Footnotes

1. The vast majority of the tables here are the work of Anton Vishio, work for which I shall be eternally grateful. However, he is not to be blamed in any way for the text ruminations that follow. Wong Jing Men Jamie, also contributed to bludgeoning the first two sections into a shape suitable for posting. My deepest thanks.

3. In order to “move things along”, we employ a number of abbreviations throughout the text, viz:

General:

H = Haydn
HSQM = Haydn String Quartet Minuets
SYM(s) = Symphony (ies)
SQ(s) = String Quartet(s)

We consider the data from two perspectives: either in terms of the components of the entire movement; or in terms of the juxtaposition, or opposition, of the three classic divisions of the entire movement, i.e. the minuet section, vs. the trio section, vs. the da Capo.

"A" = the first "section" of the "minuet" portion of the movement (i.e. up to the first repeat sign).

"B" = the second "section" of the "minuet" portion of the movement(i.e. from the first to the second repeat sign).

"C" = the first "section" of the "trio" portion of the movement (i.e. from the last repeat of the minuet to the first repeat sign of the trio).

"D" = the second "section" of the "trio" portion of the movement (i.e. from the first repeat sign of the trio to the last).

"S"(s) = Section(s); as in the total number of "A"s, "B"s etc. within an entire movement. These are usually "10-S" but there are also "7-S", "9-S", "11-S" and "14-S" .

"DL"(s) = Different Length(s); as in how many different lengths are used across "A", "B", "C", "D".

"W"(s) = Weight(s); a way to think about and to distinguish different arrangements of various "DL"s.

A note of caution:

When refering to an entire movement, we say minuet movement, or more simply, movement. M ONLY refers to that part of the movement that occurs before the Trio.

4. The choice of this edition is a personal PZ decision. In the event of discrepancies between the data presented, and some other edition, please consult the Robbins Landon. As detailed by RL, he excludes certain data-points from his SQ edition, and we follow that exclusion. As an aside, we point out that the excluded works also do not appear in the Landon edition of Symphonies. That is doubly discriminatory, but that is not our problem. The missing data points will not materially impact our overall “conclusions”, such as they are. We also remark that the op. 3 quartets are now considered spurious. Here too, the overall picture will not change were we to eliminate these data. We will also not enter in to a discussion as to if the first batches of quartets up thru op. 33 are divertimenti, or quartets.

5. Without meaning to detract from H's imagination, and/or Cage-ian penchant for syntactic experiment, the above measure-counts, while somewhat extreme in their variety, are not without precedence. For example, Tilden A. Russell, in both “The Unconventional Dance Minuet: Choreographies of the Menuet d'Exaudet”. Acta Musicologica Vol. 64, Fasc. 2 (Jul. - Dec. 1992), pp. 118-132., and “Minuet Form and Phraseology in Recueils and Manuscript Tunebooks”. The Journal of Musicology, Vol. 17, No. 3 (Summer 1999), pp. 386-419, has shown that, while H may be a bit over-the-top in terms of variety, the “standard” four-square minuet movement construct promulgated by some theorists is HARDLY the norm we have been hornswoggled into believing.

String Quartets

1. Notice how the placement of the movement (relative to the entire Quartet) changes over time, first wobbling about; then appearing to settle into the second movement slot; later on primarily appearing as a third movement.

2. Although see op. 20 where # 4 has 8/8/12/12/8/8/8/8 vs # 6 with 8/8/12/12/8/8/14/14.

3. This does not mean that there can be no 4-bar phrases embedded in a "ML" not divisible by 4. It is just that one cannot blindly count 4 bar phrases, and expect to come out “even”. Similarly, in a bar count that is divisible by 4, one cannot assume that all internal phrases are 4 bars in length.

4. As before, we stipulate that 8-bar phrases may be embedded in bar lengths not divisible by 8 (or 4). Our point is that one cannot blindly assume non-stop eight-bar phrases no matter what.

5. SQ (8) does not specify the SQM opae of the partitions, nor can we tell which M Partitions hook up to which T Partitions, or the reverse. For that, one must consult the resorted, combined tables (INSERT PROPER TABLE NUMBER!!!! these maybe Mu 21 & 22). To clarify, if you wish to know the opus, and/or specific M & T combinations for all Ms where TML = 40, consult table Mu (21), where you will find that the M lengths are highlighted, with all TML 40 grouped together (lines 2 - 8). For Ts, where TML = 40, see Mu (22) (lines 21 -- 36).

