Rage over a lost [but not forgotten] blue-book

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Dear Jacob,

In your Beethoven seminar, you ask participants to analyze the third movement of Opus 2, No. 1, which you appear to regard as structurally complex. Knowing my penchant for constructivism (or perhaps deconstruction), as well as my constant rebarrings and/or remeterings (both or which you loathe and abjure), you have, in your omniscient and imperious manner, frequently challenged me to "take the test," going so far as to present me a "blue-book" with which to do so. I have steadfastly refused to rise to the bait, as I firmly believe that for us to seriously discuss music would terminate our friendship. A seventieth birthday, however, seems an appropriate time to test your tolerance, openness, and wit and should we never again speak, I will still have enjoyed your company for more than a decade.

The work in question consists of four sections, i.e.


  1. With thanks to Charles Bernstein, poet.

A → B; B1 → C; C1 → D; D1 → E (see plate 2).2

Plate 2 Page 110

Plate 2 Page 111

A → B = 14 measures
B1 → C = 26 measures
C1 → D = 10 measures
D1 → E = 23 measures

Once you have confirmed that these bar counts are correct, you are left with:

a) finding a solution with equal phrase lengths, or
b) accepting a solution with different phrase lengths.

As regards a) — any solution which attempts to keep all phrase lengths equal will only "work" for the first three sections (23, as a prime number, is only divisible by 1 and by itself); will only allow for two-bar phrases (2 is the only whole-number divisor common to 14, 26, and 10); and will result in prime numbers of such phrases (the first section would contain 7 two-bar phrases; the second 13; the third 5). As the music clearly contains phrases of length other-than-two-bars, we opt for b) i.e. a solution with different phrase lengths.

Using the dynamic markings as a guide, the 14 bars of A → B may be partitioned as:

VIII (1+1+2+1+1+2) bars 1-8
II (2) 9-10
IV (2+2) 11-14

where Roman numerals indicate primary sub-sections, and Arabic numerals indicate internal divisions of the sub-sections (it is to be understood that both sub-sections and internal divisions thereof usually begin on what are nominally upbeats). I find this tripartite partition architecturally beautiful to contemplate, and traverse. A much more conventional approach would be to bisect the VIII into IV+IV. The justification for this would be the musical parallelism of bars 1-4 vs. 5-8. I am opposed to this bisection on the grounds that B. provides nothing (not dynamics, not change of articulation) that indicates that the parallelism should be emphasized, and I personally dislike bringing attention to such prosaic devices, especially when they are used so baldly. Another objection to the bisection is that it results in partitioning A → B into 4 sub-sections: i.e. IV+IV+II+IV. This is far more "boxy" than my original proposal of VIII+II+IV, and minimizes the shock of the shortness of the II. Another conventional approach would be to bisect the VIII, and to attach the II to either the preceding or following IV (i.e. IV+VI+IV; or IV+IV+VI). I am opposed to this on the above grounds, with the addition that it relegates the II, which is certainly the dynamic climax, to an architectonically subservient position, when in fact the II is the keystone that spreads A → B from a rather dull 12-bar IV+IV+IV construct into the 14-bar arch that it is. Furthermore, both bisection and/or attaching the II at either end reduces our awareness of B's "art of writing inconceivably long sections or mercilessly short ones," and that is a reduction I do not wish to countenance, so VIII+II+IV it is.

Again using the dynamics, the 26 bars of B1 → C may be partitioned as:

IV (4) bars 1-4
II (2) 5-6
V (2+1+1) 7-10
IV (4) 11-14
VI (1+1+1+1+2) 15-20
II (2) 21-22
II (2) 23-24
II (2) 25-26

Of course, these individual units could suffer crasis, and thereby be oozed into fewer segments. I am opposed to that on (at least) three grounds: a) except for the broadest of principles (i.e. First Amendment stuff, etc.) I am congenitally ill-disposed toward homogeneity; b) in my opinion, in most cases B's dynamic markings are coincident with his phrasal structures, and indeed, are a signal for those structures; c) I would oppose any effort that would reduce the contrast between an A → B of 3 sub-sections, each with inner sub-divisions, vs. a B1 → C of 8 sub-sections, most with no such sub-divisions possible.

