A Last Frontier (Displacing notes that aren't there) image

A Last Frontier (Displacing notes that aren't there)

This is the first of two examples. The contention of this pitch Quixote is that symmetrical structures extend themselves in our brains without being presented in actuality. In this example, the contention is that the whole tone scale completes itself in our brain. We hear the G# that …

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A Last Frontier (Displacing notes that aren't there)


This is the first of two examples.

The contention of this pitch Quixote is that symmetrical structures extend themselves in our brains without being presented in actuality. In this example, the contention is that the whole tone scale completes itself in our brain. We hear the G# that completes the whole tone scale. And then the low G natural in the guitar displaces a note that we hear only in our heads.

What to make of this? Whatever you like. My take--this is a last frontier. I'm bored with only hearing things that are THERE. I'm happy in this realm where we're counting angels on the head of a pin.And it makes for interesting tunes, regardless of who wins the argument.

Another example in the same piece: look for measure 6. The contention here is that our ears are soooo diatonic that major thirds always imply the intervening note, symmetrically in between. So here, between the F# and the Bb our brain supplies the G#.

In both examples it is G# that's not there. In fact, there are no G#s in this piece.

There's more to the last example. The complement to the prevailing D, E, F#, G, A, B is heard in measure 6, although it's mixed up with D's and other suspensions from the other hexachord.. The complement is Ab, Bb, C, Db, Eb, F (the tritone transposition of the D diatonic hexachord).

So in measure 6 we have all but the Ab, which, according to this contention about major thirds, we supply without it actually being there. Fact is, I felt this measure first as the complement. After playing the piece dozens of times the sense that I was hearing the complement became very clear.

Then I looked at the score and realized that complements don't have to be complete. Is it not the case that augmented 6 chords imply the rest of the complement? In Beethoven this seems clear enough. Even in Fernando Sor. Who needs all the rest of the tritone-transposition???

Oren & I record Genius Loci on July 18! It's an innocent little doodle, with these devilish deals going on under the table.

Another example---->

In this example there's an 8-note scale, with only 7 of the 8 notes. The line stops abruptly before the 8th note, which should be B natural. Instead, we get a Bb, displacing downwards the note that we imagine we heard, but didn't.

The passage has 0235s continually displacing implied 0245s. Before the first notes of the melody in the example below, a full Bb scale is instated, with the leading tone, A natural. The G is displaced down to Gb and the A down to Ab. Through these precedents the downward tendancy comes to be expected. This habitual downward semitone displacement helps to reinforce the sense that Bb displaces the unheard B natural. The fact that this passage comes preceded by a full Bb scale in the instruments makes this harder to argue. It (the passage) argues that a symmetry (the octotonic scale) can have more strength than an instated diatonic region. The passage also argues that the voice part had no Bbs and that each party in the conversation tallies independently. That's quite a contention, and yet there is much to it. I don't dismiss it. I'm on the fence. Nevertheless, I admire this chutzpah, and that's what it is because in this "aggregate", there is no B natural. In fact the actual appearance of that note is delayed for a long, long time. Notice also that the last downward displacement is the completion (Fb), assuming that we really did imagine the B natural. Completions are not heard in a vacuum. They are heard in the context of moves. Here he teaches us to hear 0245 being displaced by 1235s, repeatedly. The last time, the 4 is F and the 3 is Fb. We learn moves that fill holes. Hole filling for the sake of hole filling is not really the point.

Here's the whole passage--->

The Sublimation of 20th Century Grounds image

The Sublimation of 20th Century Grounds

---As Milton Babbitt's 95 birthday approaches, does anyone care about pitch classes any more? ----Yes, and there many excellent composers who are walking the walk. Many of us who cared about Babbitt's work understood that toward the end of the last century any music with the faintest scent of a …

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The Sublimation of 20th Century Grounds

---As Milton Babbitt's 95 birthday approaches, does anyone care about pitch classes any more?

----Yes, and there many excellent composers who are walking the walk.

Many of us who cared about Babbitt's work understood that toward the end of the last century any music with the faintest scent of a row would enjoy a moment of deep, widespread *unfashionability*. It must be known that we (Babbitt fans) shared in the sense of urgency for music to be able to get some distance from Schoenberg-density, and many of us have been doing something about it.

