The solution

How was it written?

Why was it written?

A compositional challenge

Some details

Musical steganography

The entropy of music

The piece encodes the first 244 decimal digits of π, including the initial 3, which corresponds to the first note of the right hand, the E a third above middle C. Every fourth note of the right-hand part (that is, the second note of each group of four 16th notes) gives another digit. After the initial 3, the digits 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, etc. can be read off the notes: middle C, the F a fourth above, another middle C, the G a fifth above, high D a ninth above, then low D, A, and G a second, sixth, and fifth above, and so forth. This rule applies throughout the piece (except for the final two bars, which have no 16ths at all), for a total of

**How was it written?**

I started by printing out the notes corresponding to the desired
digits of π where they’d have to go in the right hand
(like this); this in turn was automated
with a simple PostScript routine that took the input

to produce the first page (NB the digits -- which I got by writing a short3 9 7 9 8 5 3 5 6 2 9 5 1 4 1 3 Pi 5 9 7 2 3 8 3 3 4 6 2 6 4 8 3 2 Pi 7 3 9 9 3 9 6 1 7 9 1 4 8 8 2 0 Pi 2 9 5 4 4 9 4 7 9 0 2 8 5 0 1 5 Pi 9 8 0 2 6 8 2 6 0 4 6 1 8 7 0 3 Pi copypage erasepage

**Why was it written?**

The immediate motivation for writing the piece is that
it would let me (and any other pianist who learns to play it)
recite the first few hundred digits of π from memory --
albeit at a pace rather slower than is typical of such exhibitions.
The only time I’ve actually done this in public was at
Harvard’s Pi Day celebration in 2005,
where my recitation was not just slow but even included a wrong digit
about two-thirds of the way through (perhaps I inadvertently
inverted the octave pattern in bar 33), even though I reinforced
my kinesthetic memory by “air piano”-ing the right hand.

Still, even 100 digits is way more than I could recite before
starting this project. The first ten or so digits I memorized by
osmosis decades ago because I saw them so many times in books or
on the display of a hand-held calculator; I could certainly have
worked my way at least up to 100 with a conscious effort (it would be
no harder than committing a dozen or so phone numbers to memory),
but it felt like a pointless waste of time.
The inspiration for spending much more time to contrive and practice
the *Steganographic Étude* was a presentation
by Ben Rapoport, a fellow member of the Senior Common Room of Harvard’s
Lowell House,
on human memory and mnemonics. Several of the mnemonic techniques
that Ben described had in common the use of a story or image
to organize the information one wants to remember.
This is the opposite of how one goes about efficiently storing
information on a computer, where one “compresses” the data
to strip away as much redundancy as possible. Humans memorize by
*adding* redundancy, as well as extraneous information
(since there are usually many ways to construct a mnemonic for
a given subject, and the mnemonic realizes just one of these choices).
Presumably the human brain can’t readily harness its huge memory
capacity to store and organize a mass of patternless information,
and the mnemonic pattern eases this organizational task.

Ben’s talk was not long before March 14, which is 3/14 in the American date format and is thus celebrated as Pi Day at many American schools and universities, including Harvard’s. Our celebration includes a contest for reciting digits of π from memory. So I thought to construct a piece of music that could function as a mnemonic for these digits. The math department had recently acquired an upright piano for the common room where the contest would be held; while very far from a concert instrument, this piano is much better than nothing, and I could practice my piece on it (anyone can use the piano after normal working hours) and eventually give the premiere on Pi Day -- probably the first and only time a new composition had been premiered on that piano, and certainly the only time I’ve ever given a premiere performance on such an instrument...

**A compositional challenge**

Copland once said (if I remember right -- and I paraphrase in any case)
that when he used serial materials in his music
it was not for their musical qualities but for the
*absence* of musical quality: the materials impose
unnatural constraints that goad the composer in directions
that would not have been explored otherwise.
My piece uses the digits of π in much the same way.
Barring the kookiest kind of mathemusical mysticism,
we do not expect there to be some pattern to the sequence of decimal
digits that lends the piece a special musical coherence.
It is precisely the fact that digits are for all
musical purposes random that makes it a nontrivial challenge
to write a viable piece of music around them. The encoding must also
be sufficiently simple that by knowing the rule one can reconstruct
the digits from the piece with little additional effort.

