We introduce here the superposition of two different divisions inside each beat of the meter: triplet >< 2 eighth notes if we are in a binary meter or 3 eighth notes >< duplet if we are in a ternary meter.

This piece is in three distinct sections:

– The first one plays the superposition of groups of 2 and 3.

• Listen and move only with the treble, using body percussion (first eighth notes, then triplets); then do the same activity, replacing percussion by stepping. Listen to the other voice while following one’s own.

• Repeat the exercise, this time moving with the bass.

• With a partner: A follows the bass and B the treble. Move or clap while watching one’s partner, being aware of the interplay between the two.

– In the second section, one hears only groups of 3, and later only groups of 2. The task consists of creating the polymeter by oneself, without the help of the piano.

• Walk or clap the division that is not heard. Follow the variations of dynamics and tempo.

• Then, walk the division that is heard and clap the other.

– The third section is linked to the second by one measure of 4 beats.

It is a musical sequence on the same principle as the second section, with alternating divisions in the following form and played twice:

• Listen to this section many times and clap or step what is heard, in order to assimilate the sequence.

• Then, do the entire sequence only by clapping or stepping the opposite meter (the one you don’t hear).

• Finally, step what you hear and clap the opposite meter, and vice versa.

• Sitting or kneeling, clap groups of three with one hand and groups of two with the other. Change sides.

When practicing the 3 >< 2 polymeter at a slower tempo, the overall unit of the meter should be chosen rather than the unit of the beats. We superimpose the beats of a 6/8 meter with those of a 3/4 meter.

• Use the track no. 4 of volume 8, with 3 beats: place 2 beats on the same meter length.

The rhythm resulting from the overlay (composite rhythm) is:       

• Use the track no. 11 of volume 7 with 4 beats (that you consider and use here in 2 beats): place 3 beats on the same meter length.

The rhythm resulting from the overlay (composite rhythm) is:

The 12 eighth note groups offer a wide variety of possibilities, as they can be divided into 2, 3, 4 or 6 subgroups.

This track and the next one serve as summary exercises after the exploration of the various groupings of the 12 8th notes. The students will have already experienced and internalized in their bodies the different physical sensations of each grouping.

Here is the structure of the groupings:

6 + 6 twice
4 + 4 + 4 twice
3 + 3 + 3 + 3 twice
2 + 2 + 2 + 2 + 2 + 2 twice

Then we hear the entire series twice, as follows:

6 + 6
4 + 4 + 4
3 + 3 + 3 + 3
2 + 2 + 2 + 2 + 2 + 2

• Imagine that your trunk is a tree and your arms the branches. Make movements with the arms, drawing the branches of various sizes in space, according to the duration of each group.

• Using the first part of the track, in a group with a soloist for each grouping: the soloist makes movements during the first twelve 8th notes and the group imitates.

• This activity also works well two by two. A proposes a movement and B imitates. Change roles. Then in the second section where the entire series is heard, A moves on the groups of 6, B on the groups of 4 etcetera.

• Clap the first 8th note of each group, either in place or stepping the 8th notes. Show by one’s gesture in space the duration of the groups (the clapping gesture must last for the duration of each group).

• Individually or in pairs, roll a ball by giving it a pulse on the first eighth note of each group, without letting it stop. Spend the necessary time practicing each group, without the track, so that students experience the changes in the use of space and energy based on the duration of each group.

• Individually, step all the eighth notes and pass the ball from one hand to the other on the first 8th note of each group. Be aware of the arm gestures which decrease in amplitude as the groups decrease in duration.

• Create or assign a movement for each grouping. For example:

• Groups of 6: step semicircles, leading with an arm movement
• Groups of 4: 4 steps forwards, 4 steps backwards, 4 steps forwards
• Groups of 3: waltz step
• Groups of 2: step-point (one step with one foot, tap the toe of the other foot on the floor without putting weight on the foot).

The tasks proposed here should be practiced many times with each grouping before using the musical track.

• Students are in a circle, facing a partner.