  1. The da capo does not affect the partitioning I propose. Plate 2 is a facsimile of the first edition: (Vienna: Artaria, 1796), reprinted in Ludwig van Beethoven, The 32 Piano Sonatas in Reprints of the First and Early Editions, Principally from the Anthony van Hoboken Collection of the Austrian National Library, 5 vols. (London: Tecla Editions, 1989), 1:8-9.

The dynamic markings of the Trio (C1 → E) are simply not sufficient to posit them as structural signals. Indeed, as shown in the following table, the difference in rate of occurrence of dynamics is remarkable:3

A → B has (3 dynamic markings + 1 sf ) = 4 marks within 14 bars
B1 → C " (8 " " + 6 sf ) = 14 " " 26 "
C1 → D " (1 " " + 0 sf ) = 1 " " 10 "
D1 → E " (5 " " + 0 sf ) = 5 " " 23 "

Dynamics being too sparse to be useful for partitioning the Trio, I shall use the long slurs, an aspect which clearly differentiates the Trio from the Menuetto.

C1 → D is clearly partitioned:

IV (4) bars 1-4
IV (l.h. 4) vs. (r.h. 1+1+1+1) 5-8
II (1+1) 9-10

D1 → E is slightly more ambiguous. Using the slurs as guides results in:

IV (3+1) bars 1-4
IV (l.h. 1 + 3? or 4) vs. (r.h. 2½ + 1½) 5-8
I (1) 9
I (1) 10
II (2) 11-12
III (2 + 1 because of the pp) 13-15
IV (l.h. 3 + 1) vs. (r.h. 4) 16-19
IV (l.h. 3 + 1) vs. (r.h. 4) 20-23

It is conceivable that the two I-bar-sub-sections (bars 9 and 10) could be combined into a II (as they are both within a single crescendo), but if I override the logic of the slurs because I posit that dynamics have a higher hierarchical position, then, to be consistent, I must in turn split the III (bars 13-15) into two sub-sections (II+I), based on the change from p to pp. In either version, there will still be 8 sub-sections within D1 → E.

To recapitulate, or should I say da capo, the totality is:

Menuetto Trio
A → B VIII C1 → D IV
B1 → C IV D1 → E IV

which on the one hand satiates the symmetry golem, whilst still providing great variety, and I trust I have now fulfilled your blasted blue-book requirements!

Now none of the above is at all outside the ordinary, or is particularly outré, and from my point of view the movement only becomes interesting when explored at a level of greater detail, which is what I now propose to do.

  1. Where two "sf" occur at the same temporal point (one in the right hand, the other in the left) I count the two as one sf; if I counted the doubled sf as two the difference between the Menuetto and its Trio would be even greater.

As you know (except as a device to keep individual parts sort of together — which was the original intent), I am utterly opposed to barlines. To me, they are no more than the graph paper against which one plots data, and I have always loved Tukey's observation that one can gain "insight" by observing data-points "undisturbed by a grid." 4 Now I can hear you intoning that music isn't data. That is only a partial truth. Data can, and does, represent natural events and shapes, while not actually being the event or shape. Just so do musical notes represent the music, but never forget that the notes and their placement against a fascist and fashious grid are not the music! As Gaudí, Mandelbrot, and many others have shown, nature doesn't fit neatly into straight lines, or whole-number dimensions, and neither does music, and I remind you yet again of Busoni's statement that "what the composer's inspiration necessarily loses through notation, his interpreter should restore by his own." 5 The entire point of music, as speech, as poetry, is plasticity — the ability to shape phrases, and motives, in an endless variety of ways — and all that the barline does is shove everything into a box, whether it fits there or not, which prevents us from sensing the true shape.

