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Playing with Sums: Reconsidering Additive Rhythm in Balkan Music Daniel Goldberg Yale University The following is a draft of work to be presented at the Second International Conference on Analytical Approaches to World Music in May 2012. If you have suggestions for improvement prior to the conference, please feel free to contact me at [email protected]. Curt Sachs seems to have coined the term “additive rhythm” in his 1953 monograph Rhythm and Tempo. Subsequently, musicologists have used the phrase to characterize bodies of music from Africa, India, North America, Turkey, and the Balkans, but many of these applications are susceptible to criticism on ideological or theoretical grounds. Today I will consider the prospects for revising the definition of additive rhythm in the context of periodic rhythms in commercial recordings by two ensembles from the Balkans. Ultimately, my goal is not to prove that additive rhythm is a highly effective tool for studying Balkan music, but rather to demonstrate that a problematic concept can in principle be refined, and to encourage awareness of the analytical options and assumptions that accompany the use of such a concept. Goldberg 2 The sources of musical examples for my evaluation of additive rhythm are recordings by Biljana Krstić’s Bistrik Orchestra and Ivo Papazov’s ensemble Trakiya. The styles of both groups derive in part from musical conventions coded as folk and developed under the technological and ideological constraints of state radio institutions (Rasmussen 2002, 20-34; Buchanan 2006, 120–30), but the historical and political environments of the two ensembles differ considerably. Bistrik Orchestra was founded in 1999 by Biljana Krstić, a former pop singer now affiliated with the Serbian national radio station, Radio Belgrade. The instrumentation of Bistrik Orchestra, shown here in a publicity photo from their website, resembles that of Serbian folk orchestras from the second half of the twentieth century (Rasmussen 2002, 29–31). Likewise, their professed aesthetic objective of “translat[ing] folklore into contemporary art” echoes post-World War II Yugoslav cultural policy on arranging folk music for radio (Krstić and Bistik Orchestra 2012; Rasmussen 2002, 21). Bistrik Orchestra’s repertory consists of arrangements of songs representing much of southeastern Europe; in fact, their website includes an interactive map indexing the regional derivation of each piece. Specific sources for the ensemble’s arrangements include Radio Belgrade’s archives of historical recordings and transcriptions, as well as field work by members of the Orchestra, several of whom are ethnomusicologists. Goldberg 3 Standing at a greater remove from institutional sources is Trakiya, a Bulgarian wedding band led by Rom clarinetist Ivo Papazov. In the 1980s, Papazov achieved legendary status—and commanded extravagant fees—playing wedding music (svatbarska muzika), an eclectic and virtuosic style of folk music that formed an integral part of wedding ceremonies and other rites of passage (Buchanan 1996, 202–206). As Buchanan (1996), Rice (1994, chapter 9), and Silverman (2007) have documented, wedding music apparently connoted noncompliance with the declining communist government’s attempts to control the Bulgarian economy and ethnic identity. The division of instrumental roles in Trakiya’s adaptation of the folk orchestra is often reminiscent of jazz: guitar, bass, and drums provide continuous accompaniment as a rhythm section, while clarinet, saxophone, and accordion first present a composed melody mostly in unison, and then take turns playing improvised solos. Indeed, Papazov cites Charlie Parker and Benny Goodman as influences on his improvisational technique, and he describes his wedding music as “Balkan jazz” that combines pan-Balkan and Turkish styles with American jazz (Buchanan 1996, 203, 208, 222). The reception and style of wedding music have evolved considerably in the past two decades (Silverman 2007), and while Papazov has continued to perform and record albums, for present purposes I draw only from his first international release, Orpheus Ascending. Goldberg 4 I chose to rely on Bistrik Orchestra and Trakiya for musical examples because I have access both to their commercial albums and to additional technical information about their music. Specifically, I have met and corresponded with Krstić, and Kalin Kirilov’s (2007) dissertation includes invaluable analysis and transcriptions of Papazov’s music. The feature of Krstić’s and Papazov’s music that I will focus on today is their use of repeating rhythmic patterns that combine short and long durations in a ratio of 2:3; the examples shown here are common in much