Groups of six: take partner’s hand and walk 6 steps in a semicircle, change hand and direction for the second group of 6 to return to one’s place.

Groups of four: facing the center of the circle, 4 steps forwards, 4 steps backwards, and 4 steps to again face the partner.

Groups of three “Grand chain” (explanation in volume 1 no. 10).

Groups of two: clapping game to invent with partner.

The first stage of writing is close to the physical experience:

The second is more abstract:

This track shows the duration of the various groupings of twelve 8th notes (6, 4, 3, 2), in a faster tempo than the previous one and over a short period of time (2 series).

The structure of the piece:  two series of 2×6, 3×4, 4×3, 6×2 followed by 11 cycles of 12 “neutral” eighth notes (without grouping).

The goal of tasks we suggest for this track is to feel in the body the sensation of these various durations of groupings, and most importantly, to be able to continue to feel them when the piano no longer plays them.

The eleven cycles of 12 neutral eighth notes can be used in several ways: we suggest at first that the students continue the groupings in the same descending order as proposed at the beginning of the piece.

• Begin by giving students time for personal exploration of different movements according to the duration of each group. Then, each student chooses four movements that he/she can repeat in decreasing order according to the music, and eventually propose to the class. It is very important that each movement begins precisely on the first 8th note of the group, and finishes precisely on the last.

• Clap the first 8th note of every group. Draw the students’ attention to the appropriate use of space for each group.

• Divide the class into four groups (it would be perfect if there were 4 students in each group, and they are numbered 1 to 4). Each group takes a different percussion instrument (preferably with contrasting sound), for example:
• groups of 6 = dotted half notes: triangle
• groups of 4 = half notes: tambourine with rattle
• groups of 3 = dotted quarter notes: tambourine
• groups of 2 = quarter notes: claves

Do the entire series all together twice, where each group plays at the appropriate division.

• Then, when the music plays only the 8th notes, the triangle group starts with the dotted half notes, and keeps them until the end; after twelve 8th notes the next group (half notes) enters, and so on.

Overlaying these 4 groupings, we hear polymeter 2><3 (2 dotted half notes >< 3/2 as well as 12/8 >< 6/4); we also hear polymeter 4><3* (12/8 with 3/2).

* The composite rhythms of polymetry 4><3 can be found in the explanation of track no. 4.

• In a circle or in lines: this time, instead of using “legato” movements of various amplitudes corresponding to the duration of the groupings, each person chooses a “staccato” binary movement – the same for each grouping – that is repeated with the same amplitude and energy on the first eighth note of each grouping. For example: going back and forth with the head, or the hands; or a game of hiding one’s face/showing it etc.

One instrumentalist from each group plays on the first eighth note of the groups during the whole piece, while the others do their movements all together, following the decreasing order of the piece.

This more “robotic” version can give a variety of interesting visual effects depending on the orientation and placement of the movers in the space. Students can choose to adopt one type of movement or several; or they make all the groupings together from the beginning to the end of the piece, or they do a canon.

• Canon of 12 8th notes

The entire class chooses one only legato movement (we suggest a circular movement always starting at the same place) that everyone will use, and that will be adapted to the length of each group of 8th notes. The four groups of students stand one beside the other in a circle. Group 1 begins with this movement (groups of six); they continue with groups of 4, 3 and then 2. Group 2 begins with groups of 6 after 12 8th notes and so on.

The series of 11×12 eighth notes allows the 4-part canon to be played twice in a row.

• Each group of students stands in a line (one behind the other) and the lines are parallel. Repeat the previous activity, this time with a circular back and forth vertical arm movement (legato). Tap the thighs on the 1st of the twelve 8th notes and clap above the head on the 7th. Next, tap on 1, 5 and 9; then on 1, 4 7 and 10; then on 1 3 5 7 9 and 11. The movement between the two claps will accelerate according to the groupings, with the arms always stretched; it must be precise, regular and continuous, regardless of the speed.

