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How do you teach polyrhythms?

In my college years I encountered a recurring four-against-five pattern in a 20th-century piece, and my initial attempts to do it were not successful. My teacher recommended that I approximate the pattern ("fake it") while I learned the rest of the music. He also suggested that I first try tapping the polyrhythm away from the piano, which I did after the lesson with partial success. I found it helpful to accent the first note of each beat (the only vertical in 4:5). To bolster the independence of the hands between the beats, I also experimented with having each hand follow the designs that appear on dice. This choreography helped each arm feel where it was in the pattern.

While driving home and stopped at a red light, I continued practicing: my left hand tapping four on the steering wheel, my right hand banging away a five on the dashboard (Plymouth Dusters had very good resonance just to the right and above the speedometer). I got it! I was doing each pattern evenly in each hand without any attention to how the "inner" notes interacted with each other! I kept doing it until the car behind me honked its horn - not in celebration of my accomplishment but because the light had changed! By the time I arrived home, I was also able to reverse the pattern. The next day back in the practice room, I alternated between tapping on the fallboard and practicing a few 4:5 patterns from the piece. Achieving accuracy of the rhythm was no longer a crapshoot - I was consistently getting it. 

I don't recommend that everyone use this "auto pilot" approach to learning polyrhythms, even if your dashboard also has beautiful tone. Fortunately, both of the authors for this issue offer many fine practice ideas and solutions that are far safer!

Editor's note: For other discussions of this topic, please see the Rhythm department in the Spring 1997 and Summer 1999 issues of Keyboard Companion. 

Relationships and independence

 by Pete Jutras

A long time ago in a galaxy far, far away, I used to play piano and trumpet. When I decided to major in music, I chose to focus all of my attention on the piano, mainly because the piano's ability to play multiple notes at the same time resulted in a rich, complex, and exciting body of repertoire that was very appealing to me. While I don't regret that choice at all, there are many moments in my practicing when I jokingly think to myself, "Why didn't you stick with the trumpet? One note at a time would be so easy compared to this!!!" It is true that the piano's ability to do so much at once can be both a blessing and a curse, and this is certainly evident in the topic of polyrhythms.

When teaching and learning polyrhythms, I think there are two fundamental approaches: You can either focus on the relationship between the two hands, keying in on the mathematical proportions between the two rhythms; or you can focus on the independence of the two hands and let them each move towards the next downbeat without trying to fit them together. 

Happy together

The most basic polyrhythm is 2 against 3 (2:3), and this is typically the first one encountered by students. The simplest solution to this problem centers on the relationship side of the equation. While many students approaching 2:3 for the first time perceive it to be very difficult, it is in fact no harder than playing a very simple rhythm: 

If students can play this rhythm, they can play 2:3. Here a mathematical solution works well, since the second eighth note falls exactly between the second and third notes of the triplet. If the student still is unsure of the exactness of this relationship, divide each rhythm into six equal parts, and you can prove your point. 

This " relationship" approach is clear and direct for 2:3, but for most other polyrhythms, it is unwieldy and difficult. With 3:4, it is certainly possible to break things down into 12 parts for each rhythm (the lowest common denominator) and see where they fall, but in most situations there simply isn't time to count such small subdivisions. In The Art of Piano Playing,1 the esteemed pedagogue Heinrich Neuhaus noted that the only instance he could think of which warrants such internal counting of the divisions is the opening of the second movement of the Rachmaninoff Second Piano Concerto (see Excerpt 1). This is a beautiful example of 3:4 with clear and obvious proportions, but most passages of 3:4 move much more quickly. 

Excerpt 1 Rachmaninoff: Concerto No.2 in C Minor for Piano and Orchestra, Op. 18. II. Adagio Sostenuto, mm. 1-2

Most other polyrhythms defy a simple relationship-based solution. For example, passages that demand 7 against 3, 5 against 8, 11 against 6, or worse! In these cases, a mathematical solution is nearly impossible, and it is better to pursue an approach based on independence of the hands.2 

Worlds apart

In these cases, I teach students to learn the rhythms without relating the hands. We consciously try to avoid "fitting" the notes into each other - instead the emphasis is on each line being perfectly independent, even, and musical.

