Serial Structures as Fragmentation of the Familiar


            In this part of the Essay, I begin with a technical discussion of some aspects of the construction of the piece, Sand.  I then move on to a discussion of important motivations for constructing some of my pieces in this way.


II.i.  19 Tone Equal Temperament


I start at the level of tuning.  This piece is in 19-tone equal temperament.[1]  For me, the primary reason for this is exploratory:  interesting microtonal systems and tunings are an aspect of music that, without the computer, have been off-limits for most composers until recently.  Those who have ventured into this domain prior to the last decade or so, have been intrepid figures indeed,  and have had to devote much of their time and energy to such activities as building instruments,  developing and testing theories of intonation, and so on.  But this has changed quite a bit in the recent past.  The computer has made it possible to explore and compose for virtually any tuning or microtonal system that could be conceived:  from Just Intonation, to spectral techniques, to the exploration of alternative equal-temperaments, to un-equal temperaments (such as the predecessors to 12-tet including various forms of meantone, Werckmeister, tunings used in various world-musics, etc.),  or even sets of randomly generated scales.[2]

I decided that since I was working on a piece of  computer music,  free of the intonational limitations of commonly available acoustic instruments,  I might as well take advantage of the medium and write in a tuning not available on those instruments.

            It turns out that 19tet worked out well for me,  since I was planning a work emphasizing the the [0 1 3]12   trichord.[3]  This has always been a favorite sound of mine, the minor third is one of its intervals,  and 19tet has nearly perfectly Just-Tuned minor 3rds.[4]   My relationship to 19tet, at the point when I decided to use it, was virtually nil.[5]  In an attempt to be sort of honest about this, I took sounds that I liked from 12tet, and thought about warping them in various ways into 19tet.  Thus,  for example,  [0 1 3]12 can be interpreted, so to speak, in 4 different ways in 19tet:[6]

("≈" means (here) "sounds approximately like.")

[0 1 3]12          [0 2 5]19 or  [0 1 5]19 or [0 1 4]19 or  [0 1 3]19

            Some possible instantiations in pitch notation :

[C Db Eb]12   [C Db Eb]19  or [C C# Eb]19 or [C C# D#]19 or [C C# D]19

            The above are listed in order of decreasing similarity; [0 1 3]19  I find too harsh to use as a quasi-[0 1 3]12, most of the time.

            A chord which is perhaps the main harmonic unit of this piece, and which cotains this set,  also undergoes this "interpretive" process:

[0 2 4 7 8 10]12 ≈ [0 3 6 11 13 16]19 or  [0 3 6 11 12 16]19

                            or  [0 3 6 11 12 15]19

            In pitch notation, possibly:

[C D E G Ab Bb]12   [C D E G Ab Bb]19  or  [C D E G G# Bb]19 

             or  [C D E G G# A#]19


II.ii.  Serial Structure


I built a 19-pitch-class row saturated with the [0 1 3]12 translations:



            Out of that I built an array or compositional design.[7] 

            The following is the basic building-block of the array;  6 lines by 6 agreggates. If the row given in Example 1 is considered the "prime" form, then this building-block contains two instances of P, two instances of I, and one instance each of R and RI:



            To form a larger structure I did the following:  instantiate 19 different transpositions of this basic block (in other words, all the possible transposition levels of it).[8]  By concatenating these 19 blocks horizontally, according to a certain set of rules[9], and swapping pitches between blocks,[10] I can obtain an array that contains at least one instance of each of the 19 transpositions of each of the 4 canonic row-forms (P, I, RI, R); and where the partitioning scheme for each 19-tone aggregate is unique.[11]  That produces a 114-agreggate structure.

            For the pitch-structure of the piece, however, I use approximately half of that structure.  See near the bottom of this page for this 60-aggregate array structure.

            Now imagine that structure in canon with itself:



            The pitch structure of the piece Sand consists of  this, but with the  "dux" structure realized on the music's surface as unspatialized notes (i.e. un-reverberated, or "in your face"),  and the "comes" structure as "spatialized" notes (reverberated.)   Registrally, they are intertwined, as shown in this example:



            Again, see the bottom of this page for this complete contrapuntal pitch structure.

