|
The variety observed in our map does not indicate lack of structure. For example, blues are almost always shorter (have fewer measures) than greens, which are usually longer than yellows. Reds tend to be longer than yellows, but probably shorter than greens. The length of M (i.e. denoted by the black vertical between the green and yellow) can not be predicted from the length of the blues (i.e. as an example see lines 6 - 11 where the blue length is constant, but the greens wobble about; or lines 3 - 5, where the longest blue (line 4) has a short green; etc. etc.). While there are duplicate M lengths (i.e. the left black verticals align), there are fewer duplicate M lengths than is the case for the blues by themselves. We could continue, but in order to make progress, we require actual "ML"s.
Table SQ (1) provides these.
The leftmost “Line #” column is a line count for ease of reference. The "Opus", "No." (within an Opus) & Mov(ement)6
cols. are self-explanatory. The "Key" col. contains three entries (i.e. for line #1 -- E♭:E♭:c). The leftmost E♭is the key of the entire Quartet (in this case, E flat Major). The next provides the key of the M (the same E flat Major as the entire Quartet). The last item provides the key of the T (in this case the relative (c) minor).
The 6th col. provides H’s tempo designation as provided in the Landon edition.
The numbers in columns "A1" & “A2” etc. provide the "ML"s for both the initial statement (i.e. up to the first repeat bar), and for the repeat, of the first section of M; i.e. for line #1, Op. 0, the “A1” section contains 7 measures, and “A1” is repeated in “A2”. We use separate "A1" & “A2” columns (rather than counting both presentations as a single “A” column) as H does not always repeat each "S"; or he sometimes varies the length of the repeat. The two "B" columns provide the "MLs" for both presentations of the second section of M (i.e. up to the last bar of M). For line #1, the "ML" for both “B”s = 21. The “B” (and “D”) columns are color reversed, so as to make the table vertically clearer. A thin border col. separates “B” cols. from “C” cols. The "C" cols. provide the number of "ML"s of the first section (up to the repeat bar) of the respective T. "D" cols. provide the "ML"s in the second section of each T. "D+" and “alt” (for “alternativo”) provides space for codas, and/or other anomalies. Another border col. follows, followed by the dC (“A3” & “B3”) measures. Each Opus is delineated.
Initial impressions of SQ (1) are:
there is an almost overwhelming variety in the raw numbers;
there are many unexpected "ML"s, both odd and even;
not all minuet movements follow the same gross form (i.e. not all data slots are filled with "ML"s; in some cases, extra slots are utilized);
there is no consistent, or at least obvious, patterning of the components across cols. “A1→B3”;
there may not be many repeated patterns across “A1→B3”;
if there are repeated patterns, they do not occur within an opus.7
SQ (2) clarifies these impressions:
Column headings are identical to those of SQ (1), but the SQ (1) data has been resorted monotonically, sorting first on the lengths of the “A” sections; followed by a sort on the “B” sections; followed by a sort on the “C” & “D”. etc.
i.e., when reading down col. “A1”, one first finds those movements whose first “A” section consists of 7 bars; followed by all movements whose first section consists of 8 bars; followed by the 9 bars, the 10 bars, 11, 12, 14,16, etc. Col. “A2” (and the final “A3” col.) usually just repeat these numbers. Subsequently, within those movements whose “A” sections consists of 8 bars, one finds Ms whose “B” sections comprise 12, 16, 18, 20, 22, 24, 26, 30, 33, 42, etc. bars. Thereafter, in the group of movements with “A” and “B” sections of 8 and 12 bars respectively, one finds increasing measure counts for "C" & "D" sections; etc.
This numerically reordered table shows that only 3 schemes (marked by red vertical highlights) are repeated:
i.e. the “A1→B3” succession =
8/8/12/12/8/8/14/14/8/12 is used twice, i.e. opus 2, # 4 and opus 20, # 6; (see line #s 3 & 4)
the succession 8/8/20/20/8/8/8/8/8/20 is used three times: op. 2, # 6; op. 9, # 5; and op. 42; (see line #s 14, 15 & 16)
and 16/16/20/20/10/10/18/18/16/20 appears twice -- op. 2, # 4; op. 17, # 3. (see line #s 68 & 69)
Therefore, out of 80 minuet movements, 76 (95%) utilize different arrangements.
