxes (no RM) PERCUSSION
3 Blocks 2:13 6 blocks of sound with different dynamics, separated by silences.  The blocks each have unique combinations of "micro-rhythms", pitch bends and mutes. WINDS - BRASS
4 Direction 1:22 Blocks of layered pizz. strings and scraped percussion, with 1 "diving" horn blast, followed by 2 waves of rising/falling tremoli
INSERT: "Marienlied/folksong" near the end
5 Exchanges 0:30 11 Group attacks  WINDS - BRASS
6 Calm 4:36 Long tones with various brass muting and string bowing techniques (harmonics, col legno, etc…). RM causes beating/pulsations. BRASS
7 Vertical 0:46 2 accented blocks of trembling/scraped percussion and agitated strings PERCUSSION
8 Bowed Strings 0:34 Polyphonic layers BOWED STRINGS
9 Points 0:45 Isolated attacks with various muted brass and strings WINDS - BRASS
10 Wood (winds) 1:31 Low register trills and tremoli WINDS
11 Mirror 2:12 Short figures accompanied by a low background drone and a brief swooping trombone
INSERT: 1st measure of "La Marseillaise" occurs twice
12 Translation 1:13 Winds and pizz strings enter individually with tremoli, soon becoming "points".  Tremolo texture then returns, finally ending with metal percussion hits. PERCUSSION
13 Tutti 2:18 Lurching staccato figures, broken by tremoli, trills, long tones and silences TUTTI
14 Brass 0:39 Drone, with melodic accents in the middle section BRASS
15 Concert-Tuning 0:34 "Tune up", with one silent break in between WINDS - BRASS
16 Steps 0:49 Winds trill up/down, brass modulates with mutes, strings plucked and bowed in large intervals WINDS - BRASS
17 Dialogue 0:28 Dialogue PERCUSSION - BRASS
18 Layers 1:42 Polyphonic layers of synchronous accents, short figures, and independent "points" BRASS
19 Pizzicato 1:52 Alternating high and low plucked clusters, decelerating and then accelerating (features a violin "solo" of 1 isolated note in the accelerating section). PIZZ STRINGS
20 High C 0:30 Ascent to high C WINDS

     It's kind of interesting that the option to "switch around" the Moments (below) is not exercised very often.  In any other composition by another composer this would be the focal point for a program book, but MIXTUR already has so many pernicious technical challenges (controlling the ring modulation mix, the aleatoric nature of the score, dynamic balancing), that most ensembles play it "straight" (with the notable exception of Jonathan Nott and Ensemble Intercontemporain's performance in 2004).
This graphic from a slide presentation by Lindsay Vickery
shows the interchangeable Moments.

     MIXTUR's score uses a fair amount of "aleatoric" notation.  In other words, he sometimes provides pitches or melodic fragments, but leaves it to the performer as to when or how they are played.  Some "pointillistic" sections are literally dots on the page which the player interprets as floating noteheads.  This kind of notation is ideal for ring-modulation purposes, since it's more effective to distort textures and blocks of sound, rather than melodies.   During rehearsals, players are urged to decide how they are going to play their "free-choice" parts before the performance and notate it in spaces provided.

     Originally scored for full orchestra (Work 45), Stockhausen made a 2nd version for reduced orchestra (Work 46).  In 2003, Stockhausen decided to write out all the free-choice aleatory parts, creating a fully "realised" version.  This is called MIXTUR 2003 (Work 47).

     Except for the percussion (which is never ring-modulated), each of these 4 groups has it's own ring modulation part, labeled in the score as "~". The indications at top with the arrows (<-5->) are the number of beats for each section.   Below are some pages from the 2nd version of MIXTUR.

Trumpet, high horns, low horns and trombone
(© Universal Edition)
This page shows winds (H) and plucked strings (P).
The middle section is a melodic set from which the players can choose 10 notes to play.
(© Universal Edition)
"Direction" Page 2
Percussion (SCH) and plucked strings, with a little bit of wind and brass.  Percussion plays 1, 2 and 2 hits, indicated in the boxes.
(© Universal Edition)
"Tutti" Page 1
Graphic notation with dots
(© Universal Edition)

In MIXTUR 2003, Stockhausen wrote out all of the aleatoric notation into standard notation.  A good example can be seen in the CD cover to Stockhausen Edition CD 106, which shows the last part of the "BLOCKS" moment (pg 2 of Moment 3).
(click to enlarge, © www.karlheinzstockhausen.org)
Here is the original score passage (with a little bit of cut & paste editing for clarity)(click to enlarge, © Universal Edition)

Live Performance
2 possible stage set ups for the 5 instrumental groups.
"Sch" = Percussion, H/S/P/B are the other 4 groups.

