3.0 MusingsIs our observed SQ/SYM variety (primarily) due to chance; or are there organizational principles that help determine the madness?1 3.1 To pool, or not to pool?A single, larger, pooled data-set may provide results more robust, perhaps more revelatory; but before we can combine the two data-sets, we must determine if they are sufficiently similar to allow us to do so. Consider map #3, which shows map #s 1 & 2 rotated, and placed next to each other. Other than for SYM being wider (due to more samples), there seem few differences between the SQ & SYM maps. Heights are comparable. Internal color differences are not especially remarkable. There is even a curious, subtle, similarity between maps i.e. SQ shows (from the left) ascending green spikes that plateau around line 39 (Opus 20/5); followed by a bit of a "null"; with higher green spikes further towards the right. SYM has a similar progression, with an interim peak at line 48 (Symphony #56); another "null"; and then an advance to higher spikes. However, map #3 displays the "traditional" dC of "A" & "B" without repeats. As the addition of only one half of "A" + "B" may skew the "skyline", and as the "tradition" is questionable, ma #4 presents the previous map, with dCs removed! There are still few differences between data-sets, although the curve (of shorter lengths in the middle), while still obvious in SYM, is perhaps less so in SQ.2 But minuet movements consist of Ms and Ts. How do these compare between SQ and SYM? Map #5, contains data identical to map #4, but with M and T vertically displaced. Comparing Ms with Ms (blue/greens with blue/greens), and Ts with Ts (yellow/reds with yellow/reds) we see little difference. There are clear differences comparing blue/greens with yellow/reds (Ts are shorter than Ms; differently colored sections are exclusively in yellow/red Ts;), but we do not propose such conflation. The "skyline" curve is less obvious in map #5, its absence hinting that the curve may result from the combining of M and T. But what of "A", "B", "C", "D"? How different are they between SQ vs. SYM? Map #6 separates each color band. The two blue bands are hardly distinguishable from each other; and that is almost equally true for the two green bands. Yellow and red pairs are not as similar to each other as are blues and greens, but if one removes "outliers" i.e. the ochre monstrosities extruding from the yellow band, as well as any extra colors, things become comparable. We therefore conclude that, at all three levels of lengths (i.e. the complete movement; M vs. T; the sections"A", "B", "C", "D";), SQ/SYM data can be combined.3 * 3.2 The poolMu (1) presents a monotonic sort of the pooled SQ (2) and SYM (2) {SQ (2)} data. The three repeated SQ patterns (highlighted with red oblongs) are now on lines 5 & 6; 20 - 22; & 148 & 149. The four SYM repeated patterns (light red oblongs) are on lines 9 & 10; 34 & 35; 60 - 62; & 112 & 113. Five additional patterns common to both SQ and SYM are highlighted by dark red oblongs. These are: a SYM (line 23) is added to the already repeated SQ 8/8/20/20/8/8/8/8/8/20. Four new repeated SQ/SYM pairs are: lines 3 & 4 = 8/8/12/12/8/8/8/8/8/12 Mu (1) shows 164 different patterns (164/177 samples = 93% different). While slightly reduced from SQ's 95%, and SYM's 96% different, 93% remains an astonishingly robust number, and is (most probably) NOT due to chance.4 Our belief that, for these samples, H was deliberately avoiding gross form pattern repetition, is reinforced; and we now focus on how H might have organized this result. * 3.3 Five NOTsBy this we do NOT mean that H did not (in some fashion) use these concepts to construct varied outer forms. We mean only that these ideas are not particularly fruitful in trying to understand how H did move from form to form. 3.31 not ChronologySQ (1) and SYM (1) {SQ (1)} listed SQ by opus, and SYM by number. Neither list showed perceivable systematic changes sample to sample; i.e. given "A1" --> "B3" values for sample X, the values for X+1 were not determinable, although very occasionally, within an opus, there appear to be simple (??????) substitutions within an "A1" --> "B3" progression (for example, see SQ op. 30). Perhaps, if we could chronologically interleave at least the and SYM (1) {SQ (1)} lists, we might better understand how "A1" --> "B3" mutates over time. The idea fails, for at least three reasons: 1-- H chronology (even within an opus) is inexact, and does not allow for the creation of a precise time series. 