            The intertwining of unspatialized (the dux array) and spatialized (the comes array)  lines in each register allows for a wide variety of melodic, textural and harmonic interpretations on the musical surface.  What's most interesting for me as a composer about these intertwined lines, is to forget about what those lines mean and where they came from individually:  I spin music out of the composite, and then listen to how the fact that some of the notes are spatialized and some aren't "skews" the music for my ear, perhaps in some bizarre way.  Using the interface described in Part I, the user has the choice of perspective:  they may listen to the composite line, or break it apart into its two constituent components[12], thus enabling the perception of new and different lines, or hearing the composite line in a heretofore unnoted, dissected way.


II.iii.  Drones and Virtual Drones


            In Sand, a drone is usually present underneath (or in the midst of) the contrapuntal web of music, with which the pitches in the counterpoint  associate and form what I feel are real, perceptible, pitch relationships.  This was inspired largely by Indian raga music, where certain notes'  relationship to the drone becomes a matter of almost ecstatic importance.  An example of drones in a serial context is Stockhausen's  Licht cycle of "operas."  In those works, a drone might be a note, a dyad,  a complex chord, or even a texture.[13]

            Although throughout Sand a drone is almost always present as a way for the listener to form pitch-relations in their ear/minds, (whether or not I was thinking consciously of these relations as I composed),  in certain sections of the work, I composed  the music quite intentionally so that the counterpoint would seem to be intensively "feeling its way around" the drone.  Section 3 of the work is one example of this.

            I enriched the pitch structure even further by the use of what I call "virtual drone notes" that infiltrate the fundamental contrapuntal structures detailed in the preceding sections of this chapter.

            I began using virtual drones in two chamber music works I wrote,  where I was experimenting with working aspects of spectral music into my own musical language.  Simply put, the pitches of a certain fixed spectral chord(s) tend to "hang around," and "be available for use anytime" during a given section of a piece, while a serial counterpoint is being unfolded. 

            Here is an example from my piano trio Quiet Play of Lights.  The basic (12tet) contrapuntal array for this fragment is shown here:


violin--short  notes


A  D

violin--long   notes


F#   C#   E   G   Eb  C    F   Bb  Ab

piano --short   notes

E  G#         E   G# B


piano--long   notes



'cello--short   notes

Eb C          D# C# F#


'cello--long  notes




            The spectral chord[14] whose members are available for use is as follows:



Here is what happens in the referred passage of the Trio:



            "Virtual drone" seems an appropriate title, since the spectrum (the chord of pitches) is not actually droning, but making its presence felt constantly nonetheless.

            In Sand, the virtual drones are not usually complex spectra[15] as in the Trio,  but simply overtones of the current actual-drone pitch.  I usually stick to the first few partials--those are the notes available at any time to "infiltrate the counterpoint."

When I intertwine all of these possibilities: spatialized pitches, unspatialized pitches,  virtual drones and adjacent registers, I can create a quite flowing melodic product.   Some examples in Sand include the second half of Section 13, the middle of Section 23, the opening of Section 35, or the "squished" homage to Schoenberg at Section 62.


II.iv.  Aurally Sensitive Pitch-Class Set Composition


            Let us return to that chord I mentioned earlier, and talk a bit about harmony: 


[0 2 4 7 8 10]12          [0 3 6 11 13 16]19 or  [0 3 6 11 12 16]19

                                    or  [0 3 6 11 12 15]19


            This set, and its subsets thereof, are the "default" array parser units in Sand.

            Here is one agreggate-partition of the array:



            Here it is, along with some "virtual drone" pitches,  parsed by [0 3 6 11 12 16]19:




            A problem I have had with set-theoretic ways of thinking and working in the way they have been, for the most part, taught to me, is that all sets are treated as equal in terms of their aural recognizability them by ear.  This is not corroborated by my own perceptions.  For example, many of us can recognize an octatonic sound, a diatonic sound, or a hexatonic sound in a piece or passage almost instantaneously;[17] on the other hand, To hear the [0 1 2 5 7 t]12-ish-ness of a passage instantaneously and intuitively is quite another matter.   Other considerations are often left out of discussion as well, such as:  what's in the bass,  how chords are spaced,  the sound of a set as used melodically as opposed to harmonically, etc.