folk-inflected Balkan music. Throughout this talk, I will identify particular periodic rhythms by their sequences of short and long durations, as in “long-short-short” and “short-shortshort-long.” I will consistently notate short durations as quarter notes and long durations as dotted quarter notes, but in subsequent examples I will omit the repeat signs to avoid clutter. Periodic Rhythms with Durations in a Ratio of 2:3 long-short-short short-short-short-long short-short-long-shor t-short Goldberg 5 Sachs’s (1953) additive rhythm is one of several different concepts that have been used to account for this type of periodic rhythm (others include Bra iloiu’s [1984] aksak rhythm, Hasty’s [1997] pure unequal meter, and London’s [2004] non-isochronous meter). In evaluating additive rhythm, we should first note that use of the term is not entirely consistent even in Sach’s (1953) original formulation. Sachs (1953, 23–25) defines an additive rhythm as a repeating series of durations based on units of unequal length, in contrast with a divisive rhythm, which partitions time into a succession of durationally equal units. He initially presents these two concepts as complementary aspects of rhythm, referring to additive rhythm as configurative and divisive rhythm as regulative, and associating the former with the regular patterns of durations in poetic meter and the latter with the alternation of strong and weak accents (Sachs 1953, 23–29). This pairing is similar, though not identical, to distinctions between rhythm and meter that many theorists have endorsed (Bar-Yosef 2009, 30), as in the definition of rhythm as successions of note onsets and the durations of time between them, and meter as the predictive, hierarchically organized pattern of attention that performers and listeners use in cognizing these onsets and durations. Later in Rhythm and Tempo, however, Sachs (1953) sets up a stronger opposition between additive and divisive rhythm. Though he continues to speak at times of the coexistence of rhythmic additivity and divisiveness, Sachs (1953, Goldberg 6 92–94, 102) characterizes some musical traditions as either additive or divisive, claiming that Middle Eastern and Indian musics contain only additive rhythms, and that European rhythm underwent a shift from additive to divisive during the Renaissance. In this connection, Sachs (1953, 169– 70) identifies additive construction in other parameters of medieval music as well as in contemporaneous architecture, painting, and theater, describing the common feature of this “additive character” as the combination of independent parts in succession without a view to the “unity ... [of] a well-integrated whole.” The meaning of “additive rhythm” in musicological literature has tended toward Sachs’s (1953) latter sense of a trait for categorizing the rhythm of large repertories, in opposition to divisiveness (see, e.g., Nketia 1974, 128–31; Widdess 1980–81, 133; Morris 2004, 78). In the context of unequal periodic rhythms, Cler (1994, 202, 207) attributes additivity to meters with unequal beats and divisiveness to meters with equal beats, setting up a binary distinction that contrasts additive characteristics with divisive, notated, Western meter. Arom (2004, 12) similarly asserts that metric organization is either additive or divisive, and classifies unequal periodic rhythms according to whether their total number of fundamental values—which are equivalent to notated eighth notes in my examples—is prime, divisible by 2, or divisible by 3. Goldberg 7 Cler (1994) and Arom (2004) define additive rhythm more clearly and consistently than Sachs (1953) does, and my present work reflects the influence of their articles. However, their usage and that of other recent authors is still potentially problematic for both ideological and theoretical reasons. With regard to ideology, for instance, Agawu (2003, 94–96) argues that additive rhythm does not correspond to African musical conceptions or performance practices, and that the frequent characterization of African rhythm as additive in contrast with the divisive rhythms of European art music constitutes a “myth” that asserts a fundamental difference between African and European music. Indeed, Sachs (1953; 1960) employs the additive/divisive distinction and its link with the valueladen aesthetic criterion of unity in the context of a longstanding pattern of Eurocentric thought. As Tomlinson (2007, 342) explains, paraphrasing anthropologist Johannes Fabian, “European writers imagine societies contemporary to their own to represent the features of a temporally distant European society; they project present-day societies along a chronological axis reaching back to the primeval past.” Sachs’s (1960) posthumously published article “Primitive and Medieval Music: A Parallel” expresses this “denial of coevalness” in presenting a host of features, including additivity, that putatively link European music of the Middle Ages with various non-Western musics. Goldberg 8 Considering this conceptual heritage, we should be wary of unwittingly perpetuating such symbolic violence when invoking additive rhythm. In addition, London (2001, 286) offers two music-theoretical arguments against additive rhythm. First, he suggests that thinking of rhythms as additive might not reflect rhythmic experience, arising instead as an artifact of a limited notational system—as, for instance, in the awkward use of plus signs in the numerators of time signatures for meters with unequal beats. Second, London (2001, 286) maintains that the conceptual difference between rhythmic additivity and divisiveness depends on one’s perspective on metric hierarchy, such that rhythms on a given level appear to add together to form the durations of higher levels but to divide to create the durations on lower levels. In a hierarchical context, then, most rhythms would be both additive and divisive irrespective of the equality or inequality of their durations, so the distinction between additive and divisive would seem to be of little use.1 Considering these criticisms, if we wish to continue to rely on the concept of additive rhythm, we would do best to revise its definition. The biggest problems with the current definition—apart from unclear and inconsistent usage—seem to stem from the contrast with divisive rhythm. The conceptual 1 See Clayton (2000, 37–39) for a discussion of theoretical objections similar to London’s (2001). Goldberg 9 overlap with rhythm and meter, the loss of significance in relation to a metric hierarchy, the tendency to treat additivity as a marker of essential musical otherness, and the value-laden understanding of additivity as a lack of integration all depend on the dichotomy between additive and divisive rhythm. Thus, my approach to lessening these problems is to reformulate what it means for a rhythm to be additive without reference to divisiveness or a similar antipole. The revised definition takes the phrase “additive rhythm” literally, as a rhythm that is understood in terms of the mathematical operation of addition. First, let us assume a sharp distinction between rhythm and meter according to the definition I mentioned earlier, such that rhythm is the succession of onsets and the durations between them, and meter is a hierarchically organized pattern of attention for cognizing these onsets and durations. Addressing rhythm only, we will regard the duration from one onset to the next as a quantity extending from the first onset, rather than, say, an empty space between onsets or a different, less spatially oriented entity. This conception of duration accords with how we tend to think of written note values, and with Huron’s (2006, 200) hypothesis that we often represent durations mentally as attributes of their initial events. For present purposes, treating these durations as quantities acts as a precondition for addition by providing a series of measurable amounts that can be combined with one Goldberg 10 another. This example, of course, shows the durations as the smallest integers needed to represent the ratio of short-to-long. In contrast with some of the previous definitions, we cannot use the mere equality or inequality of such durational quantities as the basis for determining whether or not a given rhythm is additive, since 2 and 2 can be added just as easily as 2 and 3. While patterns with numerous different durations, such as a segment of the Fibonacci sequence, might be sufficient to suggest additivity within one series, the limitation to two basic durational values in the case of the periodic rhythms that I am focusing on means that additivity is not normally apparent in the characteristics of a single rhythm. Instead, additivity is implied by relationships among two or more similar rhythms, according to an understanding that treats the related rhythms as transformations of one another that occur through the operation of addition or according to the properties of addition. While these potentially additive relationships among rhythms might take many forms, I will discuss two types. The first type, involving periodic rhythms with the same total length, depends on an analogy with commutativity. As a property of mathematical operations, commutativity is by no means exclusive to addition, but in the case of addition it seems fundamental and intuitively obvious: when adding some number of quantities together, we will always get the same final sum regardless of the order in which we combine them. Goldberg 11 Commutativity among