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    Examples of Linear and Non Linear Compositions

    When I think of linear compositional development Beethoven and the fugue immediately come to mind as development occurs or unfolds from a singularity, motive,  or theme.  Symphony Fantastique by Berlioz unfolds from the idee fixe as the primary generator.  Even serialism compositions are dependent on distributed spatial relationships from a cell or row.  Although this is unquestionably a logical process for Western music which occurs in time; that is having a starting point, a middle, and an end, there should be examples of works that do not follow this pattern.  In other words a sound event that is more simultaneously global.  For instance the way the brain works simultaneously functioning in different regions to perceive and navigate consciouness.  Buckminster Fuller's symmetrical geometry comes to mind - the geometry of geodesic structure in which all tension is simultaneously equally distributed and codependent.  As a composer I have been interested in abandoning linear development because it seems antiquated as a model.  In much the same way as quantum mechanics and string theory have suggested indeterminacy and connectivity across dimensions.  Can anyone suggest contemporary or otherwise musical compositions that are not linear in origin?  Thank you for any suggestions or ideas.

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    • The difficulty with this is that music unfolds over time.  It is necessarily linear in a certain sense.  But I would look for music that somehow distorts our perception of linear time as possible candidates.

      The other element is to determine whether the music can have a logic to it, or if logic is what gives the music its linearity.  If so, what would be "illogical" music?  Certainly not Vulcan.

      To what extent could chant be considered nonlinear in its variety of structures? Antiphons and strophes that return etc.

    • Thanks for your comments Conor.  Although music unfolds over time does not mean the point of origin (or the beginning or any point in-between) has to be a process of unfolding.  The brain does not have an emotion, then a rational analytical thought, then a primary function that controls body function one process at at time . . . those processes happen simultaneously.  I suppose what I'm getting at is a musical "form" that involved crosstalk or an exchange of musical elements already structurally formed from the outset . . . or at any point of examination of linear time.  In other words, any part is just the sum of the whole at any given moment.  Perhaps I did not frame the question well.  If you played Beethoven's 5th symphony backwards does it reduce to a germ?  Maybe a piece by Varese or Zappa might be an example of music that begins from a perspective of multiplicity?  Or Stockhausen's Klavierstuecke?  But I guess to achieve what I am talking about compositionaly a super structure is necessary that generates the content or a multiplicity.  I was listening to Schoenberg's "Ode To Napoleon" last night and the piece begins with an explosion of sound, yet I'm sure the beginning is just a thematic element that will be developed.  A note on Penderecki's "Threnody" which had a featured article in the last issue of Spectrum - am I correct in making the statement that the piece begins with similar sound events that evolve?  The score and recording link:




    • This discussion, especially Conor's point about music in some sense being necessarily linear, reminds me of Judith Lochhead's essay on "Spiral Morphing" in Anna Clyne's 2004 composition "Choke" (Reconceiving Structure in Linear Music, chapter 8).  On page 165 she has a diagram illustrating her concept of spiral morphing, which includes both "loops" and "spiral events."  Clock times are marked on the diagram, as "Choke" is written for saxophone and tape, and this reinforces the sense that music—and the diagram—is temporal.  However, during the spiral events, the lines in the diagram at times move backwards.  I won't go into the various ways this might be interpreted.  But with respect to Conor's suggestion above about "music that distorts our perception of linear time," I suspect Lochhead would argue that "Choke" falls into this category. 

    • Thank you Rich, I look forward to investigating that composition and essay - hopefully there is a recording on YouTube or elsewhere.  In focusing my thoughts further I think I am referring to composition that operates from a perspective/structure of multiplicity rather than development [within linear time constraint].  

    • As for Beethoven, have you read Johathan Kramer, The Time of Music, NY: G. Schirmer, 1988? You might think a bit differently about B after that.


