Metric, geometric; even mystic

Just a curiosity, this, and as much as I can manage at a stupidly busy point in the academic year.

It comes from what is perhaps the closest thing the ancient world had to a blog, the Attic Nights of Aulus Gellius. Around the middle of the second century A.D., initially during an extended stay in Athens (hence the title), Gellius collected information on topics that interested him, presented in short, self-contained notes on word use, antiquarianism, philosophy–whatever had caught his attention. The twenty books of the Noctes Atticae, all but one still extant, preserve some precious material: something I’ve been writing recently on the Roman priests of Jupiter known as the flamen and flaminica is very dependent on Attic Nights 10.15, for example.

The last note in Book 18 runs like this (NA 18.15):

In the long lines called hexameters, and likewise in senarii (iambic trimeters), scholars of metrics have observed that the first two feet, and also the last two, may consist each of a single part of speech, but that those between may not, but are always formed of words which are either divided, or combined and run together. Varro in his book On the Arts wrote that he had observed in hexameter verse that the fifth half-foot generally ended a word, and that the first five half-feet had equally great force in the creation of a verse as the following seven; and he argues that this happens in accordance with a certain geometrical ratio.”

(In longis uersibus, qui hexametri uocantur, item in senariis, animaduerterunt metrici primos duos pedes, item extremos duo, habere singulos posse integras partes orationis, medios haut umquam posse, sed constare eos semper ex uerbis aut diuisis aut mixtis atque confusis. M. etiam Varro in libris Disciplinarum scripsit obseruasse sese in uersu hexametro, quod omnimodo quintus semipes uerbum finiret et quod priores quinque semipedes aeque magnam uim haberent in efficiendo uersu atque alii posteriores septem, idque ipsum ratione quadam geometrica fieri disserit.)

The issue here is the metre of epic poetry, the dactylic hexameter, and with less emphasis the iambic trimeter, metre of dialogue in tragedy, both lines consisting of six metrical feet; specifically at issue is where word breaks or caesuras were expected to fall in the verse line. The first sentence is essentially concerned with the convention in both the hexameter and the trimeter that a major word break falls within the third or fourth foot; or to put that another way, the convention that a word break should be avoided between the third and fourth foot, that is, a caesura dividing the line into two exactly equal parts.

The second sentence continues the interest in how a hexameter line was articulated, but takes a peculiar turn. It cites M. Terentius Varro, the celebrated polymath of the first century B.C., noting in his nine-book Disciplinae (maybe in the book on music; maybe in the book on geometry: only fragments of the Disciplinae survive) that the (Latin) hexameter was normally divided into two at a caesura in the middle of the third foot: analysed in terms of half-feet, semipedes, this break divided the line into five half-feet in the first section of the line and seven in the remainder of it. Then the mysterious further observation that although unequal in length, the first part of the line had “an equally great force in creating the line” as the longer second, and that this was in accordance with “a geometrical ratio.”

Varro’s idea is elucidated very deftly by the world expert on Gellius, Leofranc Holford-Strevens,* revisiting an explanation by Henri Weil** in the nineteenth century (online here in German and here in French). The key to understanding Varro is a long account of verse structure in the fifth book of the De Musica of St. Augustine, where it looks very much as if Augustine is following the same passage in Varro as Gellius is citing. It is an essential feature of a verse properly so named, according to Augustine, that it is divided into two unequal, and thus not interchangeable, parts. This characteristic of a verse is inherent in its very name, he claims: uersus, quia uerti non potest, “It is called a verse, because it cannot be reversed.” Considered more closely, however, these superficially unequal parts of the hexameter and the trimeter turn out to share “an amazing equivalence,” aequalitas mirabilis (De Musica 5.12.26). This hidden balance is revealed by mathematics: if the seven parts of the longer section of the line are further subdivided into three and four half-feet, the sum of the squares of 3 and 4 (9 + 16) equals the square of the five parts of the shorter section, 25. Augustine thus seems to be giving us what is unstated in Gellius: “the first five half-feet have equally great force in making a verse as the following seven,” and this is so in accordance with a “a certain geometrical ratio.” At this more esoteric level, the unequal components of the hexameter line in fact prove to be equal.

