Rhythm: THREE EXERCISES in CROSS-RHYTHMS
What is a cross-rhythm?
When an ongoing, regularly recurring periodic pulse is audibly organized into regularly repeating groupings (by the use of accents, phrasing, pitch patterns, etc.) and is heard against a second rhythm with the same underlying pulse but with different regular groupings, the result is will produce the sensation of two overlapping rhythms which cross over one another: “cross rhythms.”
That is, two different rhythms may not align with one another but, if they flow according to the same pulse, then after each of them cycles through a certain number of times, they will again meet up at their respective beginnings. Two phrases are illustrated below: the upper one has seven pulses while the lower one has only five. So after five iterations of the 7-count, its beginning will again align after seven iterations of the 5-count.
1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
So five 7s will finish at the same time as seven 5s. That is, 7+7+7+7+7 = 35, and 5+5+5+5+5+5+5 = 35.
Writing that out more simply, it might look as follows:
5 x 7 = 7 x 5 OR in words: 5 groups of 7 equals 7 groups of 5.
So, in general, it is the groupings which ‘cross’ not the individual articulations.
Writing it still more simply: <5 groups of 7 against 7 groups of five>. OR, still simpler: 5 groups against 7 groups, or 5:7 (five against seven)
In order for the separate rhythms to actually align, they must be based on a common underlying *pulse. Lacking a common underlying pulse, the conflicting groupings may be perceived as random, and they will not likely even give the impression or expectation of eventual realignment. And if there are no apparent points of realignment, musicians will neither sense it nor refer to it a cross rhythm.
*Pulse is not a fixed, technical term in musical practice and, like most tempo-related terms, it is always relative to the surrounding tempos. Pulse can refer to the presence of an ongoing pulsation (or periodicity) in the music, which others may hear variously as the beat or a subdivision of the beat. Depending on the notation, a quarter- or half-note may sound like an entire measure, as in the fourth movement (Sehr langsam) of Mahler’s Symphony #9. Alternatively, a whole measure might be sensed as a “pulse” or beat, and in that case, the average listener may not have a sufficiently quick ear to hear the smallest, fastest subdivisions, that is, the pulse flow. One example of this is the typical performance tempo of the second movement of Beethoven’s Symphony #9. The notation shows 3 beats to the measure (3/4 meter), but the beats flow by so fast, that for many listeners, the entire measure may be sensed as a beat, and many listeners typically tap their foot or hand to that “beat”, which Beethoven notated in the score as a measure of three-beats. This movement of Beethoven’s Symphony #9, marked Molto vivace, may easily prompt the listener to hear these three-beat measures as if they were triplets or like sub-beats of 6/8 meter. There are many such examples in music where musicians may sense and name different musical layers quite differently from the way they are notated.
(Working with the Cross-Rhythm and Cross-Phrasing of Two-Against-Three)
2. STEPS & GESTURES (5-against-6)
3. STEPS & GESTURES (easier: 2-against-3)
4. CROSS-PHRASING (based on the cross-rhythm 2:3)
PALM TURNING
CROSS RHYTHM AND CROSS-PHRASING
The cross-rhythm illustrated in the image below shows one method of deriving the cross-rhythm, “2 against 3.” It was derived here as a result of diminishing a phrase of three beats, specifically, fitting in two double-speed phrases of three against the original three beats. The LH was given to mark a steady beat and it was suggested that the class name the beats as “one – two – three.” When that felt established, I invited everyone to sense a diminished pulse flow (i.e., double-speed) inside each of these three beats, naming them “one-and-two-and-three-and.” In trying to establish the fact that there were simply six pulses (six discreet sensations)—and that the names were, to some extent, irrelevant—we renamed them as “one-one-one-…” or then “two-two-two-two-two-two” or “four-four…” and so on. This was done so that everyone could realize that the names of the subdivisions were arbitrary and could be changed without disturbing the inner sensation of the beat and the sensation of six subdivisions.
We then renamed them as, “one-two-three-one-two-three” and then, to confirm that there were indeed two groups of three, we renamed them (and added some accentuation): “ONE-two-three-TWO-two-three-ONE-two-three-TWO-two-three- ...” This is the simplest way to understand a cross-rhythm. It is not two beats running against three beats, but rather, two groups of three pulses running along with three groups of two pulses, where the pulses are always moving at a constant speed. The two quicker vocalized threes sounding against the single slow three of the LH arm was realized by the RH, which gently tapped on the right thigh, marking the first pulse of each of the two groups of three.