Clearly, these HSQMs are not “all the same”.
The previous tables have already shown that, for some movements, some “A1→B3” columns are blank; for other movements, additional columns are filled; i.e. the number of sections utilized per movement, varies.
This can be more clearly seen in SQ (3),
which adds a color-coded col. labeled “Sections” to highlight the number of “S”ections used in each movement; and uses horizontal gray backgrounds of differing hues to delineate the non-"10-S" movements, starting with the "7-S" and "9-S" at the top, followed by all the "10-S", ending with the "11-S".
Most SQ movements (68/80 samples = 85%) are nominally "10-S" (“A1→B3”); but there are also two "7-S" movements, (“C1C2D1D2” are “missing”, and an “alternativo” is substituted); seven "9-S" (a “C2”, “D2”, or “A2” is “omitted”); and three "11-S" (a "10-S" plus what we have labeled a "D+") .
We note that "10-S"s exist throughout the entire SQ collection. The "9-S"s begin with Op. 9 and stop at Op. 55. The "7" & "11-S"s are exclusively in Op. 74, 76 & 77.
One might assume that the designated number of “S” is incontrovertible, as “S” (as defined in terms of how many “A”s, “B”s, etc.) would appear to be one of the most obvious aspects of outward form. Actually, “S” is somewhat meaningless, perhaps even misleading, as we shall later demonstrate.
1.3 How many Different Lengths?
Previous tables reveal that some movements contain only a few different "ML"s, repeated many times across "A/B/C/D" etc. Other movements have no repeated "ML"s.
SQ (4) groups movements by their number of "Different Lengths" ("DL"s), as shown by the additional “Diff Lengths” col.
“DL”s are to be understood as follows --
consider:
line 1, where "ML"s for "A1/B1/C1/D1" = 8/12/8/8
line 10, where "A1/B1/C1/D1" = 8/12/8/14
line 40, where "A1/B1/C1/D1" = 7/21/12/20
line 78, where" A1/B1/B2/C1/D1" = 18/36/35/16/30
All these samples are "10-S", but line 1 (op. 20 #4) contains only 2 different lengths across all sections -- a length of 8 bars, utilized seven times; and a length of 12 bars, utilized thrice; hence two different lengths, or "2-DL". Note that we do NOT claim that the music of all bars is the same. We speak only of the size and quantity of the "ML"s. Similarly, op. 2 #4 (line 10) uses 8, 12 and 14 bars = three different lengths; hence a "3-DL". Op. 0 (line 40) uses four different lengths; hence "4-DL". Op. 20 #5 (line 78) would be a "4-DL" but for the fact that H elides the "B2" section with "C1", thereby causing a change in the lengths of "B1" vs "B2", which difference, in “sly circumspection”, causes us to categorize this sample as a "5-DL".
The majority (41/80 = 51%) of movements are "4-DL"s. There are 30/80 (37.5%) "3-DL"s; as well as five "2-DL"s, three "5-DL"s and one "6-DL".
Later we will use a concept labeled “W” (as in "W"eights), to identify "DL" subclasses.
Note that H varies "DL" very early in the game: opus 1 contains both "3-DL"s and "4-DL"s; opus 2 contains "2", "3" and "4-DL"s; a "5-DL" appears as early as opus 20; the single "6-DL" is a later experiment.
The choice of "DL" does not appear to depend on the number of "S" i.e. "10-S"s exist with all "DL"s except "6-DL"s; "4-DL"s appears in all different "S"; "3-DL"s do not; "2-DL"s only occur in "10-S". etc.
Heretofore, we have sorted our data based on each individual sample -- i.e. how many different “DL"s, or “S"s within a single movement. We now ask if all “A” or “B” etc., across all movements, have certain measure lengths in common, no matter the “S” or “ML” classification? In other words, what bar-lengths does H use; where does he use them; and how many times does he use them within the 790 sections of our 80 SQMs?