2-- Even were we able to pinpoint an exact date of writing, if H was working off an "A1" --> "B3" master pattern-shift plan, which plan he did not use in order, it would be improbable that we could deduce the underlying system. 3-- if the primary, perhaps only, rule of pattern shift, was to repeat as few "A1" --> "B3"s as possible, pattern changes over time could be quite arbitrary, and there would be no system to find. We must look elsewhere. * 3.32 not PermutationsPermuting ["A", "B", "C", "D"] "ML"s could provide structural variety. Mu (2) explores this by reducing the data-set to ["A1", "B1", "C1", "D1"],5 and then sorting on the sum (highlighted in the yellow "Σ" col.). As example, consider lines 4 - 9, all of which sum to 42 meas., but with different internals: i.e. lines 4 & 5 are identical, where ["A1", "B1", "C1", "D1"] = 8/12/8/14. Line 9 permutes these "ML"s (to 12/14/8/8, marked by blue verticals). In contrast, lines 6 - 8 contain "ML"s (i.e.10 & 16) not found in lines 4, 5 & 9; and lines 6 - 8 differ for each line. Only the following ["A1", "B1", "C1", "D1"]s permute, using six different "Σ"s, and only "Σ"s 58 & 60 have more than one permutation pairs.
While aspects of the above list are intriguing, given a total of only 18 permuted samples, permutation of ["A1", "B1", "C1", "D1"] seems a minor contributor to our total observed variety. * 3.33 not "Reps"Repeated patterns (aka "reps") by definition should not be generators of new forms — i.e. if an exact "rep", nothing new is generated. However, that may be too quick a dismissal? There are a total of 23"rep"s of 10 different "ML" patterns i.e.: Comparison of repeated patterns, across "DL"s
Reading down col.s, and to a lesser extent, across rows, one can envisage adumbrations (a-dumb-berations?) of a substitution schema; but as we shall shortly see (ad nauseam) far more is "going on" in our data than just substitutions of one or more "ML"s; so for now, we shall add "reps" to our list of "Nots". * 3.34 not Multiplicative ConstantsImagine two ["A1"+"B1"+"C1"+"D1"]s of 8/12/8/8 vs 12/18/12/12. Both are in the proportions 2/3/2/2. A number of such schemes, scaled to different sizes,6 would provide variety, and might be a construction technique. Mu (3) presents percentage proportions for each ["A1"+"B1"+"C1"+"D1"], and monotonically sorts on the percentages.7 While there are non-"red"-vertical samples with repeated % ["A1"+"B1"] pairs (i.e. see lines 60/61 or 75/76, etc.), as well as non-"red" repeated %["C1"+"D1"] pairs (lines 9/92 or 20/91, 78/115/145; etc.), no %["A1"+"B1"+"C1"+"D1"] (other than for "red" verticals) repeats exactly; but concentrating ONLY on exact %["A1"+"B1"+"C1"+"D1"] replications is too simplistic. Consider lines 111 & 113, whose ["A1"+"B1"+ The percentages of line 111's ["A1"+ "B1"+ "C1"+ "D1"] While one should not dismiss the idea of there being only a few architectonic templates in to which all samples can be sorted, Mu (3) is not the context in which to discuss this idea. "DL"s and "shapes" represent a far simpler, more obvious, means of differentiation bearing directly on internal proportions. We therefore reserve discussion of constant proportions across samples until 3.6 etc. * 3.35 not Sections, or, "S" is for "S"peciousWe previously stated (section 1.2) that the number of "S"ections, as in "10-S "s etc., was misleading. This is primarily (but not exclusively) due to how "A1"s are treated within "B1"s. Mu (4) sheds light on this, with lines 1 – 13 listing all SQ samples where "A1" is repeated exactly within "B1"; lines 14 – 25 showing same length (marked "s.l." in the chart), but altered samples, meaning that the "A1" within "B1" is the same (or extremely close to the) length