6. Note that we not only make no claim, indeed we affirmatively renounce the idea, that the internal divisions of TMLs always neatly coincide with 4-bar phrases; but we do note that while the internals are most frequently not-4, the large form very often sums to a multiple of 4.

7. It is easy to assume that the non-divisible-by-4-TMLs are just “distorted” divisible-by-4-TMLs, and are best explained as such. While that may sometimes be the case, there are clearly instances that can not be so easily fobbed away.

8. The number 81 includes the reversals of line 13 (10/10/14/14 vs 14/14/10/10), line 15 (12/12/14/14 vs 14/14/12/12) and line 38 (16/16/20/20 vs 20/20/16/16) as different “ordered partitions”.

9. The eagle-eyed will note that that these numbers do not correspond to the number of pairs-in-common that we find at the bottom of SQ (6) & SQ (7). Much of the difference is due to the fact that the data of SQ (6)& SQ (7) is restricted to single “A1:B1” etc. pairs. As the partitions of SQ (8) contain differing multiple copies of the internal components, there will be fewer partitions in common than if one only inspects pairs in isolation.

Symphonies

1. As with SQ, and as detailed by RL, certain works are excluded from his SYM edition. We follow those exclusions.

2. The order of these additional cols. says nothing about the order or placement of the sections when performed, i.e. SYM #30 and #51 both have second Ts, but they present differently, with SYM #30 ordered as M - T1 - T2 - dC; vs. SYM #51 as M - T1 - dC1 - T2 - dC2. Given that SYM 51 is the only movement with two dCs, we refuse to provide an individual column.

3. An additional curious aspect, most probably unrelated to the structural issues that concern us, is the preponderance of C and D major in SYM (see table directly below), which may be due to wind instrument intonation problems. This preponderance does not appear to be related to “S” or “DL”.

Musings

1. Full disclosure: we adamantly support "not due to chance", and dismiss any hint of that garbage-pail diagnosis "inspiration". Someone who wrote as much as H had to employ general work methods or procedures. The methods may not have always been explicit. They may have been, or become, subconscious; but such methods existed (and still do), and permeate music far more than most realize, or wish to, acknowledge.

2. The idea that H used a few various simple standardized (floor) plans to which he made simple adjustments seems unlikely after looking at these maps; but if our data show anything, it is that nothing is standardized, per se. The "norms" that may underlie the data would appear to have far more to do with a "concept" than they have to do with a prefabricated uniform construct. In that sense it may prove more fruitful to think in terms of Palladio, as opposed to William Levitt. Certainly Corbusier's Modulor is not of use here.

3. Full disclosure 2: With the most welcome assistance of Saul Sternberg (my deepest thanks) we ran a few statistical tests (Kolmogorv-Smirnoff – my suggestion; chi-square and Fisher Exact), and found nothing inconsistent with the idea that the SQ/SYM data can be pooled.

4. This assumption as well was not contradicted by (a very few) other statistical tests.

5. We reduce the data to remove clutter, and because including dCs may distort the sums which are the sieve for the permutation search. Note that if ["A1", "B1", "C1", "D1"] do not permute, a larger data set including "A2" etc., will certainly not permute.

6. A "Gerber Variable Scale" ruler approach to minuet -movements composition.

7. by definition, the "Σ" for each ["A1"+"B1"+"C1"+"D1"] = 100%. For line 1, the "Σ" of real "ML"s for ["A1"+"B1"+"C1"+"D1"] = [8+42+18+38] = 106. The meaning of %"A1" is that the "A1" section of line 1 = 8/106 = 7.5% of the sum of ["A1"+"B1"+"C1"+"D1"]; %"B1" = 42/106 = 39.6%. etc. In addition to being a possible construction technique, %"A1"s, etc., are of interest when thinking about the relative "mass" of a section i.e. on lines 46 - 64, real "A1" bounces about from "ML"s = 8 to 14 (a range 75%), whereas %"A1" ranges from 14.0% to 14.8% (an almost unnoticeable variation of less than 1%).

