The number of different timbres increases and decreases twice.
Tom-tom and marimba cycles occur in both iterations.
(Cycle) for solo percussionist (snare drum, hi-hat, triangle, vibraphone, gongs, güiro, marimba, bells, tom-tom drums, etc...)
1959 (9-15')

     Stockhausen wrote ZYKLUS for solo percussionists to use as a music competition piece, specifically for the Kranichstein Music Competition in 1959.  This kind of opportunity to write for would-be virtuosos to prove themselves would pop up from time to time, even up to 1997's KLAVIERSTUCKE XVII, written for the Micheli Competition (and derived from part of FREITAG AUS LICHT).  In fact MITTWOCH AUS LICHT's 2nd Act is titled ORCHESTRE FINALISTEN (Orchestra Finalists).  Sadly, it seems the competition performance of ZYKLUS didn't actually win the contest, but it was still the first major composition for a soloist on multiple percussion instruments (13 in all).

Sound Impressions
     Like MANTRA, ZYKLUS has lots of underlying theory which is not easy to hear when just listening to it (especially without the score).  Actually for most people it may be more fun to just listen without knowing the theoretical structures underneath it. In that spirit, I'll detail some broad strokes before going over the structural stuff.
     In a way, this piece can be enjoyed purely as "cartoon music" - that is, it is constantly changing, dynamics are all over the place, and there is a feeling of youthful vitality in all the rim shots and glissandi.  On another level, ZYKLUS is a kind of showpiece for different percussive timbres.  In a "straight-forward" performance (starting from page 1) it cycles through "featured" sounds in this order: snare drum, hi-hat, triangle, vibraphone (glissandi/trills), gong (& tam-tam), güiro, marimba (glissandi), bells and tom-tom rim shots.  As described in more detail below, these sounds come to a climax and then recede (also matching the rotation of the soloist inside his circular percussion set-up).
     Stockhausen here is also interested in demonstrating the difference between "determinate" (scored) figures and "statistical" (indeterminate) melodic figures.  In his first 1972 British Lecture, "Musical Forming", he expects that a listener could detect the movement from one kind of gesture to another. In order to identify the determinate parts, he suggests listening for repeating patterns or directional processes (such as accelerando/ritardando).  Personally I've found this pretty difficult, but it's worth mentioning.
     Finally, the recurring tom-tom rim shot is a nice motivic event which occurs frequently enough to stand as a kind of "speed bump".  There are 41 of them - count them all! This idea of a periodic "rifle crack" would return in TELEMUSIK.

Form Structure
     ZYKLUS is very true to its name in that it is composed of "cycles".  In fact there are actually 5 separate layers of cycles built into ZYKLUS's mere 12 minutes. 

1. Timbre Sequence:
Skeletal Structure of scored parts.
     During the work, 9 kinds of percussion timbres gradually become more dense, then dominant, and then gradually sparser (over 17 equal sections or "Periods").  The climaxes of these 9 timbre types are spaced equally through the piece (see at left).  These hits make up the "Skeletal Structure".  Since each approach/climax/departure cycle takes place over the full length of the piece, the percussion types naturally end up cross-fading with each other.  The sequence of "featured instrument" in a performance starting from Period 1 is snare drum, hi-hat, triangle, vibraphone (glissandi/trills), gong (& tam-tam), güiro, marimba (glissandi), bells and tom-tom rim shots.  However there are not an equal number of "attacks" per instrument.  As seen above, there are only 11 snare attacks (all rolls) distributed in the piece, but 41 tom-tom rim shots.  This variation in the number of attacks for each percussion type is designed to be somewhat analogous to 9 musical intervals (minor 2nd to major 6th), but is basically just a technique to create variety.

2. Aleatory Sequence
Aleatory structures. 
     On top of the Skeletal Structure, there exists the aleatory, statistical "free-choice" layer, as touched upon earlier.  Instead of 9 overlapping cycles, here there are 2 cycles in a row, and instead of a timbre approaching/receding, here the performer's "free choice-ness" (variable form) increases or decreases, based on 9 "Structure-types".  On each page there are different sections where the soloist has a choice of
  1. Which notes/phrases to play (out of 2-5 multiple choices)
  2. What order to play them
  3. When to play them within a specified duration
These 3 kinds of choices come in combinations which make up the 9 Structure-types (more detail about these is in the score explanation below).
     All of these factors are designed so that at one end of the scale every attack is scored (Period 1, Structure-type 1 only, determined), and at the other end pure graphic notation exists as different-sized dots (Period 17, Structure-type 9 only, highest free choice, or "indeterminate").
     From a listener standpoint this layer is not particularly audible, unless of course the listener knows the work so intimately that he is familiar with every choice available.  But from a performer standpoint, this basically allows a level of collaboration with the composer.  Stockhausen identifies 2 cycles here based on the number of kinds of notation techniques used, but again (sorry) from an aural standpoint it's fairly irrelevant.  He however does prefer that as many simultaneous hits as possible occur, so a "good" performance could be one where the performer has figured out how to play many aleatory parts simultaneously with the fixed parts.  Someone with 3 arms would easily win this competition.