The text below is a verbal representation of the numerical representation above.
LH: (tapping a slow three beat pattern)
X X X
Voice:
Step #1—1 2 3 (counting aloud)
Step #2—1 & 2 & 3 & (counting and sensing this diminished flow)
Step #3—1 2 3 1 2 3 (rename the six pulses)
Step #4—1 2 3 2 2 3 (sense it as One-2-3 Two-2-3)
RH: The right hand taps on the first count of each group of three.
1 - - 2 - - (right hand marks beginning of each group of 3)
The rationale and take-away:
When we speak of the cross-rhythm “two against three,” it is actually just a short-hand way of saying that two groups of three-pulses [xxx and xxx] are being regrouped over those same six pulses sense as three groups of two-pulses [xx, xx, and xx]. So many musicians are confused about this. The tendency is to think that 2:3 (two-against-three) means that there are two articulations sounded against three articulations because, for most people who learn to play a basic 2:3, that’s how they do it. But mathematically, it actually refers to groups of pulses. So 2:3 in full sentences means: 2 groups of 3 pulses are sounded against 3 groups of 2 pulses. So two-against-three means 2 groups of 3 played in the same time as 3 groups or 2. The sensation of the cross-rhythm arises as a result of accentuating the first articulation of each grouping.
You can see the grouping in many different ways but it adds up to the same thing.
[123456] as 2 groups of 3 = [X y z X y z] or [x - - x - - ]
[123456] as 3 groups of 2 = [Y z Y z Y z] or [y - y - y - ]
Once again: by the alignment of the two lines, the graphic above shows that the same six pulses are simply regrouped: first as 2 groups of 3 and then as 3 groups of 2. When the beginning of the two groupings are highlighted and played simultaneously, then one hears 2 crossing the 3. I said that, for small numbers (certainly below six), performers could most accurately understand and practice cross rhythms as two different groupings of a fixed number of pulses (like a common denominator) performed one against another. For large numbers such as typically seen in Chopin (e.g., 15:4) one should take this to be an interpretational instruction – an invitation to improvise the relationship between the lines, that is, an invitation to tastefully distribute the fifteen RH notes over the time of the four LH notes. Such use of high numbers or difficult ratios would rarely be considered as a demand to perform a true cross-rhythm.
Once you have some facility with cross-rhythms, you can move on and try cross-phrasing. It’s just the same idea, notched up one more degree of sophistication, like a crossed cross-rhythm. It adds another level of musical interest to your work without adding substantial difficulties, and it is musically very satisfying. Link to section on Cross-phrasing
2. STEPS & GESTURES (5-AGAINST-6)
While this is quite a difficult exercise, it is fun and satisfying to keep trying til you get it. Just remember to stop and take a break when you begin to feel frustrated and uncoordinated. And it can be made more accessible simply by using fewer elements—fewer foot-steps and fewer gestures. You can also make up your own gestures and stepping patterns. There is a file just above this one, which shows this same exercise (Steps and Gestures ) with a simpler four-against-three cross-rhythm (three gestures and four steps). However, one reason for presenting this more difficult form is for you to feel the need for some learning strategies and for me to share some of those strategies that I’ve brought to classes.
These are the little things that musicians actually do to learn music and acquire new skills (perhaps like an ‘inner game’ of music) and to prepare for rehearsals, recordings and concerts. They’re the things that don’t typically get talked about except in an individual lesson, and it often doesn’t even happen there. Some teachers may not want to divulge their methods but, more often, accomplished professionals no longer recall what they did to achieve fluency with their craft.
While this is a whole-body exercise, it is mainly an exercise for your attention. And it is that fact which makes the effort transferable to the rest of your music studies.
In Brief:
This exercise invites the students (and the teacher, too!) to try coordinating a set of arm positions with a simple stepping pattern. It is a very simple (but difficult) exercise for the division of attention. All musicians must be able to divide, or distribute, their attention, but it is not the same as what is meant by multi-tasking. Listening to all the instruments in a band as well as the vocalist requires a kind of intentional distribution of attention, but this is different from multi-tasking. It’s simply a question of how you spend the attention that is available.
