Music students are likely to encounter asymmetrical meters in multiple repertoires, including twentieth- and twenty-first-century Western art music, jazz, rock, and soundtracks for film, television, and video games. Lists of pieces that feature asymmetrical meters have been compiled by authors including Nancy Rosenberg (2010), Scott Murphy (2016), Betsey Stephens (2019), and Daylon Camarena (2020), and many more examples can easily be found. In light of such wide stylistic contexts, asymmetrical meter arguably belongs in the theory and musicianship curriculum of any undergraduate program seeking to provide students with the practical and conceptual tools of a well-rounded musician. Moreover, since asymmetrical meters are also prevalent in several repertoires outside the realm of Western art and popular music, teaching this topic presents an opportunity to include music from cultures that may be unfamiliar to many students. In this essay, I offer resources for teaching two contemporary folk dances from southeastern Europe, taking advantage of students' capacity for imitation and embodiment to introduce skills and concepts connected with asymmetrical meter. After a few preliminary notes about meter, I describe how an instructor could go about teaching pravoto, a dance from North Macedonia, and then suggest several options for continuing the lesson.

Asymmetrical Meter

Such a lesson could begin with a basic definition of asymmetrical meter. "Asymmetrical" is one of several imperfect terms that have been used for the subject of this essay; synonyms include irregular, complex, non-isochronous, mixed, and aksak. The defining characteristic of this type of meter is the presence of categorically different durations within a single metric layer. That is, according to the conception of meter as consisting of multiple coordinated layers of time points developed by Fred Lerdahl and Ray Jackendoff (1983), Justin London (2012), and others, an asymmetrical meter has at least one layer in which the succession of time points forms a repeating sequence of different durations. In asymmetrical meters from southeastern Europe, this sequence is conventionally notated as a combination of durations lasting for either two or three shorter, equally timed metric units. Example 1 shows this type of metric structure in annotations to part of the melody from a North Macedonian folk song. The rows of vertical lines representing metric layers in the figure are modeled after Gary Karpinski's (2000) pedagogical application of metric theory. I use series of 2s and 3s as a shorthand for identifying the metric sequence, such that the sequence in Example 1 is abbreviated as 322.

After the concept of asymmetrical meter has been introduced, instructors should highlight the point that time signatures provide incomplete information about meter because a time signature specifies only two metric layers, and more than two layers are often present. For asymmetrical meters, the metric sequence is also absent from the time signature (except in time signatures with composite numerators, such as 3+2+2/8). Accordingly, an instructor should draw attention to the metric sequence as an essential feature of an asymmetrical meter, so that students know that they will need to rely on listening or on details of notation such as beaming to identify this layer. That identification includes inferring the correct rotation of the cyclic repetition of the sequence of 2s and 3s—for instance, 322 and 223 are two rotations of the same sequence that determine two different asymmetrical meters with a time signature of 7/8, depending on which duration begins the measure. Instructors might also point out that a time signature alone is not always sufficient to indicate whether a meter is asymmetrical, since an asymmetrical meter with a sequence such as 2223 is commonly notated with a time signature of 9/8.

Dance offers a way for students to improve their fluency with asymmetrical meters through active imitation of body movements, avoiding a potentially limiting reliance on counting. A common technique for teaching asymmetrical metric sequences is to assign numbers or other syllables to the pulses within each duration of 2 or 3, as in "1-2-3, 1-2, 1-2" or "pineapple, apple, apple" for the meter in Example 1. This form of counting can certainly be a productive strategy for rhythm performance, but as Kofi Agawu (2006) argues, the shortest durations in a musical texture are rarely the most useful layer for conceptualizing meter. With regard to interpreting meter in West African music, Agawu instead recommends referring to the slower rates of motion articulated by dance movements. Considering cognitive and neural links between meter and movement (e.g., Phillips-Silver and Trainor 2008; Patel and Iversen 2014), as well as the principles of "bodily knowing" that underlie Dalcroze Eurhythmics (Juntunen and Hyvönen 2004), dance should help students internalize asymmetrical metric sequences, facilitating accurate recognition and performance of these patterns. This presumption is consistent with Arnie Cox's (2011) argument for the centrality of imitation and "mimetic motor imagery" in music performance and listening, and dance instruction might well be added to his list of pedagogical applications as an example of "cross-modal mimetic" training (par. 68).

