The 24 Shapes (in music) and why you should know them


Something endlessly fascinating about JS Bach’s 5th Prelude in the Well Tempered Clavier, you know, this prelude in D major, is the pattern on top of which it’s built.

That phrase on-top-of-which-it’s- built doesn’t mean JS Bach wrote patterned piece of music where the pattern itself was or is of the essence. As far as I know, follow the pattern was a compositional technique—for Bach and others, as well. Also as far as I know, diverging from a pattern, knowing when and how to do so, was also a composing technique for Bach—and others.

And, really, that’s that. A pattern—perhaps we might even call it a design pattern—is a concept, an abstraction, and, as such, it’s part of a circumstance or a something that happens over and over again. In other words, patterns are useful, but they’re not usually the point?

But, on, the other hand, that’s not completely true. There are plenty of well-known patterned pieces of repetoire. JS Bach’s crab canon in the Musical Offering is among the best.

Who also knew that very same crab canon fits perfectly onto a Möbius strip? That too is a pattern, the Möbius strip, that is. Watch the video—it’s all there.

Did Bach know in that crab canon he was creating the musical equivalent of a Möbius strip? Does it matter?

The 5th Prelude: There be patterns

Each measure in Bach’s 5th prelude is a sixteen-note pattern. Each measure, most of which contain the same pattern, consists of four sub-patterns, each of which is four notes in length.

Each sub-pattern has a specific name. But, for clarity, let’s switch nomenclature. Let’s now use the word group, instead of sub-pattern.

As an aside, the idea of a sub-pattern is it’s a smaller unit in a pattern. In other words, we can find and build patterns that exist within patterns. That’s old news in the world of music theory.

However, in Bach’s 5th prelude in D major, the first four-notes group in each sixteen-note pattern can be called an infra-polation. The 2nd group can be calleded ultra-polation. The third group could be called an interpolation. The last group would be an ultra-polation.

Is it really true that

Interpolations, infra-polations, and ultra-polations are things?

To clarify two points, those names that begin with infra, inter, and ultra tell us specifically how each four-note group is composed. But those names, don’t say anything about whether or not some particular group is the first, second, third, or fourth group in a pattern. They’re just patterns within patterns.

The second point is those names that begin with that begin with infra, inter, and ultra aren’t commonly used by music theorists. For example, they won’t show up in undergraduate theory textbooks and such.

Enter the inter-, infra-, and -ultra

But, yet, it’s true.

Interpolations, infra-polations, and ultra-polations are things.

Let’s now discuss those things. After that we can put them to work for us rather than the other way around.


Infra-polation is where the middle two notes are pitched lower than the first and the last two.

A quick example: with C B A D observe how the first and lasts notes spell out the interval of a major 2nd which we could also call a boundary interval. In case you program in Lisp, we could call the first note the head and the last note the tail.

We could say together the head and the tail form a boundary interval. So, head or tail or boundary interval—with those terms or that jargon—we’re talking simply and only about the first and last notes in group of notes.


Ultra-polation  describes a four-note group where the two middle two notes are both higher than the boundary interval.

The quick example: With A C D B, then C and D, the two pitches on the inside of the group, are are higher than A and B—the head and the tail that together specify the boundary interval.


Interpolation is when the two notes between the boundary interval created by the head and the tail both fall in between the head and the tail

The quick example: with A B C D, B and C lie between the head and the tail of the boundary interval formed by A and D.

Jargon summary

And that’s all there is to the jargon—it describes how a first and last note, the head and the tail, form a boundary interval. And it provides a way to discuss the the the range, high or low, of the notes that live within the boundary interval. For simplicity, we’ve used examples that have two intervening notes.

The 5th prelude: the pattern

The pattern for the 5th prelude we’ll see it in almost every measure throughout the prelude. Again, the pattern consists consists of sixteen notes divided into four groups of four. The order in which we’ll see those four groups are

Infrapolation – Ultra-polation – Interpolation – Ultra-polation

Nicholas Slonimsky

Those names, infra-polation, ultra-polation, and interpolation— that terminology, or jargon if you prefer—comes from Nicholas Slonimsky. In the early 20th century Nicholas Slonimsky, who emigrated to the US from Russia, was a very well-known conductor of and advocate for contemporary music. For example, he presented premieres of pieces by Edgard Varese, among others.

And pretty much, Nicholas Slonimsky personally knew just about everyone in music who was anyone—in the 20th century, that is. For example, an old friend, James McCartney, just told me that Nicholas Slonimsky and Frank Zappa once played together in a concert. In fact, I looked that up on Youtube and found  video where Nicholas Slonimsky speaks about Frank Zappa. Here’s yet one more link sent by James. It’s a very complete, detailed story about how Nicholas Slonimsky and Frank Zappa met.