    • No Richard, I have not read The Time Of Music.  Regarding Beethoven and his treatment of motive as the generating source of entire compositions (in all movements of the 5th Symphony and other works) and Schoenberg's exhaustive discussions and examples of Beethoven's usage of intervalic relationships extracted from germs, motives and themes as the central developmental/unifying esthetic in his work (Arnold Schoenberg, Fundamentals of Musical Composition, NY: St. Martin's Press, 1967), Schoenberg's theory of Beethoven's compositional method seems verified and cogent.  Would you care to share your thoughts in abbreviated form (or at more length) on the matter since you are familiar with the book?  In the Beethoven I have analyzed and studied there is a concentration of intervalic relationships played out in various dimensions (melodic, harmonic, rhythmic) architecture and this seems to be the concensus of Beethoven education, no?   

    • I always think of a musical performance as a story telling experience. Story telling goes back millennia and music is a form of story telling that uses abstract sounds. There is nothing real about music, but, after you have listened to enough of it, it seems to have its own logic in the hands of a master composer. So, if it is really story telling then it must be linear because the protagonist does not change willy-nilly at the whim of the composer. In film making, another story telling process, there is always a continuity editor who makes sure the story line does not get lost or wander off in all directions. So it is with the composer. The progress of the piece through time must have a continuity that the trained listener can follow. All the great masterpieces of the Western canon seem to have an integral logic that is unique to each piece. Music is a sound adventure that travels from point A to point B and arrives at the conclusion, safe and sound, at just the right time. There is something very nice about coming home after a long and dangerous trip, and sleeping in the same bed you occupied before the trip began–the recapitulation–putting your head on the same pillow, only you have changed as a result of your experience.
    • Thanks Steohen for your comments. Music being subjective for each listener is also an art is also highly organized and structured with elements of science, logic, and mathematucs.  However I respectfully somewhat disagree about development being structured in a "linear" manner.  Your point about music "telling a convincing story I agree with, but I think new areas of physics and mathematics point to a more "dimensional" and even multidimensional structural design for music and I am currently working on an example I can demonstrate.  Two examples of one dimensional surfaces are the matrix and Tonnetz diagrams.  I think the Tonnetz results would be more interesting if "the grid" was NOT super symmetrical, geodesic for example.  But I don't want to confuse that with the linear listening framework which is a universal.  

    • Has it become incongruous to speak of Schenker on SMT (;–))?

      Whatever one may think of Schenker, he certainly was "the" theorist to have considered music from a non linear point of view. His theory is fully organicist, he viewed all aspects of music –and of each musical work– as growing from a unique source as a plant grows from its seed.

      In §301 of Free Composition, Schenker publishes quotations from composers who allegedly had a "simultaneously global" (to quote Carson Farley) view of their work:

      C. P. E. Bach: "One must have a vision of the whole piece."


      Mozart [...]: "...and the work is really almost completed in my mind, even if it is very long, so that afterward I can see it in my mind's eye with a single glance, like a beautiful picture or a lovely person. And not at all in succession, as it must come later, do I hear it in my imagination, but somehow all at once. [...]"

      Beethoven: "Also in my instrumental music I always have the whole before my eyes."

      The quotations themselves may be doubtful, but they reveal Schenker's interest for a non linear conception of music. This is particularly evident in his theory of form, as laconic as it may be. (See Schenker's Formenlehre, A. Cecchi ed., Rivista di Analisi e Teoria Musicale XXI/2, 2015.)



    • That's exactly what I am trying to avoid as a contemporary composer Nicolas!  Music growing from a seed! - which is what linear means to me!  Please humor my satire.  You have inspired me to keep an open mind about Schenker and I very much appreciate and respect his work and even like his compositions.  Because I consider myself more of a composer, although I tend to think about musical theory much of the time, my reasons for distance from Schenkarian analysis is more selfish than anything else . . . I came to realize that since I am filled with my own ideas and creativity I cannot dedicate or do justice to studying other people's work and philosophy the way a professional dedicated theorist is required to.  But I do say that after studying music my entire life.  I've studied other people's ideas enough (and continue to because I cannot avoid it).  I need to produce my own work based on what I know with my own vision.  I understand the essence of Schenkarian analysis  . . . Gustav Holst remarked after leaving the Royal Academy of Music sooner rather than later that "he wanted to write music, not study it.". He certainly made his point.  I think this is a common predicament for composers, who have to look into their own abyss and turn off the exterior stimulus in order to create.  

