Symbol Matrix page (1 of 7)
(© Universal Edition)
for any number of instrumentalists
(1963, rev. 1974)   [duration open, probably well over an hour]

     Developed at the seashore on a vacation with artist Mary Bauermeister and then given as an exercise to students at his newly-founded Cologne Courses for New Music, this work basically functions as a composition assignment (complete with 35 rules, 318 "puzzles" and 7 pages of "answers").  The published score is titled "PLUS-MINUS, 2 x 7 Seiten für Ausarbeitung" (2 x 7 pages for working out, or realization).  Stockhausen himself never found the time to complete this "assignment", and no "official version" of PLUS-MINUS has ever been recorded.  In a sense, it's probably the closest one can get to attending a Stockhausen composition course in the 1960s.

     The chronological timing of the actual ideas in PLUS-MINUS is very fascinating, considering the works before, during and after it.  Around the same time as writing PLUS-MINUS, he was revising PUNKTE and finishing MOMENTE.  The following year would see the first experiments with ring modulation and live electronics (MIXTUR, MIKROPHONIE I & II), but it would not be until 4 years later (after HYMNEN was completed) that Stockhausen would revisit the idea of "plus minus" event transformations in PROZESSION.  With this chronology in mind, the use of such open-ended compositional devices and iconoclastic notation foreshadowed what would come.  However, at the time, Jonathan Harvey claims that he saw "incredulity, bewilderment and hostility in the seminar that Stockhausen gave on it at Darmstadt, shortly afterwards."

Melodic Life Cycles
     The below explanation uses terminology from an analogy which is not part of the score at all, but which I personally find easier to use.  Below are "my terms", with the original score terms in quotes:
  • Character: "Types"
  • Body: "Central Sounds"
  • Limbs: "Accessories"
  • Tools: "Subsidiary Notes"
     For the purposes of this article I'm going to take the liberty of using my own terms (apologies to score purists, but I don't feel like typing out "Zentralklang" 50 times).

     The basic idea of PLUS-MINUS is that 7 musical motifs, or "Characters", appear over and over again in different sequences (basically one Character at a time, but it's also possible to have multiple PLUS-MINUS layers happening at the same time).  Not every Character appears the same number of times on a page, but each page "features" one character more than any another (Fibonacci sequence distribution).  These Characters are "born" with a central chord ("Body") associated with 1 to 3 free pitches/noises ("Limbs").  A Character may often appear with a "Tool", which is a collection of 1, 2, 3, 5, 8, or 13 grace-notes/chords to be reordered/stacked in any way. 

     The Characters appear chronologically according to Event boxes (read left to right in rows on the Matrix page), and each consecutive Event is affected by the previous Event, in that the featured Character must act in some way "more than" or "less than" the previous Event's Character (louder/quieter, taller/shorter, talkative/taciturn).

     Often when a Character reappears, it may have either "grown" (by adding a new Limb or by cloning itself) or "regressed" (by cutting off a Limb or by killing off a clone).  The score rules often give the student composer both options of growing or regressing a Character (Ex. "+1/-1").  If a Character regresses into the negative (the degree of growth/regression goes from +13 to -13), then a "photographic negative" of the Character is introduced.  If the growth goes above +13, the Character is "reborn" into a new initial state, and if below -13, the Character is "killed off" (it is actually possible to kill off all 7 Characters before all of the Events have been played).

The Characters:
(All score excerpts © Universal Edition)
     The first step is to pair up the 7 symbol matrix pages to the 7 notation pages (chords/branching notes). Each pair of pages makes about 10-15 minutes of music.  All 7 pairs of pages should be played, but most recordings only feature 1 page.

     Each boxed symbol of a matrix page is an Event.  The sequence of the 3 main elements (Body, Limbs, Tools) is read from left to right in each Event square, and simultaneous events are stacked polyphonically (stacked parts can be layered rhythmically in this case).  There are 7 Characters (character types), listed below as combinations of Body chords (B) and Limb notes (L) (Tools are not part of the Character type, since they change from Event to Event):
  1. L - B (meaning L then B)
  2. L + B (meaning L and B stacked)
  3. B - L (etc...)
  4. L - B - L
  5. (L + B) - L (shown in Event 32, below)
  6. L - (L + B)
  7. L - (L + B) - L

     In each recurrence of a Character, the Body chord essentially stays the same, but the Limbs may be different pitches/timbre. At the same time, each Character reoccurrence should be "recognizable" even with growth additions/subtractions (this suggests that the Limbs and Tools should have some kind of Character-specific shape).  Each of these 7 Characters must be assigned one of 7 chords (I-VII, at right) from its paired notation page, and must use this chord for every one of its appearances (though it is possible to transpose the chords).

