Introduction

Turkish aksak rhythms offer an opportunity to teach students how to read and perform music in asymmetrical meters. Most rhythm-reading textbooks used in Western musical instruction cover the topic only briefly in their final chapters, primarily focusing on meters in 5 and 7 (Rogers and Ottman 2019; Hall and Urban 2019). One reason for such sparse treatment may be that asymmetrical meters are not a mainstream feature of Western music. Aside from notable examples such as Dave Brubeck's "Take Five," the Mission Impossible theme, or "Money" by Pink Floyd, asymmetrical meters are thought of primarily as "deformations'' of their more normative counterparts (as extra beats appended to 4/4), ways of introducing complexity in genres like progressive rock or jazz, or as a particular reference to Eastern European folk style in modern classical music (Fracile 2005; Vikárius 2012). This is not the case for Turkish music. Whether classical, folk, popular, or jazz, asymmetrical meters are a commonplace feature in any style of Turkish music. In this paper, I propose that Turkish rhythms are particularly effective in teaching the concept to Western-trained students. This approach has the dual benefit of teaching idiomatic asymmetrical meters and helping to educate students that the concept of metric regularity is culturally specific.

Turkish music follows a metrical organization traditionally known as usûl (analogous to Arabic īqā'), which is often translated as rhythmic mode (Marcus 2001). According to master tanbur player Murat Aydemir (2010, 209), usûl "comprises all the concepts of 'measure,' 'tempo,' and 'rhythm.'" More specifically, the term refers to the cyclical drum-stroke patterns that identify each usûl; Holzapfel (2014) has shown how the melodies of compositions are specifically designed in order to coincide with these patterns. In practice, Turkish musicians learn the patterns by reciting syllables and striking their hands against their knees. The syllables along with the patterns help musicians to experience the rhythms kinetically as they memorize each usûl. There are six basic syllable combinations in total: Düm, Tek, Te Ke, Te Kâ (the caret indicates a "y" sound before the vowel, as in kya), Tek Kâ, and Ta Hek. The basic beat patterns of any usûl can be embellished, for which there are additional syllables, but these are not necessary in this context. The meanings of each syllable with respect to note duration and the hand/knee-strike patterns are produced in the following table, derived from a textbook in Turkish conservatory training, Türk Musıkîsinde Usûller ve Kudüm (Ungay 1981, 2):

Table 1: Turkish rhythmic syllabes and strike patterns (taken from Ungay, 1981).

Syllable	Note Duration	Hand/Knee Strike
Düm	Long	Right
Tek	Long	Left
Te Ke	Short, Short	Right-Left
Te Kâ	Short, Long	Right-Left
Tek Kâ	Long, Long	Right-Left
Ta Hek	Long, Long	Left-Both Together

The above table and subsequent usûl examples are taken from this textbook. For certain usûl-s, there are occasional variations on the durations of certain drum-stroke syllables, such as short düm-s and tek-s. Nevertheless, the syllables largely correspond to the durations provided here.

In modern notational convention, measures with two staff lines represent the usûl-s. Strikes for the right hand are notated with Western note-values on the upper line while strikes for the left hand are on the lower line. The syllables are typically written between the two lines. Here is an example of one usûl, called Aksak—represented as one measure of 9/8—along with its corresponding syllables and hand/knee strikes:

This 9/8 differs from its Western counterpart in that its beat structure is organized 2 + 2 + 2 + 3. It can be thought of as four main beats, three of which are short and the last of which is long (Bates 2011, 51–60). The word aksak in Turkish means "limping" and is used to describe how the fourth beat creates a stretching-out of the measure that also corresponds to accompanying folk dance steps. Here is an example of a folk melody from Istanbul, written in Aksak:

When originally composed, the above melody would have been conceived to fit over the drum-stroke pattern of Aksak; note that the beaming for the final three eighth-note beats of each measure reflects this. The term Aksak refers both to this specific usûl and also generally to any usûl that asymmetrically combines beat structures of 2 and 3. Not all usûl-s which behave this way feature the word aksak in their names, but they are all to be understood as aksak in principle (Brăiloiu 1984).

The aksak rhythms fit into my curriculum by the fourth semester of a four-semester sequence in Ear Training. I spend the first three semesters drilling the rhythmic elements of Western music as they gain complexity within simple and compound meters. Simple meters may feature basic elements, such as two eighth notes or one quarter note. They may also become more complex, as in elements with dotted or syncopated rhythms and tuplets. Compound meters have a similar progression from basic to complex elements, and the students have by the fourth semester of the curriculum internalized them through repetition and exercise. By this point, two classroom dynamics have been achieved: 1) students have become fluent in reading meter as a combination of rhythmic elements expressed in simple and compound beat structures; and 2) they have been saturated with examples from Western music. This moment in the curriculum offers an opportunity to introduce music from Turkey as a way of explaining how musical systems may be organized in non-Western cultures, as well as to teach them a method for learning asymmetrical meters in a musical tradition for which non-isochrony is standard (London 2012, 121–42). Treating meter as cycles with specific recurring beat patterns further helps students internalize the rhythms in a way that is at once beneficial to both Turkish and Western music. This endeavor is facilitated by the fact that the rhythmic elements they had been studying to this point with respect to Western music are also applicable to rhythms in Turkish music.