Rhythms of Equal Length 3 + 2+ 2 = 7 2+ 3 + 2 = 7 2+ 2 + 3 = 7 In the present context, I suggest that if we identify multiple rhythms consisting of the same set of durations in different orders, then we may interpret the reordering of durations as reflecting the property of commutativity and thus as supporting an additive rhythmic conception. The change in ordering is analogous to commutativity in that the total length of the rhythms—i.e., the sum of their durational quantities—remains the same. In order for this equality of total lengths to be apparent, the rhythms must be in some sense equivalent; to this end, I require that the rhythms being compared belong to the same recorded track and fulfill similar musical roles. An example of this type of additivity occurs in a recording of a Serbian folk song, “Gde ima voda studena, Radule,” from Bistrik Orchestra’s 2002 album Zapisi. The track begins with an instrumental introduction led by Dragomir Stanojević on electric keyboard, using a plucked string timbre to articulate a Goldberg 12 continuous stream of onsets with equal durations. The pattern of octave leaps repeats every nine onsets, and I infer a short-short-short-long rhythm from the beginnings of the four lower notes. Here and in the following examples, my identification of periodic rhythms results from a parsing of accent and grouping informed by my limited familiarity with Balkan music. Arguably, the differentiation among onsets in my interpretation invokes hierarchic metric relationships, but I wish to distinguish the selection of a rhythm from an additive understanding of that rhythm, which need not relate to meter. In the case of “Gde ima voda,” my hearing corresponds to Bistrik Orchestra’s own notation of the passage, which is marked “2 + 2 + 2 + 3” in a short score of Stanojević’s arrangement of the song. The beginning of the introduction builds the accompanimental texture by adding several percussion instruments and then the šargija, a long-necked, fretted stringed instrument. The instrumental parts repeat, with slight variations, every one or two times through the short-short-short-long rhythm; the transcription below represents the texture once all the parts have come in. Goldberg 13 Š When Krstić starts to sing the melody of the song about 20 seconds after the beginning of the track, the accompanying instrumental pattern continues without interruption or a substantial change in texture. The underlying rhythm does change, however: the single long duration now occupies the second place instead of the fourth place in the series, as reflected, for instance, by the durations between onsets in the tambourine and bass drum parts, as well as the annotation “2 + 3 + 2 + 2” in the short score. This passage immediately follows the introduction notated above. There’s a second tambourine and a shaker in the texture that I haven’t included in the transcription. Š Goldberg 14 According to the additive framework, at the moment that the voice comes in, the initial rhythm transforms from short-short-short-long to short-long-shortshort by means of a reordering of its durations. The durations quite evidently still combine into a rhythm nine eighth notes in length, so the juxtaposition of these rhythms demonstrates the commutativity of their components and could be taken as evidence for an additive understanding. Details of the accompanimental parts can be taken as support for this interpretation. For instance, the bassist on this recording, Branko Isaković, constantly varies the pattern of pitches he plays in each measure. Goldberg 15 Two of his variations, from two measures before the melody begins and one measure after, are almost identical, except that the notes of the second and fourth durations are exchanged. Since Isaković switches his musical realizations of this pair of short and long durations, it seems plausible to regard the transformation of the underlying rhythm as a commutative reordering. This evidence certainly does not rule out alternative accounts of the relationship between the two rhythms. For instance, we could just as easily transform the first rhythm into the second by rotating the sequence of durations, that is, moving durations from the end to the beginning without changing their order. Indeed, Pressing (1983, 40– 41) regards rotation as a less disruptive kind of rhythmic transformation than reordering (he refers to the two types as “cyclic permutation” and “element permutation,” respectively). I do not intend to make general claims about which transformations seem more plausible, but only to propose that selecting reordering as our mode of description allows fro the present definition of additivity. The second type of additive relationship applies to periodic rhythms with