SQ (5) presents our data “ANEW”.
This table is read as follows: the leftmost col. is a line count. Next is a col. of 36 different “ML”s i.e. reading down the col., the 7-bar lengths; the 8-bar lengths; the 9-bar etc. etc. These numbers, and all following columns, are separated into quartiles of 9 numbers each. The “ML”s col. is followed by the usual “A1→ B3” cols. Numbers are splattered within these cols. The splatter is to be read as follows:
The number “1”, which occurs in the “A1” col. on the "7-ML"s line, means that there is a total of one such length in all “A1"s, in all the SQMs. The number “34”, which occurs in the “A1” col., on the "8-ML"s line, means that there are a total of thirty-four "8-ML"s, in all “A1”s, in all the SQMs. Cols. to the right of “A1” are read in similar fashion. Faint “0”s fill spaces where specific lengths do NOT occur in certain “A”s, “B”s, “C”s, etc.
To the right of the “B3” col. there is a ”line total” col. which, for line #1, tells us that the "7-ML"s occur a total of three times in all SQMs; for line #2, that the "8-ML"s occur a total of 196 times in all SQMs, etc.
Percentages of occurrence are given in the “% of total section lengths” column, i.e., the "7-ML"s occur three times out of 790 segments or 3/790 = 0.4%; whereas the "8-ML"s occur 196/790 = 24.8%.
The “# quartets with length” col. is a measure of ubiquity; i.e. it tells how many SQMs contain a specific "ML" -- i.e. the "7-ML"s appear in only one SQM, whereas the "8-ML"s appear in 57 different SQMs.
At the bottom of each column we provide the total number of occurrences for that col., and beneath that, a list of how many "ML"s are to be found in each col.
Odd number "ML"s, and where they occur, are highlighted in pink. The five highest occurring "ML"s are highlighted in varying shades of green.
Some immediate observations:
1-- DIFFERENT LENGTHS: the “ML”s col. shows a total of 36 "ML"s, ranging from 7 to 100 measures. 13 of these are odd numbered lengths. 5 are prime numbers. While the remaining 23 "ML"s are duple, only 11 of these 23 are divisible by 4 (i.e. in to 4- bar phrases). Put another way, out of 36 "ML"s, only 30.6% (11/36) adhere to one of the most musically-sacred purported “conventions”.8
Only five (out of 36) "ML"s (8, 16, 24, 32 & 96) are divisible by the almost-equally-sacrosanct number 8.
2-- DISTRIBUTIONS: "ML"s are not uniformly distributed across “A”, “B”, “C” & D”. As shown at the bottom line of the table (# of "DL"s per column) “A” & “C” utilize fewer "DL"s (“A” = 11 lengths; “C1” = 9 vs. “C2” = 7) than do “B” & “D” (“B” = 25 or 26; “D” = 19). Certain lengths rarely (if ever) occur in some columns. As examples, 8 never occurs in “B”, whereas “A” and “C” are heavily overw-eight. "ML"s of “B” & “D” tend to be less divisible by 4 than are the "ML"s of “A” & “C” i.e. 59/80 (74%) “A1” samples, and 56/78 (72%) “C1” samples are divisible by 4; whereas only 39/80 (49%) “B1” and 42/78 (54%) “D1” samples are so divisible.
"ML" = 10 appears most frequently in “A” & “C”, whereas when confronted by a 20-bar segment, there is a very high probability that it is from a “B” or “D”. "ML" = 24 never is a “C”, and is hardly ever an “A” or “D”, altho "ML" = 12 (i.e. half of 24) lives primarily in “A” & “C”, and somewhat less in “D”; etc. etc. Odd numbered lengths, taken as percentages of the different number of lengths in the respective columns, seem more uniformly distributed (over the 16 Quartets in which they appear) (i.e. “A1” contains 3 different odd numbers out of 11 lengths = 27%; “B1” has 5 out of 25 = 20%; “C1” has 2/9 = 22%; and “D1” 4/19 = 21%).
