On the Power of Inversion in Rhetoric and in Music

Good afternoon,

Every language has its characteristic ways of thinking, its inherent limitations, and the mental freedoms it allows.  

In every normal language, there is a difference between the cat, the definite one, and just a cat. In Latin, there is no such distinction — felis et felis una sunt. In Latin, the verb appears only at the end of the sentence, no matter how long the sentence is or how much content is packed into it. In German, as is commonly said, one may wait through 30 words before reaching the end of the sentence, where the word nicht suddenly appears and negates everything that came before it without warning. What does this say about how the mind of a Roman citizen worked? Or what does it say about how the mind of a modern German speaker still works? I am not one to say.

So without immediately jumping to the obvious conclusion — the assumption that the music of different peoples is influenced by the language they speak — let us pause at something even more basic: the difference in the way music is discussed from one language to another. The way we talk about music in French, for example, reinforces certain structures that are different from those found in German or English discourse. Even the subject I will speak about today — inversion in music — is one whose discussion requires us to begin by defining the boundaries dictated by language.

The Hebrew word hippukh , in current Hebrew musical jargon, denotes at least three different meanings. One of them — chord inversion — will not concern us today. But the term hippukh also denotes two other distinct operations, transformations that can be applied to a musical idea in order to vary and develop it — and these are two operations for which not every language offers as convenient a linguistic umbrella as Hebrew does, where both fall under the same term: hippukh.  

Hebrew hippukh includes both what is called inversion in English and retrograde. In German, these are Umkehrung, Spiegel or Spiegelung, and Krebs. The two German terms Spiegel (mirror) and Krebs (crab) have also entered Hebrew as names for these two specific types of inversion.  

Therefore, to avoid confusion, today’s musical discourse tends to use the terms mirror inversion (hippukh mar’ah) and crab inversion (hippukh sartan). Still, the general term hippukh remains shared by both operations. Incidentally, in the 1955 dictionary of musical terminology issued by the Academy of the Hebrew Language, these two operations are distinguished by separate names: hippukh is reserved for mirror inversion only, while crab inversion is referred to as nasog (נָסוֹג) and mafre‘i (מַפְרֵעִי — later spelled מַפְרְעִי).

What are these two types of inversion — mirror inversion and crab inversion?  

From the post–20th-century perspective, it’s very tempting to illustrate them using a compositional method that made use of both, and perhaps even combined them in a way that no language other than Hebrew could express so concisely.  

This method, which unites the two types of inversion, is called the twelve-tone technique — a method associated with the composer Arnold Schoenberg and his students (though its history is a bit more complex than that). In any case, a demonstration based on the twelve-tone method may actually be confusing as an opening example in a lecture, so I’ll illustrate the concept using simpler material.  

Even 300 years before Schoenberg, these two types of inversion were already considered two possible ways of manipulating musical material. The composer Henry Purcell, for example — whose music I wrote my doctoral dissertation on, and whose music I will also discuss today — wrote a single theoretical treatise, on composing polyphonic music, and in that treatise he discusses both types of inversion.  

He uses the Latin terms that were current in his day: per arsin et per thesin for the “straight” and the “inverted” of the mirror type; and recte et retro for the “straight” and “inverted” of the crab type.

Let’s take a simple musical idea from Vivaldi’s Concerto Op. 10 No. 6.

But the truth is that both Purcell and the twelve-tone technique blur the fundamental difference between the two types of inversion.  

On the musical staff, the graphic representation makes it easy to see the distinction between horizontal and vertical symmetry — the two dimensions, equivalent to each other, that make up a two-dimensional space. But music does not exist in two dimensions, at least not in the equal, reciprocal way that two-dimensional space exists on a sheet of paper.  

A musical performance transforms these two equivalent dimensions into two very different ones:  

• The vertical dimension becomes the frequencies of sound waves: a high note represents a sound wave of high frequency; a low note represents a wave of low frequency.  

• The horizontal dimension, by contrast, becomes duration! What appears to the right will be heard later than what appears to the left.

And yet, of course, these two dimensions absolutely define one another:  

We have no way to mark positions on the musical timeline except by means of sounds; we cannot define the duration of a silence without a sound before and after it. That is to say: sound—which is nothing but the frequencies of the vertical dimension—defines the horizontal one.  

Likewise, we cannot perceive a pitch frequency without a period of time. A wave’s frequency is the number of oscillations within a time span. Without time, there is no frequency.

So, the simple relationship between X and Y on the page becomes far more complex once it unfolds in time—especially  

if we assume  

• that someone is actually experiencing that time,  

• that time passes over the listener and affects them,

• and that the perception of time is subjective.  