            In my piece, I set some rules for myself, vis a vis using this 6-note sonority.  First, set-theoretically-speaking,  it only occurs in "prime form", never inversion.  Second, I try to have it appear in "root position" as much as possible.  I see it, in short, as an entity that is an approximation of a set of overtones, and it will be most audible as such when it appears un-"inverted,"  in all (including the traditional vertically rotational) senses of the word.

The [0 3 6 11 12 15]19, or [C D E G G# A#]19  version of this chord, is the closest in sound (in 19tet) to a Justly-Tuned chord of overtones  8:9:10:12:13:14.  This chord appears most obviously, explicitly Justly Tuned, in the first "drone section" of the piece (Section 4), (the "drone sections"  are occasional sections of the piece devoted entirely to "solos" of the drone layer).

            The following is a sketch of Section 5-6 of the work, showing  [0 3 6 11 12 15]19 parsings:



            Listening to these sections,  the "root-position" versions of the chord come out quite clearly.


II.v.   Fragmentation of the Familiar


In one strand of my composing, which I engaged in this piece,  I enjoy using serial methods---even total serial methods.   It's not a desire for unity or coherence that impels me to work in this way.  In fact, it's often quite the opposite--the desire to put myself in a situation where I can contradict any unifying force that may be inherent in these structures.    I am attracted to possibilities of contradictions.

            Steve Reich often[18] tells this story:  Berio was teaching him composition, and he (Reich) brought in his obligatory 12-tone piece, which was, none the less, quasi-tonal, composed out with triads, etc., and Berio said to him  "If you want to write a tonal piece,  then why don't you just write a tonal piece?"  To Berio,  (and, I suppose, to Reich),  I pose the question, "But why not write a serial piece that is 'trying to be tonal'?"

            Let us open this into a broader field;  we replace the word tonal by the concept of the familiar:  aspects, elements, or hallmarks, surface or structural, of a familiar musical language(s).  We replace the concept of "trying to be tonal"  by the concept of a music where fragments of the familiar are torn out of context and juxtaposed, glued, weaved, and sewn together often in interesting, emotionally and viscerally jarring ways, that might shed new light on the character of these fragments.

On an internet discussion group I take part in, George Secour wrote:

"Trouble is, even if one chooses to ignore the principles of musical acoustics and mathematical relationships, they are  still there.  So the pantonal [or atonal] composer is something like an architect  designing a building, but ignoring the laws of physics."[19]

I would like to quibble with the word "trouble".

Let us temporarily posit two points of view about composition, aspects of which I've seen rearing their heads now and then in the various musical communities I run around in.  The two could be crudely represented with some stereotypical quotations.  For mnemonic clarity, let us identify the two speakers as PAT and NIKY[20]:


PAT:                "Art is based on [laws of] nature." "Art should emulate nature." "art should be made following your intuition."  "The theory of ______ models intuitive instincts of a musically sensitive individual,  therefore you're wasting your time if you don't follow its dictates."


NIKY:             "All art is arbitrary."  "All art is basically artificial." "Many sounds we find sonically and musically interesting are artificial, inharmonic, stochastic, etc." "The mind can learn any algorithm if you pound it in long enough." "Art is mostly based on artificial cultural inheritances, environmentally begotten,  which are not necessarily ingrained in anyone's mind permanently.  The mind is in many (most) ways a blank slate.  If all musical systems are basically artificial, why not invent or expand on more of the same,  and get on with composing?"


Rather than being hopelessly oppositional,   I like to think of these two schools of thought as intimately bound together in a fascinating way.

The first possibility is that NIKY will conceptualize or structure his/her compositions in ways which seem to have little relation to what the ear (even the composer's ear) can realistically perceive.  But NIKY's arbitrary compositional systems leave room for the operation of intuition,  so that when these materials are deployed on the surface of a composition, to make a piece of music happen,  those natural instincts touted by PAT come in to play, desperately  attempting to deploy NIKY's recalcitrant, cognitively/acoustically disobedient materials in some way that, if only provisionally, locally, in different ways at different instants, satisfies  the desires  of those  instincts.

The result is a music where--again--fragments of the familiar---broken-off chunks of familiar progressions, forgotten harmonic entities, tonal chords, simple voice-leading progressions and so on---are juxtaposed in twisted, novel fashion, warping and changing our perspective on those familiar objects in unexpected ways.