of the original "A1", but with significant alterations in musical detail; lines 26 – 62 where "A1" within "B1" is expanded; followed by six instances where the embedded "A1" is truncated; and ending with those samples (starting line 69) where "B1" section has no "A" whatsoever.9 Now consider two "10-S" samples,one with an "exact" repeat (within "B1") of "A1"; the other without. An "A1" section, by itself, is a single component of a "10-S". Why, when embedded within a "B1", does an identical "A1" no longer count as a section? If it should count, is a "10-S" with an exact repeat of "A1" really a "13-S"? Should "10-S"s only be those without any remnant of "A1" within "B1"? But what of an "expanded A1" within "B1"? Does that become a "14.273-S" etc. depending upon the size of the expansion? The vexatiousness, and unlikelihood, of being able to assign a number reflecting reality; coupled with the inutility, plus hollowness of meaning, causes us to "sing", along with "Olaf", that, "there is some "S" [we] will not eat". * 3.4 Pay dirtSection 1.3 (and SQ (4) ,and SYM (3) {SQ (4) ) pointed out that some movements contain only 2 "DL"s, repeated across ["A1"+ "B1"+ "C1"+ "D1"]; while other movements had 3 or 4 "DL"s across ["A1"+ "B1"+ "C1"+ "D1"]. Mu (5), Mu (6), Mu (7), & Mu (8) present all samples, first grouped by "DL"s; and then, within each "DL", sub-categorized by "W"s, "regular" and "irregular". 3.41 Introducing "W"s"Rank-orders" can be used to classify and compare shapes within and between "DL" categories. We label these rank-orders "W" (as in "W"eightings). To understand the concept of a "rank-order", consider a Mu (5) lists movements whose "W"s "rank-orders" contain only 2 "DL"s. Movements with 3 "DL" – "W"s are found on (Mu (6)) ). Movements with 4 "DL" – "W"s are on (Mu (7)) "Irregular" minuet movements are batched in (Mu (8)). All "DL" categories (i.e. 2, 3, 4 and "irregulars") contain different "W"s, most "W"s having multiple, diverse samples. Changes of "ML"s within a "W" deform the "curve" of the "W", but not its underlying shape. Deformations (within "W"s) are bounded i.e. in "W " = 1423, if (for example), the "ML" for "C1" becomes greater or less than the "ML"(s) for "B1" or "D1", the "W " number will change. In characterizing "W"s, four aspects are fundamental i.e.: how many (if any), repeats? Where do they occur? For a specific ["A1"+ "B1"+ "C1"+ "D1"], where does the shortest "ML" appear? The longest? The interplay between these aspects varies as a function of "DL" size. For 2 "DL" –"W"s, longs, and shorts go pari passu with how many, and where. For 4 "DL" – "W"s, how many repeats is no longer a factor. 3.42 The 2"DL"sBy definition, 2 "DL"—"W"s contain two different "ML"s, distributed across an ["A1"+ "B1"+ "C1"+ "D1"]. Labeling short and long lengths "1" & "2", possible combinations of "ML"s are: {three "1"s + one "2"}; {one "1" + three "2"s}; or {two each of "1" & "2"}; but the combinatorics do not end there -- i.e. in a {three "1"s + one "2"}, the single "2" could (potentially) appear in any one of "A1", "B1", "C1", or "D1". The totality of all possible 2 "DL" – "W"s, will appear in Mu (11). Mu (5) shows four different 2 "DL" – "W"s. "W"s = 1211 and 1222 are both {three of one "ML" vs. one of the other}, and are presented first in (lines 1 - 8). However, 1211contains {three shorts vs. one long}; whereas 1222 is {one short followed by three longs}. "W"s with two each of "1" & "2" (i.e. 1122 and 1212) appear on lines 9 - 13. Mu (5) also shows an irregular "W" = 111111111122. As it only contains 2 "ML"s i.e. 8 & 16, it warrants consideration within Mu (5). As the order of "W" = 111111111122 is somewhat similar to that of "W" = 1122 (both start with the short length, and terminate with the longer) there is a superscript pointing out the possible relationship.10 As regards changes within "W"s, consider "W " = 1211. "A1"s, "C1"s, and "D1"s are constant (i.e. always 8 meas.). Alterations are ONLY in "B1" (from "ML" = 12, to 20, to 28). Were one to graph those changes, [8 + 12 + 8 + 8] vs. [8 + 20 + 8 + 8] vs. [8 + 28 + 8 + 8], the curve would deform (as "B1"s would vary in height), but the curve's basic shape (one "step" up; and then back down to the original level) would remain. Changes within "W " = 1212 are the result of changes in two cols., but the curve form remains, i.e. "step-up"; "step-down"; "step-up". The only change is in the height, and base level, of the first "step". All curves used by H will be discussed in conjunction with Mu (13). For all 2 "DL" – 4-digit "W"s, the short "ML" appears ONLY ONCE(!) in "B1" ("W" = 1122). This "B1" skew towards longer lengths (i.e. pink backgrounds) pervades most "W"s; will haunt us; and will "B"e a main focus of Mu (10); Mu (11) & Mu (12). Grey backgrounds (i.e. shortest lengths) will lessen as we move from Mu (5) to Mu (9). How "ML"s = 8 decline and fall from 2 "DL"s thru 4 "DL"s will be shown in Mu (14). In regards some of the less self-evident column labels on the various charts: M and T values are total measure counts, usually (but not always) twice those of ["A1" + "B1"] or ["C1" + "D1"]. For Mu (5), M is longer than T in 7 of 13 (54%) samples; equal to T in four samples; and shorter than T in two (15%) samples. This is better seen in the M - T col., where M < T samples have orange backgrounds; and M = T samples are lit in yellow. Cols. "B1" - "A1"; "C1" - "B1" and "D1" - "C1" tell the "ML" differences between the specified col.s. These differences, while not of great variety (or even interest) in this table, will become more so as "DL"s increase, and the data changes in interesting ways (see 3.51 and Mu (9)). The cerulean "Max[imum] Diff[erence] col. tells the largest difference between the "ML"s of an ["A1" + "B1" + "C1" + "D1"] . For 2 "DL"s, these differences are not large, being, with one exception, less than 12 meas. This col., too, will become of greater interest as "DL"s increase. Percentage col.s (at the right) display data similar to Mu (3) (line 1, Mu (5) is the same as Mu (3), line 142; etc.); i.e. but the data is now delimited to single "W"s within 2 "DL"s, as opposed to a general pool. Mu (5)'s disentangled sort shows the contrariety that underlies attempts to create ["A1" + "B1" + "C1" + "D1"]s using percentages. As example: "W" = 1211 shows seven samples. "A1"s, "C1"s & "D1"s "ML"s are constant. "B1" "ML" values increase. For the % col.s, short (grey) cols. vary from 22.2% to 15.4%; while the long (pink) col. increases from 33.3% to 53.8%. Not only do two col.s vary (as opposed to "B1" "ML"s only); but the variability is in opposing directions. Agreed, the percentages quantify relative proportions that may be potentially useful for "shaping" an understanding of the total movement; but percentages fail Occam's razor as the simplest way to think about pattern construction. How one might create outer forms is a matter entirely different from how one might perceive or try to convey such forms; and one should not confuse, let alone conflate, these activities. The far simpler method of minuet movement construction would be to substitute "real" "ML"s within col.s, as opposed to worrying about percentages of a total; and from now on, we eschew chewing on percentages; multiplicative and additive constants; repeated patterns, and permutations. 3.43 The 3 & 4 "DL"s, plus "irregular" minuetsMu (6) displays the 3 "DL"—"W"s. Mu (7) displays the 4 "DL"—"W"s. Mu (8) displays the "irregular" "DL" s. Remember that any perceived regularities within "W"s is due solely to the order of the data sort! A 3 "DL"—"W" contains three different "ML"s ("1", "2" & "3"), distributed across the four "slots" ["A1" + "B1" + "C1" + "D1"]. Given that the repeat locus is one of most salient aspects of 3 "DL"s, these "W"s are displayed by placement of the repeated "ML"s, (enclosed in green oblongs), starting with cols. ["A1" & "B1"] ("W" = 2213); then ["A1" & "C1"] ("W"s = 1213, 1312 & 2321); ["A1" + "D1"]; etc.; ending with repeats in ["B1" & "D1"], and ["C1" & "D1"] (the null space between lines 42 & 43 signifies that there are no "W"s with repeats in ["B1" & "C1"]. A 4 "DL"—"W" contains four different "ML"s ("1", "2", "3" & "4"), distributed across the four "slots" ["A1" + "B1" + "C1" + "D1"]. Given no repeated "ML"s in a 4 "DL"—"W", these "W"s are now arranged in Mu (7) by placement of the shortest and longest measure lengths, first in col.s "A1" & "B1"]; then ["A1" + "D1"]; ["C1" + "D1"]; and ["C1" + "B1"]. "Irregular" Minuets are ordered by minimum difference from one to six. Mu (6) shows 11 different "W"s; plus six "irregular" 3"DL" -- "W"s at the bottom (superscripts mark possible interconnections). Unlike 2 "DL" -- "W "s (which allowed two possible types {three of one "ML" vs. one of the other} vs. {two of each "ML"}), 3 "DL"s are of one type only i.e. in a 3"DL", one of the three "ML"s MUST repeat once. The remaining two "ML"s may not repeat. While Mu (5) shows no permutations, Mu (6) shows six permutations (white letters a through f), in small blue boxes, placed between the "D1" and M col.s). Mu (7) shows only six different "W"s. Given fourteen possible 2 "DL"s (see Mu (10)), vs. thirty-six 3 "DL"s, vs. twenty-four 4 "DL"s, these nominal differences appear large; but from a percentages perspective, they are not, i.e. H uses 36% (4/14) of all possible 2 "DL" "W"s; 31% (11/36) of all 3 "DL" "W"s; and 29% (7/24) of all 4 "DL" "W"s. Mu (8) groups the 22 so-called "irregular" minuets. Ten of these can be easily associated with standard four-digit "W"s (see the rightmost col. labeled "W"). An additional seven "irregulars" are either 5 "DL"s (lines 16 - 20), or 6 "DL"s (lines 20/21). Five "irregulars", ostensibly of 3 or 4 "DL"s, cannot be associated with standard "W"s i.e. lines 2, 3, and 8, all with "alt"s, and with line 8 crossing the 3 - 4 "DL" barrier; the oddities of lines 12 and 13; and line 10, a nominal 4 "DL" that fills all the slots of line 1 (a 2 "DL"), but unlike line 1, wobbles up and down in terms of section lengths. Clearly, 22/177 "irregular" samples (12.4%) is insufficient for us to toss out the 2 - 4 "DL"—"W"s concept; but 22 samples is a sufficient number to cause us to wonder if there might be an alternative method that better categorizes all minuet movements (for which, see section 3.6). * 3.5 A more subjective approachMuch, much, time could be spent considering changes within, and between, "DL"s, and their concomitant "W"s; but the detail would suffocate, and to what advantage? It would be better, even imperative, to try to determine IF anything of "DL" or "W" might have actually meant something to H. That we have found a method, or methods, that appear to arrange the data in fairly rational categories does NOT mean that this is how H thought. Nature is full of phenomena where it is unlikely that the object of surveillance has a clue as to what the observer imagines! The succession of the various chambers of the Nautilus tends to approximate a Fibonacci series; but one has to doubt if your average Nautilus has ever heard of Mr. Fibonacci (I exempt the college-educated Nautilus, of course). FULL DISCLOSURE: while I stipulate that it passeth (my) understanding how anyone could believe that H would be totally unaware of using four different types of "DL" arrangements, let alone the internal "W"s utilized, for the sake of argument, assume that H was always catatonic when writing minuet movements. So the question is: can we find aspects in the "internals" of the "DL" or "W" data that provide additional support for the idea that H may have been thinking (in some way) about "DL"s or "W"s? For these, we turn to: 3.51 DistributionsIn regards "DL" internals: Inspect Mu (9) — appropriately clad in black and white striped prison garb. Starting from the left of table, after the line count, there is a col. of "ML"s. There are then four alternating white/black/white/black panels (each of four col.s). The first (white) panel contains all the data for all "A1"s ONLY; the second (black) panel contains all the data for "B1"s; then "C1"s (white panel), and "D1"s (black). Within each panel are col.s for 2 "DL"s; 3 "DL"s; etc. Minimal summary statistics are found at the bottom. We know that "ML"s for "B1" and "D1" tend to be longer than "ML"s for "A1" and "C1". What we may have missed is that, as "DL"s increase, there is a