Each person in the same group does the same movement, and each group is starting twelve 8th notes after the previous one.

• Repeat this activity, but this time, the canon is done within each group, which requires more autonomy from each student:

The first person in line begins, the second enters twelve eighth notes later, etc., so the canon is shown in every group.

The two previous exercises on the different divisions of the 12 eighth notes introduced the 4><3 polymeter by superimposing groups of 3 eighth notes (12/8 measure) and 4 eighth notes (3/2 measure).

This track serves as a summary exercise of the 4><3 polymeter after a thorough exploration of the groupings of 3 and of 4 8th notes. The students will have already experienced and internalized in their bodies the different physical sensations of each grouping.

• Follow the music and experience the feel of beats that last 3 or 4 eighth notes, by stretching, following the beats of each meter given by the bass (and sometimes by both hands). Be aware of the difference in sensation between the two meters.

• Still following the music, specify the movement by proposing a spatial instruction; for example: use the vertical plane to orient and spread the gestures in 3/2 and the horizontal plane for the gestures in 12/8. We can also associate words to the gestures, saying or singing “ver-ti-cal” or “ho-ri-zon-tal”.

• We can also use images to inspire the movement; for example, for the 3 do a gesture evoking the movement of a jellyfish (vertical plane) and for the 4 in the horizontal plane, the movement of a dolphin. Enrich this activity with your own ideas.

Beginning with gestures which engage the whole body – in each meter separately – is a preparation for later work using hands and feet. One logical way of working on a polymeter is to walk what we hear in the bass and clap what we hear in the treble. But there are other ways to approach polymeter.

The following exercises progressively introduce the polymeter.

• Students are in a circle. Follow the bass throughout the piece (sometimes, both hands play the same meter). For the 12/8: step the first step forwards – or backwards – and the other three from side to side; for the 3/2, “patch in the air” in front of the body (movement with the hands, placing the palm against an imaginary wall in front of us). Do each of these activities separately, several times.

• Half the circle follows the 12/8 meter and the other half the 3/2 meter, regardless of whether in the soprano or in the bass. Try to continue even when you no longer hear your meter, using the feel of the eighth note pulse. Watch the other half while doing this, in order to be aware of the interaction of the two groups creating together the polymeter, even if one is not doing it by oneself. Change roles.

• We can also address this polymetry aurally by using for example a type of vocal sound such as [daung] for 4 and a whispered sound [tsch] for 3.

• When placing 3 on 4 (12/8) step the beats (e. g. 4 steps forwards, 4 backwards) and superimpose the jellyfish movement in 3/2.

• To practice 3><4 (12/8) slowly, use vol. 7 no. 11 (4 beats only).

• When placing 4 on 3 (3/2), step the 3 in the shape of a triangle (1st step forward, 2nd step to the side and 3rd step back (notice the alternation of feet on each first beat). The hands make 4 wave-like (dolphin) but precise gestures on a horizontal plane. Add your own creativity to our suggestions.

• To practice 4>< (3/2), use vol. 8 no. 4 (3 beats only).

Become aware of the very different perception and sensation of the rhythm created by the polymeter, depending on whether it is played in 12/8 or 3/2, the tempo of the eighth notes always remaining the same.

• Step whatever is in the bass (4 or 3) while clapping the other meter. The musical track is conceived in an alternation 12/8 – 3/2. The first change of meters is anticipated by 2 bars of transitions in 12 eighth notes shifting the feeling from 12/8 to 3/2. Continue the activity when the piano no longer plays the polymeter.

The subsequent changes are done without the transition measures. Choose how to take advantage of the two transition measures that have been placed to help prepare for the first metric transformation.

• Analyse and write the structure.

To pass easily from one meter to the other, keep feeling the pulse of the eighth notes inside.

• Start by stepping the five beats and swing arms back and forth in a continuous movement, one measure/one movement.

• Then, turn your attention to the slow division (in two beats): two by two, facing each other, hold two bamboo sticks with your partner (one hand pulls and the other pushes). Feel the duration of this meter while listening to both meters in the music.