Unfashionabilty doesn't keeping us from admiring the work of composers
who dared to stick to their guns. I'm thinking of brilliant composers like Gunther Schuller,
Richard Wilson, and Brian Fennelly, and many others. I'd also like to mention
Robert Martin & Robert Morris, who are onto things courageous, new and interesting
that is not 12-tone, but might be confused with 12-tone music-- doubly damned!

Many Babbitt followers celebrated the courage of people like Steve Reich, John Adams and Philip Glass for building sonic levers to pry us out and into the future. What is less widely understood at this point is how Babbitt's work showed another way into the 21st Century. I know that he couldn't give a damn about the fashionability of, let's say, "Schoenberg-density", but his way of working in his most recent music got away from that density. I want to try to show how all this went down. I'll rely on some help from friends who are onto these questions--Frank Brickle, whose work amplifies these developments in Babbitt's latest work; Robert Morris, who has come up with an original and powerful sublimation of 20th C. grounds; and Jon Dawe, who injects fractal viruses into tonal superstructures.

I think of the 20th Century regrounding of music in rows, rather than scale degrees. Toward the 21st C. we see some composers who can do without rows *without reverting back to grounding in scale degrees*--Bob Morris calls this an "open space", after Ben Boretz' zine.

I see Babbitt coming to this by arpeggiating through his arrays, and through is counterpoints of arrays.

morphine array

A *faithful* realization of such an array can be less musically
significant than a selective culling of moves suggested by the chart.
Gradually, Babbitt, with Brickle hard on his heels, started to look at the larger
*unordered* collections suggested by their charts. For example--a section treating one
vertical slice followed by a section based on a slice that nicely displaces a
pitch or two in the first slice. The array becomes a broad source for universes of
such moves--more moves in one array than is needed for any one work.

The vertical elements in such arrays can be arpeggiated for any length of time, with minute, Feldmaneque transformations as any one "voice" might proceed to its next note while the others stay put.

Arpeggiation through arrays shifted our focus from ordered to unordered collections,
with enormous consequences.

Pick one of the vertical tetrachords and think of it as a focal collection, much as the tonic triad is a focal collection. The stuff between each recurring focal tetrachord is a chromatic extension (a non-diatonic *prolongation*) of the focal collection. The focal collection continually reappears in rotations as the array progresses from left to right.

Much like Schenker: Think of an imperfect cadence with 3 in the
soprano, followed by a perfect cadence with 1 in the soprano.

Babbitt's music shows a continual distancing from the voice-leading prescriptions implicit in the literal left-to-right progression through an array. According to Brickle, it was in the early '90s that Babbitt ceased to think of arrays as prescriptions for left to right realizations.

Brickle: I Can't forget that it was an offhand remark of Ben's in--what, 1972?--
that struck deep and eventually became the fundamental clue for my own
developments. It had to do entirely with what Milton's music would be like if he
worked with an additional level of middleground between his charts and his scores.

Babbitt's Cavalier Settings for tenor and guitar, and his Soli e Duettini for two guitars still seem to show the left to right realizations of the arrays, but each instrument has its own array. *Counterpoints of arrays* are being heard. The arrival at less literal readings of the arrays certainly had much to do with the way counterpoints of arrays encourage a gutting of the specific voice leading in each aggregate in favor of larger features of the individual arrays that arise in the counterpoint of aggregates. As this process unfolds there is still a critical relationship of the work to the arrays in which they are grounded. The arrays continue to serve as a (non-diatonic) ground, but with less micromanagement. (Let's fight about this!)

Simpler Composition

I now see the ultimate sublimation of all this as follows:

1--Dispense with arrays altogether.
2-- Write a tune that lays down a focal collection.
3--Try to understand the difference between diatonic and non-diatonic prolongations of this collection.

This describes a lot of music that we care about. This nicely outlines the nested harmonic arcs in Harold Meltzer's highly successful Cygnus piece, Brion.

Counterpoint: Instruments converge on the focal collection from different places,
approaching by different intervals. The various routes in the various instruments
might relate by retrograde, inversion, M5, *as if* they are each realizing a different array.

This sublimation is a broad guideline for improvisation. You write the pitch you hear and you honor the 20th Century as much as you want or can. To the degree that one understands or cares, one can preserve the distinction between diatonic and non-diatonic prolongations. And prolongation is a middle-ground thing, describing how to move without moving. I'm not attempting here to describe how one local flavor displaces another over entire sections.