This exercise presents one way to realize a cross-rhythm, and it may give you insight in that regard, but it is not a ‘musical’ exercise per se.
Setting up the exercise: (for the teacher/leaders)
Five Arm Gestures
Begin by inviting someone to make a gesture with one or both arms and invite everyone in the group to imitate it. Others are then invited, one at a time, to create a different arm gesture until there are five different gestures, and everyone practices taking all five in the sequence in which they were given. Each arm gesture lasts for only one beat, (which is the same tempo as each foot-step). In the graphic representation below, they are shown as G1, G2, G3, G4, G5.
While the gestures should be uncomplicated and easy to learn quickly, it is good if there are some details for the hands, elbows, or other nearby body part. Specific details help some people establish a stronger image since they require a more active attention. Examples: both hands are placed on the hips with elbows akimbo; a finger is placed in front of the lips making a “shushing” gesture; the right hand forms a military salute gesture on the forehead; the right arm is raised vertically overhead with the index finger pointing upward; both arms are folded in front of the chest with the left palm facing the chest while the right hand forms a fist; and so on. And any of these can be accompanied by a simple head movement turning upward, downward, to the right or left.
Six Steps
Organize everyone to form rows and files so that they can move forward and backward as a group without bumping into each other. The group repeatedly takes six slow steps forward and then six backward. They try to take the same size steps* and, using their peripheral vision, the group tries to keep the rows and files straight. (This is good practice for maintaining awareness of your musical ensemble.)
When they’re ready, they can each try to combine the gestures with the steps—one gesture for each step. It is quite difficult, and it is quite likely that no one will be able to do it. So here are some strategies for learning to do it.
The leader might suggest adding just the first gesture when they count “One,” and then continuing the verbal counting without the gestures for the remaining four counts. Then they can try to add the first two gestures, then the first three, and so on.
An important learning strategy in real-time work (which pretty much includes all music work), is to decide what is most important at any stage of your work or in any practice session. Here, for example, I would prioritize steady continuity rather than trying to get everything right. So in this case, the movement of the feet should be the essential priority. And in addition to adding only one new gesture at a time (as you might carefully add cayenne to a soup), it is good to suggest that the moment things start to fall apart (and they will!) to maintain the stepping pattern but to abandon the arm gestures. When they can learn to make such an intentional fall-back position, they may find that they do not get lost so easily. And then when they do lose the thread, their feet can more readily keep their work on track.
BIG HINT:
Six can be an unexpectedly large number for a musician to internalize, that is, to sense a count of six without actually numbering them. Many times it’s practical for a musician to short-cut the counting in order to count to a smaller number. So just as 4/4 time is often sensed and counted in two longer beats, the six steps can be sensed and counted as two groups of three or three groups of two (1 2, 1 2, 1 2) or [1 & 2 & 3 &]. So, for a few cycles of the exercise, take a decisive step on the odd numbers, and only a much smaller step for the even numbered steps. The most organic way to do this is to count only the more decisive step with the foot that begins, with the even numbered steps simply considered as a joining of together of the feet. So the right-left, right-left, right-left, can be sensed as RIGHT, join, RIGHT, join, RIGHT, join, or, in numbers: One & Two & Three &. It’s much simpler to count three steps than six.
Further, the mind may soon leave the numbers entirely and sense the three main steps by inwardly characterizing them. “One” is the launching or initiating of the displacement. “Two” can be felt as continuing the journey forwards. “Three” can be sensed or characterized as concluding the forward movement.
* There’s a very interesting exercise on this site called “Pace and Stride” that focuses on stepping. It’s located in the group of exercises called, “For Children and Ensembles.” You can use the search bar to locate it.
Help Using Notation
We can notate the Steps Forward as [Sf] and the returning (backward) steps as [Sb]. So the full twelve steps would look like:
Sf1, Sf2, Sf3, S4f, Sf5, Sf6, Sb1, Sb2, Sb3, Sb4, Sb5, Sb6.
It’s easier to visualize the relationship between the upper and lower body (the torso and legs) if we align them in the manner of two musical lines in a piano score, with the arm gestures written on a line above the feet:
The colours may help you more easily see that six groups (cycles) of five gestures will re-align after five cycles of the six steps.