different total lengths. In this case the inference of additivity depends on chains of inclusion relations in which adjacent rhythms may be transformed into one another by the addition or subtraction of one duration. For example, adding another short duration to the end of a long-short-short rhythm transforms that Goldberg 16 rhythm into a second, longer rhythm that includes the entire long-short-short series; we can extend the chain by repeating the same additive procedure. Additive Inclusion among Rhythms of Different Lengths 3 + 2+ 2 3 + 2+2+ 2 3 + 2+2+ 2+ 2 Another Serbian folk song recording released by Bistrik Orchestra in 2007, “Nišnu se zvezda,” demonstrates a chain consisting of only the first two of these rhythms. The shorter rhythm predominates throughout the recording, but the stable local periodicity is temporarily interrupted whenever Krstić sings the words “Jane more.” I’ve omitted a few details from this transcription, notably the backup vocals that come in the second time through the repeated section. Goldberg 17 The orchestra’s short score specifies that the change at the beginning of the second system interpolates two iterations of the long-short-short-short rhythm in the middle of the song, likely in imitation of a 1974 field recording that Krstić cites. Again, the accompanimental texture is consistent with an additive understanding: the rhythms in the guitar and bells in the two longer measures include the those of the following measure, not only in the sequence of longs and shorts but in the exact rhythmic pattern, just as if the final short duration has been added to the end of the longer measure. Goldberg 18 In my attempt to strip the definition of additive rhythm of its dependency on contrast with divisive rhythm, I have arguably replaced the original meaning with a different, more limited concept. Additive rhythm now refers not to an aspect of all musical rhythm or an inherent characteristic of rhythm in an entire repertory, but rather to a procedure that might describe a small subset of the rhythms in a given piece. Even if the new definition represents a theoretically viable possibility for conceptualizing certain rhythms, though, it is not necessarily of interest for describing rhythmic techniques in Balkan music; whether we choose to invoke additive rhythm will depend on our analytical goals. My remaining musical examples illustrate analytical perspectives for which additive rhythm might prove useful. In the case of “Jana i turcin,” another folk song recorded by Bistrik Orchestra, additive rhythm facilitates an account of the ensemble’s procedures for arranging folk songs. As in “Nišnu se zvezda,” the melody that the group took as Goldberg 19 the source for their arrangement includes an interruption of periodic rhythmic regularity. Their transcription data sheet shows a short-short-long pattern with a single instance of a long- short rhythm midway through, created by a type of vocal interjection common in Eastern European folk singing. In this particular song, the interjection functions onomatopoeically in imitation of the cooing of doves or pigeons. Like the other two songs we’ve heard, Bistrik Orchestra’s arrangement of the song is credited to their keyboardist, Dragomir Stanojević. This arrangement adds a third periodic rhythm, long-short-short, which occurs during instrumental interludes between the song’s verses. This addition produces exchanges between the long-short-short and short-short-long patterns several times on the recorded track. The following transcription shows the first switch from the short-short-long rhythm of the source melody to long-short-short in an instrumental interlude. š Goldberg 20 The long-short-short rhythm creates an additive connection between the two patterns in the initial melody: it includes the single long-short rhythm, and relates to the short-short- long rhythm by commutative exchange. As a derivation from and extension of the source rhythmic material, we can interpret this change in the arrangement as a technique contributing to the ensemble’s goal of “translat[ing] folklore into contemporary art.” This analytical perspective relies on the common music-theoretical assumption that the terms in which we describe a stylistic feature need not match the creator’s conception in order to be profitable and accurate. In the recordings by Bistrik Orchestra, we’ve seen additive relationships linking only two or three rhythms at a time. The other ensemble that I introduced at the beginning of this talk, Ivo Papazov’s wedding band Trakiya, offers an Goldberg 21 example of a more extensive set of connections on a track entitled “Kopanitsa,” from the 1989 album Orpheus Ascending. A kopanitsa is a Bulgarian folk dance defined in part by its