Students often find this difficult at first, and they can hardly be blamed for taking such a position. For their entire pianistic lives, they have been continuously asked (or perhaps badgered!) to keep their hands together and related at all times. This sense of relationship becomes an ingrained part of their muscle memory, so much so that students who rely heavily on muscle memory often have trouble starting from an odd spot with just one hand.

Asking students to abandon this comfortable feeling of togetherness and enter a foreign world of un-relationship is a big change, and it is one (like most concepts) that will proceed more smoothly with good sequencing and preparation from the teacher. Here are some fundamental steps to help the student, using 3:5 as an example: 1) Establish a strong steady pulse. Have the student move and internalize the beat. For consistency, you may wish to use a metronome or other external reference.

2) Count the 5s while moving to the beat and accenting the downbeat. "1- 2 - 3 - 4 - 5 - 1 - 2 - 3 - 4 - 5 ...."

3) Count the 3s in similar fashion to the same beat: "1 - 2 - 3 - 1 - 2 - 3 ...." 

4) Repeat until each rhythm is secure, then alternate rhythms without pause. Keep feeling the main pulse.

5) Transfer the rhythm plus the following downbeat to a simple technical pattern on the piano. For this one, I've chosen a minor five-finger position for the 5s and a minor triad for the 3s (see Example 2). Other patterns that can be useful for different groups include diminished seventh chords (for 4s), I-V-I single notes (for 2s), whole-tone scales (for 6s), and diatonic scales (for 7s). Larger groups can be handled with chromatic or scalar runs.3 

Example 2

6) Repeat each pattern hands alone ad nauseam, continuing to feel a strong downbeat. Repetition is essential here, as the hands need to be able to go on "autopilot" with each pattern. 

7) After each pattern is secure, start alternating between the hands without pause, still feeling each downbeat. 

8) The final step is a leap of faith. After many repetitions of alternating hands, close your eyes, don't think too much, and let the hands go at the same time, concentrating on the main beats. If you've practiced each rhythm enough, the hands should do their own thing and arrive together on the downbeat. Initially, the brain may try to interfere by thinking too much, and the rhythms will fall apart. At first, let the hands be free and automatic. After some repetitions, we'll invite the brain back in to listen to each part independently. 

9) For most students, the rhythm will not be completely secure and reliable after their first time through step 8. It is often necessary to alternate step 8 with step 7 for some time, allowing the student ample opportunity to reinforce and solidify the rhythm. 

Once those steps have been mastered with a simple technical pattern, they can be repeated with a passage from the repertoire. Doing them first on a separate technical pattern allows the student to focus completely on the act and execution and not worry about the actual notes. Another advantage to this approach is that if mistakes do arise in the first attempts, the student isn't learning the actual passage in the repertoire incorrectly and making wrong first impressions that could crop up in future performances.

As students play the polyrhythm with both hands, encourage them to listen independently to alternating parts - first focusing on one hand and then the other, listening for smoothness and flow to the rhythm. If necessary, move from practicing one instance of the polyrhythm to multiple iterations - scales practiced in polyrhythms are a great tool for practicing extended sequences of the polyrhythm. Just remember that if the LH is playing the faster rhythm, it will always have to start two or more octaves away from the RH or the hands will collide!

Which approach is better?

Artistically, I think the flow and seamlessness of the independent hands approach leads to a superior musical product. As a teacher, I can't deny the usefulness of getting students started on 2:3 by showing them the simple proportions. In my experience, this gives them a more direct and concrete start, and this is important. In an ideal world, they would quickly move from this mathematical beginning to a musically flowing, independent approach to 2:3 - no one would argue that feeling every "One Two and Three" of a 2:3 passage would usually result in a plodding and pedantic sound. Keeping the hands independent is challenging, but the musical result is worth it!

Avoid incidents!

