Estimating Probabilities and Alliteration Frequencies in Old English Verse
In an often-quoted paper on "Philological Probability Problems," A. S. C. Ross has calculated the probability of random alliteration linking consecutive stressed syllables in Old English as about one in twelve. This number is cited as authoritative by Bliss, and (so far as I know) has not been superseded in the fifty years since Ross's article was published.  This brief essay will attempt to solidify our understanding of various kinds of alliteration rates for Old English poetry, and to suggest some ways in which non-primary alliteration sometimes plays a functional and artistic role in Old English verse.
First off, it must be noted that Ross's numbers really do not show what he claims, since he bases his statistical conclusions on the frequency distribution of initial sounds in "stressable" words of Old English prose (33, note). Ross's focus on words collected from prose sources may have seemed appropriate at the time, since oral-formulaic analysis had not yet reached Old English studies: the extent of formulaic influence on poetic word choice may not have been fully appreciated. Ross's failure to take into account the existence of a specialized poetic vocabulary, however, is harder to explain. Moreover, basing the analysis on stressable words is also problematic as, in verse, many words (such as forms of wesan and weorðan) are seemingly stressed or unstressed based solely on position, rather than on any quality inherent in them as linguistic entities. 
Table I, on the next page, attempts to rectify these problems in Ross's method by counting the stressed words and syllables in a thousand-line portion of Beowulf, taken from Dobbie's edition in the ASPR in order to facilitate comparisons with other poems.  As the numbers indicate, the distribution of sounds participating in primary alliteration differs from the distributions of sounds in non-alliterating stressed positions in both the a-line and b-line.
As is immediately clear from the table, the rates often differ from one another in quite remarkable ways. For the Beowulf-poet at least, vowels are the most frequent carriers of primary alliteration, but they are quite rare as stressed but non-alliterating elements in the a-line. Other alliterators have the opposite distribution (see the s-clusters, especially, but also c, n, þ).  Ross's calculations, which failed to take into account the structure of verses, cannot account for these differential distributions.
The data in Table I can allow us to estimate the probability of several important (or potentially important) features of alliteration in Beowulf. For example, we can calculate the probability that two consecutive lines will share the same alliterating element by summing up the squares of the frequency (in the first column) for each alliterator. The sum in such a case is .0969; in a thousand lines, we would expect to see about 97 such pairs of consecutive lines with the same alliterating sound. In fact, between lines 101 and 1100, I count only 23 such pairs, and we can conclude that such consecutive alliteration between lines is not randomly distributed, and is in fact actively avoided (although not completely so) by the poet.
Table I: Alliteration Rates in Beowulf 101-1100
(Initial Sound: Primary Alliterators | Non-Alliterating Stressed Elements a-line | b-line)
- Vowel: 156 (.1562) | 11 (.0161) | 89 (.0756)
- b: 60 (.0601) | 45 (.0657) | 67 (.0569)
- c: 13 (.0130) | 28 (.0409) | 72 (.0612)
- d: 29 (.0290) | 48 (.0701) | 59 (.0501)
- f: 105 (.1051) | 57 (.0832) | 63 (.0535)
- g: 86 (.0861) | 56 (.0818) | 47 (.0399)
- h: 117 (.1171) | 77 (.1124) | 120 (.1020)
- l: 41 (.0410) | 54 (.0788) | 73 (.0620)
- m: 83 (.0831) | 46 (.0672) | 86 (.0731)
- n: 18 (.0180) | 31 (.0453) | 44 (.0374)
- p: 0 | 0 | 1 (.0008)
- r: 16 (.0160) | 25 (.0365) | 24 (.0204)
- s: 116 (.1161) | 51 (.0745) | 94 (.0799)
- sc: 13 (.0130) | 26 (.0380) | 56 (.0476)
- sp: 1 (.0010) | 6 (.0088) | 4 (.0034)
- st: 4 (.0040) | 14 (.0204) | 33 (.0280)
- t: 6 (.0060) | 10 (.0146) | 24 (.0204)
- þ: 24 (.0240) | 27 (.0394) | 76 (.0646)
- w: 111 (.1111) | 73 (.1066) | 145 (.1232)
Clearly, avoidance of this sort of consecutive alliteration had a positive value for the Beowulf-poet. Presumably, variation of primary alliterators may have been valuable as it helped to define line-boundaries; regardless, it seems that such variation was intended (and probably interpreted) as artistically and poetically effective. Notably, not all poets shared the Beowulf-poet's avoidance of consecutive alliteration. The poet of the Metrical Psalms clearly had a different perspective. In the 107 lines of Psalm 68, for example, there are twenty cases of consecutive alliteration: twice as many as an estimate of randomness would predict (and eight times as many as would probably appear in a similar passage from Beowulf). Whatever else we might conclude, it is clear that the Psalm-poet and the Beowulf-poet have very different ideas about the poetic value of consecutive alliteration. Both, it is interesting to note, appear to be paying some attention to alliterative patterns at a poetic level larger than the full line.