Another possibility is the case of "hard-core algorithmic" music, where some kind of process or generating system is set up, seeded, and the piece, as it were, writes itself. . . . . in this case, it is in our ears, the ears of the listener, where this recognition of the familiar takes place.

In either case, the way we parse through such pieces, looking for some vague cognitive signposts--gradually expanding registers, step-wise bass lines, serendipitous fragments of tonality bridged by walls of impenetrable chaos--this ad hoc parsing is fascinating for me--to experience as a listener, and to try to facilitate in my music, as a composer.

In a way, these kinds of pieces are written in a higher-level language, in the computer-science sense of the term,  because these little bursts of the familiar, these fragile  handholds that we grab onto to "survive" as listeners, are often like signifiers, standing for whole passages of music in more traditional musical environments.

Late in Babbitt's 2nd Piano Concerto, (mm. 507-510)  a swirling chaos of notes seems somehow to magically coalesce into C# Major,  so strongly that the passage seems to me to be the "climax"  (a very little hill in this very flat piece) of the work.

At another point in this piano concerto, (mm. 203-204)  there is a series of what sound like  chromatically descending clusters, with the pitches distributed around the ensemble, in the midst of other activity--this chromatic descent, as something vaguely familiar or simple to me, is what I "grab onto" in this part of the piece.[21]  I enjoy the feeling, while listening, of not knowing quite what to do or what to grab onto, mentally and emotionally, in what I'm hearing.

These experiences were probably not intended by the composer (though Babbitt's music in particular is filled with triadic arpeggios and other tonal puns, which are so numerous and ubiquitous that intention seems inevitable), but they have been very suggestive to me as a composer.

This is why I continue to write [some of my] music serially  or with other "arbitrary" or "unnatural" systems or materials.  Rather than trying to avoid octaves and tonal triads and subconsciously internalized voice-leading rules-of-thumb, I do want to use those elements, re-contextualized and juxtaposed in odd, bizarre, and unexpected ways that shed new light on the objects themselves.

As a composer, then, I'm interested in systematic composition, like serialism, but I'm not so interested in serialism as a language in and of itself, but rather in using it as a fragmenter of the familiar.  Likewise, to the person who asks, "Well, if you want to use triads, why don't you just write tonal music?"---again, the point is that I want exactly to "refer" to tonal music from something else.

Perhaps it's more complex than that:  I think the best serial pieces from the past were written with this state of mind going on, probably subconsciously---as it was customarily dogmatically insisted that serialism was its own language---but sub-consciously, the reason that the best of those pieces work well,  is that the composers were thinking this way--i.e., desperately trying to "fake" a familiar, "cognitively correct" syntax in the midst of this other,  artificial syntax that they were trying to be true to.  And that is the state of mind, as a composer, that I want to be in as I write certain of my pieces:  I'm literally interested in "pretending" that I believe in the purity and goodness of artificial systems in and of themselves, but behind my own  back, I'm sneaking in their usage as fragmenters of the familiar.

What I am talking about is different from, say, the system recently elucidated in a Perspectives of New Music article by Ciro Scotto,[22] that attempts more of an official marriage between tonal syntax and pitch-class-set composition.  Rather than a marriage, that forms an official system that successfully integrates the artificial with the natural (i.e. tonal in this case) compositional languages,  I prefer to throw the two together in their pure  states---and, metaphorically speaking, watch them fight, make love, decorate the house, milk the cows,  and maybe enjoy an occasional ice cream cone together.

For a specific example, In Scotto's system, there are specific ways of elaborating the basic underlying structures.  To write a neghbor-note melodic configuration, there are certain notes one selects to use as neighbors in a particular case--extrapolated from the underlying structures themselves.  These may or may not be "neighbors" in the traditional sense (i.e.  200 or fewer cents away from the pitch being elaborated.)  In my way of working, on the surface of the music, there are what seem to be neighbors, arpeggiations, and any number of other familiar formations--but these formations are not re-defined as in Scotto's system, but are re-contextualized.

