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Polyrhythmic Etudes In Canon Form

by Rudi Seitz

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1.
6:5 01:24
2.
7:2 01:08
3.
7:3 01:09
4.
7:4 01:09
5.
7:5 01:09
6.
7:6 01:09
7.
8:3 01:09
8.
8:5 01:10
9.
8:6 or 4:3 01:10
10.
8:7 01:10
11.
9:2 00:45
12.
9:4 00:45
13.
9:5 00:46
14.
9:6 or 3:2 00:46
15.
9:7 00:46
16.
9:8 00:46
17.
10:3 00:37
18.
10:4 or 5:2 00:37
19.
10:6 or 5:3 00:38
20.
10:7 00:38
21.
10:8 or 5:4 00:38
22.
10:9 00:38
23.
11:2 00:29
24.
11:3 00:30
25.
11:4 00:30
26.
11:5 00:29
27.
11:6 00:30
28.
11:7 00:29
29.
11:8 00:30
30.
11:9 00:30
31.
11:10 00:30

about

This project is a survey of the 31 polyrhythms that can be created with divisions of 11 parts or fewer. I wrote these exercises as part of my own study as a composer, as a way to expand my understanding of various rhythmic juxtapositions. The collection may also be valuable for keyboardists who could use it for hands-on practice with a wide range of polyrhythms, and also for listeners who want to sample some exciting rhythmic phenomena that are not commonly showcased.

This project fills what I saw as a gap in available resources for studying polyrhythms. Certainly there is no shortage of listening examples, given that polyrhythms are abundant in so many kinds of music including various percussion traditions of Africa, some progressive rock, much Afro-Cuban music, some classical music from North and South India, as well as 19th and 20th-century Western classical music ranging from Chopin to Stravinsky to Carter and beyond. On the pedagogical front, it is easy to find or construct rhythmic exercises that can be tapped or performed on percussion instruments (see, for example, “Polyrhythms: The Musician’s Guide” by Mike Magadini). And then there are some studies from the Western classical repertoire (like no. 2 from Chopin’s Trois nouvelles études and Saint-Saëns’ Étude, Op. 52, No. 4, as well as many of Ligeti’s études) where various polyrhythms occur in a context of sometimes-elaborate melody and harmony. What I could not find was a comprehensive survey of polyrhythms where the focus was on isolating each rhythm itself, with just enough melodic variety to aid in the perception of rhythmic groupings, but not enough melodic or harmonic complexity to provide an interpretive challenge in its own right. In other words, I wanted something with more melodic interest than a straight percussion exercise, but considerably less than a Chopin etude, so that the primary focus would remain on the rhythm, but yet the potential for fusing these complex rhythms with simple melodic ideas would be revealed.

I came upon a simple framework for building canons that can be used to illustrate a wide range of polyrhythms. The basic idea is to start with an outline where the first beat of each measure alternates between a simple octave and a compound octave, with these octaves moving up or down by some interval. Imagine playing two Cs an octave apart on the piano and say we decide to move by fifths. To begin the next measure, the high C moves up by a fifth and the low C moves down by the inverse interval, a fourth, forming a compound octave on G. Next, the low G moves up a fifth and the high G moves down a fourth, forming a simple octave on D to begin the next measure, and the process repeats, now from a starting point that's a major second higher. This outline can then be elaborated so that the larger jumps (in this case the fifths) are each filled in with a chromatic run that fits into one measure (in this case, a septuplet of quarter notes in 4/4). The smaller jumps (in this case the fourths) can then be filled in with a melodic figure that contrasts against the chromatic run, with its beats also equally-spaced across the measure through some kind of tuplet. By varying the number of beats in the contrasting figure, we can visit a range of rhythmic juxtapositions; for example, if we use a three-note contrasting figure, we engage the three-against-seven polyrhythm.

In these pieces, the rhythms constantly alternate between hands so that we have A-against-B in one measure, and B-against-A in the next. Another aspect of the framework used here is that the intervalic sequences are reversed midway through the piece, so an ascent reaches a climax and turns into a descent, and we then hear the melodic inversion of the preceding figures.