3.Steps & Gestures (easier version)
The Cross-rhythm of Three-Against-Four
While this is a tricky exercise, it is fun to keep trying til you get it. You can also make up your own gestures and stepping patterns. In this folder there is an example of the same exercise but which uses the more difficult cross-rhythm of five-against-six. While it is a body-based exercise, it is mainly an exercise for your attention. And it is that, which makes the work transferable to your other music study.
A suggestion for organizing the exercise in a class setting:
Three Arm Gestures
Begin by inviting someone to make a gesture with one or both arms. Invite everyone to imitate it. Then invite two others, one at a time, to create a gesture with the upper body. Everyone then practices taking all three in the sequence in which they were given. Each arm gesture lasts for only one beat. In the graphic representation below, they are shown as G1, G2, G3.
The gestures could include the head and torso but it should remain quite simple. Examples: one arm is raised vertically and the index finger points upward; both arms are folded in front of the chest; hands are placed on the hips; a finger is placed in front of the lips making a ‘shush’ gesture. And any of these can be accompanied by the head turned upward, downward, to the right or left.
Four Steps
Everyone is organized in rows and files so that they can move forward and backward together without bumping into each other. The group repeatedly takes four steps forward and four backward. Then they all try to combine the three gestures with the four steps—one arm position for each step.
That is the whole exercise. The rest of this file has material to help you formulate strategies for learning how you might approach complex tasks. There is a file nearby for older students which offers the same exercise but with a cross-rhythm of 5:6 (five-against-six). It is not intended as a trick move but simply as work for the musical attention.
For help with your practice
We can notate the Steps Forward as Sf and the returning steps (Steps Backward) as Sb. So the full eight steps would look like: < Sf1, Sf2, Sf3, S4f, Sb1, Sb2, Sb3, Sb4 >.
We can put them together (below) in the manner of two musical staves, with the Gestures written on a line above the Feet:
G1———G2———G3———G1———G2———-G3———G1———G2———G3———G1———G2———G3———||G1——— etc.
Sf1———Sf2———Sf3———S4f———Sf1———Sf2———Sf3———S4f———Sf1———Sf2———Sf3———S4f———||Sf1———etc.
The image below might be helpful, because colouring the patterns may make their “crossing” somewhat clearer. In the graphic below, you can more clearly see that the Gestures (top row) occupy three boxes (or three beats) and the Steps Forward occupy four boxes (or four beats). At the end of twelve beats (boxes) they are both ready to begin together at their beginning. You can see that the 4 steps take one more beat than the 3 gestures. They will eventually begin again together (realign) after 3 cycles of 4 steps and 4 cycles of the 4 gestures, but it is difficult to actually maintain them both at the same time. It can be interesting for younger (and even older) students to actually draw the patterns to see when they will both begin together again.
It should be obvious, mathematically, that the shorter cycle of 3 arm positions will finish before each group of 4 foot-steps, so these two sequences–arms and legs–will overlap for several cycles before beginning again together (i.e., before aligning again). So now everyone can be invited to try combining the 4 steps and the 3 gestures. This is a big challenge but there is a section included below with helpful hints for those who really want to try to do it.
By the way, it won’t feel or sound like a cross rhythm (as when we beat two-against-three with two hands), because it will be too slow (and quiet) to feel the two different phrases, but it is 4 against 3 (4:3). Note for musicians: the ‘4’ here is not the 4 steps but the 4 repetitions of 3 gestures, and the 3 does not refer to the 3 different gestures but rather to the 3 repetitions of the 4 steps.
More help:
If you find that the gestures distract you from counting the steps then you can try differentiating a step to signify the fourth or final step. So if you begin with the right foot then the first three steps can be forward but then make the final step by joining the left next to the right. That way, that fourth step will feel like a conclusion, and the body can begin to have the confidence to count four steps.
While this probably seems like a no–brainer, the idea will be particularly useful when trying the version that works with five-against-six or any other longer pattern.
Moving on to harder stuff:
There is another file following this one that has this same exercise in a much more demanding form. It presents the cross-rhythm five-against-six, also with stepping and gesturing. This is intended for older students and there are many additional suggestions with graphics and, most importantly, where to place your attention while working.