Composers choose    or    much as one chooses graph paper. You find a grid that more or less will portray the data so as to best reveal the result you wish to show, but finally, it is always a compromise, often a matter of convenience, and especially of convention, and NEVER do you, or should you, confuse the grid with the actual underlying "structure." Musicians have elevated the grid, or the barline, to a position of such preëminence, that the vast majority of baroque and classical music performances are ineffably dull, i.e. to paraphrase a certain governor of California, you've heard one iamb, you've heard 'em all.

In my world, barlines stand accused of imprisonment, of distortive rhythmic inflections, and of preventing one from sensing the smallest possible unit(s) out of which motives or phrases have been constructed. As panacea (or in your case, should I say emetic?) I present you the movement in question WITHOUT BARLINES (example 1).6

Viewed in this manner, our ability to see, and sense, each item as itself, but also in relation to others, is maximized. Each item comes into focus, while at the same time, overall ambiguity is increased. What is downbeat? What is upbeat? What is the meter? Is there a constant one? Is the first note the beginning of a group of three (1+2) (YES) and only later (especially with the play of the sf) do we discover that what was downbeat is now upbeat, and up now down? Note that the original beaming has not been changed from plate 2! Imagine how this would seem if the plate 2 ( ) beaming were replaced with one that corresponded to the trajectory of the phrase. For example, the run of eighth notes, starting on page 119, system 2, would be a   , slurred to 1 quarter-note, the whole equaling   , and this revised beaming would focus our attention on the similarities and dissimilarities of this run vs. the eighth-note runs in the Trio. In short, "the central purpose [of this tool] is to liberate the mind from serfdom to tradition, and to exhibit man's natural powers in their creative capabilities when expanding in the open-air-of-the-spirit-of-responsible-freedom." 7 This version without barlines refreshes our collective memory that "Of course it is extremely difficult nothing more so than to remember back to its not being beautiful once it has become beautiful." 8



  1. John W. Tukey, Exploratory Data Analysis (Reading, Mass.: Addison Wesley, 1977), 42.

  2. Ferrucio Busoni, Sketch of a New Esthetic of Music, trans. Theodore Baker (New York: G. Schirmer, 1911), 16. Note that the grid is equidistant primarily because that was the convention of the time, and not because B. could not conceive of changing meters. Indeed, I have always found it remarkably dichotomous that you (and you are hardly alone) on the one hand raise B. to such (in my view unwarranted) heights while you are simultaneously perfectly content on the other to perceive him as a metrical moron — as if not until Stravinsky (for whom you have no use — another point of disagreement between us) did or could composers think of changing meters. Strange — especially when one of your great twentieth-century heroes himself ascribes having learnt from B. "Rhythm: the displacement of figures to different beats," and if you preserve the same accents and impulses while displacing to different beats, you have quite clearly shifted the barline, and that equals changing meters. Do reread my article in the Journal of the Arnold Schoenberg Institute 15, no. 1 (June 1992); see page 166 and especially the example on page 170.

  3. You (and many others) will say that one can convey both the " -ness" as well as the cross-meters. This is not the place to go into extensive detail, but unfortunately that idea is wrong — one simply cannot, with one voice, do both. The abbreviated argument goes as follows: In order to have listeners perceive groupings (of 3 or 4, etc.), performers slightly distort (on the order of approximately 10-25 msec.) the mean duration between attacks. The rules are fairly clear — i.e. first and last interattack times in a group are lengthened relative to the mean; intermediate interattack are shortened relative to the mean; (groupings by 2 are a special case). Assume a bass in 3 with constant simultaneous embedded cross-meterings of . It is agreed that two or more players can each produce their own grouping pattern, and that in this case both groupings can be perceived. It is conceivable that the two hands of a pianist can be trained to provide the subtle dichotomous lack of simultaneity that would convey both groupings. What cannot be done is to have one player, or an entire orchestra, when playing only the embedded group, accent both the embedded group, as well as the meter. The arithmetic doesn't work. At many points in the succession of attacks, the elongation or shortening of the interattack times of one grouping will completely contradict the elongation or shortening of the other grouping, resulting in either nullification, or gross distortion. At best, you can choose to emphasize one grouping or the other, and it is when the performer chooses to stick with the metric that one encounters the cloddish stupidity of repeating a melodic pattern while each time accenting it in different places. I point out that in "hemiola" the "you-can-do-both-proponents" rarely follow their own advice, preserving their metric oecumenicism only for those cross-meterings that have a pontifical designation.