There are many pieces that have what I would term "incidental" polyrhythms - one small instance of 2:3 or 3:4 that lurks unsuspectingly in an otherwise rhythmically innocuous piece. A great example of this is the first movement of Mozart's Piano Sonata in F K. 332, Mvt. 1. The student plays along happily, lining up twos and fours and ones, and then BAM - in measure 49, 3:2 appears (see Excerpt 3). Similar spots lie in ambush in measure 67 of the second movement of Beethoven's "Pathetique" Sonata in C Minor, Op. 13. After an entire movement of normal relationships between the hands, Beethoven calls for an isolated instance of 4:3.

If a student is unprepared, these become prime spots for mislearning - one hand is likely to become uneven, or there will be an abrupt change of tempo, or both. As an isolated spot that gets little repetition, it may be hard for the student to really get a feel for the flow of the polyrhythm, and this spot is likely to remain a proverbial bump in the road for a long time. These pieces, and others like them, should not be assigned unless the student is comfortable and confident with the required polyrhythm.

This is an example of what is called "incidental learning" - learning of a concept, often a peripheral concept - that occurs only in an isolated section or two. In music teaching we do this all the time, particularly with things like music vocabulary terms, unusual accent marks, and other score markings. With rhythms however, we are dealing with more than just a cognitive concept - we are dealing with a feeling, and the feeling can't be learned incidentally.

I believe that when students learn a rhythm, it should be something they learn wholly and completely - they should hear it, move to it, feel it, and reproduce it in a variety of settings at a variety of speeds. Once it is ingrained this way, it becomes a part of the student's natural music vocabulary, and they can use it effectively and musically in practice and performance. For this reason, I think pieces like Debussy's Arabesque No.1 and Chopin's Fantasie-Impromptu in C # Minor are excellent vehicles for teaching 2:3 and 3:4, respectively. Despite their warhorse status, they provide a lot of repetition of the concept, and they force the student to learn the polyrhythm wholly and completely. Pedagogically, this is much more productive, and it will help the student handle pieces that do have incidental polyrhythms with security and confidence. 

Mozart: Piano Sonata in F Major, K. 332. 1. Allegro, mm. 47-50

Endnotes

1 Neuhaus, H. (1993). The Art oj Piano Playing. London: Kahn & Averill. (Original work published in 1967. First English translation published in 1973.)

2 It should be noted Chopin often wrote such passages (known as fioriture) - in his repertoire. These polyrhythms do not need a strict mathematical interpretation, but instead can be played with a degree of rhythmic freedom and flexibility. In the work of later composers such as Scriabin, one should try to be exact with these rhythms.

3 I am forever indebted to one of my former teachers, Mr. Alfred Mouledous, for illustrating the creative possibilities for practicing odd rhythms using simple technical patterns, helping me to solve a problem that to that point had always been troublesome. 

Stocking the musical pantry

 

by Sonnet Johnson

A few years ago a very musical student transferred to me after her teacher moved to another city. This young lady wanted to "finish" the Debussy First Arabesque. Unfortunately, the inequality and distortion of her polyrhythms had become an ingrained habit. It's obviously much easier to prevent such problems than it is to repair the damages.

How do I teach polyrhythms? First, I begin early, before polyrhythms appear in the student's repertoire. Second, I teach the whole body by initially establishing the patterns in the larger muscles. Third, I reinforce frequently by using polyrhythms in multiple ways prior to introducing them in the repertoire.

I often use cooking metaphors in my teaching. When putting together a big holiday dinner, many preparations can be done well in advance. Polyrhythms are like that. I like to think of them as one of those staples that should always stay in our musical pantry, ready for use anytime they are needed. 

Students are ready to begin stocking their musical pantries with polyrhythms as early-intermediate students. As soon as a student is able to feel and count a steady pulse, and can subdivide that pulse into duplets or triplets, a teacher can begin the introduction of both at the same time. I think the student should be comfortable practicing with a metronome prior to the introduction of polyrhythms so the metronome can be used as a means of assessment.

Polyrhythms are spices and seasonings that create wonderful synergy. As the two opposing rhythms "rub" together, they form a unique, rhythmic interrelationship. Since they generate such interesting musical qualities, polyrhythms become increasingly more common as the piano literature advances. They can be a joyful, puzzle-solving adventure for both students and teachers.

I believe every student needs multiple approaches to polyrhythms. However, the choice of where to begin may vary, based upon a student's age and learning style.