• The class is divided into two, facing each other; one half moves with five beats, the other with two beats. Change roles at the teacher’s signal.

• Repeat this activity, this time tapping five or two beats on different percussion instruments (for example tambourine and claves, whose sounds can be easily differentiated). Change roles.

• Repeat these two activities in pairs. The added difficulty is that there is no group support. This time, A follows the bass and B the soprano, adapting to the changes of meters in the music.

• Individually, step or clap only the five beats or only the two beats, whilst listening attentively to both voices.

Then follow the bass by stepping and the soprano by clapping; change when music tells.

• Practice placing 2 and 5 together on tracks 7, 8 and 9 of this volume or on track no. 8 of volume 1 (accelerando/rallentando).

The “andante” tempo of this track allows for an instinctive approach to polymeter, based on the bodily sensation of the globality of the groupings. We recommend giving the theoretical rhythmic information after having experienced it.

• The fastest meter is the easiest to rely on: concentrate first on groups of five and clap five beats in a horizontal line to the right, and back to the left next bar; you can also do it vertically, five up and five down; then follow the groups of three by clapping repeatedly in three different places, designing a triangle.

• Repeat the same exercise by stepping: lines for groups of 5 and triangles for groups of 3.

This preparation will help feeling the globality of each measure from the first beat to the next.

• In order to be able to follow both voices with the same musical awareness, divide the class in two groups, group A moving on 3 (a triangular movement) and group B stepping the 5 beats either in a line, or around group A, clapping the first beat in their hands.

• Develop your skills by following the changes of meter heard in the bass (group A) and in the soprano (group B).

• Two by two, A is clapping five beats and B three beats, changing roles when music tells (bass/soprano); to better integrate the two different speeds, find appropriate words to speak out loudly e. g. “five-is- flow-ing-through” and “ge-ne-rous” (expressing the longer note values of the three). All movements should become harmonious by using the appropriate space and energy relating to the duration of the beats.

• Eventually, do the polymeter by yourself: step the five while clapping the three in a triangular movement.

• Do the same tasks, using the tracks with groups of five only in various tempi (tracks 7, 8 and 9 of this volume and track 8 – with accelerando/rallentando – in volume 1).

• Move the five beats and clap the downbeat.

• Listen to the bass and sing the long values (at the beginning of the piece) to feel the duration of the group; show with a gesture the slow movement which lasts five pulsations.

• Clap the five in a line: up and down, sideways, in circles, invent as many ways as you can, showing the line of five, without making a rhythm.

• Step the bass and clap the treble; change over when the music tells.

• Walk towards someone (one measure), walk together (two measures), walk alone (one measure) and repeat.

• Participants are in a circle. Walk 5 steps forwards and 5 steps backwards. While walking forwards, each participant in turn claps on 2 out of the 5 pulsations (for example 1 and 4) and during the 5 steps backwards the class repeats the proposition. Choose also more than 2 pulsations out of the 5.

• Use these propositions and move with the whole body to the rhythm thus produced. For example, clapping 1st and 4th beats gives dotted quarter note and quarter note as rhythm to move with the body. This exercise can be done either individually or in pairs.

• Using the same idea of forwards/backwards steps, each person in turn proposes a rhythm of 5 beats, which is repeated by the class. These rhythms may be created by using subdivisions (8th or 16th notes or triplets) or multiples (half note, dotted half note) of the quarter note.

As mentioned in volume 9, this track lends itself also to the more complex study of polymeter, namely superimposing 2 or 3 regular pulsations on groups of 5.

See the rhythms resulting from these overlays in exercises 5 and 6.

The interest and the goal of the work on polymeters is to express the two meters in a fluid and musical way. Following two voices with equal attention is as beneficial for the mastery of global motor skills and laterality as it is for the quality of the instrumental playing.