Matrix Event Squares
1 square box is 1 Event
(this is Event 32, which features Character 5)
These are the individual parts of an Event square (above):
  1. Body (circle): Assign the Character chord (possibly arpeggiated) from the notation page.
  2. Limbs (triangle/diamond/spot/?):  Assign short/med/long/indeterminate free pitches/noises (same register as the Body chord), each Limb occurrence following as fast as possible, but with the durations clear.  In Event 32 above, the 2 diamond Limbs above mean 2 medium-length Limb notes/sounds, with one during and one after the Body chord.
  3. Tools (eighth-note stem without notehead): The number here indicates which group of grace-notes (1-6 below) on the paired notation page to use.  The grace-notes/chords in each group can be permutated or stacked freely.  The flat/sloping line indicates fastest speed (-), accel. to fastest speed (/), or ritard (\).  In Event 32 above, Tool 5 is indicated, so Group 5's 8 notes must be used, with a ritard.
    A set of 6 Tools (branched notes) used on 1 matrix page
  4. Flags: When a Flag containing a positive/negative number appears above a Character, it indicates an addition/subtraction to the number of repetitions of the Body, Limbs, or both (the Tools notes may or may not increase proportionately, since more hands should have more hammers, right?), and stays in effect for all recurrences of this Character until a new Flag appears.  Also, the additive effect is additive and not exponential, so adding 1 means adding an original Character part, not a clone group.  The Flag only takes effect from the 1st recurrence (not the original flagged Event).  In Event 32 above, the composer is given the choice to either add or subtract 2 limbs, bodies or whole characters (cloning/killing). 
    • If the number becomes 0 or negative, Limbs and Bodies are expressed as 2 "negative bands", with the number value represented by pauses.  The negative bands should be like a "photographic negatives" or "ghosts" of the original sound.  
    • If a Character reaches 13 iterations (13 repeating clones), it starts again with 1 of the original Character, but with a new interpretation. If it reaches -13, it disappears and is skipped for the rest of the composition (is "killed off"). 
  5. Hairpins (</>): these symbols contain numbers indicating changes in lowest pitch, register, dynamic, or duration of the event (< means more, > means less), compared to the previous Event.  The numbers refer to degree of change, and the number can also be spread over more than 1 parameter.  These degrees of change can be amplified and spread over a "section" (ie - the composer creates a form structure of sections based on musical trends in the Events).  In Event 32 above, the Event must be greater than Event 31 by 1 degree in a parameter.
  6. Transition: symbol indicating whether the Event is followed by a short/long pause, segues immediately into the next Event, or overlaps into the next Event.  In Event 32 above, the "v" indicates a short rest before the next Event.
  7. Articulation:  "Damping/muting", accents (with echoes), reg/irreg. rhythm, or K for "combination" (above).
  8. Timbre: These solid and dashed lines indicate the timbre of the Event parts (divided into combinations of soft/hard sounds(pitches)/noises)).  Sounds, noises and sound-noises are analagous to vowels, consonants and semivowels.  In Event 32 above, the timbre is to be a mixture of soft sound-noises.
  9. Downward Arrows: More than 1 Layer (up to 7) of a PLUS-MINUS realization can be played at the same time, each with.a "signature" sound characteristic.  The thick downward arrows are used for coordination of Events between multiple Layers (longer arrow = more unison Layer Events transitions).  In Event 32 above, the Event begins simultaneously with many other Events in other Layers (if any).
  10. Staircases: The bottom right "staircase" figures control the degree of harmony or dynamics between layers by exchanging/deleting notes between simultaneous Events and controlling swelling/fading envelopes.  In Event 32 above, the indication is to replace this Event's pitches with the pitches from a simultaneous Layer's Event.
     Some Events have thick borders and some are empty (first, fourth and last Events on top row of Matrix page above).  The thickly-bordered Events are "time windows" (such as used in MIKROPHONIE II), inserted into the blank Event boxes on other pages, depending in the upper page corner arrows.

     If more than 1 Matrix page of Events is being realized, the next Matrix page will continue the "saga" of the same 7 Characters (if they haven't been killed off), but their Body chords and Tool sets are newly-assigned from the new Matrix-page's complementary notation page.  Stockhausen asks that an interpreter send the publisher the details of the last Event realized in his/her version, so that the "next" realization can continue on from it....

