Bounded Triplets in WTC 1

I have begun running simulations for what I’m calling (at Gemini’s suggestion) “Bounded Triplets”. These are the coincidence of three different MQs sharing some combination of boundary points—meaning where the groups start/end—where Groups A and B are in ratio C; the ratio naturally is x1000, so the MQ of Group C is seen to act as a “marker” of the ratio. An example:

423/1269=.33333333333… (being exactly 1/3), the MQ of 333 from the same boundary is the [rounded and x1000] “marker”.

There are 6,222 of these triplets in the WTC1 set, which initially seems like an incredibly high amount; but for perspective, there are 1176 unique contiguous groups in a set of 48 numbers, 270,372,200 ways to combine them into groups of three, and 3,408,440 combinations which pass the “shared boundary” rule. So only %.18 of the WTC1 set’s shared boundary groups satisfy our conditions, which is not nothing, but not quite so spectacular as 6,222!

In a random set (18-115), there were on average around 5,200 (%16 fewer). The highest result (in 100 tests) was 5,729; so the odds of randomly generating 6,222 are quite close to 0. I have a fuzzy understanding of this simply being a consequence of the WTC1 set being very much non-random (as established in previous entries); repeated MQs naturally produce more of these bounded triplets. So this test is fundamentally another way of showing that the set is non-random.

My theory is that some of these triplets are intentional, and the ratios are those between simple combinations of shapes, as one would expect in any architecture. So if we test for occurrences of said ratios—1414 for the square root of two, 1732 for the square root of three, etc—we find that nearly two hundred occur in the WTC1, while only an average of around 80 occur at random. Accounting for the difference in total, the percentage occurrence is still about half—%3.13 to %1.6—and no simulated test got close the WTC1 percentage, so again, probably a chance close to 0.

Here we have to consider that there is absolutely no reason that this set of geometric ratio “markers” would occur more often than chance in the WTC1 set, even taking into account the repeated quantities (which might have an otherwise compositional excuse): the set of geometric ratios as a whole has no inherent “pattern” that could be related to MQ placement outside of intent.

Remaining skeptical, we can shuffle the actual WTC1 set, and a previous suspicion is confirmed: the average number of bounded triplets is right around 6200: their relative abundance is caused by the repeated MQs, not necessarily Bach’s ordering. Comically enough though, the number of occurrences of geometric ratios drops to %1.4.

Since the “rounding” operation for the irrational ratios is offensive to some, we can look at only the set of simple “harmonic” ratios of 1:2, 1:3, and 2:3, which can occur exactly, as in the example above. There are 31 instances across all three ratios in the WTC1, so about .5%. At random, there are .11%, and in the shuffled set, .15%. Notably, in the shuffled set, only 14% of the runs had results across all three ratios. One notes though that this subset probably is influenced by the repeated MQ patterns previously observed (for their compositeness), but, as before, that’s begging the question.

A test for simple occurrences of the geometric ratios as contiguous MQ sums (not considering bounded triplets) finds 34 instances in the WTC1 set, and from 100,000 runs:

Test 1: Random Generation (18-115)
Average Matches Found: 16.95
p-value: 0.0002
Test 2: Constrained Shuffle
Average Matches Found: 19.30
p-value: 0.0011

So there’s about a 1 in 5,000 chance that 34 constants appear at random, and a 1 in 1,000 chance that they occur even if you shuffle Bach’s chosen MQs.

It occurred to me that perhaps in this test the WTC1 order was simply more likely to produce MQs in the range of the constants, and one might argue that the constants themselves were cherry-picked, even though they are a tangibly related/externally meaningful set. So I ran a control simulation testing for hits in the range (+/- 9) of each, which Gemini called “Fuzzy Targets”.

--- Final Statistical Summary (Fuzzy Targets) ---

Observed Matches in WTC Data: 329

Test 1: Random Generation
Average Matches Found: 282.75
p-value: 0.0014
Conclusion: The count is STATISTICALLY SIGNIFICANTLY HIGHER than random chance.