• This can be done by:

• Walking one meter and clapping the other in your hands, then vice versa.
• Walking in 5 beats and beat time in a 2-beat or a 3-beat meter; reverse.
• Beat time in the 5-beat meter with one arm and the 2-or 3-beat meter with the other arm, then reverse.

• Work also in pairs with two different percussion instruments (for the diversity of the timbres).

Working with polymeters is a challenge even for experienced people, so it is important to be able to practice them a lot. The use of various materials, for example, bamboo sticks or elastics, hoops, balls, etc., provides many opportunities to practice polymeters by generating interest and sustaining student focus. It is clear that some materials lend themselves to working in pairs or groups while others can be used individually.

Refer to exercises 5 and 6 by adapting the polymeter to the tempo of tracks 8 and 9.

The interest and the goal of the work on polymeters is to express the two meters in a fluid and musical way. Following two voices with equal attention is as beneficial for the mastery of global motor skills and laterality as it is for the quality of the instrumental playing.

• This can be done by:

• Walking one meter and clapping the other in your hands, then vice versa.
• Walking in 5 beats and beat time in a 2-beat or a 3-beat meter; reverse.
• Beat time in the 5-beat meter with one arm and the 2-or 3-beat meter with the other arm, then reverse.

• Work also in pairs with two different percussion instruments (for the diversity of the timbres).

Working with polymeters is a challenge even for experienced people, so it is important to be able to practice them a lot. The use of various materials, for example, bamboo sticks or elastics, hoops, balls, etc., provides many opportunities to practice polymeters by generating interest and sustaining student focus. It is clear that some materials lend themselves to working in pairs or groups while others can be used individually.

Refer to exercises 5 and 6 by adapting the polymeter to the tempo of tracks 8 and 9.

The changes of meters in this track occur irregularly, but on a 4 measures basis or its multiples (8 and 12). This exercise develops the overall memory of the form, appealing first to an instinctive response (which may turn into analysis).

• Listen to the meter, notice when it changes, do you hear the moment where a change could happen? Feel the length of the measures by using a scarf: for example, toss up the scarf on the first beat and catch it on the last beat.

• Find different ways to use the scarf which you feel correspond to the inflection of the music and dance with it. Show through your movements the strong beat of each measure and the line which links one measure to the next. You might hold the scarf in the other hand each time the meter is changing.

As the body is articulating the different meters it helps the student internalize the inner movement of the music in order to reproduce it on his/her instrument.

• Identify the different meters and beat time from where you are standing in the room, looking at someone; look at someone else and adapt the gesture when the meter is changing.

• Students are in two groups, A = dancers, and B = conductors. The role of the members of group B is to “conduct” the dancers, each in turn, so the dancers’ movements are coherent with the measure. The conductors need to see their role as inspiring a beautiful quality of movement by not necessarily conducting metrically, but by indicating clearly the length of the measure in various different ways in the space, always guided by the character of the music.

• In this activity there would be one conductor (group B) at a time, conducting the whole group A. The conductors could “pass the baton” to the next person with the change of measure, or when they wish, with an appropriate anticipatory gesture so the next person knows it is his or her turn.

• Work in twos (A and B): B conducts A’s movement for a certain time, and then they change roles. Each pair creates an interaction between dancing and conducting: the dancer’s style and energy influences the conductor, and vice versa.

Another option: work in groups of 4-5 people and a piece of elastic approximately 4-5 meters in length, with the ends tied together. The group all hold the elastic with both hands and take turns to “make the elastic dance” (each one for one measure) according to the length of the measure. The resulting effect is a sort of undulating wave of movement in the elastic, but movement of different amplitude, resulting from the time available to do.

This is a walking piece (not a dancing one) with changing meters from 2/4 to 5/4; the beat is the same all the way through.

The changes are made every four or eight meters, except for an alternation 2/4 – 3/4 every measure towards the end of the track. They follow a certain formal structure but are not systematically announced by a modulation or another obvious musical signal.

Until one knows the music well, one will always be surprised by a change – this is to be expected! But if one cannot react instantly, one can react as quickly as possible.