            There are many  examples of fragmentation of the familiar in serial music of the past.  Some have been discussed extensively.  For example, the fact that Stravinsky's characteristic "sound" (which, in the area of harmony, relates to such familiar objects as the diatonic and octatonic scales, poly-tonal harmonies, and so forth) was retained through several changes of musical language is widely discussed throughout the literature.  For an example we turn to the "Bransle Gay" of Agon, in which the following curious chord appears:



            This beautifully misterioso sonority could be heard as A-mixolydian, i.e. diatonic,  but, perhaps because of a prominent C-Bb motive that occurs repeatedly, earlier, and later, in the bassoon (i.e. the "bass line") in this short movement, one might hear it as an octatonic subset.



            Abraham and Isaac is a most uncompromisingly "atonal" work, making its way very flatly and ascetically through a set of winding dissonant lines, voicing a sacred text.  Suddenly, right at the moment where God, in the nick of time, prevents Abraham from sacrificing his son (m. 173),  an electrifying set of chords knocks one out of one's seat:



            The "electricity" of this moment is due partly to the fact that these are chords;  loud, medium-high register, secco chords, in the strings, a textural formation rare up to this point in the work;  but it is no doubt contributed to by the fact that these chords are tonally reminiscent as well, as indicated in the example.

            In Schoenberg's String Trio, we find a more subtle example, where, to my ear, the voice-leading seems to very much suggest a tonal V-I cadence, but it's realized in an incomplete,  fractured way, with many of the key leading tones left unresolved.



            In Sand, there are places where the fragmentation of the familiar undoubtably happens unconsciously, or accidentally.  In some places it was more or less intentional.  Here are two examples:

            The first is an explicit echo of a triadic progression.  Sometimes I scan through an array partition, looking for some pattern of notes that strikes my eye as something familiar that might be made to "pop out" of the texture.  So, in the following array segment:



            I noticed several triadic entities peeping out of the structure:



            We can hear this in Section 12 of the piece.

Another familiar signifier I found on several occasions was the idea of a chain of suspensions.  In Section 25 these manifest themselves as 2-3 suspensions (2-3 in the traditional sense of 2nd-3rd.)   In another passage, (in Section 26)  I find much "narrower" suspensions:  minor 2nds (or augmented unisoni, the two being different intervals in 19tet)  "resolving" to major 2nds, and so on.

Indian shenai-like (nasal-timbred) melodic passage-work briefly rears its head in Sections 20 and 23.

These phenomena are usually worked into the texture at hand---meaning that they are somehow related to the default parser-unit---[02478t]12  in its various 19-tet interpretations, and their subsets.

            Again, "the familiar" is certainly not always "tonal music",  or triads, but also phenomenon like  "parallel motion," or "suspensions", or "clusters"  which need not be in a tonal context of any sort to retain their significative power.  In all of the examples I've pointed out here, it has been fairly clear what, exactly, was the familiar phenomenon being evoked, but there are many instances where a passage may  invoke a  familiar sensation, but this reaction may be quite complex and vague. . .we may not be able to figure out how it's "familiar" to us. 

I should also point out that context is very important--more important than I have implied.  Some of the examples I've presented here have been isolated from their contexts, and as such, they lose a lot of expressive power.  A V-I cadence, especially a fragmented, impoverished one, is interesting in itself, but in the context of the ending of the String Trio, it's an even more expressive gesture; the context being both the atonal, harmonically ambiguous formations that lead into and out of the excerpted passage; and the 12-tone structure, the motivic structure---all of the aspects of the music that we ordinarily look into.  I don't want to discount these latter, but simply to assert that the things I have been discussing here are an essential part of the musical experience, especially with this kind of music, and therefore I choose to treat them with respect.

One might argue that as a de-contextualizer of the familiar, random notes, or other algorithmic schemes, could also work just as well as serial ones.  I have, in fact, tried a variety of ways of disciplining myself for these purposes, but I  find that I prefer serial methods. There are doubtless a number of reasons for this that I'm not even aware of,  but something I can articulate is that I like the way that serial structures themselves can, on occasion, "pop out" quite obviously onto the music's surface, in spite of themselves.  They can, in short, "act at various distances from the surface of a piece."[23] This adds another layer of richness to a work.


II.vi.   conclusion


I like to feel that I am working with these methods to get out of them something that they are not--or something they would, at first glance, seem to be unlikely to produce.  The very idea that Berio laughingly dismissed is the cornerstone of my composition aesthetic here.


[1] For now on, equal temperaments will be abbreviated as "tet", e.g. 19tet = 19-tone equal temperament.