8. As 28 is 2.33 times 12, the new ["A1"+"B1"+"C1"+"D1"] "ML"s would be 28.00 + 51.33 + 18.64 + 46.66, respectively. As the "Σ" for line 111 = 62, one measure, equaling the smallest possible variation in percentages, is 1/62, or 1.6% (in the rightmost col.). The %"A1"s of lines 111 & 113 are identical. % "B1"s are 35.5% vs 36.1%; % "C1"s are 12.9% vs11.1%; and % "D1"s 32.3% vs 33.3%. Note that if the smallest quantum was two meas. (and as we shall later show, for most cases it was), the smallest percentage variation in this case would have been 3.2%, which would swamp the 1.8% difference between

9. While an "exact" repeat sample vs. a "none" is clear-cut, we stipulate to (mild) fuzziness between other categories i.e. someone's "same length altered" might conceivably be another person's somewhat "expanded" or "truncated". That being said, the table provides a reasonably fair picture. While the table is restricted to SQ data only, SYM data can be similarly categorized, but is not, as the entire matter is too much of a digression.

10. One might also argue that "W" = 111111111122 references "W" = 1222, which also starts with the short length, and terminates with the longer. Furthermore, as the last 22 of 111111111122 is a repeat of a single section, one might argue that long string is an attempt at a mirror image of "W" = 1222, which starts by repeating a single "A", and then has a series of constant repeated longer values, as opposed to 111111111122 which starts with a series of short values and ends with long values. No matter how distant, perhaps even outrageous, such a relationship may appear, note that there is no such relationship between "W" = 111111111122 and "W"s = 1211 or 1212.

11. The "ML" s of line 12 are "8" & "20". Adding 6 to each produces "14" & "28", i.e. the "ML" s of line 13.

12. The case for 3"DL" —"W"s is more balanced, with real interplay between repeats vs. longs and shorts; but as 4"DL"s have no repeats, "W" structure resides only in the gross shape of the lengths; but note that "irregular" 4"DL"s allow repeats within a 4"DL" context!

13. We tried creating similar tables for the samples missing any "ML"s = 8, but could never find any dispositive reason to choose one minimum value over another.

14. From old French i.e. a ragout, hash, 1. a hodgepodge, made up of the remnants and scraps from the larder. 2. Any inconsistent or ridiculous medley.

Endnote

1. By this I mean that while most everyone uses a cell-phone, the average user could not explain the physics or math of how it actually works; whereas any performer of H’s era, in as an example, a discussion of musical tuning or temparement, had to have some basic acquaintance with the physics of pipes and strings

2. From a letter to Goldbach, 27 April 1712, quoted in O Sacks, The Man Who Mistook His Wife for a Hat (1985).

3. A tangent: speaking of the 12th century, Taruskin writes: "Training in composition would henceforth be basically training in polyphony -- in 'harmony and counterpoint,' the controlled combination of different pitches in time -- and such training would become increasingly 'learned' or sophisticated. Combination, the creation of order and expressivity out of diversity or even clash, became the very definition of music (or, to be more precise, the primary musical metaphor)." (Oxford History of Western Music, Vol 1, p148

4. "A needless Alexandrine ends the song,/ That, like a wounded snake, drags its slow length along." (Pope, Essay on Criticism, 1. 356)

5. A complete M, with an "A" within "B", all repeats taken, results in the following form:

M = "A1"+ "A2" + "B1"+ "A3" + "B2"+ "A4"

T = "C1"+ "C2" + "D1" + "D2"

dC = "A5"+ "A6" + "B3"+ "A7" + "B4"+ "A8"

i.e. two long arms (the Ms) embracing a centerpiece T;

In addition, there is a hierarchy of repeats, or perhaps more properly, boredom i.e. hearing an "A" 8 times; vs. a "B" 4 times; vs. a "C" and "D" twice each, HAS to increase one's interest in the things heard the fewest times.

Moreover a "traditional"

dC = "A5" + "B3" + "A6"

clearly imbalances the full bracelet.

6. "The beginnings of the general bass, according to a mathematical method of teaching, and which are most clearly advanced by means of a machine invented for this purpose," by Lorenz Christoph Mizler von Kolof

7. In Dissertatio de Arte Combinatoria (1666, and reissued in 1690), Leibniz' proposes that all concepts are combinations of a relatively small number of simple components, just as words are a combination of letters. Using permutations, this alphabet of thought creates a logic of invention as opposed to one of demonstration.

8. For an edition of which see:
http://imslp.org/wiki/Der_allezeit_fertige_Polonoisen-_und_Menuettencomponist_(Kirnberger,_Johann_Philipp)

9. A facsimile of which was found here: http://purl.pt/28871/1/index.html#/1/html

pour un Haydn

Footnotes

Introduction

String Quartets

Symphonies

Musings

Endnote