# Analysis Section ( 2/5 ) - Character statistics

## 2.1 Introduction

This page will first list a number of observations about the Voynich MS character statistics that may be found in the printed literature, and then concentrate on more quantitative analysis results.

## 2.2 Observations in the printed literature

#### Tiltman (1967) (1)

(Note: Tiltman treats f as a variant form of k and p as a variant form of t. In the following, characters or sequences in parentheses represent such variant forms).

• cKh (cFh) and cTh (cPh) appear to be infixes of k (f) and t (p) within ch. The variant symbol represented by m appears most commonly at the end of a line, rarely anywhere else.
• Paragraphs nearly always begin with k (f) or t (p), most commonly in the second variant forms, which also occur frequently in words in the top lines of paragraphs where there is some extra space.
• y occurs quite frequently as the initial symbol of a line followed immediately by a combination of symbols which seem to be happy without it in any part of a line away from the beginning. Otherwise it occurs chiefly before spaces very frequently preceded immediately by d. Hence my belief that these two have some separative or conjunctive function. (I have to admit, however, that y also seems sometimes to take the place of o before k or t (though rarely, if ever, after q); this is particularly noticeable in some of the captions to illustrations in the astronomical section of the manuscript - these most commonly begin ok (of) or ot (op) and it is here that we occasionally see yk (yf) or yt (yp).
• o, which has a very common and very definite function in "roots" (2), seems to occur frequently in "suffixes" in rather similar usage to a, but nearly always as or and ol.
• l, usually preceded by a or o, is very commonly followed by d, much less commonly by t (p), with or without a space between.
• Speaking generally, each character behaves as if it has its own place in an 'order of precedence' within words; some symbols such as o and y seem to be able to occupy two functionally different places.

#### Currier (1976) (3)

• These letters: o, d, y, s all seem to start with a 'c'-curve. [...] The forms all have counterparts starting with i: j , l , r, etc. We also have a = e + i.
• The final letters (that is, the ones I call finals, although they can also occur elsewhere) are in two series, one preceded by a and the other by o, giving a series of sixteen: n in iin iiin, l (il) (iil) (iiil), r ir (iir) (iiir), m im (iim) (iiim).
The ones in parentheses are very low-frequency; the others all occur with respectable frequency. In addition, these combinations of symbols which appear as finals may occur separately — 'unattached finals,' as I call them. A large number of unattached finals is a characteristic of 'Language' B, and not 'Language' A, by the way.
• You can pick out resemblances between Latin abbreviations and other alphabets for most symbols except for the series t , k , p , f. [...] The last two [...] appear 90-95% of the time in the first lines of paragraphs, in some 400 occurrences in one section of the manuscript.
• One might conclude that p , f are an elaborate form of t , k , with the same value. [...] But, not true! These two letters p , f are not the same as those two t , k , as the statistics show. The letters t , k are followed anywhere in a 'word' by our little friend e about half the time (say 750 out of a total of 1500), including initially. These two, p , f, are never, ever, anywhere in the manuscript, followed by e. [...] Therefore, p , f are not aberrant or variant forms of t , k

Currier's first observation has been noted independently by several people, and was taken up recently by Brian Cham, who developed the >>curve-line system out of it.

#### D'Imperio (1978) (4)

• The split gallows seem only to occur on first lines of paragraphs, and in labels.
• The same "word" may be repeated two, three or more times
• Many "words" differ by only one character and are found in each other's vicinity
• Certain symbols occur characteristically at the beginnings, middles or ends of words, and in certain preferred sequences
• Certain symbols appear very rarely, and only on certain pages
• There are very few doublets. Primarily: e or i and occasionally also y, d and o.
• There are very few single-letter words in the running text, primarily s and y.
• Prefix-like elements are found in front of certain "words" that also occur commonly without them. Such prefix-like elements are: qo, o and y
• The symbol q almost always precedes o, connected to it by an extension of the crossbar of the 4. This combination is found almost always at word start.
• On most herbal folios, the first paragraph usually starts with t, k, p or f, usually immediately followed by ch, Sh, o, y, aiin or dy.

## 2.3 Character frequencies

Oddly enough, there is no consolidated set of this most basic statistic, due to the use of different transliteration alphabets and different transliteration sources. Several examples may be found in different sources.