tendency for "ML"s to also increase, both in range and value. Notice how, as "DL"s increase within each panel, the data "splatter" (within each panel) and dribble further down the bib; i.e. "A1" maximum "ML"s progress (across "DL" col.s) from 14 to 26; "B1" maximums increase from 28 to 58; "C1"s proceed from 16 to 26; and "D1"s from 26 to 48. So the questions are: Does this "qualitative difference" truly exist? Or is this just a (mistaken) impression on our part? And if true, why might that be? To answer the former question, inspect the two tables below. The first compares median "ML"s across "DL"s (medians are preferred to means, as medians are less influenced by data "outliers"). The second compares how many different "ML"s are used. In regards the first table, as "DL"s move from 2 to 4, "A1" medians increase from 8 to 10 to 12 to 13. "B1"s, "C1"s and "D1"s also increase with "DL"s ; and means (see Mu (9) at the bottom) generally confirm the medians.
Comparison of Median "ML"s vs. "DL"s
Why should median "ML"s differ across "DL"s? Differences in "ML" ranges, medians and/or the number of "ML"s, can NOT be ascribed to differences in the number of samples within 2, 3 or 4 "DL"s. Ranges, medians, and values can be independent of the total number of ["A1" + "B1" + "C1" + "D1"] samples. There is no reason why 2 "DL"s (as example) must draw "ML"s from a narrow pool. I.e. if 2 "DL"s utilized "ML"s selected from a large range, median 2 "DL" "ML"s could be as large as those of 4"DL"s. Therefore, the proper question may not be why do "ML" medians increase with "DL", but rather, why are there no 2 "DL"s with massive "ML" spreads? What is it about how "ML"s are chosen that tends to keep smaller "DL" "ML"s more tightly "packed"? Our next table shows that, as "DL"s move from 2 to 4, the number of different "ML"s increase. How many different "ML"s used in:
Why? We most probably greatly overestimate how many different "ML"s are needed to form (65) 3 "DL"s, or (77) 4 "DL"s. Five different numbers, each used once, taken 4 at a time, allow for 120 permutations; but we can not just permute all four numbers, as "B1"s and/or "D1"s are usuallylonger than "A1"s and/or "C1"s. A less simplistic model is required. Assume a group of six long "ML"s (used in "B1"s and/or "D1"s ONLY); + three short value "ML"s(for "A1"s and/or "C1"s ONLY). The 6 longs, taken two at a time, no repeated "ML"s allowed within a single pair, provide 30 total combinations. The three short values, taken two at a time, no repeated "ML"s allowed in a single pair, provide 6 combinations. Thirty long combos (for "B1" / "D1"), paired with six short combos (for "A1" / "C1"), allows for 180 varieties of 4 "DL"s. In other words, a total of nine different "ML"s suffice to create more than twice the number of 4 "DL" ["A1" + "B1" + "C1" + "D1"]s of Mu (7). Why, then, does H use so many more than 9 "ML"s in the 4 "DL"s ? 3.52 Why so few "W"s?Turning now to our question (posed just prior to 3.51) regarding H's use of "W"s. Twenty-one "W"s fashion our data, but combinatorics tell of 74 conceivable "W"s. Why did H use so few? And what of the "curious incident" of the "W"s that "did nothing in the night-time"? (pace Silver Blaze). Tables Mu (10), Mu (11) and Mu (12), all follow the same format i.e. they first show all possible "W"s within a specific "DL". We then enclose in blue oblongs those "W"s used by H. We propose six "rules" and see which provides a "best fit" to explain why H might have used the "W"s he chose. Mu (10) shows the 14 possible 2 "DL" – "W"s i.e. two lengths (labeled 1 and 2), taken four at a time. As we know, these are of two "kinds", and are displayed as such. Mu (11) shows the 36 potential 3 "DL" – "W" combinations; i.e. three lengths, with one length repeated; and Mu (12) shows the 24 possible 4 "DL" – "W"s, each of four lengths, each length used once. The six rules (identical for all three tables) are: I. B always greater than A: i.e. there is a rule for every possible AB, AC, AD etc. pair, arranged in some "guess" as to the likelihood of "success" for the rule. To help clarify, for each rule, the color