And this is how to listen to Babbitt--you shouldn't care about his charts. His point is not to realize *faithfully* the chart. The chart guides and enriches the improvisation. The improvisation is about listening to things. In the very opening of Swan Song, listen to what's naughty and what's nice.

The opening 6 pitches are quite nice, and most memorable in those
opening 6 pitches, perhaps, are A-C#-G#; which comes back amplified
in a big way. What is naughty in the opening? What crunches?---the 0127s.
The crunchy 0127s crunch and resolve into diatonic collections. It's all so clear.

Looking at things in this way makes me even more grateful for the wonders and riches that arrays or rows can bring to this. The array can give so much--enough and more than enough. Brickle: "The elaborations of the aggregates/arrays go into sketches of actual musical passages -- specific notes and rhythms. It's these *passages* (not chart-like entities or collections) that constitute the middle-ground references for the actual surface of the piece."

What we see Babbitt doing, on the other hand, is awfully hard to understand and harder to explain.

Brickle puts it this way:

"I think he's vividly and acutely aware of all of the information and associations and cross-references at every point in his rows, arrays, and superarrays, and how they are implicit in the work as a whole. At the same time, any individual passage is clearly an improvisation informed by the unfoldings that interest him in his source arrays."

The next issue I had to clarify for myself---what's the difference between this and Haydn? What is the difference between a diatonic prolongation and a non-diatonic (atonal) prolongation? It's only the difference between pitches and pitch classes, but it was nevertheless, a distiction that I fought for years.

I underestimated the difficulty of this question, for honest reasons,
but also due to an overlong period of smug complacency and denial.

I am still struggling with this issue.

Tonal extensions of a focal collection are *diatonic* extensions.

In tonal music the focal collection is usually, but not always the tonic triad. Mertz &
Sor both offer examples where tetrachords can, arguably, be taken as their focal collection
(Sor Study in D [Re-Fa#-Si-La, etc.]; Mertz' Lied Ohne Worte is based on two tetrachords;
Brahms Op. 88--the focal collection is the F major 7 chord.)

[047] is scale degrees 1,3,5, which may be prolonged via movement through scale degrees 2,4, 6,& 7. No need to hit all those scale degrees in this prolongation, but it's hard to avoid. If secondary dominants occur we could see the entire chromatic scale in a tonal passage without ever obscuring the scale-degrees.

Music in the minor keys offer us weird collections that cannot be taken as collections, so powerful are the pitches rooted in the scale degrees.

A favorite move of Fernando Sor: [F,C#,E], where the F is en route to the E.
It is a wonderful non-diatonic collection, but the scale degrees explain it to us so well.
Minor third symmetries and the augmented triad-- our sense of them *as collections*
Is compromised by how well we hear them as mixed modal entities, scale degrees.

The Collection An Sich

as in "ding an sich"-- the thing in itself
The collection an sich is the
collection without its relation to scale
without its habitual grounding in the diatonic scale.

21st Century composers may have a focal collection which may or may not be diatonic. That can't matter. The 21st C. composer will prolong the flavor of her focal collection using notes from the complement at her pleasure. No need to hit all those notes in this prolongation, but it's hard to avoid. We can honor the aggregate as a structural unit or not.

Brickle continues to think in very slowly unfolding aggregates,
but sometimes his aggregates are missing pitches.

Note, if the composer wants to stick with diatonic flavors, said composer can avoid the taste of high 20th C. 12-tone "Schoenberg-density". The extensions of said composer's focal collection can involve endless common-tones to smooth over the hard edges that we find in music of Schoenberg-density. (This is another thing that came out of arpeggiation through arrays.) And, with enough care and understanding, said composer can learn the difference between a (perhaps chromatically inflected) *diatonic* extension and a 20th-Century-inspired 12-tonesque prolongation that knows only the peculiarities of the focal collection and knowingly avoids the diatonic extensions of that collection. And in fact, the game may involves some punning between the two kinds of prolongation.