      A non linear composition to me (and pardon if my definition is incorrect) is one in which any individual part is simultaneously structurally part of the whole and generated from that perspective.  The Solar system is a good metaphor: planets are in orbit around a star at different velocities and distances/ellipticals.  Yet the whole system works as a whole governed by gravity.  Otherwise random in motion relative to position, at times planets will align or converge mathematically at specific points only to repeat any reoccurances after the entire cycle is completed - like the isorhytmic motets of the Ars Nova period.  Those covergences or "collisions" as I call them generate change in some manner.  This is a different way of thinking about sound for me.  And I have written the composition already called "Orbits."  Harmonic structure, rhythm, amplitude, velocity, change, modulation are all controlled by the system relationships, not the individual parts.  I'm currently working on version 2 in order to refine the rather complex procedure.  I actually think it's a good composition and I'm my own worst critic!

    • Carson -- You're a very inquisitive person, which is terrific.  But if you're going to ask all these questions on SMT Discuss, I suggest you work on resolving your false composer vs. theorist dichotomy!  Nicolas's comment is brilliant.  Unfortunately you have no idea what he's talking about, and go on to say that you basically don't care, since you're mostly a composer.

      Stop commenting for a minute and go read Chapter 1 of Free Composition.  Here's a teaser from the first page: "Origin, development, and present I call background, middleground, and foreground; their union expresses the oneness of an individual, self-contained life."  (Background, middleground, and foreground refer to different levels of structure, in case you're not aware.)  Definitely non-linear! 

      Thank you Nicolas for a great post on this topic!

    • Rich, thanks for your comments.  If you are a real composer you don't spend your time listening to other music because you have music in your own head that demands your attention.  The same goes for other forms of information.  When is enough information enough? When I know as much as you know? Maybe you don't know what I'm talking about.  I don't need to go running to Schenker for all of my definitions for music - I'm a creative person. I understand Schenker, I studied Schenker at the UW (University of Washington). I prefer Schoenberg's perspective and philosophy on compostion and theory - you know he did write many books on the subject that are available for you to read. To say that I don't care is untrue.  I have studied music my entire life and continue to study it in detail - cello studies, jazz studies, compostion studies, theory studies, history studies, ethnomusicology studies AND I am a professional musician.  I care.  I ask questions all the time, that's how I learn.  

    • Carson, you mentioned the matrix and Tonnetz as being planar.  Would it be possible for you to develop a 3D Tonnetz in say SketchUp, as perhaps a step toward what you are seeking?  Am I correct in understanding that these musical devices establish a whole already in existence, which are then explored through the composition?  I think 3D modeling could help create superstructures that don't have the potential to form linear constraints.

    • Carson, when you write

      Music growing from a seed! - which is what linear means to me!

      I am afraid that we don't understand each other – and that you don't understand Schenker correctly. The "seed" that would produce linear music seems much closer to Schoenberg's somewhat cryptic idea of the Grundgestalt than to Schenker's organicism. Organic growth does not mean that a "seed" is repeated up to the point when a whole work develops linearly through time, but that music grows by developing branches, twigs and shouts, all from the same origin.

      Schenker did see repetition as one important aspect of a musical logic, but he refused to see repetition as the essence of the logic. He also refused any idea of musical narration (this is an answer to Stephen Jablonsky) because he considered that narration belongs to the logic of ordinary language, (unnecessarily) imitated in music only when it cannot develop its own logic. (He said this mainly in Der Geist der musikalischen Technik, 1895.)

