I mentioned in a comment on Koen G’s recent post that I thought that Voynichese benched gallows (i.e. gallows that have a ch glyph struck through them) may well be nothing more complex than a different way of writing gallows+ch; and that I thought this was much more likely than the alternative notion that it was a different way of writing ch+gallows.

When Koen asked me what evidence I had for this, I thought that I ought to write a brief post explaining how I got there (i.e. rather than cramming my “truly marvelous demonstration” into a Fermatian margin). So here goes.

Yes, It’s Contact Tables (Again)

The evidence I’d point to is from (you guessed it) contact tables for glyphs following benched gallows. The notable feature of these I mentioned recently on Cipher Mysteries (though the obeservation is, of course, as old as the hills) is that benched gallows are only very rarely followed by -ch.

Here’s a simple parsed count example (Takahashi transcription), showing how very rare benched gallows + -ch are as compared to both -e and -ee:

Baseline: (ch 10652), of which (che 4138), (chee 742), and (chch 18)

Furthermore, as I noted in that post, almost all of the places where benched gallows are followed by ch seem to be Takahashi’s transcription errors (sorry Takahashi-san).

Compare and contrast with the contact tables for the preceding glyph, where the ch- instance counts hugely outnumber the counts for e- and ee-:

Baseline: (ch 10652), of which (ech 143), (eech 33), and (chch 18)

As a sidenote, the interesting things in this particular table are (a) how rarely benched gallows are preceded by ee- (far less than by just e- or ch-), and (b) how frequently benched gallows are preceded by ch- when ch itself is very rarely preceded by ch-.

So, What’s Going On Here?

I think it’s safe to say that there is probably a really basic reason why benched gallows preceded by ch- are found so much more often than benched gallows followed by -ch. But what might that reason be?

For me, the suspicion is simply that c+gallows+h is just a different way of writing gallows+ch. The contact tables I quote above certainly don’t seem to offer anything to support the alternative scenario where c+gallows+h is a different way of writing ch+gallows.

To my eyes, replacing benched gallows with gallows+ch would match the statistics baseline for che/chee/chch far more closely than replacing benched gallows with ch+gallows would match the statistics baseline for ech/eech/chch. That is, the benched gallows right contact tables (i.e. the contacts that benched gallows have with glyphs immediately following them to the right) seem to me to broadly match the ch right contact tables, but the benched gallows left contact tables don’t obviously match the ch left contact tables.

The big issue here – as always, though – is one of proof. It’s all very well my speculating that it would be better to replace benched gallows with gallows+ch rather than ch+gallows, but how can this be made stronger?

Though I’m not sure that it would be possible to turn this gallows+ch replacement hypothesis into a smoking-gun proof, I do suspect that it could be tested much more rigorously. Perhaps CM readers will have good suggestions about how to carry out a suitable test (or three). 🙂

Finally: Might ch Be Enciphering U?

To me, Voynichese’s various families of shapes and glyph behaviours look (much as John Tiltman suggested) like a grab-bag of contemporary cipher tricks. As a result, it would make a lot of sense to me if the distinctive benched gallows was simply one of the set of slightly older cipher tricks that were artfully combined to form Voynichese.

Along these lines, I’ve previously floated the idea (based mainly on the look of the benched gallows, but also on my long-held suspicion that e/ee/ch/sh might somehow be vowels) that Voynichese ch might in fact encipher plaintext U/V. This is because I can easily imagine that c+gallows+h may have begun its life as an early 15th century steganographic trick used to disguise or visually disrupt QU patterns before being absorbed into the Voynichese Borg mind.

Replacing benched gallows with gallows+ch would be entirely consistent with this idea (though note that the gallows need not necessarily be enciphering Q, even if the trick started that way), so it’s possible that both ideas might turn out to be true simultaneously.

Incidentally: in “The Curse of the Voynich” (p.177), I mentioned a strikethrough trick that appeared in an “otherwise unremarkable” 1455 cipher (Ludovico Petronio Senen) to encipher the Tironian-style ‘subscriptio’ shorthand sign (e.g. that turns “p” into “p[er]”). My speculation here is therefore that the strikethrough trick may have first emerged in this general era, though instead used to visually disguise plaintext U’s.