This is a fascinating line of thinking, but (it hardly needs saying) thoroughly unhinged. It isn’t entirely certain that Augustine’s idea can be blamed on Varro. It suits Augustine’s project in the De Musica as elsewhere, “to demonstrate the presence of an organizing principle functioning in every aspect of reality,”*** very closely indeed, after all: even by studying poetic metre we can rise from the disorder of the corporeal realm to the perfection of the spiritual. But we also have Gellius’ heading for this chapter, which seems to characterise Varro’s original observation as highly peculiar: Quod M. Varro in herois versibus observaverit rem nimis anxiae et curiosae observationis, “That Marcus Varro noted in heroic verses something requiring excessively anguished and painstaking observation.” That does sound like Varro also was dealing in squares. It’s also not obvious what else Varro could have meant by “a geometrical ratio,” or at any rate what he could have meant that would have drawn this interest (and this heading) from Gellius.

But what does any of this matter? Not a lot, for sure. But let’s assume that Varro did believe that the hexameter, in particular, metre of the highest poetic forms, possessed this remarkable character, that beyond its superficial imbalance it embodied a near-mystical perfection. Varro’s voice was an influential one, and not only on later figures like Gellius and Augustine. So we can’t exclude the possibility that Virgil, for example, a younger contemporary of Varro, when he described Dido, Queen of Carthage, in a hexameter of perfect elegance, regina ad templum, forma pulcherrima Dido (Aeneid 1.496), the line disposed into two parts of five and seven half-feet, felt that he was wielding a metrical form that was itself of ineffable beauty.

____________________________________________

 

*L. Holford-Strevens, “Parva Gelliana,” Classical Quarterly 44 (1994), 480-89, at 483-6;

**H. Weil, “Die neuesten Schriften über griechischen Rhythmik,” Jahrbücher für classische Philologie 8 (1862), 333-51, at 336-7; idem, Études de littérature et de rythmique grecques (Paris, 1902), 142-4;

***P. d’Alessandro, Varrone e la tradizione metrica antica (Hildesheim, 2012), 101-146, at 132.

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About Llewelyn Morgan

I'm a Classicist, lucky enough to work at Brasenose College, Oxford. I specialise in Roman literature, but I've got a persistent side-interest in Afghanistan, particularly the scholars and spies and scholar-spies who visited the country in the nineteenth century.

4 responses to “Metric, geometric; even mystic”

  1. Jonathan Jones (@nmrqip) says :

    There’s a more obvious way to relate this to geometry and music, which the Pythagoreans would have been familiar with.

    The relevant question is “how does the number 1.5 split up the interval between 1 and 2?” The obvious answer is that it comes half way between the ends, because (1.5-1)=(2-1.5)=0.5. This answer is based on addition and subtraction, and is what we would now call an arithmetic interval. But there’s another sense in which 1.5 is more than half way along, because (1.5/1) is bigger than (2/1.5). This answer is based on multiplication and division, and is what we would now call a geometric interval.

    If we use the geometric picture then how far along the line from 1 to 2 is 1.5? Well using logarithms to base 2 we have log_2(1.5)=0.584963, a number which is very close to 7/12=0.583333. So we can say that 1.5 splits the interval almost in the ratio 7:5.

    This pattern will be familiar to any musician as the way in which the perfect fifth (frequency ratio 3:2) splits the octave with seven semitones below it and five above it.

    The weakness with this argument is that it seems at first sight to suggest that the line should be split seven and five, not five and seven. But it still makes a lot more sense that Augustine’s ramblings.

  2. The Shaved Poet says :

    Marvellous article, thanks! If I was in one of your Oxford tutorials, I’d say “Yes Sir”, but since you dangle these things in public, I’ll take the bait and run with it. I don’t agree with ‘thoroughly unhinged’ and neither, I think, do you. Unhinged from an English point of view, yes of course, but you know perfectly well that Vergil has some kind of numerology going on in his Georgics, and that Pindar often represents poetry as architecture, a branch of knowledge where mathematical proportions are significant. So I have no doubts that some classical poets would have experimented architecturally with numerical ratios, and since they thought geometrically, the squares you mention could have been understood as virtual squares, possibly with 5 half feet in 5 consecutive lines, 3 in 3 lines, and 4 in 4 lines – making three squares of different sizes, each functioning as a unit with some kind of allusive or coded meaning built into it. Of course, verse like that is the least likely to survive transmission, once the code or key is lost or forgotten, but maybe Vergil has got something somewhere along those lines.

    Another thing that strikes me, is one of those dumb things even professional scholars sometimes say – not you, I hasten to add, but a professor of English literature, who once wrote somewhere that English verse settled on the iambic pentameter as its main verse form because somehow it is a natural unit for our language – more feet tend to break apart, fewer are too few for argument’s sake. But as your article points out, where the line breaks is part of the dynamics of verse. I think iambic hexameters would be just as effective in English as they were in classical verse, especially in dramatic poetry. Anyway, thanks again for the stimulus.

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