Robert Morris' Apres Vous, for solo guitar, derives collections that are not founded in scale degrees. They are grounded solely in canonic processes. Locally, the music's canonic unfolding is very clear, and he uses many tricks that he picked up from his work with 12-tone procedures to clarify the canonic elements. For example, the canonic strands can be wonderfully intertwined, yet distinguished by register or dynamic level or timbre (harmonics!). On the large scale, the canonic processes can be made to steer toward collections whose flavors preside over large sections. Here it is the canon an sich. The collections that result are rooted solely in the canons.

Jonathan Dawe composes fractal Baroque operas. His fractal procedures are numerous--rotations, nesting, and cellular automata. He leaves his Baroque model almost intact, particularly the cadence points. He interpolates fractally, with a counterpoint of fractal procedures all converging on the original cadence points in his Baroque model. What results can remind us of a Babbitt convergence. In Babbitt, his instruments, each unfolds its own array (and remember how much arpeggiation and freedom from strict order can be afforded with arrays, as opposed to rows). The instruments converge on a target collection, a collection as large as a hexachord, or a smaller collection that acts as a signature for a hexachord, a hexachord that displaces another. With Dawe, the tonal convergence remains--and that's what tonal counterpoint does, the intstruments converge on a triad, andl at the same time the tonal elements are gutted by his convergence of fractalized elements all aimed to coalesce at the same point where the tonal cadence occurred. Dawe's example is less a sublimation of 20th C. grounds, but rather an entirely new and original relationship to tonal grounds. Through his usurpation he can sound like Babbitt, Ligeti, or Kurtag, but he really always sounds like Dawe.

I'm finished for now, but I will be adding some examples of collections an sich.


Ives approaches the collection an sich, despite the key signature. To Schoenberg a new grounding was vital, and Babbitt & Brickle broadened that effort. What we see in Ives, perhaps, is how fuzzy the distinction is. Brickle plays the collection an sich against our habitual scale degree consciousness with the clear hindsight of how the ground had been shifted, and with respect for the importance of the distiction. Ives, perhaps, didn't feel the distinction needed to be made so overtly. I am more like Ives, when I, and perhaps most of us are.

Ives' Cradle Song--he cares about keeping a sound going, he cradles us in that opening (Lydian) harmony. He keeps it alive linearly in the soprano's lydian moves. Brickle admired this in Ives and took it to heart, he does what Ives does, but with attention to slowly unfolding aggregates which may even relate to arrays or superarrays. Brickle is Princetonian to the end, and yet his work shows that the high 3/4 Century Princetonican thinking could adapt itself into a more mainstream dialogue.

cradle song resize

Brahms approaches the collection as sich in a kind of mirror image of the way Brickle flirts with scale degree associations. In Op. 88, the tune's first 4 notes are elaborated throughout the 4 movements of the work, elaborated as a collection. In the Sarabande at the end of the slow movment, the collection rises to the surface, and also makes explicit that the collection is the major third key relationships (background) superimposed upon the middle-ground's traditional, habitual dominant-tonic realationship:

F--C; A--E

Opening tune: C--F E-F-GABCA-

the other leg of the background augmented trid:

F--Db; F--C

And note that the opening tune is harmonized in the traditional manner, we hear the tune as the collection [CFE] only by *hearing through* the harmonization.

A Response!--->

Frank Brickle wrote:

....in the case of Brahms that would amount to a full-up analysis of the whole piece. Have you looked at Ben's [Boretz, that is...no I haven't] analysis of the opening of the 4th symphony?

Far as Ives is concerned, it would be a similarly long and detailed
analysis of his (leit-) motivic technique, which is awesomely powerful
and brings together quite a few different strands, so that he can vary
one strand at a time and leave the others essentially intact. [!] This is
most obvious in the two piano sonatas, but also informs most of the
chamber music as well. Not so sure about the orchestral music.

Hieronymus Kapsberger

In this insidious theorbo solo by Kapsberger, 034 gets more explicit *as a collection*, until, in measure 14, it's right there in your face. The re'entrant strings make this piece sound like it's being played on both manuals of a double manual keyboard. And the re'entrant tuning also makes it possible for the F# and Fnatural to overlap. The tablature calls for them to be on different strings. I don't mean to suggest that the collection here transcends its imbrications with scale degrees. I suggest it as an example of a collection gathering force as a collection, on its way to trumping scale degrees.

kapsberger corrente

More examples to follow!

A brief note on Babbitt & Teleology

–William Anderson