3-- FREQUENCY: certain lengths appear far more often than others. Some numbers that we might expect to appear very frequently, do not. For example, while myth (aka theory) has it that the eight-bar phrase undergirds everything, "ML" = 8 only constitutes approximately 25% of the entirety, although "ML" = 8 is distributed across 57 of the 80 SQMs.9
"ML" = 16 appears far less than "ML" = 8. "ML" = 12 appears more than "ML" = 16, both in terms of percentage and distribution across SQMs. "ML"s = 10, 14 & 20 are comparable to "ML" = 16, in terms of frequency of use. 32s are not worth discussing; etc. etc.
4-- DISPERSION: We noted above that, as a percentage, "ML" = 8 appear not to be all that ubiquitous. One could, however, imagine a different measurement of ubiquity. Rather than the preponderance of a number, we could ask about its dispersion -- i.e. instead of asking the total times X appears in every section of every movement, one could ask how many times X appears in any movement. The rightmost col. of this table provides these numbers, and they tell a tale far different from the total number of appearances. For example, "ML" = 7 appears only once in the 80 SQMs, whereas "ML" = 8 appears at least once in 57 (out of 80) different quartets. This is 71%, and not the ca. 25% we saw above in the basic frequency count. This may help account for our impression of the prominence of the eight-bar phrase. One might also consider (as an example) 22-bars, which, while in only 3.5% of total sections, is dispersed among 12 out of 80 quartets (i.e. 15%), etc.
However, while all of the above distributions are of great interest, and perhaps use, it is problematic to think of these lengths in isolation, as one normally needs to pair lengths in order to form an M or T; so we next ask -- are there observable patterns in how H pairs these lengths?
1.5 “A1:B1” and “C1:D1” pairs
SQ (6) (for Ms) & SQ (7) (for Ts) present distributions of the pairs that form an M or T, the data reduced to “A1:B1” and “C1:D1” pairs, only.
SQ (6) &SQ (7) are to be read as follows: the leftmost column provides "ML"s for “A1” (SQ (6)) or “C1” (SQ (7)). The top row provides "ML"s for “B1" (SQ (6)) or “D1” (SQ (7)). A number within a cell at the intersection of a line and a column provides the number of times a particular pair of values is used. For example, the pair 8:8 never appears in SQ (6) "A1:B1", but is used 9 times for “C1:D1” (SQ (7)). For both tables, line sums (at the far right of the table) replicating respectively the “A1” and “C1” cols. of SQ (5), indicate how many times a length appears as the first term of a pair i.e. for SQ (6), "ML" = 8 occurs 34 times as the first term of an “A1:B1” pair; "ML" = 10 appears eleven times etc. Column sums (at the bottom of the table) replicating respectively the “B1” and “D1” cols. of SQ (5), indicate how many times a length appears as the second term of a ratio. For example, SQ (6) shows that "ML" = 14 is used once as the second term of an “A1:B1” pair; whereas 24 is used ten times as a second term.
Colored cells provide ratio “boundaries” -- i.e. cells located between green and aqua colored cells have ratios between 1:1 and 1:2. Cells located between aqua and pink colored cells have ratios between 1:2 and 1:3. Put another way, if an “A1:B1” ratio falls between the green and aqua “boundaries”, “B” has a length longer than that of “A”, but that longer length is less than twice as long. If an “A1:B1” ratio falls between the aqua and pink “boundaries”, “B” has a length at least twice as long as that of “A”, but less than three times as long.
There are 50 discrete “A1:B1” pairs vs. 43 “C1:D1” pairs. 20 of these pairs are common to both SQ (6) & SQ (7), leaving 30 “A:B” vs. 23 “C:D” unique pairs.
The maximum number of repeats for any “A1:B1” pair is 4 times (see line "ML" = 8 for several examples). The maximum repeats for “C1:D1” can be as many as 9.
The most frequent in-common pairs across SQ (6) and SQ (7) are "ML"s = 8+12, and 8+16. The most frequent “A:B" not-in-common-with ”C:D” pair is 8+18 [4], followed by 10+24 [3]. The most frequent “C:D" not-in-common-with "A:B” pair is 8+8 [9], followed by 8+14 [4].