Hence one thing I have been meaning to do recently is to trawl carefully through Mark Knowles’ fascinating haul of 1400–1450 Northern Italian ciphers to see if there is any indication there that a strikethrough trick was ever used in one of those ciphers to disguise the U in QU pairs. You might have thought that encipherers would have added a special token for “QU”, or might have simply chosen to omit the U after Q: but neither of these options typically seems to have happened in this general timeframe (outside of the most complicated syllabic ciphers).

I recently mentioned in a comment that my working hypothesis was word-initial EVA l- was a different token to EVA l elsewhere: and Emma May Smith asked me what evidence I had for that statement. So I thought I’d post a few stats to throw onto the fire.

The Evidence

Just to be clear, though: because I’d rather not mess up my stats with line-initial EVA l- stats, all the following figures relate to word-initial (but not line-initial) stats. And to keep everything as clear as practical, the comparisons are solely between words beginning l-, ol-, and al-.

So, here are the raw instance counts according to the Takahashi transcription for word-initial (but not line-initial) l-, ol-, and al-. For example, there are 1267 word-initial (but not line-initial) l- words, of which 58 are just EVA l (on its own), along with 433 word-initial (but not line-initial) words beginning with lk-. (Note that the “(-)” line is an estimate, my app unfortunately couldn’t calculate it.)

To compare these three columns, we now need to turn their values into percentages. What this following table is saying, then, is that word-initial (but not line-initial) l- is followed by k 34.18% of the time, t 2.68% of the time, etc. (Note that I didn’t try to capture all of the values.)

In short, this table is trying to compare the contact tables for three word-initial (but not line-initial) contexts: l-, ol-, and al-. So… what does it say?

Though the +f and +p rows are broadly the same for all three contexts, I think just about every row presents significant differences. For example:

• Only one word in the VMs begins with EVA alt (on f72v2, Virgo)
• Comparisons between the ch and sh lines seem to imply that tehre is vastly more similarity between ch and sh (ch seems to occur 3x more often than sh) than between l-, ol-, and al-.
• l- is typically followed by k (34.18%) and ch (23.13%), but this is quite unlike ol- and al-.

However, the biggest difference in all these counts is where l, ol, and al form the whole word (the “(-)” row). So here’s the last table of the day, which is where the whole word counts are removed from the totals, i.e. word-initial but not line-initial and also not word-complete:

Even though taking out all the word-total instances has damped down some of the larger ratios, there are still plenty of big ratios to be seen.

Perhaps the most surprising is the comparison between ly- (1.08%) and aly- (14.48%). (Interestingly, all but one of all the places where the ly and aly instances occur in the text are at the end of a line or butted up against a mid-line illustration. Which I think points strongly to ly and aly being abbreviated in some way, but that’s an argument for another day.)

The Conclusion

For me, I simply can’t see anything systematic or language-like about the comparisons between any of the three columns. When their contact tables are so different, what actual evidence is there that l-, ol-, and al- are all presenting the same (right-facing) linguistic context? Personally, I simply can’t see any.

My conclusion from the above is therefore that l-, ol- and al- are (without any real doubt at all) three different tokens, i.e. they are standing in for three different underlying entities.

Since posting about Voynichese’s strange single leg gallows behaviours a few days ago, I have continued to think about this topic. On the one hand, it’s clear to me how little of any genuine substance we actually know about how they work; and on the other, I’ve been wondering how I can start some broadening conversations about them (by which I mean ones that ask more questions than they answer).