      Schenker refused to see musical form as a temporal concatenation of parts. His theory of form is puzzling because the forms that he describes are the same as those of any traditional theory. But he decribes their origin and growth differently, not as a temporal development, but as a development in the depth of the levels. He wrote:

      With all of this the cohesiveness of the total content of a piece is provided and established as a unity between the depths of the background and the breadth of the foreground. [Das Meisterwerk in der Musik III, Ian Bent's translation, p. 8.

      which I commented, in the Schenker's Formenlehre book that I already quoted, as follows:

      The “unity of the background-depth and of the foreground-breadth” is an essential concept in Schenker’s Formenlehre. The “foreground-breadth” is that in which the various parts of the form follow each other, forming the linear, temporal development which we usually associate with the very idea of form. The “background-depth”, on the other hand, denotes the link that even the most individualized forms maintain with the primal unity of the Ursatz. It is this depth that determines the function of the parts at the surface level and their hierarchy.

      I fully understand that you, as a composer, cannot spend much of your time in such abstractions. Be convinced, however, that there is way too much in Schenker that cannot easily be grasped and that requires a lot of study. This is why you, as a composer, and I, as a theorist, could complement each other.



    • I think the following illustration will help us to get our arms around the subject that Carson has presented.

      Break up the song "Row Row Row Your Boat" into 4 components:

        [a] row row row your boat

        [b] gently down the stream

        [c] merrily merrily merrily merrily

        [d] life is but a dream

      One could recompose this song as an endless random sequence of these 4 components, e.g.

         c a b b d a c d b b c a d ...

      This would be a reasonable piece of music.

      It does not have any "long line linearity" (it's not moving toward a goal or endpoint or anything like that)

      However, the recomposed-song does exhibit a degree of linearity (I would prefer term "connectivity") between each two components. Some component-pairs connect a little more naturally than other.

      So I think my recomposed-tune is an example of what Carson is looking for, about as much as this is possible.

      To summarize in slightly technical terms:

         I think that all music unavoidably exhibits at least a small degree of "linearity" or "connectivity" between  adjacent components. This is because of the role of time, and how human beings naturally experience music. So in that sense, there is probably no piece of music that is perfectly non-linear.

         However, there are plenty of example like my recomposed-song etc. where linearity across the entire item of music is minimized; and where the feeling of connectivity between adjacent components is minimized.


      Isaac Malitz, Ph.D.




    • Thank you for your insight Nicolas.  As noted before, I have an open mind about Schenker.  I have even gone out of my way to listen to his music.  I did take a 400 level course in Schenkarian Analysis (of my own choice and not required in my program) because I am very interested in different perspectives on music theory.  Here's an interesting quote from theorist John Rahn from his book BASIC ATONAL THEORY - 


      To analyze music is to find a good way to hear it and communicate that way of hearing it to other people.  Probably there is no single best way of hearing any given single piece, but some ways are generally recognizable as being better than others.  Each person makes his own decision as to the degree and quality in insight communicated to him by a particular analysis of a particular piece.  If these individual judgments did not tend to converge into clumps, music theory would be even more anarchical than it is.

      As essential ingredient for this clumping process is willingness to pursue actively someone else's truly alien insight.  Without such willingness no music analytical conversation (however heated it may become) can occur, no mutual accommodation, no synthesis of two formally separate and alien viewpoints; no learning.

      - BASIC TONAL THEORY, John Rahn, Schirmer Books, NY, 1980.

      For the sake of civility and after a productive interchange of perspectives from SMT members I admire and respect for their intellectual and academic heft, I will end my comments because I am satisfied with what I have learned.  And I hope that my "alien insight" was willingly and actively pursued in the same spirit.  

      The original intent of my post emanates from the technique of the First Viennese School AND Schoenberg's theories . . . and stimulated thought about that process in tandem and as a result of interest in Buckminster Fuller's book and ideas on synergetic geometry and if there are other ways of conceptualizing "development" in musical structure, henceforth the use of said terms linear and non linear.  

      Thank you all for sharing your ideas, information, and perspectives.