Rhythmic variations: a mathematical-musical explanation (part 2: syncopation)

This post is a follow-up part 1 here, in which I laid out a musical theme and the different possible rhythmic combinations one could make by "choosing" different "empty" beats. To recap, here's the musical theme, which you can play back by hitting the play button in the upper left corner, along with all examples embedded in this post! (Although warning, they load pretty slowly, it seems.)

Since it has 9 notes over 12 beats, we can vary the rhythm by changing which 3 beats will be without a note, which is equivalent to choosing which 9 beats will have a note. Therefore there are "12 choose 9" or "12 choose 3," or 220, possible rhythmic variations for the theme. At the end of the post, I claimed that 191 of those possibilities are syncopated. So what does it mean for a melody to be syncopated, how did I arrive at that number, and why does it matter that there are so many more syncopated possibilities than non-syncopated?

In this post I'll define syncopation more carefully with supporting examples, explain the basic algorithm I wrote to detect syncopations in a rhythmic sequence, and explain why it's important that basically any rhythmic pattern has more syncopated variants than not.


What is Syncopation: inverting the hierarchy of beats
In music, the passage of time is organized into beats. Or really, it might be more accurate to say, as humans, we naturally and inevitably organize musical events into beats. Repetitive cycles of beats set up a perceptual hierarchy where some beats are "strong" and some beats are "weak." Syncopation, loosely defined, is when a musical event in one manner or another inverts that perception, placing an emphasis on the weak beat over the strong one.

This is, alas, more easily felt than explained, though of course, we can arrive at a more complete understanding with a detailed explanation. Here is a simple motivating example of a melody everyone knows. Example 1 is the usual, non-syncopated version, example 2 is a syncopated version. ​

The hierarchical nature of musical beats, with some strong, and some weak—likewise easily felt but not easily explained—has to do with the brain's natural affinity for organizing events. In "Mary Had a Little Lamb," you can most likely feel that the syllables "Ma -" of "Mary" and "lit - " of "little" occur at stronger points than "-ry" and "-tle". Not coincidentally, this mirrors where you would hear the accents or stressed syllables in if someone were to just speak the words "Mary had a little lamb."

A quick funny story about accents and hierarchies in speech to further illustrate how the human brain works! A friend of mine's boyfriend is named "Nemanja," a fact that she originally relayed to me by text message. I could tell that the "ja" is pronounced like "ya", but I had to ask which syllable amongst the three gets the accent or stress. She replied that the three syllables are equally accented, but in fact this is impossible! The reason is not that it's impossible to say the three syllables with equal emphasis, but rather that it's impossible for us to hear three equally accented syllables and not "feel" the stress on the first syllable. That's because we organize the three syllables of the name into rhythmic beats (though there are many possible ways to do this, see examples), and it's impossible not to hear those beats in a hierarchy. When you don't add any external stresses or accents to any one syllable and say them in three equally-spaced time chunks, our brains default to hearing the first beat, which is always strong, as the first syllable. 

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In the case of Mary Had a Little Lamb, each level of the hierarchy is divided into two parts, where at each level, the first unit of the group is strong and the second is weak (see illustration below, ex. 4). The top level of the hierarchy occurs over two whole measures; the first measure is strong, and the second measure is weak. Each level of the hierarchy subdivides into two, so the next level is one bar, where the first half is strong, and the second half is weak. And so on. Typically we feel the hierarchy extending down to the lowest common subdivision of the beat, so in the simplest example, four levels suffice for our perception of the beat hierarchy (two bars, one bar, half bar), but in example 5, each bar is divided into 16 sixteenth notes, so our hierarchy has two more levels:

In general, melodic syncopations occur by agogic emphasis, which is a fancy way of saying that longer notes are perceived as more accented, or stronger, than shorter ones. If a note on a weaker beat of the hierarchy is longer than the preceding note on the strong beat, or if a note on a weak beat extends over the strong beat so that there is no articulation on the strong beat, then we feel a syncopation.