Two against three and three against two

Generally, the first polyrhythms I teach are 2:3 and 3:2 (the first number refers to the LH and the second refers to the RH). These are the easiest polyrhythms because the rhythm sounds like "quarter eighth-eighth quarter," making assessment relatively easy.

Since rhythm is much more physical than mathematical, I usually introduce these by rote, having the student copy my gestures. We tap our knees with large arm motions or tap on a closed fallboard or even a drum, if available. Sometimes I use numbers to help organize these tapping gestures.

1  2  3  4  5  6 

1  2  3  4  5  6

Next, I use diagrams to organize the gestures visually and to rep- resent divisions of time. I typically use box charts to form a visual image of 2:3 and 3:2. 

Example 4

In Example 4 each square represents one unit of the underlying subdivision. The RH taps with the X's in the top row and the LH taps with those in the bottom. 

Example 5

Example 5 uses a series of dots (representing subdivisions within the polyrhythm) with stemming showing the RH and LH gestures. This is the diagram that I use most frequently, especially for more complex polyrhythms, since it's so easy to draw. Then I add words, literally speaking the directions for the tapping motions: "Both--, Left, Right, Left--" or "Together--, Left, Right, Left--." I also reverse the pattern. Example 6 shows both diagrams. Verbalizing the hand divisions ingrains the physical and mental patterns more securely. 

Example 6

When these gestures are comfortable, I use my favorite saying for both 2:3 and 3:2: "Not Dif-fi-cult." The words are psychologically encouraging and also function as a mnemonic (or memory aid) for retaining the pattern. The mnemonic helps with listening assessment, especially when spoken with precision and energy. 

Example 7

Example 7 on page 44 shows both diagrams with "Not Dif-fi-cult." I have students tap only one hand while saying the words so they learn to listen to only one rhythm of the pattern at a time. In my experience, students who verbalize the hand divisions and speak the mnemonic with the polyrhythm are able to retain these patterns more easily.

Depending upon the student, I cover these steps in a few lessons or over several months. I prefer to reinforce polyrhythms a little longer than may be absolutely necessary to ensure security. Each polyrhythm is drilled at many different speeds. 

The ultimate goal is to assimilate the polyrhythm into one pulse. Students can walk while tapping the polyrhythm on their body and say, "Not Dif-fi-cult." This will establish a whole-body feeling of containment within a pulse, all done away from the piano using big gestures. With polyrhythms, it's important to have mastery in the large muscles first, because small muscles do not retain what large muscles do not understand. 

Polyrhythms in technique and keyboard skills

To introduce these polyrhythms at the piano, I assign them in various ways into the student's technique routine. For example, I integrate 2:3 by assigning polyrhythmic scales with the RH going up and down three octaves, and the LH two octaves, as in Example 8.

Excerpt 8

For those students who really like a challenge, the scale assignment can be polyrhythmic and polytonal at the same time! The teacher could also play triplet scales and have the student play duplet scales against the same pulse.

It's possible to include polyrhythms in keyboard-harmony drills by assigning familiar, five-finger-pattern pieces which have melodic eighth notes, like Little River Flowing, with a triplet accompaniment using tonic and dominant chords (see Excerpt 9). Later, the hands can be reversed.  

Excerpt 9

Occasionally, I have students improvise a brief series of polyrhythms within specific boundaries (like five-finger patterns or on groups of black keys) during their lessons. In some cases, it makes sense for the student to use a polyrhythmic passage as a transposition exercise, as in the third movement of the Haydn Sonata in Major, Hob. XVI:7, shown in Excerpt 10. Sometimes I use the actual notes of the polyrhythm (taken from a piece to be assigned later) as the basis for a sequential pattern exercise. It's always fun to create exercises combining polyrhythms with various elements of technique. 

It's practical to isolate specific polyrhythm examples from several different pieces and have a student practice just those sections. By this time, the student should have little trouble performing this specific polyrhythm within their repertoire. With sufficient groundwork and preparatory repertoire, even the complexities of the 3:2 rhythm in the Grieg Notturno, Op. 54, No.4 become manageable. Editor's note: For a complete article on this piece, see the Summer 1999 issue of Keyboard Companion, or go to our website to hear audio clips from that article. 