One example is found in D'Imperio (1978) (see note 4), Fig. 28 on p.106, from several sources but none covering the entire MS text.

I have produced some graphics of the single character frequency distribution of the entire MS, for four different transliterations. These may be found in the beginning of this page.

As a very short summary, the single character frequency distribution in the most important transliteration alphabets is largely similar to that of texts in normal European languages, thought the drop in frequency appears to be marginally steeper.

## 2.4 Entropy

The concept of entropy has been explained in the introductory page and the reader should have read this introduction in order to properly appreciate the following.

The entropy of the Voynich MS text was first analysed in detail by the Yale professor William Ralph Bennett Jr. (6). He developed the concept in many easy steps and in more detail than in the above-mentioned introductory page. He first analysed texts in common European languages and then addressed the Voynich MS text, which he transliterated using his own transliteration alphabet (7). He writes (8):

[...] the statistical properties of the Voynich Manuscript are quite remarkable. The writing exhibits fantastically low values of the entropy per character over that found in any normal text written in any of the possible source languages (see Table 5). The values of h1 [i.e. first order entropy - RZ] are comparable to those encountered earlier in this chapter with tables of numbers. Yet the ratio h1/h2 is much more representative of European languages than of a table of numbers alone.

His computed values are as follows (9):

Entropy order Normal languages Voynich MS
First 3.91 - 4.14 3.66
Second 3.01 - 3.37 2.22
Third 2.12 - 2.62 1.86

He finally identified one language with a set of similarly low entropy values, namely Hawaiian, but he also pointed out that this is not likely to be significant.

More statistics related to entropy calculations may be found in an on-line paper by >>Dennis Stallings: understanding the second-order entropies of Voynich text. This basically confirms the results of Bennett. His descriptions will be useful for those who have no access to a copy of Bennett's book.

I have re-done the calculation for first- and second-order entropy for a larger number of languages, using the text of the Universal Declaration of Human Rights (10). This analysis will be described in more detail in a dedicated page, and for the moment I just show some of the results. In this analysis, the space character has not been interpreted as a character, but as a separator between words. The first plot below shows the (conditional) second-order entropy plotted against the first-order entropy, for a number of modern European languages, also including the results for the Voynich MS. The Voynich MS statistics are those computed by Bennett (left-most point) and those computed by Dennis Stallings for Herbal-A and Herbal-B using the Currier alphabet. The meaning of the legend is shown in a table below the figure. It is clear that none of the languages shows a similar behaviour to the Voynich MS text.

Code Meaning Languages
ROM Romance languages Latin, French, Spanish, Italian, Portuguese, Catalan, Galician, Occitan (Auvergnan), Corsican, Friuli, Maltese
GER Germanic languages English, German, Dutch, Frisian, Afrikaans
SCA Scandinavian languages Swedish, Norwegian (Modern and Bogmål), Danish, Icelandic
SLA Slavic languages Russian, Polish, Czech, Slovak, Croatian, Bulgarian, Macedonian, Belorus, Georgian
GAE Gaelic languages Scottish Gaelic, Irish Gaelic, Breton, Manx

The calculations have been repeated for a number of other languages from around the world. These are listed in the following table:

Code Meaning Languages
EUO European, other Albanian, Basque, Finnish, Hungarian
AFR African Ethiopian (Amharic), Swahili, Hausa, Edo, Somali, Bari
IEU Indo-European Greek, Estonian, Latvian, Lithuanian, Farsi, Hindi, Nepali, Urdu
ASI Asian Turkish, Armenian, Turkmen, Kurdish, Hebrew, Arabic, Azerbaidjani, Bengali, Minjiang (a Chinese dialect, spoken vs. written), Tibetan, Mongolian, Japanese, Korean, Thai, Laotian, Burmese, Cambodian, Vietnamese, Indonesian, Tagalog, Cebuano, Hawaiian

The result is shown in the following plot, where the points for the "European" languages have been repeated in grey.