Test 2: Constrained Shuffle
Average Matches Found: 319.57
p-value: 0.2039
Conclusion: The count is NOT statistically different from a constrained shuffle.

This shows that the WTC1 set itself does tend to produce MQs in the range of the geometric constants [when shuffled]; but it does not tend to produce the exact things. This would seem to indicate that there is no reason the geometric constants could be so present in the WTC1 other than conscious ordering.

-jtr

Solstice, Castor/Pollux, Combinatorics

I’ve been talking to a robot all day: Google’s Gemini 2.0. It has struggled mightily to add double-digit integers, but it has apologized, and I’ve not choice but to accept, as I need it to generate Python code for simulations, which to this untrained eye it has done quite well. Google’s Colab has been running millions of simulations off these scripts without yet requiring me to shovel coal into a furnace to power it.

Conversations at this year’s Bach Network Dialogue Meeting showed me that people are still quite skeptical about any kind of numerical design in Bach. So, I am tackling one of the most essential and unassailable numerical traits of WTC1, the set of component measure quantities (MQs), meaning how many measures to which each piece in the collection totals. Points about the set:

WTC I Component MQs.
Sorted by size, repeat values colored.

  1. There are 33 unique values: 10 are repeated at least once; and 25—over half of the set—belong to this “repeated” category. If choosing at random [from the range 18-115], one would expect around 38 unique values; a simulation returned 33 (or less) ~2% of the time.

  2. The majority of the repeats occur in the lower MQs: 17 out of 25 (68%) are below the median of 36. This occurs in a simulation ~6% of the time.

  3. Two MQs repeat 4x, one repeats 3x, and seven repeat 2x. MQs repeat [at least] as often in a simulation ~.1% of the time.

  4. There are three noticeable “gaps” in the MQ range: there are no values from 59-69, 75-85, or 88-103. Three gaps of [at least] this size (not limited to the exact ranges) occur ~.02% of the time in random simulation.

We might thus conclude that Bach did not merely “arrive” at these MQs in a process of random free composition, which should surprise no one, as Bach was no robot. The question is to what degree Bach was choosing the numbers, as opposed to them being some kind of emergent property of his compositional process and style. For instance, the MQ of 24 repeats 4x, and only in preludes. This could easily be explained as a structure of 2x12, 4x6, etc; preludes being perhaps more likely than fugues to have such an arithmetic structure. On the other hand though, easily subdivided MQs like 32 and 36 (which are very near the median of the set) are not used at all.

The above points all deal with the set of MQs in isolation. To assess whether or not Bach was choosing MQs, we might consider how they are actually arranged in the work:

WTC I Component MQs
Repeats colored.

  1. The MQ of 35 repeats 4x—twice in prelude and twice in fugue—but only in major; the odds of an MQ occurring 4x but only in a single mode are ~1%.

  2. There are 7 repeated MQs in a row at 31-37 (among a larger cluster of 11 out of 12). There are also voids at 3-8 and 20-25. These occur together about .2% of the time in simulation. As the distribution of repeated MQs across Mode and Prelude/Fugue is essentially equal in both cases—12:13—there is no clear reason why groups of repeated MQs would cluster.

  3. Combinations of 35/27, 19/34, and 29/41 all repeat. Only one of these is not in a P/F pair. Disregarding the P/F tendency, picking at random, this occurs .03% of the time; shuffling the actual WTC1 set improves the odds to .35%.

  4. The MQs that repeat 4x—24 and 35—each have equidistant instances. The first three instances of 24 are nine components apart, and the first-second and third-fourth instances of 35 are each seven components apart.

I cannot imagine a reasonable explanation for all of these that does not surreptitiously accept that Bach was aware of the numbers as an abstraction apart from the consequence of composing more or less measures; at which point Bach chose and arranged MQs as a structural device is a short and convenient stumble away.

I will attach the code for all simulations referenced at some point in the near future; and will likely continue to edit and revise this post if I find errors in the script or my assumptions.

-jtr