As today’s experimental contribution, I’m going to write a post listing a load of the questions I have in my head to do with single leg gallows but without really trying to answer any of them. I can’t tell how this will work, but here goes regardless. 🙂

Incidentally, for anyone who wants to run their own statistical experiments on single leg gallows, I would strongly recommend using Herbal-B + Q13 + Q20 as their basic test corpus, because I’m acutely distrustful of any Voynich stats that combine Currier A and Currier B. Even though I’m basically doing the latter here. 😉

Questions: final flourish

Rather than finishing with a second vertical leg on the right hand side, single leg gallows instead cross over the left hand leg and finish with a slight flourish to the left. This final flourish can be (1) short, (2) long and straight, or (3) long and curved (i.e. finishing with something like an EVA c-shape).

1. Have the variations in the finishing flourish of single leg gallows been catalogued and/or transcribed?
2. Are these variations found uniformly throughout the manuscript, or are they strongly correlated with the various scribal hands (as recently proposed by Lisa Fagin Davis)?
3. If they have been transcribed, is each flourish type statistically associated with any neighbouring textual behaviours (e.g. contact tables, etc)?

Questions: followed by EVA e?

One huge difference between single leg gallows and double leg gallows is that non-struckthrough single leg gallows are very rarely followed by EVA e. If you count strikethrough gallows separately from normal gallows, the statistics are quite, umm, striking:

• k:ke = 9758:3809 = 39.03%
• t:te = 5802:1748 = 30.13%
• p:pe = 1383:5 = 0.36%
• f:fe = 416:3 = 0.72%
• ckh:ckhe = 876:242 = 27.63%
• cth:cthe = 905:190 = 20.99%
• cph:cphe = 212:64 = 30.19%
• cfh:cfhe = 73:14 = 19.18%

Moreover, looking at the eight instances in Takahashi’s transcription where EVA p and EVA f are followed by EVA e, I suspect that many of these may well be transcription errors (i.e. where Takahashi should have instead written EVA pch / fch).

Hence it seems to me that Voynichese has a secret internal rule that almost completely forbids following EVA p and EVA with EVA e. This is a massively different usage scenario from EVA t / EVA k (which are followed by EVA e 39.03% and 30.13% of the time respectively).

OK, I know I said I was only going to ask questions in this post, but looking at these numbers afresh, I can’t help but speculate: might it be that EVA p/f are nothing more complex than a way of writing EVA te/ke?

1. Has anyone looked closely at the eight places where pe/fe occur?
2. Why is there such a huge difference between pe/fe and the other six gallows?
3. Might this be because EVA p and EVA f are optional ways of writing EVA te and EVA ke?
4. Has anyone considered this specific possibility before?
5. How similar are the contact tables for EVA te/ke and EVA p/f?

Questions: Followed by EVA ch?

Similarly, comparing the stats for instances where gallows are followed by the (almost identical looking) EVA ch glyph reveals more differences:

• k:kch = 9758:1074 = 11.01%
• t:tch = 5802:975 = 16.80%
• p:pch = 1383:733 = 53.00%
• f:fch = 416:190 = 45.67%
• ckh:ckhch = 876:5 = 0.57%
• cth:cthch = 905:3 = 0.33%
• cph:cphch = 212:1 = 0.47%
• cfh:cfhch = 73:0 = 0.00%

Here, we can see that both p and f are followed by ch about half the time (53% and 45.67% respectively), which is significantly more than for k and t (11.01% and 16.80% respectively).

At the same time, the dwindlingly tiny number of places where strikethrough gallows are followed by ch (only nine in the whole manuscript) again raises the question of whether these too are either scribal error or a transcription error.

As an aside, I previously floated the idea here that c + gallows + h may have simply been a compact (and possibly even playful) way of writing gallows + ch, which would be broadly consistent with these stats.

1. Is there anything obviously different about Voynichese words containing EVA kch / tch and Voynichese words containing EVA pch / fch?
2. Has anyone looked in detail at the eight instances where strikethrough gallows are immediately followed by EVA ch?
3. If you remove paragraph-initial p- words from these stats, do the ratios for p:pch and f:fch settle down closer to the ratios for k:kch and t:tch?
4. How similar are the contact tables for EVA tch/kch and EVA cth/ckh?
5. How similar are the contact tables for EVA tech/kech and EVA pch/fch?