     (The above paragraphs don't include many other more detailed rules from the score, which have to do with transposition conditions, etc...)
    Tool Trends
         There are 7 pages of Matrices, and 7 notation pages of chord/grace-note elements, to be matched one to a page.  Below are some general characteristics of each of the 7 pages of Body chords and Tools groups.  For the most part, the kind of Tool set (out of the 7 below) used in a Matrix page can give a strong identity to that particular page.
    1. wide chords, chord clusters
    2. wide chords, with a couple clusters, diads and a few single notes
    3. wide chords, with a couple clusters, major and minor 2nd diads
    4. wide chords, with a couple big clusters, wide grace note leaps
    5. medium-spaced chords with a couple big clusters, range descending downwards, scalar figures and single notes
    6. wide chords with a couple big clusters, 2-note tremolo figures
    7. medium-spaced bass chords, single notes
         The above paragraphs describe what the resulting composition must be but leaves the steps of how to actually do it up to the composer.  If I ask myself how I would proceed, this is a possible answer:
    1. Pair up the Matrix and notation pages.
    2. Identify all of the Character 1 types and fill them with one of the 7 notated chords for each page.
    3. Fill all of the Limb parts for Character 1 and and give them some kind of unified signature shape.
    4. Fill in the Tools for Character 1 (also maintaining a unified shape if possible)
    5. Apply the Flag growth/regression marks.
    6. Repeat for the remaining 6 Characters.
    7. Starting from the first Event, and going one by one, fill in the timbre articulation markings, hairpin transformation rules and transition markings.
    8. If more than 1 layer, apply the "staircase" and "down arrow" coordination markings.
    9. Move the Time Windows into the appropriate pages.
         There are 2 commercially-available versions of PLUS-MINUS on CD.  Elizabeth Klein performed a 1-page, 1 Layer, solo piano version, realized by Nils Holger Petersen.  This recording is a very straightforward representation of the growth of the 7 Characters (actually listening to this version inspired the "Characters, Body, Limbs, Tools analogy).  Peterson only used 1 matrix page for this version and it lasts about 13 minutes.  In 2002, the Ives Ensemble recorded a 2-Layer version, with each Layer given to a mixed ensemble (left and right in the recording).  One Layer was realized by Christopher Fox and the other by John Snijders.  This one also has a strong periodic character, and for its "negative band", it uses taped excerpts of ambient rehearsal noise.  It's a full 51 minutes, so I wonder if it's a complete double realization of all 7 Matrix pages.  What makes this interesting is the arranging of the character elements over multiple instrumental timbres, as well as the mixture of synchronous and asynchronous Event transitions.  The article that Christopher Fox wrote about his experience creating a performance score out of PLUS-MINUS ("Stockhausen's PLUS MINUS, More or Less: Written in Sand", Musical Times, 2000) is a very interesting read and also gives a description of several other versions, including one by Gavin Bryars which used excerpts of Tiny Tim and field recordings of pissing elephants for the "negative band" sounds.  Stockhausen did not approve of this...
    (from "Stockhausen's plus minus, More or Less: Written in Sand", Courtesy C. Fox)
    (from "Stockhausen's plus minus, More or Less: Written in Sand", Courtesy C. Fox)
         Because of the reiterative nature of this piece, the result usually starts out with a wide variety of short melodic motifs, but as the parts multiply in repetitions, a kind of periodic structure results.  Theoretically, it could be possible to get a maximum of 13 repeating Body chords and 39 Limb notes from just a single 3-Limbed Body (13 + (3 x 13)), which of course, will sound pretty "groovy".  This is also reinforced by the instruction to allow polyphonic parts to be expressed rhythmically.  In general, this work usually doesn't produce a "points and groups" kind of piece, but instead possibly creates something closer in shape to STIMMUNG, which would come 5 years later.  Come to think of it, a multi-Layer a capella version of PLUS-MINUS would be a good companion piece to STIMMUNG.

         Apparently many versions were created in Stockhausen's composition class (some of them by now well-known composers), but few recordings of this work exist, and there is no "official" version in the Stockhausen Complete Edition CD catalog.  The openness of the score allows for a "co-composer" to imprint his or her own style on it to a very high degree, so maybe he felt that any realization/recording would have enough of the co-composer's contribution to make that one essentially "official".  Or perhaps, had Stockhausen lived long enough, he would have eventually worked out his own version, and would have then released that as an official recording.

         Another possibility is that after he heard the results he was getting from his students, he lost interest due to the unavoidable appearance of so many regularly-repeated melodic motifs.  His next "plus-minus" work would be PROZESSION, which would use excerpts from more "thorny" works like GESANG DER JÜNGLINGE , MIKROPHONIE I and KLAVIERSTÜCK XI to act as initial character types - keeping the idea of transformation, but jettisoning the iterative elements.

    Puchase the Score
    Wiki Entry
    "Stockhausen's plus minus, More or Less: Written in Sand" (Christopher Fox, Musical Times, 2000) 
    Some Recordings: