Perfect Intervals, Made Obscure

Over the last few days, I’ve begun slowly working my way through the second volume of the English translation of Proclus’ commentary on Plato’s Republic. It’s slow going, both because the arguments Proclus offers in the text are challenging in their own right, and also—perhaps moreso—because of the huge barriers and impediments to understanding which the translators add in by their own choices.

The translation is a collaborative project by Dirk Baltzly, John Finamore, and Graeme Miles, the present volume of which was published earlier this year by Cambridge University Press. This post is not meant as a complete book review. It’s rather an attempt (a) to share some interesting philosophical and historical points from Proclus’ text, and (b) to note some of the pervasive problems with the choices and attitudes that Baltzly, Finamore, and Miles have brought to the work.


In Essay 7 (with which this volume begins), Proclus is discussing the relative proportions between the three different parts of the soul, as famously presented by Plato in the Republic. Our translators give fairly typical English renderings of the names of the three parts: reason, spirit, and appetite.

Here Proclus is discussing the virtue of moderation, as it appears in the well-ordered tripartite soul:

Now surely if it is reason — since it is to both [the other parts] ruler and cause of their reversion upon it and of their becoming subject to persuasion — that contains the first principle of the measure it give to those [other parts], then this would be moderation: originating from reason, it finds its completion in appetite through the intermediary of the spirited part. It is thus also a harmony that runs throughout (dia pasôn) since it is composed from three terms: reason, spirit, and appetite. Since spirit is the one in the middle, taken with the one it makes the concord of the musical fourth (4:3), while taken with the other it makes the fifth (3:2). The concord between spirit and reason is the fifth, while that between spirit and appetite is the fourth. At any rate, the Pythagoreans called the latter a ‘syllable’ insofar as it is not a perfect concord, while the fifth is more a concord than this one.

212.21–213.4, pages 41–42 in the translation of Baltzly et al.; all parentheses and brackets as given by the translators.

At the end of this passage, the translators add the following footnote:

See the note by Winnington-Ingram at the conclusion of Festugière[’s French translation of this same text] vol. 2. It is unclear why Proclus thinks that the musical fourth is imperfect in any sense or less of a harmony than the fifth.

p. 42, n. 41; brackets mine.

Such dismissive comments, of the style “it’s unclear why Proclus would think this,” are pervasive throughout this volume, as they were in volume 1. From the accumulated mass of such remarks, the reader of the translation could be forgiven for thinking that Proclus is simply a muddled, unclear thinker making the sort of wild, disconnected assertions which are unworthy of our serious consideration as philosophers.

Surely, though, if they instead chose to apply a principle of charity in their interpretation (and chose to treat Proclus as a serious, even profound philosopher, whose careful and meticulous thought should have the power to challenge us), our translators could have given this just a little bit of thought, and gone beyond this dismissive note. Lacking ready access to an academic library, I have been unable to consult the remark in Festugière volume 2; nonetheless, if it did provide any clarity to the issue, our translators could (and should) have said more in their own note than simply “it is unclear.”

So, what’s going on here? After reading the passage and the dismissive note, I came up with three reasonable possibilities, the last of which seems to me the most important.

(1) The elegance of the mathematical ratios. Fractions or ratios like 1,347:296 are, in some fairly intuitive way, less elegant, less simple (recall the importance of unity to the Pythagoreans and the Platonists, including Proclus!), arguably less perfect, than ratios like 2:1 or 3:2. So too, 4:3 is just a bit farther from that elegant simplicity.

(2) Musical overtones: the higher notes which emerge of their own accord, when a properly tuned interval is sung or played. The intervals to each of these additional notes, implicit in the original, are typically thirds and fifths, but never the fourth.

(3) The standard (“Pythagorean”) tuning of the classical and post-classical Greek world. As I learned as an undergraduate philosophy student being introduced to the history of Pythagorean thought, you can’t tune the octave, the fifth, and the fourth perfectly all at the same time. The small discrepancy, discovered by some of the earliest Pythagoreans (a discovery which allegedly drove some of them to suicide!) is known as the “Pythagorean comma.” In our modern system of tuning, known as equal temperament, the solution is to tune the octave perfectly, but then to fudge all the other intervals within the octave, just a little bit, by the same amount (hence the name). On this approach, the fourth and the fifth would indeed be equally imperfect. But in the Pythagorean system of tuning, as the very first sentence of the Wikipedia article tells anyone who bothers to read it, the solution was to tune (from the tonic) both the octave and the fifth perfectly, and then to fudge the other intervals (including the fourth) a bit more than we do today, to get them to fit. While this Pythagorean tuning may not be The One Objectively Right Way to Tune a Lyre™, it is the way that the music of Proclus’ world was standardly made.

And so, contrary to the translators’ dismissive remark in the note, we have quite a few reasons why Proclus might have made the assertion that he did, all of them grounded in the Pythagorean mathematical and musical theory that is well-known to be one of Proclus’ major concerns.