[2] See, for example, Marcus Bittencourt's KA, a radio-play, whose pitch structures are based on a number of non-octave "modes" selected randomly out of equal-division-of-non-octave-interval scales.

[3] From here on,  a set-class designation with a proceeding subscript will indicate the given set-class in the subscript's et.  Hence [0 1 3]12  indicates "[0 1 3] in 12tet."  [0 1 3]17  would indicate "[0 1 3] in 17tet."


[4] The .02 cent error in the size of the 19tet minor third, relative to Just, is the minimum size of error for any Just interval approximated by any et less than 50tet.


[5] Some aural experimentation quickly reveals a few things about 19tet.  First of all,  a diatonic scale can be extracted, which sounds quite like that in 12tet.  Folks inexperienced with equal-temperaments (as I was), often are somewhat amazed that relatively familiar-sounding (i.e. diatonic) scales can be extracted from most ET's, though they will be "mistuned" to various degrees.  The ear, in an effort to hear the familiar in everything, will try to fit what it hears into a familiar template.  Thus 5tet or 7tet will sound like slightly-out-of-tune pentatonic or major scales--though the step-sizes are obviously all equal in these cases.  For more on ET's and their relations to diatonic scales, see Easley Blackwood, "Modes and Chord Progressions in Equal tunings," Perspectives of New Music 29, no. 2,  (1991): 166.


[6] An explanation of the notation for 19tet appears here.


[7] The reader unfamiliar with arrays and compositional designs is urged to consult Joseph Dubiel, "Three Essays on Milton Babbitt": "Part One:  Introduction, 'Thick Array / of Depth Immeasurable,' " Perspectives of New Music 28, no.2;  or Robert D. Morris, Composition with Pitch-Classes,  (New Haven and London: Yale University Press, 1987),  esp. chapters 5-6.

[8] Andrew Mead refers to a portion of material containing all transpositions of a row as a "hyperagreggate."  See Andrew Mead, An Introduction to the Music of Milton Babbitt, (Princeton: Princeton University Press, 1994), Chapter 1 and elsewhere.


[9] Specifically, the next block chosen should always be T5 of the previous block.  The new block's lines are re-arranged according to a certain pattern.  These two factors allow for a maximum of swapping possibilities.


[10] See Morris, Chapter 5, for more about "swapping" to create varied partitional shapes.


[11] This will not use up all of the possible partitionings of 19 tones into 6 parts, so it is not an "all-partition" array, but it is an "all-different-partition" array.

[12] A surface melodic line or other heard formation may also play across boundaries between registral lines, involving 4 (or more) constituent lines of counterpoint.


[13] For example, the "Invisible Choirs" from Donnerstag aus Licht, is a drone (continuous background texture) that may also be presented as a separate piece in its own right.

[14] analyzed from a refrigerator drone.

[15] An exception to this is the "Armageddon chord," (my own colorful programmatic label), which makes itself heard now and then throughout the piece; for example, in the background of sections 7-12.


[16] This is further elaborated into Section 9 of the piece.


[17] All of these have a high degree of symmetry, which may in part account for their  increased recognizability.


[18] I've heard this at 3 talks that I've seen him give.


[19] George Secour, e-mail post #38218, to the Tuning e-mail list group,  available at  http://groups.yahoo.com/tuning, posted 26 June 2002.


[20] These are exaggerated, extreme versions of various more nuanced views that I've seen posed in e-mails, journal articles, books and so forth.  I exaggerate them in order to make my point here as clear as possible.

[21] Originally, I heard this moment without knowing its scored representation.  In fact, the process of finding this moment in the score was quite difficult--it doesn't "look like" what it sounds like at all, due to the fact that the descent's pitches are distributed between horns, piano, marimba and clarinets, all of which look, on the score, as if they are doing their own thing.  There are no visual clues that the instruments might coalesce in this way.  This is an example of why, as I stated in Chapter 1 of the essay, I actually find it inhibitive to examine a score before I've heard a work many times.


[22] Ciro Scotto,  "A Hybrid Compositional System: Pitch-Class Composition with Tonal Syntax,"  Perspectives of New Music  38,  no. 1, 169.

[23] Milton Babbitt, Words About Music,  (Madison: University of Wisconsin Press, 1987), p. 27