The SQ (6) ratio bands shows far fewer pairs between the 1:1 and 1:2 bands; i.e. far fewer pairs are to the right of the pink cells, than is the case for SQ (7). SQ (6) also shows far more pairs (a total of 25) whose ratios fall on or beyond 1:3, than is the case for SQ (7). In other words, the length differences between “C1” & “D1” are often less than between “A1” & “B1”.
There may be a tendency for “C1:D1”s to use more “simple” ratios i.e. there are 21 “C1:D1”s that fall directly on a colored cell, as opposed to 15 “A1:B1”s.
Not every “A1” or “C1” is paired with every “B1” or “D1”. Indeed, the majority of cells are empty. i.e. for SQ (6), out of 275 cells, only 50 (18.2%) cells are filled. For SQ (7), only 43 of 171 cells (25.1%) cells are filled.
Given that the majority of SQ (6) & SQ (7) cells are empty, one could argue that having the maximum number of “A:B” or “C:D” cell combinations is NOT H’s primary priority. Can we envisage a system that would account for why certain combinations seem to be avoided?
1.6 Partitions Galore
The "A1:B1" or "C1:D1" lengths, plus their specified repeats (or lack thereof) of SQ (6) & SQ (7) , sum to form Total Measure Lengths (TMLs) for an M or T.
What are these TMLs? What are their frequencies? If used more than once, are these TMLs partitioned in the same way? If not, what Partition Types (PTs) exist, and what are their frequencies?
SQ (8) provides answers.
The first col. is a line count. TML (Total Measure Lengths) are the number of total measures for either an M, or a T; i.e. on lines #6 & 7 where TML = 40, we find all Ms or Ts whose partitions sum to 40. TMLs highlighted in “gold” are common to both Ms and Ts. ΔM is the difference between the TMLs for M; i.e. the difference between 40 and 44 measures, is 4. ΔT is the difference between TMLs for T.
"Occ" i.e. occurrences, provides the number of times the specific TML is utilized. i.e. TML = 40 (lines #6 & 7), occurs 11 times -- in five Ms and six Ts. The highest “Occ”s (16, 11 & 10) are highlighted in pink.
PT (Partition Type) provides the number of different partitions used for the specific TML; i.e. TML = 40 is partitioned in only 2 ways (10/10/10/10; 8/8/12/12). The greatest “PT”s (6, 5 & 4) are highlighted in pink.
The M and T partitions cols provide specifics i.e. TML = 40 (for Ms), is partitioned once as 10/10/10/10; and four times as 8/8/12/12. The parenthetical (4,3) next to the 8/8/12/12 tells that the four 8/8/12/12 partitions are paired with three different T partitions. The T partitions col. uses the same scheme -- i.e. for TML = 40 Ts (6,6), there is only one T partition (8/8/12/12) utilized six times, each time paired with one of six different M partitions. As PT 8/8/12/12 is used for BOTH Ms and Ts, we highlight that aspect with thin vertical purple bars. Partitions are ordered by their difference from an equal-value partition i.e. TML = 56 (lines 18 - 22) first displays the equal value partition 14/14/14/14 (used only once, as a T), and then displays a succession of bifurcations, each bifurcation becoming less “balanced”.
Most PTs follow a “standard grouping” of two equal values, followed by two other equal values. However, this is not true for all PTs, and the exceptions are highlighted i.e. equal value PTs (such as line 18) are highlighted by white text on a black background. PTs that have fewer or more than 4 parts have been given a light yellow background. “Standard grouping” PTs that contain an odd number are on a light blue background. A PT printed in light blue text placed on a light yellow background contains a odd number and a “nonstandard” number of parts.10
1-- HOW MANY TMLs: The TMLs col contains 37 different lengths, 29 in M; 25 in T. 17 TMLs (gold backgrounds) are common to both M & T.
