Researching Some Music in Ligeti's Musica ricercata

The composer György Ligeti was born 93 years ago today. Here is a relationship between pitches and meters in some music from Ligeti's Musica ricerata that requires a little investment upfront.

1. A measure like 2/4 can be represented by the ordered series of positive integers 2-1, where 2 is the strong beat and 1 is the weak beat. Likewise, a measure of 3/4 can be represented as 2-1-1. A measure of 4/4, instead of being represented by 2-1-1-1, is often thought to have a metrical accent halfway through that is not as strong as the downbeat, but stronger than the weak beats: this could be represented as 3-1-2-1.

An uneven measure like 7/4 often breaks down as 4/4 + 3/4, but instead of simply concatenating 4/4's 3-1-2-1 with 3/4's 2-1-1 to make 3-1-2-1-2-1-1, a representation of 4-1-2-1-3-1-1 reflects the measure's initial division into 4/4 + 3/4 by making the beginning of the 3/4 measure into a "second-rank" downbeat.

When the quarter-note beat of one of these meters is subdivided into eighth notes, this inclusion can be represented by adding one to each of the numbers and then placing a 1 after each of them. For example, 3/4 (2-1-1) with eighth notes would be 3-1-2-1-2-1. 7/4 (4-1-2-1-3-1-1) would be 5-1-2-1-3-1-2-1-4-1-2-1-2-1.

2. Now, given a certain series of numbers, its cumulation is a series of numbers that sum the original series starting from the left and up to that point. For example, the cumulation of 2-1-1 is 2-(2+1)-(2+1+1) or 2-3-4. The cumulation of 3-1-2-1 is 3-4-6-7.

3. Lastly, a cumulation mod 2 of a series is a series's cumulation where every even number is replaced by 0 and every odd number is replaced by 1. For example, whereas the cumulation of 2-1-1 is 2-3-4, the cumulation mod 2 of this series is 0-1-0. The cumulation mod 2 of 3-1-2-1 is 1-0-0-1.

Below is an 7/4 example from the Overture to Leonard Bernstein's Candide -- placed into time signatures commensurate with the discussion above -- that shows 1) the 7/4 meter as a series of positive integers, 2) the culmination of the series, and 3) the culmination mod 2 of the series.

Below is a video with a score and recording of the second movement of Ligeti's Musica ricercata; attend to the first four measures.

I've re-notated these four measures to make a steady stream of eighth notes, and analyzed the music using the methods outlined above.

Notice how the alternation between 0s and 1s in the culmination mod 2 of the first measure, the second measure, and the third and fourth measures as a single measure of 7/4, corresponds exactly to these measure's alternations between the two pitches that open the movement. Each fermata both resets the metrical hierarchy and toggles the pitch-meter mapping.

Thanks to Nick Shaheed for encouraging me to think about this music.