Pravoto

After introducing the concept of asymmetrical meter, I suggest teaching one of the most widespread patterns of dance steps in North Macedonia. For current purposes I refer to this dance as pravoto (pronounced with emphasis on the first syllable), which could be translated as "the straight [dance]." 1 Pravoto is commonly performed by attendees of weddings and similar celebrations, and although the dance has likely been in existence for a long time, the version that I teach is based on contemporary practice.

To learn pravoto, students first need to form an open circle or line. This type of formation, in which everyone faces the center and holds hands (or in some cases shoulders or belts) with the person next to them, is a defining characteristic of the class of traditional dances called ora (singular oro). For many ora, including pravoto, dancers should bend their arms so that their hands are at shoulder height and the pair of arms between two people resembles the shape of a W. The palm of each person's right hand should face upward and the palm of their left hand should face downward. Although some ora include hand movements, most of the activity is concentrated in steps and other gestures with the feet. Dancers usually keep their upper bodies stable except for bobbing up and down slightly in time with the meter, which is accomplished largely by bending the knees or lifting the heel while the ball of the foot is on the ground.

Some ora have lengthy patterns of steps, but the steps for pravoto are quite simple and follow a repeating cycle that lasts for three measures of music. Like several other dances from southeastern Europe, the timing of pravoto steps can be adjusted to fit music in different meters, including pieces with equal metric durations so that the cycle lasts for three measures with a time signature of 2/4 or 6/8 (see Rice 2004). The 322 sequence with a time signature of 7/8 in Example 1 above is a common asymmetrical meter for this dance, and the melody in that example is from an instrumental interlude in "Eleno, ḱerko" ("Elena, Daughter"). This recording of the song features Kevser Selimova and Ǵorǵi Želčeski, a Macedonian folk singing duo who first gained popularity in the 1970s when the country now called North Macedonia was part of Yugoslavia. 2

Each of the three measures in the pattern of dance steps for pravoto includes two movements with the feet, marking the beginning of the 3 and of the first 2 in the 322 sequence as illustrated in Example 2. In the first measure these two movements are steps, first with the right foot and then with the left foot. Dancers take these steps to the right while facing the center of the open circle, such that the second step involves crossing the left foot in front of the right foot. In the second measure of the pattern, dancers take another rightward step with the right foot, and then move the left foot forward and touch the ball of the foot to the floor without transferring body weight to this foot. Dancers move the left foot back under the body and transfer weight onto it on the downbeat of the third measure, and finally repeat the gesture of touching the floor during the second half of the third measure, now using the right foot instead of the left foot. This pattern repeats over and over again for as long as the music lasts—which might be well over half an hour in the context of a party, during which musicians often string together a succession of numerous songs without pause.

To encourage students to focus on the dance movements rather than the meter, I normally avoid mentioning the time signature or metric sequence until after students have learned the dance. Consistent with the goals of the exercise, this strategic withholding of information shows students that they can synchronize with an asymmetrical meter without relying on counting, just like most dancers do in southeastern Europe. Thus, rather than talking about the 322 sequence or showing Example 2 to the class, I describe the pattern of dance movements without bringing up the timing of the steps and demonstrate the steps without music, while students imitate my movements and then practice looping the cycle of steps simultaneously with me.

The first part of Video 1 demonstrates the steps using techniques to facilitate imitation, in the form of facing away from the audience and wearing mismatched shoes. After practicing the cycle a few times, students should be ready to perform the dance with the recording along the lines of the latter part of the video, although the small circle that I move in when dancing with the music would be much larger with a line of multiple dancers. As mentioned above, the dance line forms an open circle, and since the dance moves to the right, the person on the right end of the line is the leader. Ideally the instructor would initially fill this role when dancing with music, modeling the dance movements and guiding the line around the room. After providing feedback about students' dancing and continuing the dance practice as needed, the instructor can lead students to connect the pattern of movements with the 322 metric sequence and 7/8 time signature. To offer students an image of the dance in a typical setting—an important part of the lesson in light of Susan Foster's (2005, 88) argument that "conceptualiz[ing dance in] a pure space" shorn of cultural particularity serves colonialism—I recommend sharing this video of dancers performing pravoto at a wedding, with music performed by Amajlija Bend ("Amulet Band") from Probištip, North Macedonia.

The basic pattern of steps and its coordination with music are perhaps the most important features for students to grasp in relation to its meter, but a few points about dance style are also worth noting. Beginners sometimes have a tendency to exaggerate the movements of the dance, making the steps larger than necessary. It may be helpful, then, to encourage students instead to keep the dance movements small and relaxed, so that, for instance, a dancer only travels about two feet to the right during each three-measure cycle. Variations of the dance movements are also possible; for instance, lifting the foot and briefly suspending it above the ground may replace the touches in the second halves of measures 2 and 3.