There’s more than one choice of encoding scheme that one could use.
I chose an association of digits with numbers that almost every musician
knows well, via scale degrees. Using the piano’s home key of C major,
and Middle C as the canonical starting point, this means
that the scale notes CDEFGABC correspond to 12345678. Most of these
correspondences are also reinforced by the name of the interval from
Middle C to the scale note: D is a second above Middle C,
then E is a third above, F a fourth, etc. (The upper C is an octave above,
and “octave” is basically the Latinate form of “eighth”;
likewise the neutral interval between middle C and itself is sometimes called
a “prime”, i.e. “first”,
though “unison” is more common.)
Naturally then the D a ninth above Middle C had to correspond
to the digit 9. As for zero, that had to be the B *below*
Middle C, to continue the progression consistently.
*
[That association is alas less natural in common-practice
music theory, where we count starting from 1 even when
it ought to have been 0; that’s for instance why two octaves
add up to a 15th rather than a 16th: arithmetically
all the interval numbers should be reduced by 1,
so octave+octave=15th is really 7+7=14.
But I digress. Twelve-tone music theory does count starting from zero,
using the digits 0 through 9 for the notes C, C#, D, Eb, E, etc. up to A;
but this encoding would make the mnemonic harder to use for most
Western musicians, since we’re trained to think of (say) the
Ode to Joy tune as 334554321123322
(see e.g. the “Scale Degree” window at
themefinder.org)
than as 445775420024422 --- to the extent that we think about it
numerically at all.]
*

OK, so we have
this sequence of several hundred notes.
This is a somewhat well-known “tune”
(which confirms that my choice of encoding is natural --
indeed one of the other competitors in the recitation contest
*sang* the first few dozen digits of π to those notes).
But it’s musically worthless as it stands:
the notes are of necessity as “random” as the digits of π
that they encode, and do nothing musically other than stake out
the key of C major. (Note that this encoding scheme not only restricts
the notes to the C-major scale but also overrepresents C’s, and the
neighboring tones of D and B: a random digit distribution yields
20% C’s, 20% D’s, and 20% B’s,
and only 10% of each of E, F, G, and A.)
So the challenge is to embed this sequence of notes into a
musically meaningful mnemonic.

I chose to make the π notes appear on every fourth note of a
“perpetuum mobile” figuration -- 25% was about
the largest proportion of forced notes I felt I could accommodate.
I made my task easier by putting those forced notes on the second
16th-note of every group of four, an unstressed position and thus one of
less structural significance in the larger-scale shape of the piece.
The left hand would usually play a chord on every beat under the stream
of 16th notes in the right hand. This texture, plus the contour of the
opening phrase or two, make the beginning of my piece reminiscent of
Bach’s D-major prelude from
Book I of the *Well-Tempered Klavier*;
but the two pieces go in different directions, and overall my piece
feels closer to *perpetuum mobile*
(last movement of his first piano sonata) than to Bach’s prelude.

But I’m getting ahead of myself. First I had to decide how long the piece would be; I figured it would be reasonable to aim for 50 bars or a bit more, so that the piece would encode a respectable 200+ digits but not wear out its welcome. I then found a point in the digit sequence that suggested an appropriately decisive concluding gesture for the piece. That still left about 700 notes and their accompanying chords to write. Again, the challenge is not just to get from one forced note to the next but to do so in a way that makes musical sense on larger scales than four 16th-notes at a time -- usually in the familiar doubling sequence of beat pairs, bars, two-bar and four-bar phrases, etc.

**Musical steganography**

The resulting piece took the form of an étude --
a short musical composition that the performer can use to
practice a specific technical challenge. (Creating the piece
could also be regarded as a compositional study or étude.)
I could have titled it a “mnemonic étude”
for its intended function, or an “acrostic étude”
because the notes that encode the digits of π
appear in the same place in each beat,
as do the letters forming the extra message in an
acrostic text. But it turns out that
spotting this message makes for a surprisingly good puzzle:
most people have a hard time finding the “message” in the score,
even when told of its existence and given various hints on its nature.
[I’d be most impressed if somebody could *hear* the acrostic
in real time!] This suggests that the étude is also
an example of musical *steganography*, whence my choice of title.
(See this Wikipedia page if you’ve never heard of
“steganography” before.)
Musical steganography seems to be a natural idea which
hasn’t been previously considered as such -- e.g. the
Grove Dictionary
has an extensive entry on musical cryptography, but none on
steganography, though some of the cryptographic examples
might be regarded as instances of steganography as well.

And yes, I did consider the alternative title “transcendental étude”, punning on the fact that π is a “transcendental number” in the technical mathematical sense that it satisfies no nontrivial polynomial equation with rational coefficients (Lindemann’s theorem). Alas this title is already firmly associated with Liszt’s series of twelve etudes. My étude, though not trivial to perform, is easy as π compared with Liszt’s...