Other calculations can be readily made as well. The occurrence of purely random "linking alliteration," connecting the last major stress of one line with the primary alliteration of the next, can be estimated by multiplying the numbers in the first and third columns and adding the resulting products. Such a calculation gives a probability of .0735.  In a thousand lines, we ought to expect about 73 or 74 cases of such linking alliteration, and by my count there are 72 examples between lines 101 and 1100. At least in this stretch of the poem, the frequency of such linking alliteration seems consistent with the hypothesis that it is a random occurrence. A similar calculation and comparison suggests that the rate of alliterative linking between the final stresses of two consecutive lines is also consistent with the hypothesis that such alliteration is random in this sample.
A more complicated calculation can be used to estimate rates of "cross alliteration," in which a non-alliterating but stressed element in the a-line shares an initial sound with a stressed element in the b-line.  The calculation is, in fact, far more complicated because the number of available stressed elements varies. By my count, the possible candidates for cross alliteration in the a-line in the sample from Beowulf include the following 632 lines:
a-lines with one non-alliterating stressed element: 580
a-lines with two non-alliterating stressed elements: 51
a-lines with three non-alliterating stressed elements: 1
Four b-lines have only one clear stress (.004); 177 have two non-alliterating stresses (.1773) and three have three non-alliterating stresses (.003). A rough estimate of probable occurrence of cross alliteration would go like this:
Candidates for cross alliteration:
1 stress in a-line and 1 in b-line: 473
1 stress in a-line and 2 in b-line 103
1 stress in a-line and 3 in b-line 2
2 stresses in a-line and 1 in b-line 42
2 stresses in a-line and 2 stresses in b-line 9
Lines without a second stress in b-line: 3
Using a somewhat complex (or perhaps merely tedious) calculation based upon the likelihood of cross alliteration dependent upon the primary alliterator, we can estimate a general probability for alliterative linking of single non-alliterative a-line and b-line elements as .071 (i.e. about 14 to 1), and given the multiple chances for cross alliteration in some lines, a total probable occurrence for cross alliteration can be estimated at about 57 examples in 1000 lines. This result compares relatively well to the observed 43 examples of cross alliteration in these lines, although the difference may be significant, indicating that, in these lines, the poet is slightly biased against cross alliteration. 
This is not to say, however, that the occurrence of cross alliteration is always random. Consider the first 100 lines of Beowulf, left out of my examination so far. Of these 100 lines, 57 are candidates for cross alliteration. We would expect (from the estimate and observations above) only four or five examples of cross alliteration in such a sample. Yet there are no fewer than 13 examples of cross alliteration in Beowulf's first hundred lines: between two and three times the expected number.  It is unlikely that the density of examples of cross alliteration here is due to chance. Instead, we should probably conclude that, while cross alliteration is not used for effect throughout the entirety of the poem, it is used at the beginning of the poem, presumably to provide a heightened poetic effect involving denser than usual sound-patterning. That is, cross alliteration might be best considered as a useful (and usable) secondary poetic effect in Beowulf, and presumably in other poems as well.