I am interested at the very beginning of a new work in creating my own sounds.
And creating my own sounds means mixing, and mixing with the traditional
instruments means, superimposition of different instruments, which results in complex
sounds that cannot be analyzed anymore. So what I really want is that, when a
percussion player makes his own version of Zyklus, he creates sound complexes that
are his own, the result of the superimposition of several instruments, and you cannot
analyze how he made them. . . . and by this create fantastically mysterious sound
complexes. . . .
 - Stockhausen, from “An Interview with Karlheinz Stockhausen.” Michael Udow, Percussive Notes Research
Edition 23, no. 6 (1985), (found in Stuart Gerber's superb performance thesis, link at bottom)

3. Instrument Density in the Aleatory Parts Sequence
     In the aleatory parts (but not hits that are part of the Skeletal Structure), each percussion timbre comes in and then departs after 4 cycles (tom-tom and marimba actually get 2 cycles).  The picture at the top of this article shows the timbre distribution over the 17 periods.  The sequence of timbres matches the circular placement of the drum surfaces around the performer, so the soloist ends up slowly rotating 1 time (360 degrees) during a single performance. 

4. Tom-tom Rim-shot Intensity Levels Sequence
     The rim shots divide ZYKLUS into 41 segments, each with its own dynamic range and "intensity-form".  The intensity-form basically comes from 5 types of dynamic distribution (all loud, all soft, mixture, crescendo, etc...).

5. Vibraphone/Marimba Glissandi Range Sequence
     Each vibraphone and marimba glissandi determines a range of pitches to be used until the next glissandi (with a few exceptions).  The pitch range basically shrinks and expands 1 time over the entire piece (see below).

     ZYKLUS is probably Stockhausen's most graphically interesting score.   Each of the 17 periods gets its own page (actually 1 and 17 are split on 1 page).  On each page there is 1 central timescale divided into 30 beats.  Above and below this central timescale are the aleatoric elements, usually in boxes, triangles or brackets.  The percussion instruments are indicated by graphic symbols both before and after the staff it is on. 
Set up of the percussion and the notation guide.
These are the instructions on how to interpret the 9 aleatory Structure-types (click to enlarge):
(© Universal Edition)

Period 3
(© Universal Edition)
     In the first large box with the triangles, the soloist must play the 3 sounds (tom-tom, cowbell, cymbal) in any order, but in the spots connected to the hi-hat timescale.  Then, the 3 bracketed parts (2 above and 1 below the timescale) means that the soloist must choose 1 of the 3 to play in that section.  The 3rd section is played as it is written with no aleatory elements.

Period 12
(© Universal Edition)
     Here the soloist can choose 1 of 2 boxes to play in the first section (either marimba/vibraphone or log drums).  Either way the soloist can play the 3 attacks in any order and at any time during the period adjoining the box.  However it is preferred that the attacks coincide with the güiro, snare and marimba sounds as much as is technically possible. In the next section everything in the 2 boxes must be played but all of one box must be completed before the other. In Section 3 the 6 attacks can be in any order at any time.  Section 4 is an extended box, basically more stuff to play during the marimba-dominating timescale.

     The score can also be played backwards or upside down.  That's why each staff has symbols and clefs on both sides. It can also be started from any page, but must continue until all pages are completed.

Live Performance

Version by Chris Sies

Sound samples, tracks listings and CD ordering
Buy the Score
Stockhausen's Solo Percussion Music: A Comprehensive Study (Gerber, Thesis)  
A Performers Approach to ZYKLUS (Kearney)
ZYKLUS' 9 Aleatory Structure-types (B. Michael Williams, PDF)
Sonoloco Review of ZYKLUS
ZYKLUS (Aleksander Wnuk, Live YouTube)
Interview with Max Neuhaus discussing preparations for ZYKLUS