Here we see a number of points among the group of Asian languages that lie in the relative vicinity of the Voynich MS text. The lowest and leftmost point is Tagalog (Philippinian). The two points to the right of this are the spoken and written version of Minjiang. These text have been written in the Latin alphabet without indication of tones. Hawaiian, the language named by Bennett, is the lowest point directly above those for the Voynich MS.

A more recent study by Luke Lindemann and Claire Bowern of Yale University (11) covers some of the same ground, and analyses even more different languages, including historial corpora. It comes to the same conclusion and summarises:

The character set size and frequency of characters is conventional, but the characters are combined in an extremely predictable way, as indicated by an unusual conditional character entropy that is distinctly lower than any of the 316 comparison texts. This discrepancy is not attributable to the transcription system used to encode Voynich, although decisions about the compositionality of glyph sequences can have a significant effect on entropy. Nor is it the result of conventional scholarly abbreviations of the historical period or the absence of written vowels. Rather, it is largely the result of common characters which are heavily restricted to certain positions within the word. Voynichese most closely resembles tonal languages written in the Latin script and languages with relatively limited syllabic inventories.

An alternative method to compute entropy is the so-called 'commas' method, which has been used by Jim Reeds and later by Gabriel Landini. This will be included here at some future time.

Jorge Stolfi has set up a tool to visualise the number of bits of entropy per character in the following location: >>Jorge Stolfi: where are the bits?

Furthermore, I have addressed the question how it is possible that the character and bigram entropy of the Voynich MS text is so much lower than that of, say, Latin, while the word entropy (about which something will be said here) is similar. This is addressed at this page: From bigram entropy to word entropy. I realise that this page is rather hard to understand in its present form, and I will re-do it. The short summary is that, counting from the start of each word, the entropy per character in the Voynich MS starts off lower for the first two characters, but is higher for the remainder of the characters, when compared to normal languages.

While the entropy values are single values derived from a frequency distribution, more can be learned by looking at the detail of these distributions, for which see here. This discussion exemplifies even better how much different the Voynich MS text is from 'regular languages'.

There is a critically important conclusion to be drawn from the first- and second-order entropy values reported by various authors. As already mentioned in the analysis section introduction, the entropy values do not change when one consistently replaces characters by others, i.e. in a simple substitution cipher. This tells us something about the possible plain text of the Voynich MS.

1. It could be that the text is meaningless, i.e. there is no plain text language, and the anomalously low entropy is the result of whatever process was used to generate the strings of characters
2. If there is a plaintext that was encoded using a simple substitution, then this plaintext must have the same anomalously low entropy values. This then excludes most of the typical languages that might be assumed for a European MS of the 15th Century. In fact, no candidate plaintext language could yet be identified. Hawaiian, the one identified by Bennett, does not match for other reasons (as will become apparent in later pages). Some languages like Hebrew, the various Arabic languages, Persian, Armenian etc. have not yet been tested quantitatively, to my best knowledge.
3. If there is a plaintext in one of the known languages used in European MSs of the 15th Century, then this text must have been modified by some process changing the statistics quite drastically. This change is indeed so drastic that it is no longer possible to identify the plaintext language from the Voynich MS text, and any attempts of this nature will be invalid.

In general, and quite briefly, any attempt to translate the Voynich MS into something meaningful in Greek, Latin, English, etc. using a simple substitution must fail. As this is the first thing most people will try, we can begin to understand how the MS has resisted all translation attempts.

However, there is much more to this, as we shall see in the following (12).

## 2.5 Vowel/consonant detection

An algorithm for detection of vowels and consonants was designed by B.V. Sukhotin, and Jacques Guy has experimented with this in the 1990's. He published a first English summary of the algorithm in Cryptologia (see note 5). Results indicated that the characters that look like vowels (a, o, y) also appeared statistically like vowels, though the confidence of the result was not very high. There is also a more recent >>internet blog entry related to running Sukhotin's algorithm on individual pages of the MS.

Significant further work on vowel/consonant detection has been done using Hidden Markov Modelling, which is summarised in the next section.

## 2.6 Hidden Markov Modelling

This technique has been introduced briefly on a previous page. There are not many publications that discuss its application to the Voynich MS. Following is a summary.

Jeffrey Krischer
Techniques similar to HMM processing wer initially classified, and publications of its application have been restricted. D'Imperio mentions (see note 4) a first application of this in a paper by Jeffrey Krischer. He also published an unclassified summary (13), which does not give any details about the method and the results.