It is important to note at this point that the calculations presented here are really only estimates. They derive from a section of Beowulf and really apply only to Beowulf itself. A quick count of the primary alliterators of Guthlac A (again skipping the first 100 lines), for example, yields a different distribution, in which vowels are slightly more frequent (.1741), and w somewhat less frequent (.1031), while g (.1031) and h (.0780) have (roughly) a reversed distribution. The shift of g and h may simply reflect the names of major characters in the two poems; the differences in vowels and w may reflect poetic habit or preference. Yet the estimates provided from the Beowulf data can be used to make at least rough estimates for other poems, although such estimates should obviously be used with caution.
Indeed, given the estimates based on Beowulf that I have made here, it is worth considering a few other poems and passages. In Azarias, for example, only 84 of 190 complete lines (44%) are candidates for cross alliteration, a significantly smaller proportion than in Beowulf. Further, only three of these 84 lines actually feature cross alliteration, and to find even three, we must count the example of "transverse alliteration" in Azarias 39: "swa waroþa sond ymb sealt wæter."  With far fewer candidates for cross alliteration and examples of cross alliteration appearing at a rate of only one in 28 candidate lines, it seems clear that the Azarias poet avoids cross alliteration to a degree somewhat greater than that of the Beowulf poet. Riddle 3 shows a similar pattern: 25 of 74 lines (34%) are candidates for cross alliteration, which appears only once (line 71). Of the 46 candidates for cross alliteration in Judgment Day I's 119 lines (41%), only one actually has cross alliteration (line 96). The general avoidance of cross alliteration in these poems seems to provide confirmation that cross alliteration was indeed a perceptible poetic effect, one which could be avoided by poets inclined to do so. Even this negative evidence, then, suggests that cross alliteration was not a purely random and coincidental phenomenon.
Other poets clearly used non-primary alliteration for effect, as does the poet of Judith. At the moment Judith stands over the drunken Holofernes, sword poised to strike, she begins to pray:
"Ic ðe, frymða god ond frofre gæst,
bearn alwaldan, biddan wylle
miltse þinre me þearfendre,
ðrynesse ðrym. Þearle ys me nu ða
heorte onhæted ond hige geomor" (ll. 83-87).
Line 86a must be scanned as an E-verse (/\x/) and 86b as an A-type verse (/xxx/x); with careful scansion of the remaining lines, it is clear that the first four lines of Judith's prayer feature cross alliteration. Since this occurs at such a key moment in the narrative, it is hard not to conclude that the cross alliteration here serves to provide a rhetorical flourish, drawing Judith's prayer into high relief by its formal alliterative density. 
In addition, it is interesting to look closely at the end of the poem now called The Order of the World, because its final lines are mostly hypermetric, and thus allow the poet a good deal of room for extra alliterative linkages:
Forþon scyle mon gehycgan þæt he meotude hyre;
æghwylc ælda bearna forlæte idle lustas,
læne lifes wynne, fundige him to lissa blisse,
forlæte heteniþa gehwone sigan
mid synna fyrnum, fere him to þam sellan rice. (ll. 98-102)
Again, if we include finite verbs as being somewhat stressed even when not participating in primary alliteration, then we see forms of cross alliteration in 98 and 102, linking alliteration connecting 99-100 and 101-102, and three lines linked by the alliteration of l (99-101).  At the end of this poem, concerned (as its modern title suggests) with the order of the world, to find such a highly ordered and ornamental array of supplemental alliterative patterns again seems both remarkable and significant: the poet here phrases his final injunction to piety in a highly ordered and alliteratively-patterned fashion, the poetic sign of the divine order which spurs on the injunction in the first place.
The corrections I have made to Ross's original estimates may seem small here. But the significance of my findings lies, of course, in the attention which must be paid to how a word's position within a verse or line relates to its alliterative qualities. In a formulaic poetic tradition, such as we have in Old English, words (and initial sounds) clearly have different frequencies and distributions than they have in prose. In order to investigate questions about accidental or purposeful alliteration, a more precise calculation than Ross's has long been needed.