  4. Louis Sullivan, Kindergarten Chats (Cleveland, 1901-2); reprinted in Witterborn Documents of Modern Art; Vol. 4 (1947), 15.

  5. Gertrude Stein, "Composition as Explanation," Selected Writings (New York: Modern Library, 1945), 515.

And yet, even in its unbarred form the music has still not been distilled into its irreducible elements. To highlight this microstructure I propose a further manipulation.

Imagine a modern poetic with a restriction of one word per line, but with a variable syllable count for each line, with irregular stanzaic breaks. Greater vertical space between lines and/or stanzas will serve as a proxy for greater temporal space when scanning or reading. If music were written in this fashion, and a "word" were defined as either an isolated note, or anything under one slur, the Menuetto might look as it does in example 2.9 Now we clearly note the rhythmic retrograde augmentation of lines 5 and 6 (4 + 2 ) vs. lines 4 and 3 (2 + 1 ), etc., etc., and this presentation also forces upon our attention the issue of articulation — the separating out of the smallest units. In turn, this separating draws our attention to the choppiness of some sections vs. the more mellifluous ones, which in turn reveals that the microstructure has a similar erraticism that we earlier observed in the macro (or phrasal) structure.

All of the above provides no solutions. It is not intended to. It only provides a set of lenses, which when used with consistency, can help to more closely approach original intent. How one puts Humpty Dumpty back together again I leave to you. I am not a pianist and that isn't my problem!

As a close to this homily I can do no better than to remind you of a stanza from John's Composition in Retrospect:

Happy Birthday, and many more,

Paul Zukofsky
November 1997

Originally published in
Essays in Honor of Jacob Lateiner,
Festschrift Series No. 15,
Pendragon Press, Hillsdale, NY

  1. I have distributed the eleven sub-sections of A → C across twenty-two pages, with each sub-section on two facing pages. The layout on any two pages is a function of the number of "words" within a sub-section. Note that rests are also "words" — i.e. the third sub-section (pages 128-29) contains eight "words" although two of them are silent. This is not mere sophistry. Even in B. a silence is not an extension of a previous sound, and it would have been much simpler for B. to write a half-note, as opposed to a quarter-note followed by a quarter-note rest. The fourth sub-section is on one line as I consider this to be an inverted canon which overlaps the quarter-note rests. It is, however, possible to see this fourth sub-section as four "words," in which case it should have four lines.

    The use of twenty-two facing pages, while a luxury, is a structural device, and perhaps an imperative. This arrangement emphasizes the breaks between sub-sections; assigns equal weight to each sub-section; and illuminates the differences in word and "syllable" count between and within sections. Note that, except for centering each "word"; leaving as much space as possible between sub-sections; and slightly adjusting the vertical spacing before and after sforzandi, I have not manipulated horizontal and vertical "word" placement.

    Should you feel my "poetics" hopelessly too modern, and/or my arrangement on the page too fanciful, please note that it is 100 years since poetic stanzas ceased to uniformly phalanx down a page, and it is precisely because of those breaking ranks that, in 1897, Mallarmé's "UN COUP DE DES JAMAIS N'ABOLIRA LE HASARD" was such a clarion.

  2. John Cage, Composition in Retrospect (Cambridge, Mass.: Exact Change, 1993), 27.
Plate A

Plate B

Plate C

Plate D

Plate E

Plate F

Plate G

Plate H

Plate I

Plate J

Plate K