In cognitive terms as well: the X and Y axes of the staff are processed in one area of the brain, while tracking pitch and sequence — the order in which notes occur — is handled in other areas, responsible for temporal perception, memory, and pitch hearing.

Later in the talk, I’d like to focus on the more challenging dimension — the horizontal one — and discuss retrograde inversion, the “crab” or “retrograde” inversion: a reversal of the order of sounds or broader musical ideas. In other words, inversion along the time axis.

* * * 

Let us speak of reordering, and of the retrograde inversion, the "reversal", the "crab" inversion — as it appears in rhetoric.

Inverting the order of words in a sentence is a well-known expressive device; a familiar trick in which, within a sentence, the words switch their places.

This trick is known as chiasmus — chiasmus is the name given to this rhetorical stunt.

Its weak point, for those who use it, is a lack of variety; and lack of variety is, for those who use it, its weak point.

One solution to the problem is the use of so-called 'Synthetic Parallelism' where one changes the order, but also advances the argument:

Choose — certain words — at the beginning of the sentence;

And at its end — after other words — let your eyes wander...

however,

A speech written entirely in parallelisms is exhausting; and a full lecture in chiasmus is wearisome.

Still, I hope its rhetorical power is clear to you… It is no coincidence that it appears throughout the Bible, far and wide.

One of the best-known examples is in Genesis 9:6. Listen to the following six words:

Shofekh dam ha'adam; ba'adam damo yishsfekh

[litteraly: Whoever sheds the blood of man through man shall his blood be shed]

The 1st and the 6th relate to each other -- shofekh/yishafekh

The 2nd and the 5th relate to each other -- dam / damo

The 3st and the 4th relate to each other -- ha'adam / ba'adam

The book most saturated with parallel structures — including chiastic parallels — is, of course, the Book of Psalms. Psalm 20, for example, contains a cluster of chiastic parallels between verses 3 and 6:

3May He send your aid from His sanctuary, and may He support you from Zion.  

4May He remember all your meal offerings and may He accept your fat burnt offerings forever.  

5May He give you as your heart [desires], and may He fulfill all your counsel.

6 Let us sing praises for your salvation, and let us assemble in the name of our God; may the Lord fulfill all your requests.

Naturally, it’s not easy to preserve these features in translation. In verse 6, for example, the verbs (sing praises / assemble) are external actions. 'For your salvation' and 'in the name of our God' are abstract ideas.

In the original Hebrew (נְרַנְּנָה, בִּישׁוּעָתֶךָ--    וּבְשֵׁם-אֱלֹהֵינוּ נִדְגֹּל) there’s a reversal here: verb–object / object–verb. That’s a chiasmus. This verse exists in more than one English translation. In the King James Bible from the early 17th century, it is rendered chiastically:

We will rejoice in thy salvation, and in the name of our God we will set up our banners.

Here too, the actions (rejoice, set up our banners) are external, while thy salvation and the name of our God are internal. As in the Hebrew original: verb–object / object–verb. But in another translation — one set to music by Matthew Locke, one of Purcell’s teachers — the chiasmus disappears:

“We will rejoice in thy salvation, and triumph in the name of the Lord.”

This yields: Rejoice → Salvation; Triumph → The name

So: verb–object / verb–object — still a parallelism, but not a chiasmus. There is no reversal.

Interestingly, there are also cases where translators introduced a chiasmus that wasn’t in the Hebrew original. Take Psalm 27:5, for example:

“For He will hide me in His shelter in the day of trouble; He will conceal me in the secret of His tent.”

In the Hebrew (כִּי יִצְפְּנֵנִי, בְּסֻכֹּה בְּיוֹם רָעָה: יַסְתִּרֵנִי, בְּסֵתֶר אָהֳלוֹ;), this is not a chiasmus — just a standard parallel.

But in the King James Bible, it becomes a full-blown chiasmus:

For in the time of trouble he shall hide me in his pavilion: in the secret of his tabernacle shall he hide me.

Another example I’m especially fond of comes, in fact, from the Gospel according to John, chapter 19. 

Pontius Pilate is crucifying Jesus, and in verse 19 he places above the crucified man a sign reading: Jesus of Nazareth, King of the Jews. The Jews object to the inscription, and ask Pilate to amend it so that it is clear that it was Jesus himself who had said, “I am the King of the Jews.” In verse 22, 'Pilate answered and said, “What I have written, I have written.”' I do not know Greek, but the standard Latin translation — the Vulgate — reads: respondit Pilatus: quod scripsi scripsi. Why do I bother mentioning a sentence that is evidently not a chaismus? Not in English, nor in Latin... 

