In Common Time, Ain't No Onset on a Strong Beat When Threes Unfold, Unless There Is

A couple years ago, Richard Cohn wrote an article exploring about music in "pure duple" meter — that is, music that divides time into units of powers of 2 — interacts with a rhythmic pattern that evenly divides time into a succession of 3s. There are no integers x and y such that 2^x = 3y. This means that, if a pulse with successive onsets separated by 3 unit durations -- say, sixteenth notes -- starts on the first downbeat of 4/4 music, no onset will fall on metrically relatively important moments such as beat 2 of m. 1 (4 sixteenth notes later), beat 3 of m. 1 (8 sixteenth notes later) downbeat of m. 2 (16 sixteenth notes later), downbeat of m. 3 (32 sixteenth notes later), downbeat of m. 5 (64 sixteenth notes later), and so forth. Relatively metrically important can mean, among other things, that a change of some musical aspect, such as harmony or form, is more likely to occur at these moments. The notation below demonstrates this initial stages of this pervasive non-coincidence: never is an onset from the bottom part synchronized with an onset from the top part.

The relationship between these two parts can be inverted: if in the top part, sixteenth rests and sixteenth notes are converted into one another -- producing the complement, or negative image, of the original rhythm -- then always is an onset from the bottom part synchronized with an onset from the top part, as shown below.

In his article, Cohn spends some time with Bill Withers's song "Ain't No Sunshine." The first two verses of this song each unfold over an eight-bar span, which I hear as a shortened form of the twelve-bar blues structure, with each bar in 4/4. Instead of a third verse, Withers chains together twenty-six instances of "I know" in a single breath, each sung to what would be notated as an sixteenth-eighth rhythm to match my proposed eight-bar-verse notation. The notation below shows two possible metrical readings of this music.

Cohn puts forward Reading #1. This works out quite well for many reasons:

• the "I know" chain begins a quarter of the way during the eighth measure of the second verse's span, exactly as each of the two previous verses begin a quarter way during the measure that precedes each verse's span
• the strings fade out at this reading's beginning of the third-verse substitute
• the last "know" falls on a power of 2
• the meter falls right in line with the fourth verse to come

However, I can also hear Reading #2, which corresponds to the second example above. In fact, I find that sometimes I have to work against Reading #2 in order to hear Reading #1. Something about the new musical idea prompts a resetting of the meter for me. Or perhaps Reading #2 compels, especially in retrospect, because it continuously reinforces the pure duple's junctures with onsets. In Reading #2, the word that falls on a power of 2 toggles back and forth between "I" and "know." This means one cannot stop the pattern on any power of 2, unlike Reading #1, where the word that falls on a power of 2 -- if there is a coincident onset at all -- is always "know."

One can generalize this phenomenon beyond 2s and 3s. There are no integers x,y, and z such that 2^x = zy, and z is not a power of 2. Cohn's article, and the discussion above, concern the situation when z = 3. The next largest z would be 5. I have in mind a 23-second passage in a well-known song by a progressive rock band for which z = 5 would be appropriate. However, it neither continuously avoids pure-duple moments (like my first example) nor continuously articulates them (like my second example). Rather, it inhabits a happy medium between these two extremes, creating both a pulling away from stability and a push toward resolution, all within a single perpetual process. I will blog about this music next February.