2-- DIVISIBILITY OF TMLs: ΔM shows fairly consistent 4 measure increments. ΔT is less consistent; but the Δs of TMLs common to both M & T are, with only one exception, always 4-bar increments. Of the 37 TMLs, 24 are divisible by 4. These 24 divisible-by-4-TMLs are 4 times the numbers 8, 9, 10,11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 & 39.11
Of the 17 (gold backgrounds) common TMLs, 16 are divisible by 4.12
3-- FREQUENCY OF TMLs: The “Occ” col shows that, of the 37 TMLs, fourteen occur only once in M or T; six occur twice.
The 12 most frequently used TMLs (in descending order of use) are:
TML = |
Used X Times |
56 |
16 |
40, 48 & 64 |
11 |
32 & 68 |
10 |
60 |
9 |
52, 72, 80 & 84 |
8 |
88 |
6 |
These 12 different TMLs account for 116 (72.5%) of the 160 M & T segments. All of these most frequently used lengths are divisible by 4. With the exception of TML = 32, all are used for both M & T.
4-- HOW MANY PTs: SQ (8) shows 81 different partitions.13
51 different PTs are used in M; 48 PTs are used in T. 18 PTs are common to M & T.14
5-- FREQUENCY OF PTs: of the 81 PTs, 53 are used once; eleven are used twice.
The 10 most frequently used PTs encompass 67(out of 80) samples.
These 10 most frequent PTs are:
| TML = |
X times |
partitioned as |
40 |
10 |
|
32 |
9 |
|
48 |
9 |
|
56 |
7 |
|
56 |
6 |
|
60 |
5 |
|
64 |
5 |
|
44 |
4 |
|
52 |
4 |
|
64 |
4 |
|
72 |
4 |
|
The TMLs having the greatest number of PTs are:
| TML = |
# PT |
| 84 |
6 |
| 56 |
5 |
| 52 |
4 |
| 64 |
4 |
| 68 |
4 |
| 72 |
4 |
| 80 |
4 |
| 88 |
4 |
6-- "FLAVORS" OF PTs: Of the 81 PTs, two (TML = 96 & 100) are single-units (lines 64 & 68).
Seven (lines 2; 3; 4; 8; 9; 49; 61) (TMLs = 32, 36, 42, 84, 94) contain three-unit PTs. Two of these three-unit types are for TML = 42. Note that while TML = 84 is twice the length of TML = 42, the components of TML = 84 are not the double of TML = 42.
Lines 27, 55 & 56 (TMLs = 63 and 88) are five-units types.
The above twelve partitions occur once each. In this table, each appears on a yellow background.
The remaining 69 PTs are all of 4 units, but of different types. While most of the 69 are “standard groupings” consisting of two equal values, followed by two other equal values (i.e. 8/8/12/12), there are three single numbered PTs (i.e. 8/8/8/8 = TML = 32; 10/10/10/10 = TML = 40; 14/14/14/14 = TML = 56); and there are those with 4-units, but three different values (8/8/42/41 = TML = 99; 18/18/36/35 = TML = 107; 24/24/58/57 = TML = 163); leaving 63 “standard grouping” PTs of x/x/y/y.
Of these 63 “standard groupings”, 59 are short/longs i.e. 2 equal shorter values followed by two equal longer values.
The 4 exceptions are:
12/12/10/10 = TML = 44;
14/14/10/10 = TML = 48;
14/14/12/12 = TML = 52;
20/20/16/16 = TML = 72.
Many of the “standard grouping” partitions follow a pattern of “symmetric bifurcation”. Consider TML = 56 (line 18). 56/4 = 14; and line 18 displays a partition of 14/14/14/14; but we also find (on lines 19 - 21) 56 partitioned as 12/12/16/16 (each number is 2 measures + or - from the equal-division partition); 10/10/18/18 (an additional + or - 2 measures); 8/8/20/20; and even a 7/7/21/21.
As an additional example, consider TML = 44 (line 11). Here, sadly, one would hardly expect a partition of 11/11/11/11; but one does find 12/12/10/10 (+ or - 1 measure from the equal-division partition); and 8/8/14/14 (+ or - another 2 measures). As the shortest ML = 7, there can be no 6/6/16/16. H may therefore be exploring rule-bound partitions of a TML number, usually using 2 measure lengths to adjust internal proportions.