Excerpt 10

Three against four

Similar processes are appropriate for teaching 3:4. There are now twelve subdivisions, with three notes in the LH and four in the RH (see Example ll).

My favorite saying for 3:4 is "PASS the GOLD-en BUT-ter" (see Example 12) which I learned from Professor Harold Heiberg one summer at the American Institute of Musical Studies in Graz, Austria. I also use the phrase "DRILL the PO-ly- RHY-thm." These sayings have the underly- ing subdivision of a triplet; they are not reversible because the accents in 4:3 fall differently.

The trick is to have the R-L and L-R motions fall very close to each other as the arrows in the example indicate. Very slow practice is beneficial in order to get the adjacent notes close enough together. A typical error students make is to play the second note in the RH too soon, creating six relatively even beats. Many students benefit by focusing their drills on certain portions of the pattern, such as "Pass the Gold" or "Golden Butter." 

Excerpt 11

The Chopin "Minute" Waltz in D-flat Major, Op. 64, No. 1  and the  Fantasie-Impromptu, Op. 66  use a 3:4 polyrhythm, as do many of the  Nocturnes.  Another famous example of this polyrhythm is in the first movement of Beethoven's "Moonlight"  Sonata, Op. 27, No. 2.  Some scholars recommend playing the sixteenth as one sixth of a beat, which is too late and too short to be a true 3:4. We know from Czerny that the sixteenth note is not to come with the final triplet, as in an 18th century "accommodated" rhythm. Charles Rosen warned, "the problem lies in making the sixteenth note different from the triplets eighths, and yet not so short as to be trivial."Although the Adagio is written in alla breve, student performances are often too slow, using the quarter (or even the triplet eighth) as the pulse, instead of the Adagio half note. I personally prefer it as a true 3:4 polyrhythm because of the character. I have heard European teachers correlate this rhythmic, repeated-note motif with the same rhythmic/ melodic idea in the grave- yard scene of Mozart's opera, Don Giovanni. 

Four against three

The 4:3 polyrhythm has an underlying subdivision of four. Maurice Hinson (in B eethoven Selected Easiest Sonata Movements for Piano, Vol. II, edited by Maurice Hinson. Alfred Publishing Company) gives a useful saying: "BACH MO-zart BRAHMS and LISZT" (see Example 13).

Here again, the difficulty is in keeping the R-L and L-R motions in close proximity rhythmically. This polyrhythm is found in Debussy's "The Snow is Dancing" from Children's Comer.

Over a period of time, I intentionally assign repertoire containing these four com- mon polyrhythms and I select pieces that become gradually more complex. When stu- dents are given an in-depth and multifaceted process in their basic experience with polyrhythms, they develop problem-solving skills which carry over to other more complicated polyrhythms. 

Example 13

Other polyrhythms

With all polyrhythms, I multiply the number of notes in the LH rhythm pattern by the number of notes in the RH rhythm pattern, which expresses the least common multiple. This gives the total number of subdivisions within the polyrhythm. For example, 3:5 would have fifteen subdivisions (see Example 14). Once more, the inner patterns of L-R followed by R-L obviously need to be close together.

The key to solving any new polyrhythm is (1) working out the right/ left order and (2) determining the places where the hands fall in close proximity. I usually diagram new polyrhythms in lessons and mark arrows showing which notes fall close together. I also review the various ways of practicing, always experiencing the patterns in the large muscles first. 

Conclusion

Polyrhythms are so commonplace in the advancing piano literature that serious problems will develop unless these skills are mastered. When the basic polyrhythms are introduced in terms of whole-body experience, technical proficiency, and sufficient repertoire, students become independent learners free to explore the vast realm of polyrhythmic puzzles in the piano literature. With a solid foundation of understanding in body and mind, students can solve polyrhythms of ever greater complexity and perform them with freedom and emotional expression. ...

Endnote
Rosen, C. (2002). Beethoven's Piano Sonatas, A Short Companion. New Haven and London: Yale University Press, p. 157. 

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