These situations do not arise as often in pre-1900 Western music as they do in post-1900 Western music, but, when they do, classical composers more often offer another solution: make the underlying meter yield to the pulse, rather than the other way around. I call this temporal process "trickling up": a pulse at a relatively quick frequency that is incommensurate with the prevailing meter begins on a relatively strong part of the meter (beginning-synchrony), and then the meter stretches to include some multiple of this pulse's frequency, setting up an end synchrony within this new meter. Here's one example of trickling up from the last movement of Haydn's String Quartet op. 76 no. 4, nicknamed "Sunrise." Below are this movement's first 21 and 3/4 measures.
The opening establishes a pure-duple meter on multiple levels, such as common time (4/4), two-measure hypermeasures, and four-measure phrases. A one-measure extension in m. 19, which uses an ascending-third melodic sequence to lift the top voice to the precarious ninth of the dominant harmony, breaks from the two-measure pulse. However, it appears the two-measure pulse returns with the repeat of m. 20 as m. 21 and the new idea in m. 22. Yet, this new idea begins a repeating three-eighth-note motive (G-F-Eb in Violin 1) that initiates a three-eighth-note pulse, although the bowing playfully continues the quarter-note pulse of the pure-duple meter. A post-1900 pop-music response to this situation might be something like the following:
But Haydn does not do this. Instead, he stretches the two-measure unit to a three-measure unit. This allows the accent scheme of the three-eighth pulse to be both beginning- and end-synchronized, while also cleverly dovetailing the end of the G-F-Eb motive with the pick-up of the recapitulated main theme (marked as A).
In this example, the threes trickle up by a factor of 8, which is 2 to the power of 3 (2^3): the repeated motive is three eighths long, and the three-measure unit that accommodates this repeated motive is eight times as long as this motive.
"Even Less" is the opening song on Stupid Dream, the fifth album by the progressive rock group Porcupine Tree, released in 1999.
Below is a diagram -- click it to zoom in -- that maps out the proportions, metric organization, and form of the seven-minute version of the entire song; Porcupine Tree later released longer versions of the song. The colors represent similarity and difference of material. (Curiously, what I've labeled as the chorus has no singing, but otherwise it functions a lot like a chorus.) I am considering 1 as the beat, around 124 bpm. The asterisk indicates the ordering of the metrical hierarchy: X*Y means that there are Y instances in a row of a X span.
Much can be said about temporal proportion in this song. For this post, I will focus on the 192-beat solo section from 4:33-6:06. This solo section divides into two equal spans of 96 beats. The first span features a loud guitar-based three-beat pulse, a more distorted guitar solo, and an accompaniment with fewer layers in its texture. The second span features a switch to a softer bass on the three-beat pulse, cleaner guitar solo, and an accompaniment with more layers in its texture. Unlike my analysis of all other spans of the song, each of these two solo spans is labeled in two different ways: 2^5*3 and 3*2^5. The 2^5*3 means that the 96-beat span is divided into three equal parts by the repetition of some pitch pattern. The 3*2^5 means that the 96-beat span is divided into 32 equal parts by the three-beat pulse. Rather than continue the pure-duple meter of the chorus into the solo section, and amend the three-beat pulse using one of the first two methods shown in the first two examples above, the 64-beat span is stretched into a 96-beat span. This allows the accent scheme of the three-beat pulse to be both beginning- and end-synchronized, creating satisfying metric resolutions at both 5:20 and 6:06.
In this example, the threes trickle up by a factor of 32 (2^5), four times that of the example from Haydn. This is the highest trickle-up factor I have come across; please post a comment if you know of one that is greater.