Because in the German translation — the one used in the most famous musical setting of the Gospel of John, Bach’s St John Passion — the phrase reads:

Pilatus antwortet: Was ich geschrieben habe, das habe ich geschrieben.

And that, ladies and gentlemen, is a magnificent chiasmus! When Bach sets this phrase to music, it sounds almost as if he’s thinking about the chiastic structure as he composes:

In fact, Heinrich Schütz, Bach’s great predecessor, also composed a St John Passion, and we can see him playing with retrograde inversion there as well — even if not exactly.

Exact musical retrograde certainly exists, but it’s extremely rare. The few examples that do exist are quite famous:

• Bach composed a short canon in which a melody can be played against its own retrograde.

• Schoenberg, in one of the Pierrot Lunaire songs, has the ensemble accompany the singer through the first half of the song, then instructs them to play the second half in reverse.

• Paul Hindemith ends his Ludus Tonalis, a cycle for piano, with a closing movement that is an exact retrograde of the opening prelude.

But exact retrograde in music is very different from chiasmus. Chiasmus deals with components of a sentence, reversing their order — often with synonyms or antonyms involved.

In fact, chiastic structures appear on a much larger scale in literature. The biblical flood narrative, for instance, is arranged chiastically:

Events at the beginning correspond to those at the end; events just after the beginning align with events just before the end — and at the center of the story is Genesis 8:1: “And God remembered Noah.” This kind of structuring is clearly mnemonic — a memory aid — and similar chiastic structures appear in many folk epics from other cultures as well.

Chiastic structure exists at the level of letters or individual tones; at the level of words, phrases, or ideas.

It can be exact — like in the words “noon,” “deed,” or “racecar” — or free, shaping the sentence or story as a whole.

Can we find musical equivalents to this kind of freer, large-scale inversion? We can, for example, identify musical forms that clearly fall into an A–B–A shape. A well-known example is the children’s song Yonatan HaKatan (originally 'Hänschen klein'). And even more loosely, three-part forms like the sonata form or minuet–trio–minuet all follow an A–B–A or A–B–A’ structure.

But what about more complex forms — A–B–C–B–A, or even A–B–C–D–C–B–A? These are known as arch forms, and they are famously associated with the great Hungarian modernist Béla Bartók (1881–1945). So, in contrast to the Flood Narrative, which takes us back 2,000 or 3,000 years — back to a time when literary theory was, well… nonexistent — Bartók uses formal tricks like chiasmus with full awareness. For him, the shaping of his compositions according to specific forms was a central priority. Indeed, Bartók’s works are often described as ones in which the climactic moment falls exactly on the golden section.

Is it as thrilling to discover the golden section in Bartók as it is to discover it in the spiral of a seashell or in ancient Assyrian architecture? Well, perhaps less so — if only because unlike the seashell, Bartók could deliberately plan his music around the golden section. He could choose to design his music in an arch form from the very beginning.

My own small contribution to the study of chiastic structures is the discovery that the English composer Henry Purcell, whom I mentioned earlier, used them extensively. I’d now like to play a short excerpt from one of his sonatas for two violins, cello, and bass. The main musical idea here — [sing motif] — is passed between the different voices. Even for those who don’t read music, I hope you’ll be able to follow along with the colored annotations I’ll add to the slide. Every time this musical idea appears, I’ll highlight it and indicate which instrument it appears in — first violin, second violin, or cello.

To what extent did Purcell do all of this consciously? In my humble opinion — he absolutely knew what he was doing.

First, unlike crop circles, such complex structures don’t just happen by accident. Second, some of the chiastic verses from Psalms I mentioned earlier were not randomly chosen. Some of them were set to music by Purcell himself — meaning there's no doubt he was familiar with chiastic parallelism, since he was composing music to texts built on that very structure all the time!

To sum up:

Inversion, both in rhetoric and in music, can serve multiple roles. It can function as a rhetorical flourish. It can serve as a mnemonic aid. It can also be an architectural device that conveys meaning — like placing “And God remembered Noah” at the center of the flood story. That’s more than just a memory aid — it’s the heart of the story’s message. 

And what we see in Purcell, in my view, is simply a working method.

In light of all this, we begin to see inversion not merely as a stylistic curiosity, but as a powerful and versatile tool — one that crosses the boundaries between rhetoric, literature, and music. Whether used consciously, as in the works of Bartók, or embedded more intuitively, as in the music of Purcell, chiastic structures offer a way of shaping time, meaning, and memory. They can highlight central ideas, draw connections across distances, and lend a sense of symmetry or inevitability to a piece. Above all, they remind us that form is never just an afterthought — it is often the very vessel through which content gains its deepest resonance.