After devising this framework, my main effort as a composer was to write the contrasting figures for each rhythmic grouping in a way that balanced clarity and interest. In some cases, it was simple to do, as in the 7:5 study, where the fourths from our example above were filled in with simple chromatic runs of five notes (forming a quintuplet) against the seven-note runs in the opposite direction; in other cases it was more complicated, as when a sextuplet was needed instead of a quintuplet, so as to explore the 7:6 polyrhythm – what is the best melodic shape for the six-note figure?

To find all polyrhythms that can be created with subdivisions of size 11 or less, I simply took all pairs of integers between 2 and 11 that have no common divisor greater than 1. If we look for such pairs where 3 is the larger number, the only possible pair is 3:2. Likewise, if 4 is the larger number, the only possible pair is 4:3. If 5 is the larger number, we have 5:2, 5:3, 5:4. If 6 is the larger number, we have 6:5. If 7 is the larger number, we have 7:2, 7:3, 7:4, 7:5, 7:6. If 8 is the larger number, we have 8:3, 8:5, 8:7. If 9 is the larger number, we have 9:2, 9:4, 9:5, 9:7, 9:8. If 10 is the larger number, we have 10:3,10:7, 10:9. If 11 is the larger number, we have 11:2, 11:3, 11:4, 11:5, 11:6, 11:7, 11:8, 11:9, 11:10. There are a couple of outliers in this set of 31 possibilities that aren’t immediately compatible with the framework I described above, since the framework will always have a skip of at least a perfect fifth (7 semitones) that needs to be filled in with a chromatic run. (A skip of a tritone, 6 semitones, doesn’t work because it leads to an infinite bouncing back and forth with no ascent or descent.) How can a polyrhythm like 3:2 be fit into that framework where one of the subdivisions has at least seven parts? The solution is to render 3:2 as 9:6, using the framework with a skip of a major sixth (9 semitones) against a 6-note contrasting figure. Similarly 4:3 can be rendered as 8:6, and 5:2, 5:3, and 5:4 can be rendered as 10:4, 10:6, and 10:8 respectively.

I find that these pieces occupy an interesting place on the border between purely mechanical and aesthetically intriguing. They are all variations on the same basic idea but I find their effects differ widely. Some of them tire me out after one or two listenings while others fascinate me and draw me back for repeated grappling. For example, while I acclimate quickly to 7:2 and find that this particular study goes on a bit longer than I’d like, the 7:5 study, which is actually the same length as the 7:2, challenges my ear in a way that makes me want to hear it again right after it stops. 11:2 seems straightforward, but 6:5 is disorienting in a way where it seems to be simultaneously speeding up and slowing down at every moment--I've never heard anything quite like it. I hope other listeners might find some of the same variety in the collection and come across a few examples that are genuinely perception-expanding.

The tracks offered here are software renderings where the software “performs” the piece. My role in the rendering process was to painstakingly configure the virtual instrument (actually the recordings are meant to sound like a piano duo with one instrument per line) and make decisions about tempo and articulation. The tempos used here are only one set of possibilities and different aspects of the polyrhythmic interaction will come to the fore if the pieces are played slower or faster, which I encourage performers to experiment with. From the standpoint of keyboard fingering, these pieces are all at roughly the same level of difficulty, as they all have two voices consisting of chromatic runs and other simple chromatic figures. However, the difficulty of playing the rhythms accurately varies widely from piece to piece, with some being quite approachable and others requiring an extreme level of rhythmic virtuosity.

In most contexts, I’d prefer to offer performances by a real (human) musician, but these etudes are one case where the rhythms are so complex that it’s actually desirable to have a software rendering as a reference for how the pieces sound when played with absolute precision (as the player piano made possible in Nancarrow's studies). So, while I hope to have live performances of these studies available at some point, I think the current software renderings will retain their value as a standard for accuracy if not expressiveness and variety. Musicians interested in working with these pieces are welcome to contact me for scores.

credits

released August 29, 2016

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Rudi Seitz Boston, Massachusetts

Rudi draws upon his passion for the counterpoint of Bach and Renaissance masters, his delight in the expressive poignancy of Schubert and Chopin, and his fascination with jazz and the musics of North and South India to craft compact works in which every note counts. Along with composing, he sings and plays guitar. Rudi lives in Boston, MA. ... more

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