Can I say with indubitable certainty that any or all of these specific considerations were on Proclus’ mind when he composed the paragraph in question? Of course not. But I’m quite confident in mentioning them, and in seeing their relevance—especially that of the third point—for bringing a rather large amount of clarity to Proclus’ text.


This careful consideration is the sort of thing our translators could have done, too, whether by recalling their own long-ago undergraduate lessons, doing a tiny bit of research in publicly available sources, or simply asking their colleagues in the music departments of their respective universities, “hey, what do you make of this?”

Lest my own reader think I’m making mountains out of molehills, the passage in question is not merely an offhand remark in Proclus’ text. Proclus goes on to use the differences between the different ratios as an illuminating metaphor (or more than a metaphor) for the different relations that obtain within the soul: The spirited part is situated closer to the appetite than to reason (since both spirit and appetite are irrational, and so found together in nonrational animals, etc.). Yet when the soul is properly tuned or harmonized, the consonance between spirit and reason is greater than that between spirit and appetite. And all three parts are gathered together in the most perfect interval of all, the octave (made of adding a fifth and a fourth).

And if my third point is even plausibly implicated here—as it clearly seems to be—then we find Proclus weighing in on what was literally a life-or-death issue within the Pythagorean tradition for more than a thousand years, in a way that transforms a discovery which once destroyed people’s confidence in a rational cosmos, into something that, through that very same imperfection, profoundly illuminates the condition of the human soul, and the role of reason within that soul!


This example is just one of many in the present volume (and in the first volume by the same team of translators), which I selected because I think the musical points are especially interesting in their own right. I won’t belabor the issue by quoting and expounding others. But the mere fact that I can easily spend 800 words unpacking the meaning that’s brushed under the (very lumpy) rug by our translators’ dismissive note may start to show why this book has been slow going, for both of the reasons I mentioned at the outset.

And of course, that meaning seems to be one that, once pointed out, is worthy of a great deal more careful meditation and attention.

So while on the one hand I’m glad that this important philosophical work is finally becoming available in English, on the other hand, I’m rather saddened and frustrated that such a needlessly problematic translation will be the only one available in English for the foreseeable future, throwing unnecessary impediments in the way of anyone who might want to take seriously the careful, systematic rigor of Proclus’ philosophy.

I’m certainly appreciative of the work the translators put into preparing this volume. Both Proclus’ arguments and his Greek can at times be extremely challenging, and those challenges are compounded later in the volume, due to the poor condition of the manuscript. But at the same time, I feel compelled to ask: Do the translators want Proclus to be taken seriously as a worthy philosopher, or do they want to brush him aside, consigning him to a dark, rarely-visited storeroom in E.R. Dodds’ “dusty museum of metaphysical abstractions”? While I cannot read their minds to say for certain, a great many indications throughout their work point, tragically, to the latter.

2 thoughts on “Perfect Intervals, Made Obscure

  1. “Plato compares sophrosūne or harmonía (Rep. 430e, 431e, cf. 442c). He does not (does he?) actually call it a harmonía ðià pasōn, but he uses the term ðià pasōn (with sunáiðontas, 432a) as a kind of metaphor to describe the effect produced by sophrosūne upon the constituent members of the state. Proclus, or some predecessor, has taken the metaphor as a precise analogy and has sought to work it out in detail, in relation not to the state but to the parts of the soul. It was a commonplace that the octave (ðià pasōn, in the ratio 2:1) could be analyzed into the fifth (ðià pénte, 3:2) and the fourth (ðià tessharon, 4:3), with the result that there are three terms (hōroi), the middle of which stood in the relation of a fifth to one of the extremes and of a fourth to the other. The octave was regarded as the perfect consonance (v. infra); the fifth was ‘more of a consonance’ (māllon…sunfonìa) than the fourth, because its ratio was simpler i.e. involved smaller numbers. Proclus therefore gives the less imperfect sumfonìa to the relationship between lógos and thumós, since they are in closer harmony than is thumós with hepithumía. He realizes however that, while the consonance of the fifth is more perfect than the fourth, the distance between the terms is larger; and he finds a reason for this too (213.7). (Proclus should perhaps have reflected that, if hepithumía stands in the octave relationship to lógos, it should be in closer harmony with it than either with thumós!)”

    That’s the first paragraph of the note of Winnington-Ingram. As for the approach of Baltzly, Finamore, and Miles, I’m not surprised. Many of them were involved in the later volumes of the Timaeus translations that started out in V. I with Tarrant comparing Proclus and his circle to reactionary neoconservatives fighting progress in the intro (on page 4). Kudos to them for all of this hard work, but they (and, sadly, many translators nowadays) are only marginally better than Dodds when it comes to hostility.

    Liked by 1 person

    1. Thanks for this! It looks like the Winnington-Ingram account — “its ratio was simpler, i.e., involved smaller numbers” — is equivalent to my option (1) from the list of three I gave in the post.

      And because I’m feeling snarky today (at all of them, not at you!), as to that “commonplace that the octave could be analyzed into the fifth and the fourth”… umm, yeah, that’s still Music Theory 101. Like, the first week of it.

      Liked by 1 person

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