Thesaurus of Scales and Melodic Patterns

Among Slonimsky’s accomplishments, and really the reason he’s been at the centre of this post, was his book. The Thesaurus of Scales and Melodic Patterns. Actually, compiled, is the verb that most aptly describes how he wrote the book. That’s because Slonimsky’s Thesaurus is simply and only an exhaustive list of symmetrical patterns generated by breaking an octave, or two octaves, or more octaves into smaller pieces.

So the book is the exhaustive list of scales. Among other things, it demonstrates that Slonimsky had a gift for finding obscure things and naming, organising, or taxonising them.

About that gift, when Arnold Schoenberg saw The Thesaurus of Scales and Melodic Patterns his comment was to the effect of “Yes, it is complete.” That wasn’t meant as a complement.

One more thing about Slonimsky is many jazz musicians know Slonimsky’s Thesaurus, although few really use or used it—with the lone exception of John Coltrane who famously worked with it for years. But, given those previous two videos with Nicholas Slonimsky and Frank Zappa, perhaps that should be amended to say

Few worked from it with some noteable exceptions being John Coltrane and perhaps Frank Zappa.

But, anecdotes aside, the important point is Slonimsky made up his own terminology to describe the patterns he generated for the Thesaurus, all, as already described. based on

Infra-polation, ultra-polation, and interpolation.

Patterns in practice

Once you see the pattern work in the 5th prelude—how it structures the sound of the prelude measure by measure—you may also find editorial fingerings in your edition that don’t make a lot of sense but do make for much practice. They make for much practice because they’re not always that practical.

So, that’s one use to which Slonimsky’s terminology can be put to used—which is to say, once we find a or the pattern we’re usually better prepared to deal with other instances of the same pattern.

In this case, dealing with the pattern, so to speak, means finding the most appropriate fingerings to play the prelude. To spell it out, if we find those appropriate fingerings we’ll do less practicing to learn the prelude than we would otherwise using fingering that’s, say, more clumsy.

Why fight with the piano when we can befriend it?

Patterns in theory

We can make up melodies with patterns. How exactly we do that is up to us. As one possible example, any jazz musician or improviser who’s practiced some particular lick over and over again is working with a pattern. For that matter any aspiring musician who writes counterpoint over a cantus firmus is working with a pattern.

We can break whatever patterns we make, anywhere and everywhere, with impunity, if our ears tell us to do so. Although in the case of the cantus firmus, that’s not usually done.

But stylistic issues aside, it’s useful to be able to find patterns and to figure out how we might use them. And, to say it again, patterns themselves, once we begin to work with them, don’t have to govern how we use them.

That’s because sometimes it’s enough just to know they, the patterns, are there. That in turn, reminds me of a book by Laurence Dreyfuss called Bach and the patterns of invention.

Patterns in the wild

We can all find patterns at the level of a scheme or a schema in many of Bach’s pieces, which is part of what Dreyfuss writes about in his book. Mozart’s Alberti basses, the arpeggiated figures in the left hand, are of course, patterns. They’re almost always interpolations.

More details

Jerry Bergonzi, a fabulous jazz saxophonist and educator, wrote a series of books about improvisation. One of them, Volume V, is The Thesaurus of Melodic Patterns.

I’m sure Jerry Bergonzi’s book is a response of sorts to Nicholas Slonimsky’s book. And although I really have no idea, it’s probably safe to add Jerry Bergonzi to that list of Slonimsky’s Thesaurus users—the list with John Coltrane and possibly Frank Zappa and now, likely, Jerry Bergonzi.

With four notes we can make twenty-four different groups where each group has it’s own individual identity in terms of the order the pitches it contains. The math for that is simple. It’s 4!, four factorial and that equals 24.

In other words 4 * 3 * 2 * 1. Any four-note group consists of twenty-four different orderings.

 For completeness, the first time I wrote up all of the above is in a part 1 and a part 2. The context is a little different from this particular blog post but, if you’re interested, those links are there.

More about Nicholas Slonimsky: You might previously have come across hims through his wonderful book, The Thesaurus of Musical Invective. It’s a collection of music criticism spanning several centures. It features off-the-wall bonkers writing by music critics invecting about well-known pieces by well-known composers.

You can’t make this up!

Speaking about invectives, I wrote a a chamber opera that’s since been transformed into a film. From start to finish it was an consuming project with ups and downs and all things in between.

I mean, returning to my lodging at 4am after late-night mixing sessions entailed watching very seriously for bear and mountain lions! But that’s normal, bears and the mountain lions, in the Canadian Rockies.

But, about invective. I remember a wonderful passage from a review of the opera’s premiere which took place at the Portland Museum of Art in Maine in 1994—several years before the film was made.

A critic for the Portland Press Herald wrote

The vocalist could be heard above the din.

I just now remembered that phrase—the din. I didn’t compose anything vaguely tonal for many years after that!

The din!