Questions: Double Leg Parallels?

Some researchers (perhaps most notably John Tiltman, if I remember correctly) have wondered whether EVA p / f might simply be scribal variations of (the much more common) EVA t / f.

1. Beyond mere visual similarity, is there any actual evidence that supports this view?
2. I would have thought that the pe/fe stats described above would have meant this was extremely unlikely, but am I missing something obvious here?

Questions: Paragraph-Initial?

Yes, single-leg gallows (mainly EVA p) are very often found as the first letter of the first word of paragraphs. But…

1. How often do single leg gallows (and/or strike-through single leg gallows) appear in the first word of a paragraph but not as the very first letter of the word?
2. Do these these paragraph-initial -p-/-f words show any pattern?
3. Are there structural similarities between paragraph-initial p-/f- words and other paragraph-initial?
4. Might there be some kind of numbering system embedded in paragraph-initial p- words (particularly in Q20)?

Questions: vs Double Gallows?

Yes, single-leg gallows are to be found mainly in the top line of paragraphs, but that’s imprecise and unscientific.

1. Are the number of gallows characters (whether single or double) per line roughly constant for both the first lines of paragraphs and for the other lines of paragraphs?
2. Do these statistics change between sections?

And Finally…

Please feel free to leave comments asking any other single leg gallows questions, I’m sure there are plenty more that could sensibly be added to this page. 🙂

All answers happily received too. 😉

Anyone who proposes that Voynichese works in a ‘flat’ (i.e. straightforward) way has a number of extremely basic problems to overcome.

For a start, there are the Voynichese’s ‘LAAFU’ (Emma May Smith’s acronym for Captain Prescott Currier’s phrase “Line As A Functional Unit”, though she now prefers to talk about “line patterns”) behaviours to account for. These relate to the curious ways that letters / words work both at the start of lines and at the end of lines, many of which are discussed by Emma May Smith here:

• Line-first words have a quite different first-letter distribution from the main body of words’ first-letter distribution
• Line-first words are slightly longer than expected
• Line-second words are slightly shorter than expected
• Line-final words frequently end in EVA ‘m’ / ‘am’

At the same time, there are also some odd PAAFU (“Paragraph As A Functional Unit”) behaviours to consider. The most famous of these is the way that the first letter of a paragraph (and even more so of the first paragraph on a page) has a significantly different distribution from elsewhere, one that strongly favours gallows characters (and in particular the single leg gallows EVA ‘p’).

But the other major PAAFU behaviour is that single leg gallows glyphs appear predominantly on the first line of paragraphs, and only rarely elsewhere (these are known as Tiltman lines, after my hero John Tiltman). You can see this throughout the Voynich Manuscript, right from Herbal A page f3r…

…to the Herbal B page f43r (which has an extra single leg gallows, but the remaining ones all sit on the first line of their respective paragraphs)…

…to the Q13 Balneo page f76v (where there are two extra single leg gallows, sure, but the rest of the page slavishly follows the pattern)…

So, even though the internal structure of Voynichese words changes significantly across the different sections (and that’s a separate topic entirely), this single-leg-gallows-mainly-on-top-lines-of-paragraphs Tiltman behaviour seems to remain essentially constant throughout them all.

This is an issue that has been floating round for decades, and I would be surprising if it had originated even from John Tiltman. More recently, Rene Zandbergen discussed it on voynich.ninja back in 2017, pointing out that this behaviour appeared – in his view – to be inconsistent with any model for Voynichese that was inherently uniform (which I call ‘flat’ here), whether linguistic, cryptographic or whatever.

So, the challenge to anyone trying to come up with some kind of theory for the Voynichese text is simply to explain why this unexpected behaviour is the way it is. What kind of mechanism could be behind it?

Q20 Paragraph-Initial Glyphs

For the rest of this post, I’m going to restrict my discussion to the twenty-three Voynich Q20 (‘Quire 20’) pages, simply because their lack of drawings make them particularly easy to work with.