MIXTUR, Opus 16 1/2 (Forward Version):
Ensemble DIAGONAL, Director: Jürg Henneberger, Live Electronics: Cornelius Bohn
Hochschule für Musik Basel in collaboration with the Elektronischen Studio Basel ESB, Volkshaus Basel, 20.03.2013

Do It Yourself Ring Modulation
     Many sound editing software programs have a ring modulation effect which can be applied to a sound file.  I use Adobe Audition (3) and in the group effect "Generate" there is an effect called "Tones".  If no audio is already imported, you can first create a steady pitch, for example 440 (A).  Then by clicking Tones again and choosing the "Modulate" option, it is possible to enter in a control sine-wave in the "Base Frequency" box.  A sine-wave glissando can also be created by entering in Initial and Final values (after "unlocking" the base frequency settings).  Use of the Preview button is pretty handy for typing in different values to hear their effect immediately.

Stockhausen Edition CD 106: MIXTUR 2003
     The release of MIXTUR 2003 on Stockhausen CD 106 is in 2 single tracks (Forwards Version and Backwards Version).  Here are my approximate timings for the individual Moments for these 2 tracks.

MIXTUR 2003:
Track 1
Mixture 0:07
Percussion 0:28
Blocks 1:54
Direction 4:28
Exchanges 5:57
Calm 6:39
Vertical 11:12
Bowed Strings 12:10
Points 12:44
Wood (winds) 13:34
Mirror 14:30
Translation 17:13
Tutti 18:30
Brass 21:12
Concert-Pitch 22:09
Steps 22:43
Dialogue 23:37
Layers 24:10
Pizzicato Strings 25:35
High C 27:21
MIXTUR 2003:
Track 2
High C 0:07
Pizzicato Strings 0:42
Layers 2:25
Dialogue 3:46
Steps 4:21
Concert-Pitch 5:16
Brass 5:52
Tutti 6:49
Translation 9:18
Mirror 10:40
Wood (winds) 13:23
Points 14:15
Bowed Strings 15:01
Vertical 15:34
Calmness 16:33
Changes 20:57
Direction 21:37
Blocks 23:06
Percussion 25:50
Mixture 26:50

Sound Impressions
     The very first time I heard MIXTUR I actually thought there was something wrong with the record.  It really sounded to me like the vinyl grooves had deteriorated in some way.  Over time I got used to it and became very fascinated by the new textures obtained from the ring-modulated instrumental groups.  However I personally feel the effect works better in the later piano pieces, such as in MANTRA or MICHAEL's REISE.  Somehow the timbral complexity of the orchestral instruments almost makes the additional ring modulated sounds almost too complex for me.  However what I probably enjoy even more than the ring modulation is the orchestral writing itself.  In fact I think MIXTUR could stand perfectly well on its own even without the electronics.  Stockhausen uses a real smorgasbord of textural composition techniques to get these 20 Moments to sound independent and have their own identity.  In fact I may like this collection of Moments even more than the primary "moment form" work, MOMENTE.  In any case, there are many depths to plumb in this piece.

     Update: Since hearing the new CD 106 recording of MIXTUR 2003, I've decided that the ring modulation actually works quite well and, at least in this recording, the modulation successfully creates a new sound-world of unknown textures ("Calm" is certainly a highlight).  In this version, the percussion part also sometimes seems to sound ring-modulated somehow... (to great effect).

Sound samples, tracks listings and CD ordering
Purchase the Score
Stockhausen's Notes on MIXTUR for Large Ensemble (PDF)
Stockhausen's Notes on MIXTUR for Small Ensemble
Stockhausen's Notes on MIXTUR 2003 (PDF)
Sonoloco Review of MIXTUR
Albrecht Moritz on MIXTUR
MIXTUR 2003 Report by Robert Worby 
Serial and Non-serial Techniques in the Music of Karlheinz Stockhausen from 1962-1968 (Jerome Kohl)