I begin by introducing students to the beat structure of each usûl (i.e., 2 + 2 + 2 + 3) and having them learn the drum-stroke patterns using the traditional syllables and tapping the beats on their knees. Students repeat this until they can execute the drum patterns with confidence. Then, they learn to sight-read and perform the examples of Turkish music on Kodály syllables while continuing to tap the usûl on their knees. This is similar to pedagogical practice in Turkey, where musicians often learn songs by singing melodic solfége on fixed do while tapping the usûl. The use of Kodály or some other rhythmic solmization system—such as takadimi, counting, or simply "ta"—is necessary to perform the rhythms because Turkish music has no system in place beyond the usûl for the vocal performance of surface rhythms alone. In fact, the Kodály method is adapted from Hungarian musical instruction for children whereby the lyrics of songs are replaced by play words and the play words are then ultimately substituted with rhythmic solmization syllables (Houlahan and Packa 2015, 157–58); this application is equivalent to substituting the Turkish song lyrics with rhythmic solfège. Performing the drum-stroke patterns of the usûl enables the students to embody the asymmetry of the aksak meters. Continuing to tap out the usûl while performing rhythms on Kodály syllables helps students learn individual rhythms while retaining the rhythmic cycle in their minds.

One way I have used to help students bridge the gap between learning the usûl patterns and then performing rhythms on top of them is doing a small group exercise (2–4 students). Half of the group performs the usûl syllables while tapping their knees and the other half uses the Kodály syllables to perform the rhythms of the examples, also while tapping their knees. The group maintaining the usûl helps to account for any mistakes the group performing the rhythms will make until the latter group is able to perform both at the same time. Then, I reverse the roles of the two groups until all the students are able to tap out the usûl on their knees while performing the rhythm on Kodâly rhythmic solfège. The beat tapping in this case functions similarly to Western conducting patterns, only it adds a percussive element to the exercises. Using Turkish music and its traditional methods of instruction helps students engage with asymmetrical meters in musical contexts in which they are normative. As Western-oriented textbooks like Rogers and Ottman (2019, 313) continue to refer to these kinds of meters as "less common," it becomes especially important to demonstrate to students the centrality of asymmetry in the beat structures of other musical cultures.

The Aksak Meters

Like Western music, in Turkish music the aksak meters are combinations of two- and three-beat structures beginning with 5/8. There are two usûl-s for this meter, the most common of which is Türk aksağı, or "the Turkish limping rhythm." It is represented as follows:

This particular usûl breaks down into two unequal beats with a structure of 2 + 3. Here is an example of a song in Türk aksağı with relatively basic rhythmic elements:

An important feature of sight reading this example is to maintain focus on the 2 + 3 beat structures of each measure; two eighth notes and their equivalents fall on the first half of the measure, while three eighth notes occupy the second half. This can be especially challenging at the end of measure 4. Here, the flow of sixteenth notes carries rhythmic momentum into the next measure, suggesting an elision with measure 5; the linked performance emphasizes this. Lerdahl and Jackendoff (1983) identify this phenomenon as a grouping overlap. To help students maintain a sense of the 2 + 3 beat structure it can be shown that measure 5 is the return of the music from measure 1, and so this is a two-part phrase grouping that resembles a parallel period. The first quarter note in measure 5 is thus the beginning of the next iteration of the usûl coinciding with the beginning of a second phrase.

From the perspective of the rhythmic elements, a more difficult example in the same usûl would be the following song, which contains thirty-second notes and sixteenth-note triplets. The faster note-values in this song coincide with a slower tempo than the previous example, and this can impact how students experience the 2 + 3 meter:

The effect of the long second beat becomes especially palpable in measure 5, where the first and second half of the measure contain the same rhythmic motive, yet the second half is extended by one pulse using the same motivic content. Despite their differing levels of rhythmic complexity, the aksak feel will help students to conceptualize and perform both examples.