Mary D'Imperio
Mary D'Imperio made her own analysis, which was published in an undated paper that was also classified, but released a few years ago (14). It is a rather lengthy paper, which is hard to summarise, but this will be attempted here at some point.

Jim Reeds
Jim Reeds has made similar analyses, but these have not been published as far as I know. I have some information about this from personal communication with him. He summarised that, while for known languages the HMM tends to separate vowels and consonants, for the Voynich MS text the analysis brings out a circular pattern of characters. This may be the same as what D'Imperio found and published.

Reddy and Knight
One of the analyses reported in Reddy and Knight (2011) (15) is the application of a two-state bigram HMM to the characters in the Voynich MS words. They report that normally, in alphabetic languages like English, the clusters correspond almost perfectly to vowels and consonants. They find, however, that with the Voynich MS the last character of every word is generated by one of the HMM states, and all other characters by another, i.e. as if the vowels are at the end of each word. They speculate further on this point in their paper.

Later work
A recent paper by Acedo (16) describes a single application of a two-state HMM to the Takehashi transliteration of the MS and tentatively concludes that this successfully shows a separation into vowels and consonants.

I have started a more comprehensive approach myself, which is still under work, and which is summarised in this page. At the present stage, this seems to indicate that the separation into vowel- and consonant states is not nearly as clear as for known plain texts.

## 2.7 Other material

#### 2.7.1 Line-initial/final and word initial/final character properties

Following observations are paraphrased from Currier's papers (see note 3).

• In those pages where the text is presented linearly, the line is a functional entity. The following three bullets clarify this general observation
• 1: the frequency counts of characters at beginnings and endings of lines are markedly different from elsewhere. There are some characters that may not occur initially in a line. There are others whose occurrence is about one hundredth of the expected
• 2: the ends of lines seem to contain meaningless symbols: little groups of letters which don't occur anywhere else (17). There is one symbol (18) that occurs at the end of the last word of a line 85% of all of its occurances.
• 3: there is not one single case of a repeat going over the end of a line to the beginning of the next
• Skewed frequencies at beginnings of lines may be illustrated by the two letters ch and Sh. If its occurrence as an initial were random, we would expect it to occur one seventh of the time in each word position of a line. Actually, it is a very infrequent word initial at the beginning of a line, except when there is an intercalated o. This applies only to 'Language' A.
• The 'ligatures' [ cKh cTh cFh cPh ] can never occur as paragraph initial, and almost never line initial.

The obsernation of Currier that the line appears to be a functional unit was further analysed in 2012 by Elmar Vogt, for which >>see here. One of the most obvious features he shows is that, when using the Eva alphabet, the first word tends to be on average 1 character longer than the second and following words.

#### 2.7.2 Location of gallows (and other) characters

Julian Bunn highlighted the positions of the gallows characters on each folio of the MS in >>a page at his blog, in colour coded graphics. They show a peculiar vertical pattern, which may be related to the observations of Andreas Schinner in his 2007 cryptologia paper (19), which is discussed in a later page.

The following page by Sean V. Palmer gives a very visual representation of the feature that many characters have very preferential positions inside the words of the MS: >>Voynich MS glyph position stacks.

## Notes

1
See Tiltman (1967).
2
For Tiltman's roots and suffixes, see here. Additional observations are listed there.
3
See here or here.
4
See D'Imperio (1978).
5
See Guy (1991), also >> online here.
6
See Bennett (1976), Chapter 4, pp. 103-198.
7
For which see here.
8
On p.193.
9
Combining Table 5 on p.140 and Table 12 on p.193
10
This was inspired by a recent paper by Hauer and Kondrak. The texts may be found >>here.
11
See Lindemann and Bowern (2020)
12
For example here: here.
13
Krischer (1969).
14
See D'Imperio (undated)
15
Reddy and Knight (2011).
16
Acedo (2019).
17
This statement is not fully understood, and it seems worthwhile to understand what he means. The only character that typically occurs at line ends is mentioned separately, so perhaps he means character combinations or groups.
18
Currier almost certainly means m
19
See Schinner (2007).