of each 4-digit "W" changes to reflect whether a "W" is, or is not, in accord with the rule. If the "W" is in accord (i.e. for 2 "DL" – "W" = 1112 rule I: as "B" is NOT greater than "A","W" = 1122 appears in an almost invisible "grey-black"; whereas for 2 "DL" – "W" = 1211 rule I: "B" IS greater than "A", that "W" = appears in "black" type)."W"s used by H always remain encased in blue no matter if the "W" is, or is not, in accord with whatever rule. Finally, for each specific rule, we show how many violations or omissions are to be found for that rule i.e. for 2 "DL" rule I: we show 1 violation (as H uses "W" = 1122, which is not allowed by the rule (as in this case B = A)); and 1 omission (as H does NOT use "W" = 1221, which IS allowed by the rule). As the table below shows, no rule works anywhere close to 100%, but some clearly work better than others, i.e.:
but we are not looking for a perfect fit. We only need show that overall "W" choices are not random; and the table above clearly shows that aplenty. After sorting our data by "DL"s and "W"s we find H uses different ranges per "DL", with different numbers of "ML"s, in ways that can not be explained only by the number of samples used per "DL"; and he limits his use of different "W"s across different "DL"s. Those facts allow us to suggest that at least at some "visceral" level, H had a "feel for" "DL"s and "W"s; and therefore "DL"s and "W"s are valid concepts for the better understanding of our data. 3.53 The Scattering of "W"sThe table below shows that perhaps the 2"DL" –"W"sdo not lead neatly in to the 3"DL" – "W"s, nor they, in turn, to the 4"DL" – "W"s. This is surprising, given the prominence of what appear to be substitutions found within individual "W"s.
Monotonic Comparison of "W"s, across "DL"s
Perhaps there is an alternative method to arranging the "W"s that is less strictly quantitative, e.g. 1212 and 1324 could both be instances of short-long-short-long? Note that we are not rejecting the 2, 3 & 4 "DL" classification. That taxonomy is obvious, and clearly descriptive. However, there are other "forces" at work beyond "DL"s & "W"s. But before all hope is lost, Mu (13) may show that all of this was not quite an exercise in futility. * 3.6 Step-Functions; an alternativeAnother method for grouping minuet movements is based on the overall movement's "shape". Mu (13), in two columns, shows four different "step functions", labeled: "Step Ups"; "Single Silos"; "Slopes"; & "Double Silos" (irregular "W"s are NOT included!!!!). Numbers to the left of a function show the digits of a "W". Labels are fairly self-evident i.e. a "Step Up" starts at a low level (1), and moves up (to level 2). "Single Silos" have a bump, or depression, someplace in the middle of the "step function". "Slopes" imply the ability to draw a descending tangent across three segments of a movement. "Double Silos" have more than a single bump or depression in the middle of the "step function". One thing that intrigues is that, with the exception of "Step Ups", all other "function"s cross the barriers of 2 - 4 "DL"s. In other words, "Single Silos" contain both 2 & 3 "DL"s. "Slopes" contain both 3 & 4 "DL"s. "Double Silos" contain 2, 3, & 4 "DL"s!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! In terms of distributions: the great majority of "W"s are in the two types of silos (15/21 = 71%); and the number of samples (149/157 = 95%) is even more lopsidedly distributed i.e. Distributions of "W"s vs. Samples
The three most frequently used "W"s in Double Silos, i.e. 2413 (36 samples); 1423 (23 samples); and 1312 (22 samples); account for 81 (63%) samples. Even though some of these include "reps", these three "W"s are either H's default shapes (I have to write a minuet in a hurry and do not have time (or patience) to experiment); or H actually liked the shapes; or perhaps some other set of numbers (such as the sums of M & T) needed "filling in" (for which see 3.8). Viewing "W"s as "step-functions" obviously provides supplemental, perhaps even a fundamentally different perception of the "W"s. For example, we have heard (a * 3.7 The Decline