Further, I believe I have provided powerfully suggestive evidence that cross alliteration and consecutive alliteration (of all the secondary alliterative patterns noted here) were far from random occurrences in many cases. Actively avoided in some poems and just as actively employed in others, cross alliteration could function to provide a denser, more rhetorically powerful kind of alliterative patterning. The use of cross alliteration at the beginning of Beowulf, at the opening of Judith's prayer, and (with other alliterative devices) at the end of The Order of the World should urge us to pay attention to how Old English poets sometimes used alliteration beyond its primary manifestation. The employment or avoidance of consecutive alliteration likewise seems to be a meaningful variation in poetic practice. A failure to pay attention to such patterns, when they are being used to effect, is a failure to understand the art of these poems.
 A. S. C. Ross, "Philological Probability Problems," Journal of the Royal Statistical Society B xii (1950): 19-59; A. J. Bliss, The Metre of Beowulf, 2nd ed. (Oxford: Blackwell, 1962). To take one more recent comment, E. G. Stanley suggests that "complex statistical analysis would be required" to draw any conclusions about the intentional use of "enjambment of alliteration" (which I call linking alliteration below), citing only Ross's essay for statistics (In the Foreground: Beowulf [Cambridge: Brewer, 1994]: 137).
 For instance, "wæs" is unstressed in Beowulf 102a: "Wæs se grimma gæst" (a B-type verse) but stressed in 140b: "þa him gebeacnod wæs" (also a B-type verse). Such alternations are frequent in Old English verse.
 George Philip Krapp and E. V. K. Dobbie, eds, The Anglo-Saxon Poetic Records, 6 volumes (New York: Columbia UP, 1931-52).
 Much of the difference between primary-alliterator rates and rates of non-alliterating stresses can probably be ascribed to formulaic usage; consider the following sets of formulas found in my sample of lines from Beowulf:
170b: wine Scyldinga, 229b: weard Scyldinga, 371b, 456b: helm Scyldinga, 500b: frean Scyldinga, 778b: witan Scyldinga;
132b: last sceawedon; 204b: hæl sceawedon, 843b: trode sceawedon, 983b: hand sceawedon;
230b: healdan scolde, 832b: þolian scolde, 1065b: mænan scolde, 1070b: feallan scolde.
As such examples suggest, the relatively high rate of "sc-" appearing on non-alliterating stressed positions in b-lines can be, to a great degree, accounted for by formulaic usage. Many of the other distributional differences seem likely to be due to similar influences.
 The lacuna at 389b-390a leads to only 999 alliterating lines. Note also that (due to another lacuna at 403b) there are only 998 b-lines in my sample.
 Note that I am generous in my estimates of stressed elements: I include secondary elements of compound names and finite verbs (other than wesan and weorðan) in the first foot, even when they do not alliterate. Other metricists may well disagree with me about the level of stress on these items.
 The estimate here is actually a bit high, since consecutive alliteration on the same sound will preempt linking alliteration. We can estimate the downward correction factor as the rate of such non-consecutive alliteration: 976/999 = .977, giving a probable occurrence of linking alliteration as .0718.
 To be more precise, cross alliteration involves the co-alliteration of syllables scanned with primary or secondary stress across the caesura which is separate from the primary alliteration.
 The examples of cross alliteration in the lines under discussion are, by my count: 131, 201, 208, 209, 236, 282, 288, 305, 343, 350, 365, 374, 396, 397, 418, 490, 506, 525, 530, 566, 589, 591, 614, 653, 690, 699, 730, 755, 779, 803, 829, 907, 919, 925, 938, 971, 987, 1016, 1017, 1023, 1044, 1065, 1066.
 Examples of cross alliteration in Beowulf's first hundred lines include: 1, 18, 19, 32, 33, 34, 39, 49 (treating garsecg as a full compound), 64, 65, 88, 93, 98.
 The other examples of cross alliteration are Azarias 49, 125.
 Of course, the Judith-poet is also very sensitive to other sorts of ornamental effects, as in his extensive use of rhyme.
 One should probably also note the supplemental alliteration on "l" in verse 99b and the internal rhyme in verse 100b, as both seem to add to the density of sound patterning in this passage. For a familiar example of finite verbs which apparently alliterate in a secondary fashion, consider Ruin 25: "Crungon walo wide, cwoman woldagas." Considering the Ruin-poet's care in employing dense poetic effects, the alliteration of these verbs is hard to dismiss as accidental and unimportant.