Once the students are more familiar with the piece, they can identify the changes together and note them on the board.

• Listen carefully, walk with the music and identify the strong beat. Snap it. The arms can move to indicate the direction of the walk. They can also express the duration of the groups by designing circles or paths in space; they can also sway backwards/forwards or sideways with the down beat. Change direction on the first beat of each measure.

• Step the beats while conducting the measures. Show clearly the meter as well as the phrasing.

• Individually, pull with one hand an imaginary thread of cotton during one measure. Take it with the other hand at the other “extremity” for the next measure.

• Same activity in pairs, alternating for each measure and so forth.

• Individually, express the duration of each meter by stretching movements either in a free use of space or by associating each type of meter with a particular plane, for example: 2/4 deep (front-back), 3/4 oblique, 4/4 vertical and 5/4 horizontal.

• Repeat the same activity in pairs in mirror, palm to palm or at a distance.

• Still in pairs, repeat this activity, this time without associating the planes of space with particular meters nor mirroring, but developing movements in complementary spatial planes: A defines a plane during the current series of meters (e.g., the vertical plane) and B moves simultaneously, in a complementary space (horizontal plane); when the meter changes, B initiates the movement in a new plane and A adapts using a complementary plane.

• Listen to the whole canon several times while stepping the beats and beating time.

• Identify and analyse the various rhythms heard, notate them in your mind or on a board, and if necessary, turn off the recording, step the beats and clap each rhythm several times.

In a continuous canon the mind must be able to represent two things simultaneously: the present measure to which one is listening, and the previous measure whose rhythm must be clapped, even if it is no longer being heard.

One way of being successful in this exercise is to be able to rapidly notate the rhythm in one’s mind so that it is available even while listening to something else.

• To quickly visualize the notation of a rhythm, practice with interrupted or continuous canons and also repeated rhythms from previous volumes (vol. 6 no. 9, vol. 7 no. 8, vol. 3 no. 9 and 10): listen to the rhythms and figure out their notation in your mind.

• Step the beats and clap the rhythm one measure later. Follow the musical line you hear while processing the one before. Keep listening globally: it is better to miss a detail than lose the feeling of the whole measure.

• Step the rhythm in canon, one measure later.

This is a quick reaction and memory exercise: sing back melody and rhythm one phrase later. Be careful not to accelerate the tempo when singing back quick notes. One of the main challenges in this exercise is the steadiness of the tempo, so one needs to interiorize the pulse.

• How do we do this? Firstly, beat time without walking.

• Step the beat and beat time. It is important to have a solid, stable step, and the feeling of the feet being connected to the floor. Sing the phrases in your mind either while only beating time, or also stepping while beating time.

This piece is a kind of melodic dictation in echo, where all the phrases are anacrusic, but with upbeats of differing lengths. It is, therefore, important to be very aware of the measure. Some anacrusis are on a beat while others are syncopated. Beating time is essential to become aware of the starting point of each anacrusis and to know at each moment where we are in the meter. This is what allows the student to be able to start singing back at the right time.

This track is an alternance of measures at 4/4 and 6/8.

• Begin by showing the beats on various parts of the body (body percussion).

• Clap the beats, expressing the difference between ternary groups and binary ones.

• Conduct intuitively with the whole body.

It is important to show the strong beat, as the music begins with an anacrusis of 2 beats in 4/4.

• Draw the phrases with a clear direction at the beginning of each phrase, taking into consideration the anacrusis. Change arms for each phrase.

• Work in pairs (A & B): A moves for the first long phrase of 4 measures, while B claps the beats; B moves on the second phrase while A claps the beats. Repeat the activity changing after 2 measures – this time, B could imitate A’s movement and vice versa.

• Beat time on the spot. Then step the beats while beating time.

• This music also lends itself to dancing. Create dance steps, using as a basis the idea of a skip for the ternary groups and steps for the binary groups.

At a certain point, there is a part played more legato in the high register; adapt to the change of dynamics by finding a contrasting step and use of space for this section.