Example 2 then is a relatively strong syncopation at the 4th level of this hierarchy, the quarter bar or quarter note, where the emphasis is displaced from quarter note beats (for instance, the third beat of the first measure and the first beat of the second measure) to the eighth-note beats immediately preceding. The emphasis shifts because the eighth note immediately preceding is lengthened and holds across the bar line. Examples 5 and 6 below, in contrast, make a somewhat weaker-feeling syncopation because it occurs one level higher, at the level of the half note, because the emphasis is shifted from the third quarter note of each bar to the second beat, by virtue of the elongated second beat of each bar that suspends through the stronger third beat.

Examples 7 and 8 provide two more useful examples, example using 4 hierarchical rhythmic levels, but not syncopated (in contrast to example 2), and example 8 using 5 levels of hierarchy, including syncopations at the 4th and 5th levels.

A more specific definition of melodic syncopation: Weak beat articulation followed by an empty beat 
We can extend this intuitive definition of syncopation into a more specific one that a computer program can understand and detect: a weak beat articulation or note, at any level of our metric hierarchy that isn't followed by a subsequent note or articulation, always results in a syncopation at that level of the hierarchy. As illustrated above, higher levels of the hierarchy generally result in weaker feelings of syncopation.

The program I wrote only analyzes one level of a metrical hierarchy, and so it's limited in scope (the next step in the algorithm would be to make it recursive so that it proceeds from the higher levels of the hierarchy to the lower ones). Nevertheless it makes for a useful example of the theme from the last post, in which the meter is 6/8, and the most salient hierarchical level is 3 beats of a half bar, where the strong beat is the first of 3, and the remaining two beats are weak. 

Two examples suffice to illustrate the correctness of this definition specifically for 6/8 time. In example 9, the emphasis shifts forward from the strong fourth beat of the measure to the weak fifth beat, in this case by virtue of the fact that weak beat 5 is longer (quarter note) and has no articulation immediately following on beat 6. Similarly, in example 10 the emphasis is shifted backward from the first beat of the second bar because the last beat of the first bar is articulated and the following beat is not articulated.

The code for the algorithm in python is posted here on github​. The last part of the algorithm, easily modified, asks the question, "for a 9-note melody in 6/8 over 2 measures (12 beats), how many of the rhythmic variations are syncopated vs. unsyncopated?" The answer, as stated earlier, is 191 vs 29.

Why does it matter that most rhythmic variations are syncopated?
I came from a classical music background, so the emphasis in my musical education typically didn't dwell on rhythm. Although syncopation is not uncommon in the canon and history of Western classical / art music, it is comparatively rare, and the types of syncopations you hear are typically of the weaker, higher-level hierarchy variety. This is well illustrated in common folk tunes as well, which, like "Mary Had a Little Lamb" are typically not syncopated in their traditional arrangement.

But the point of the preceding experiment (or algorithm, whatever you want to call it) is that syncopation opens up a world of rhythmic possibilities, possibilities that the last century of popular and dance genres have used to great effect, without necessarily devising new harmonic or melodic "schemes" to force new styles of music into being.

Perhaps if you're familiar with the history of classical music in the 20th century, you'll recognize in the last sentence my personal scorn for the twelve-tone, atonal "systems" of music that arose early in the 20th-century, which in my humble opinion relegated art music to a relative cultural obscurity that continues to this day. In particular, Arnold Schoenberg's (failed) musical revolution, which tried to force a new system of harmony and melody down the throats of the public, considered traditional harmony and melody to basically be "used up" and already exploited to the fullest extent possible, which, when you consider the possibilities of rhythmic variation noted in this post, was a truly absurd contention, given that it came before or at least at the very beginning of a rhythmic revolution in all kinds of genres of music, from ragtime, to jazz, to rock, to pop, and so forth.