The first thing to point out is that we have two single leg gallow behaviours (very frequent at paragraph starts, and very frequent on the top line of paragraphs) which overlap somewhat.

For example, f103r (the first bound page of Q20), has 19 starred paragraphs, of which 9 begin with the single leg gallow EVA ‘p’ (i.e. 47.3%). And if you count all the paragraph-initial p’s and f’s in Q20, you get:

The values for Q20 as a whole are remarkably consistent: there is a 161/292 = 55.14% chance that a paragraph starts with EVA p, and 16/292 = 5.48% chance that a paragraph starts with EVA f.

Given that ‘p’ makes up 1.03% of the glyphs in Q20 (‘f’ makes up 0.19%), ‘p’ is ~55x more likely to appear as the first glyph of a Q20 paragraph than it is to appear in any other glyph position: even ‘f’ is 28x more likely to appear paragraph-initial than elsewhere. That’s striking, and not at all flat.

Q20 Tiltman Lines

Q20 contains about 10700 words across about 1100 lines (I don’t have the exact figures to hand): 643 of these contain a single leg gallow, i.e. the raw chance any given Q20 word contains a single leg gallow = 643/10700 = 6%.

But whatever the explanation for p being so strongly biased to this paragraph-initial position, I think we should try to separate the single-leg-paragraph-initial behaviour from the single-leg-top-line (Tiltman) behaviour.

So if we remove the 292 paragraph-initial words, the raw chance that any non-paragraph-initial Q20 word contains a single leg gallow goes down to (643-292)/(10700-292) = 3.3%, which is our baseline figure here.

But what of top-line-but-not-initial Q20 words? Given that Q20 has 292 paragraphs, each with a first line containing (say) ten words, and we are removing the first word, we have 292 x ~9 = ~2628 top-line words of interest. Of these (by my counting), 353 contain a ‘p’, and 80 contain an ‘f’. Hence the probability that any given Q20 paragraph-top-line-but-not-initial word contains a single leg gallows is 433/2628 = 16.5%.

Similarly, the probability that any given non-top-line Q20 word contains a single leg gallows is roughly (643-177-433)/(10700-292*10) = 0.4%. So if we discount all the paragraph-initial words, words containing single leg gallows are about 16.5%/0.4% = ~41x more likely to appear on the top lines of paragraphs than on the other lines.

Q20 Neal Keys

One of the interesting things that has been noted about these single leg gallows on the top line of paragraphs is that they seem to often appear in adjacent words. This is something that Voynich researcher Philip Neal first mentioned in a Voynich pub meet a fair few years ago that he had noticed: at the time, I christened them Neal keys.

But even though this is a visually striking thing, is it statistically significant, particularly if we remove all the paragraph-initial single leg gallows first?

For non-paragraph-initial-top-line words, the raw (expected) probability that a pair of adjacent words both contain a single leg gallows would seem to be 16.5% x 16.5% = 2.7%.

My counts for the actual number of pairs of adjacent non-paragraph-initial-top-line Q20 words both containing single leg gallows (i.e. ignoring all paragraph-initial words) were 5/5/6/1/8/12/7/6/7/4/5/0/8/6/3/5/9/4/12/5/1/5/2 = 126 instances out of (353 + 80) = 433.

So, of the 292 x (9-1) = ~2336 potential adjacent pairs (discounting the end word of each top line), 126 instances points to a chance of 126/2336 = 5.4%.

So my conclusion from this is therefore that the phenomenon of Neal keys (pairs of words containing single leg gallows on the top line of paragraphs) is, while visually striking, only 2x the expected value.

To be clear, the phenomenon is definitely there, but the main factor driving it appears to be the very strong tendency for single leg gallows to appear on the top line of paragraphs, rather than the adjacency pairing per se.

Verification

I’ve done a lot of this manually, because I didn’t have sufficient automated tools to hand. So can one or more other Voynich researchers please verify these figures?

• I used the Takahashi EVA transcription
• I counted ch / sh / ckh / cfh / cph / cth as individual glyphs
• I didn’t count space characters in the percentages