Incorporating dance into classroom activities involves a few practical considerations. Instructors may need to plan on clearing chairs out of the way or relocating to a different classroom to ensure that students have enough space to move freely while dancing. If the type of movement that pravoto calls for is outside of the range of physical abilities of some students in a class, the activity could be adapted—for example, performing the dance movements with the hands or fingers would still serve the main pedagogical purpose. At the time of writing, public health circumstances call for social distancing, and a manual adaptation of the dance might also be suitable in situations when students cannot hold hands. The dance could also be taught remotely, with students dancing individually at home; indeed, for the sake of social distancing many folk dance hobbyists in North America have been using video conferencing to hold remote dance lessons in lieu of in-person meetings.

Next Steps

Depending on the learning goals for the course, there are several options for continuing a lesson after students have learned to dance pravoto. For instance, an aural skills instructor might draw attention to musical characteristics of the recording that students have just danced to, leading a discussion of how rhythmic patterns in the music helped students to time their steps or an exercise in transcribing music in asymmetrical meter. 3

The lesson could also continue with analyzing and contextualizing the meter that students have just learned. Most music that accompanies southeast European folk dance has two- and four-measure melodic grouping, meaning that the three-measure cycle of pravoto shifts in and out of alignment with the melody in the manner of a hypermetric grouping dissonance (see Krebs 1999). An exercise in which students map the dance cycle onto musical phrasing in a recording would lead them to discover the dissonant three-against-two pattern. 4

Students might note further that the rhythm of the dance steps in pravoto articulates a pattern in which the two 2s in the metric sequence are combined, effectively grouping the seven eighth notes in each measure as 34 instead of 322. As Kalin Kirilov (2015) explains, this tendency to combine pairs of 2s is common in meters from southeastern Europe, and could be considered an additional metric layer as illustrated in Example 3. See Goldberg (2019) for discussion of this type of layer in relation to theory of meter. 5

One more option for material following pravoto is to expand the lesson to asymmetrical meters with other sequences besides 322. An exercise in aural identification of the series of 2s and 3s in various recordings with asymmetrical meters could serve to develop listening skills and to draw students' attention to the range of styles in which asymmetrical meters occur. Alternatively, the instructor could teach another southeast European line dance with a different meter, such as a widely known dance that can be performed to music with a 2223 sequence and a time signature of 9/8. In Bulgaria, one of the names for this dance is samokovsko horo (after the town of Samokov; pronounced with emphasis on the first syllable of samokovsko and on the last syllable of horo). As for pravoto, the steps of samokovsko horo follow a three-measure cycle, the timing and movements of which are illustrated in Example 4 and Video 2. Other general features of the dance are also similar to pravoto, except that dancers hold hands with their arms straight downward to form a V shape instead of the W with arms bent normally used for pravoto. 6

In the video, I first demonstrate the basic pattern of dance steps, and then elaborate the steps by adding grapevine-like crossing steps and a subtle lift of the body weight that delays slightly the last step in the first measure. The latter gesture divides the dotted-quarter-note duration into an eighth followed by a quarter, matching a rhythm that occurs frequently in the percussion in the recording that I use. If time is limited, teaching the dance with only the initial, simplified version of the steps would likely still be effective for the purpose of learning to move with the meter. Either way, the most challenging component of the movements for many students to master occurs at the ends of the second and third measures, when dancers need to remember not to transfer their weight onto the foot that they put down at the end of the measure because they will take a step with the same foot in the opposite direction at the beginning of the next measure.

The recording that I recommend for teaching samokovsko horo is "Byala roza" ("White Rose"), a song written and performed by folk singer Slavka Kalcheva that has remained popular in Bulgaria long after its release in 2000. This video shows a performance of the dance by members of the dance club Ogledalo ("Mirror") from Varna, Bulgaria. A relatively recent phenomenon, this kind of dance club resembles an exercise class and is how many Bulgarian folk dance enthusiasts now learn and perform dances (see Ivanova-Nyberg 2016).

As many instructors work to expand the cultural scope of undergraduate music programs, I hope this essay will allow anyone to incorporate asymmetrical meter into theory and aural skills curricula and to add to the ways in which students learn to engage with music. No single instructor can become an expert in every musical style, but if many instructors share their expertise in different areas, we will have a rich collection of pedagogical resources to draw from.