and Fall of 8Table Mu (14) shows an initial line, every slot filled with "ML"s = 8. There follow the 2 "DL"s, in their two configurations (lines 2 - 11). Starting line 12 thru line 46 we find the 3 "DL"s; followed by the 4 "DL"s, (lines 47 - 125), and the fact that the majority of samples without ML = 8 occur in the 4 DLs only reinforces this idea of a "qualitative" difference in the 4 DLs. That leaves 52 (out of 177) or 29% of total samples without any "ML"s = 8. (The details are: three 2 DLs (21%); + 19 3DLs (27%) ; + 26 4 DLs (31%) ; plus 4 irregs not counted elsewhere = 52. Remember the table arrangement is purely a result of the sort. It is not indicative of the order in which H made changes to the numbers of "ML"s = 8. That being said, the table is very provocative in showing how "ML" = 8 becomes less and less prominent. In addition, the table hints at a substitution game; i.e. if "ML" = 8 is no longer the shortest value, what is?; and is H just plugging in different substitute values for "ML" = 8, resulting (obviously) in expanded M &/or T lengths?13 If we wish to compare repeated "ML" = 8 in 2 vs. 3 "DL" – "W"s, we can not count raw numbers of total "ML"s = 8 (as the rules have changed from 2 vs. 3 "DL"s); but we can ask how many individual lines show repeated "ML"s = 8. For Mu (5), that number is 10/13 lines (77%). For Mu (6) it is 34/65 lines (52%). In order to reach the 2 "DL" 77%, we must include the 16 lines of Mu (6) containing a single "ML" = 8s. That raises the total to 50/65 lines, or the same 77%, having at least one "ML" = 8, again suggesting the possibility of "ML" substitution. So we are faced with our usual chicken and egg problem of: did H substitute (for ML = 8) other MLs in order to increase M &/or T? Or was H doing something different? 3.8 ExhaustionAlong the way, we have come across various aspects calling in to question the way we have been looking at things i.e. was H truly building things "up" from ["A1"+ "B1"+ "C1"+ "D1"]s? Or thinking (primarily) first in terms of various sized "DL"s; and then "W"s? Or thinking large shapes only? In addition, the unadulterated truth is that minuet movements are made up of two large sections aka an M and a T. Are we certain that H was not using a "top down" approach i.e. first decide what total length for M; then a length for T, in some fashion taking in to account the ratios or relations between the two; and then deciding the internal partitions of M and T. Mu (15) addresses this question. In regards M and T values, in Mu (6), we find M is longer than T in 52 of 65 (80%) samples; and shorter than T in 13 samples. M and T are never equal. Compared with 2 "DL" – "W"s, M and T values in 3"DL" – "W"s skew towards longer Ms. "A1" < "B1" for 97% of samples. Why is M so often longer than T? Various possibilities come to mind: H deliberately chose to make M > T to counterbalance the short : longs of "A1" : "B1" and "C1" : "D1". Or perhaps M > T results from "B1"s almost always being the longest section. 3.81 M and T pairingsOur final three tables are primarily for "bookkeeping" — i.e. if you wish to know what length M is paired with what length T; please see Mu (16). For what length T is paired with what length M; please see Mu (17). For sorts on M + T, Mu (18). 3.9 SummaryWe have presented a highly combinatoric taxonomy of H minuet movements. As indicated by the epigraph from the introduction2, our Gallimaufry14 provides only some of the materials needed to finish our building; but we have uncovered certain salient aspects that can be grouped under the headings: What do we know? What do we think we know? What do we know that H did not do? What can we surmise H might have done? What will we probably never know? And so we come to the end of this (mis-)adventure, in the hope that at least a few will realize that, far from being "all the same", the humble "boring" Haydn minuet was in fact an incredible laboratory of experimentation. If only the performance today would remove the blinkers of "schlumpy" tradition, and wake up to that fact. Paul Zukofsky |
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