The issue of "performance" brings up an ethical question of using Turkish music solely for the purpose of rhythm-reading exercises. A glance at the key signature of these melodies, for instance, indicates that there are microtones in these excerpts, a topic that never gets covered in the melodic or harmonic portion of the typical Ear Training curriculum. As a practitioner of Turkish classical music myself, I find it strange to perform the rhythms without getting into aspects of performance practice intrinsic to the appreciation of this music, like melodic structure (makam) and ornamentation. For this reason, I can envision expanding this exercise by having students sing along while I accompany them on the Turkish lute (oud). As previously mentioned, Turks use melodic solfège on fixed do for sight reading, and I could try to have students do this. Hearing the melodies along with the rhythms would create a more accurate cultural experience with the added benefit of training students' ears with microtones. The curriculum in my courses might in the future be adjusted to include the melodic portion of Turkish music in a chapter of a textbook on asymmetrical meters and microtonal melodies. However, for those not familiar with performance practice of Turkish music it is acceptable to use the rhythms alone. This is because microtonality and ornamentation in Turkish music are specialized concepts that require an advanced level of expertise in order to demonstrate properly. The aksak rhythms, however, provide enough overlap with Western music that non-experts in Turkish music could teach the topic using the information provided in this article.

Beyond usûl-s in 5, examples of aksak rhythms in Turkish music are to be found in several other meters. For the remainder of this article, however, I would like to focus on two more examples: one in 7/8 and another in 10/8. There are two usûl-s in 7/8 that in different ways combine two beats of two eighth notes with one beat of three eighth notes. One of them is Devr-i turan and it places the two eighth-note beats at the beginning of the measure, making for a beat structure of 2 + 2 + 3:

An example would be the following instrumental composition:

Performances of this piece tend to be quite fast—at least 360 bpm using an eighth-note pulse—and so this seemingly simple exercise becomes quite challenging when attempted at high speed. While the repetition of the sections alleviates the difficulty somewhat, the variety of quarter- and eighth-note patterns complicates matters in measures 2–6. The striking of the beats on the knees in this case also helps accent the beat structure in measure 5, which otherwise could be performed as 7 undifferentiated eighth notes. It is especially important for students to accent beats 1, 3 and 5 in this measure.

After 9/8, the Aksak meter already discussed, there is Aksak semai, a meter in 10/8 which is described as consisting of two oppositely arranged Türk aksağı (5/8) meters put together. Its beats are structured (3 + 2) + (2 + 3); the second two-beat usually strikes both eighth notes—Düm Düm—as part of a popular variant of the usûl's drum-stroke pattern. This rhythm is most strongly associated with a form of Turkish instrumental music known as the saz semaisi. Its cyclical pattern is as follows:

An example of an instrumental composition (saz semaisi) in Aksak semai is as follows:

This example is noteworthy for the variety of rhythms it uses. The Aksak semai beat structure is clearly represented in each of the first three measures, where the Düm Düm syllables of beats 5–6 coincide with four sixteenth notes. Measure 4 is an attempt at a melodic cadence on A, and so the rhythm at the analogous moment changes to feature two sixteenth notes and an eighth note. It can be tricky to strike the Düm Düm pattern for a fourth time and perform this new rhythm over it.

Even more difficult is how the beat pattern of the cycle is undermined due to an evaded cadence in beats

8–10 of the same measure. Here, instead of a quarter note plus an eighth note, the surface rhythm is flipped, creating a kind of syncopated feel; the syncopation is especially pronounced as the quarter note enters before the final Tek of the usûl's beat pattern. Measure 5 is an extension of the phrase and is so marked using ten consecutive eighth notes; as in the previous example, care must be taken to accent beats 1, 4, 6, and 8 despite the monotonous surface rhythm. Measure 6 is a varied repetition of measure 4, however with the beat structure corrected at the end to reflect the cadence. The compositional logic behind the rhythmic choices over the Aksak semai beat pattern will allow students to feel rhythm as it is connected to formal rhetoric in Turkish music, and from this to be able to handle shifting rhythmic emphasis over a stable and repeating structural beat cycle.

Turkish aksak rhythms provide an alternative to the unidiomatic exercises in asymmetrical meters found in textbooks focused on Western music. The stage of the Ear Training curriculum that covers asymmetrical meters presents an opportunity to introduce Turkish music in a way that both fits with the sequence of topics and teaches students about rhythmic organization in other cultures. Asymmetrical meters are not a mainstream feature of Western music, so using Turkish music at this point gives students access to a musical culture in which these meters are more common. Reading and performing in asymmetrical meters in Western music is often thought of as difficult and composing in asymmetrical meters that sounds smooth to the Western ear is considered a challenge. The rhythms used in Turkish music also present useful solutions to this problem. The traditional way of learning usûl-s with cyclical beat patterns is an effective pedagogical tool for getting students accustomed to the aksak principle: developing an innate feeling for combinations of two- and three-beat structures in a single measure. To that end, I have found stepping outside of Western pedagogical models on this topic to be a fruitful endeavor. I have found that through studying this material students gain an important perspective about different approaches to music and culture and master asymmetrical rhythms using techniques not available in Western musical training.