§ § §
Steps and Gestures:
Five-against-Six [5:6]
Link to: “Steps & Gestures [ 5 : 6 ]”
§ § §
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Cross-Phrasing
Once you have a basic understanding and some practical facility with cross-rhythms, you can move on to the more complex exercise of “cross-phrasing,” that is, playing higher level cross rhythms. This just means playing a cross-rhythm “over” another cross-rhythm. Some musicians call this “cross-cross-rhythms,’ but there is no formal terminology for it. On this web page, you’ll only see it illustrated with the cross-rhythm, 2-against-3, but it can be done with any cross-rhythm. It’s not really more ‘complex’ to do, but the cumbersome verbal explanation may make it seem so.
The easiest way to approach it is to try substituting different material while repeating the cross-rhythm. Substitutions can consist of different pitches, different articulations, or anything … while maintaining the cross-rhythm. When that new material alternates in some pattern with the original material, a new kind of cross-rhythm emerges, and that is what is meant here by a cross-phrase.
Before describing the two-hand cross-phrasing exercise, a simple piano realization of cross-phrasing is presented to make the idea more concrete.
The Set-up:
When beginning this exercise, it’s best to continue using the same hand gestures as you used in the Palm Turning exercise, where the palms are turned either down [D} or up [U] . In that exercise, the LH gestures were [Down/Down/Up] while the RH showed its two-beats as [Down/Up].
Now, what would happen if the LH had a pattern that was not a multiple of threes, such as Down/Up/ or Down/Down/Up/Up? You can see that in the music notation above, where the LH is playing a repeating pattern of four beat. And what if the RH played a pattern that was not a multiple of twos, such as [Down/Down/Up]? You can also see that in the notation above. The first thing you’ll notice is that it ‘upends’ the two-against-three’ cross-rhythm. It’s obvious that this new pattern takes longer for the two hands to re-align, but the change is more than just the length of the pattern; there is also an unexpected difficulty. This is because, while the arms still maintain their original cross-rhythm of two-against-three, the cycle of hand-turning gestures have a different cycle, and their pattern will not be completed when the cross-rhythm is completed.
The image below, with palm-turning, shows the gestures in parity with the cross rhythm. That means the pattern of palm turning is the same length as the cross-rhythm itself. That is, the pattern of hand turning finishes when the cross-rhythm finishes.
D = palm down; U = palm up. The dashes between represent unarticulated pulses. So one full pattern of either hand occupies the same six pulses: |D – D – U – | for the RH and |D - - U - - | for the LH. Notice that the LH is shown above the RH, and note that it takes a single iteration of the LH shape to arrive together with one iteration of the RH pattern. This is because the LH is playing a phrase in three and also shows the same number (three) of gestures: [Down Down Up]. The RH is playing in phrases of two and shows the phrase in two gestures: [Down Up].
LH|| D – D – U – |D – D – U – |
RH|| D - - U - - |D - - U - - |
But now, what will happen if the hands ‘break’ parity, that is, if the length of their patterns does not match … does not realign every six pulses? This will not only create a longer pattern but also new spaces which may invite different kinds of variations and other content. To demonstrate this breaking of parity and the resultant cross-phrasing, there are many different patterns we could use, but the one below is already familiar and quite simple: two-against-three.
In the diagram below, the LH still shows the 3 articulations of the 2:3, but by its expression of “Down-Up-Down-Up” it is now shaping a phrase in fours. The RH is still playing the “2” of the 2:3 cross-rhythm, but its phrases are now shaped in threes since it repeatedly plays “Down-Down-Up.”
In other words, it takes four iterations of the full RH movement to arrive together with nine iterations of the complete LH movement. Thus, while the cross rhythm of the articulative pattern is 2:3 (the fundamental relationship of the beat of the Left arm to the Right arm) there is a higher crossing pattern of 6:9 (i.e., the Right hand against the Left hand). It is that latter crossing rhythm crossing that is being given a separate name: cross phrasing. It is just another level of cross rhythm but, since we hear it more as an aspect of form and not as a fundamental physical rhythm, I think it seems well worth giving it another name to signify another kind of experience.
This image is a full expression of the cross-phrasing cycle described above.
The idea, that you can produce higher level (more complex) shapes by means of substitution, is one of several strategies for accessing more complex material without a significant increase in “maintenance” energy, i.e., with only a marginally greater demand on your attention. In this exercise, we created substitutions by employing different combinations of palm gestures. However, this is all much more interesting if we were to incorporate changes in pitch, harmony, texture, articulation, or any other musical materials.