Someone directed me to an article in Smithsonian Magazine, “Secret Mathematical Patterns Revealed in Bach’s Music” (February 16, this year). It reports on a study that attempted to quantify the “information entropy” or “surprise” in some of Bach’s compositions. But the study’s way of describing musical surprise is limited, and it will identify as surprising certain kinds of musical passages that really are not.

Here’s a summary from Smithsonian:

[Physicist Suman] Kulkarni boiled down 337 Bach compositions into webs of interconnected nodes and calculated the information entropy of the resulting networks. In these networks, each note of the original score is a node, and each transition between notes is an edge. For example, if a piece included an E note followed by a C and a G played together, the node representing E would be connected to the nodes representing C and G.

Chorales, a type of hymn meant to be sung [hymns by definition are meant to be sung—ed.], yielded networks that were relatively sparse in information, though still more information-rich than randomly generated networks of the same size. Toccatas and preludes, musical styles that are often written for keyboard instruments such as the organ, harpsichord and piano, had higher information entropy.

The problem is that there are kinds of musical surprise that cannot be reduced to the sounding of a new note, and there are new notes that can sound without surprising you.

A better method would need, first, to take into account chords and harmonic function (the relationships, in a given key, between a chord and other chords). Let’s take the notes mentioned above. E, C, and G, if played together, form a C-major triad (let’s make C the lowest note). Or part of a C-major tetrad. Or part of a minor-seventh chord on A in first inversion. Or other possible things. You can hear these notes in sequence and have no sense that you have moved to — been “surprised” by — a different chord, because you may not have been; and you can hear the notes together without hearing any clash, as would not be true of two different chords.

But now suppose we go from E, C, and G to F-sharp, A, C, and D. F-sharp and D do not belong to any of the chords I mentioned above and indeed clash with them. F-sharp and D could belong to, say, a first inversion of a dominant-seventh chord on D. That could lead to a G-major chord, and thence back to a C-major one. The study is blind to this entire “are we on a new chord now?” domain of musical surprise.

It even gives false positives. Nothing could be more unsurprising than to hear someone play a rapid C-major arpeggio up and down four octaves. Each note is different from the preceding one, yet you always know what’s next and they’re all part of the same chord.

A second limitation of the study’s method is that it does not recognize the existence and separateness of voices. The chorales, mentioned above, are set for four human voices to sing, each moving harmoniously with but independently of the other three. A similar principle applies in most instrumental composition.

A third element of musical surprise the study must overlook is everything from counterpoint, i.e. the quasi-dialogical play of independent voices as they develop a motive or “subject” according to complex rules (as in a fugue or canon). If you go to the keyboard and play a chunk of The Art of Fugue, you’ll recall that the fugue subject is sometimes made proportionally longer (augmentation), sometimes proportionally shorter (diminution). Sometimes it is played “upside down” (inverted). Sometimes it is played in reverse order (retrograde). Between the statements of it in different voices there are certain recurring intervallic and harmonic relations. All of this varied specificity in and between groups of notes would register in the study as merely so many new individual notes — just like the notes of our rapid but boring C-major arpeggio. Again the surprising is represented in a way that makes it look banal.

There is more-frequent harmonic change and more independence of voices — and in respect of those things more musical surprise — in a Bach chorale than in the arpeggio- and scale-like passages in Bach’s toccatas (although the toccatas have other passages that are chorale-like or contrapuntal).

One could include these missing domains of musical surprise in a form of description something as follows: A possible chord is a column of compossible pitches. The actual chord is specified by assigning to each possible pitch in the column a value, “sounded” or “silent”; the sounded pitches make up the chord. Relationships between chords are evident from the series of columns, since new chords are columns with different compossible pitches from those in the preceding column. And to perceive different musical voices, we draw lines as necessary from pitch to pitch in column to column. Chords are vertical piles of notes; melodies one or several are horizontal lines between notes in the piles.

A C-major arpeggio will look like this: The same chord-column, over and over, with but one note sounded each time, and with no lines between notes (for there is not really anything that qualifies as melody; it is never stepwise and singable, but simply the scattered elements of a single chord, reduplicated in successive octaves). All of this will be predictable as we listen and can go on indefinitely: C, E, G, C, E, G, up and up and up and up, and then C, G, E, and C, now G, now E . . .

Not much musical entropy.

A chorale will have (usually) four notes in each column; most columns will be different from the preceding and following ones; and we will see four distinct sets of lines, corresponding to the four voices, that go from chord to chord forming a multilayered and much less predictable tracery.

More musical entropy, if fewer notes.