Interest in TMLs leads to our next (and final) question of this initial overview i.e. how do TMLs pair to form an M+T? In other words, is there a “higher level” equivalent for the previously considered "A+B" and "C+D" pairs?
Just as SQ (6) & SQ (7) showed (respectively) how lengths "A" paired with "B" (SQ (6)), and "C" paired with "D" (SQ (7)), SQ (9) shows how M pairs with T.
SQ (9) is read as follows: the leftmost column provides the Total Measure Lengths (TMLs) for M. The top row provides the TMLs for T. A number found at the intersection of a row and column tells how many times a specific pair is utilized. For example, the M:T pair 40:32 is used once. Line sums indicate how many times a TML is used for M i.e. TML = 56 occurs seven times as the first term of an M:T pair. Col. sums indicate how many times a TML is used for T i.e. 44 occurs 4 times as the second term of an M:T pair. Colored cells provide ratio “boundaries”.
Some observations are
There are 29 lines vs 25 cols., for a total of 725 “cells” or intersections. Numbers appear within only 68 (9.4%) of these cells. These 68 distinct M:T pairings compare with 50 distinct “A:B” pairs, and 43 distinct “C:D” pairs. In other words, H repeats "A:B" & "C:D" pairs far more than he repeats M:T pairs.
Distinct Pairs |
M:T |
“A:B” |
“C:D” |
68/80 =85.0% |
50/80 =62.5% |
43/80 =53.8% |
Only 8 TML M:T pairs are used more than once. These are:
TML for M =
|
TML for T = |
used X times |
partitioned as |
40 |
44 |
3 |
M = 8/8/12/12, T =8/8/14/14, twice.
M =10/10/10/10, T =8/8/14/14, once. |
52 |
32 |
2 |
M = 10/10/16/16, once. M = 12/12/14/14, once.
For both cases, T = 8/8/8/8.
|
56 |
32 |
3 |
M = 8/8/20/20, T = 8/8/8/8, thrice. |
64 |
60 |
2 |
M = 12/12/20/20, twice.
T = 8/8/22/22, once; and 10/10/20/20, once. |
68 |
68 |
2 |
M = 8/8/26/26, T =14/14/20/20, once.
M = 10/10/24/24, T = 8/8/26/26, once. |
72 |
56 |
3 |
M =16/16/20/20, T = 8/8/20/20, once.
M =16/16/20/20,T = 10/10/18/18 twice. |
84 |
40 |
3 |
M = 10/10/32/32, once. M = 12/12/30/30, once.
M =14/14/28/28, once.
T = 8/8/12/12 for all cases. |
100 |
64 |
2 |
M = 8/8/42/42, T = 8/8/24/24, once.
M = 18/18/32/32, T =10/10/22/22, once. |
In regards what TML appears with the most number of other TMLs -- for M, the answers are:
TML for M = |
used X times |
# of diff. T |
| 56 |
7 |
5 |
| 64 |
7 |
6 |
| 68 |
7 |
6 |
| 52 |
6 |
5 |
| 72 |
6 |
4 |
| 84 |
6 |
4 |
| 40 |
5 |
3 |
for T:
| TML for T = |
used X times |
# of diff. M |
| 32 |
10 |
7 |
| 56 |
9 |
7 |
| 48 |
7 |
7 |
| 60 |
7 |
6 |
| 40 |
6 |
4 |
In regards the ratio bands, far more M:T pairs fall to the left of 1:1 than was the case with SQ (6) & SQ (7). Also unlike SQ (6) & SQ (7), no pairs fall to the right of the blue 1:2 ratio band. This reversal of the short:long of “A:B” and “C:D” is so extreme that we are able to insert a 2:3 ratio band on SQ (9). Note that only one sample falls to the right of this band.
Only 3 samples fall directly on the 1:1 ratio band -- more than the single occurrence of the “A1:B1” pairs SQ (6), but far fewer than the 12 “C1:D1” samples SQ (7).
And with this, we end our SQ overview.
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