short- short-long-short-short periodic rhythm. However, as Kirilov (2007, 156–57) explains, this particular rendition by Trakiya is an otkrivane, a medley-like composition intended to showcase the band’s virtuosity rather than to accompany dancing. As such, the first three minutes of the track consist of a pre-composed series of phrases that employ many different periodic rhythms, and only subsequently does the ensemble settle into an extended section featuring the typical kopanitsa rhythm and improvised solos. Though unusual in Papazov’s recorded output, the otkrivane and similar types of pieces were common in live performances by wedding bands in the late ‘80s (Kirilov 2007, 157). We can organize the rhythms in the composed section into two chains of inclusion relations; with the exception of a short segment near the beginning of the track that does not employ a pattern of long and short durations (mm. 7–9 in Kirilov’s [2007, 370] transcription), these two chains represent all of the periodic rhythms in the recording. In the collection of vertically aligned series on the left, labeled (a), each rhythm can be transformed into the rhythm below it by adding a short duration to the beginning of the series (or a long duration in the case of the longest rhythm). The bracketed rhythm in this example completes the pattern but Goldberg 22 does not occur in the recording. The collection of rhythms labeled (b) shares one rhythm, short-long-short- short, with the set of rhythms at (a), and generates three longer rhythms by successive addition of a long duration to the middle of the series. (a) (b) 2 4 4 6 7 8 5 3 1 Papazov’s “Kopanitsa” serves to demonstrate two more analytical motivations for additive rhythmic explanations besides the characterization of compositional style that we saw in “Jana i turcin.” First, the two sets of inclusion relations could support a creative analysis intended to enrich a listener’s experience of this particular recording. My example of such an interpretation is dramatized by the expectations of a moderately informed listener with respect to the title of the track: since there is no paratextual indication that the recording is an otkrivane, a first-time listener might well be surprised that the piece does not Goldberg 23 open with the short-short-long-short-short kopanitsa rhythm, and subsequently expect the eventual emergence of that rhythm. The bold numbers to the right of each rhythm in the two chains trace the sequence of rhythms over the course of the otkrivane by indicating the order of appearance. For a transcription of the pitched instruments in this recording, please refer to Kalin Kirilov’s (2007, 370– 76) dissertation. The periodic rhythms that I have identified in the two chains of inclusion relations usually correspond closely to the written time signatures and the durations in the bass part, but at the beginning of the piece, I have interpreted the alternation of measures in 7/8 and 11/8 as a single rhythm 18 eighth notes in length. The first part of the piece systematically works its way through the rhythms in (a), beginning with the longest rhythm and moving to the shortest, next to the second-longest, and then to the second-shortest. This sequence targets the kopanitsa rhythm at the center of the chain of inclusions, but instead of playing out in its entirety, the pattern stalls on the short-long-short-short rhythm. This rhythm predominates in the remainder of the pre-composed portion of the recording, serving as a jumping-off point for introducing the three longer rhythms in (b). Only after this excursion, which concludes with a virtuosic phrase juxtaposing the middle two rhythms in (b) and adumbrating the kopanitsa rhythm, does the ensemble finally settle into the long-awaited kopanitsa section for good. Goldberg 24 Considering that Trakiya performed versions of this composition with other dance types besides the kopanitsa, we might be inclined to doubt this goaloriented trajectory. Granted, as in the previous example, the interpretation does not depend on establishing explicit compositional intention or a single, inherent meaning; we would be free to construct different pathways through the rhythmic materials to suit other renditions of the piece.2 Still, skeptics might prefer an alternative analytical approach to the sets of inclusion relations in “Kopanitsa” in terms of rhythmic properties of a larger repertory. Instead of positing rhythmic relationships as part of musical experience, this approach regards the co-occurrence of rhythms on the recording as a function of the rhythmic vocabulary that the performers draw upon. Here we might emphasize the differences between the rhythms in the two chains of inclusions with respect to a broader musical context. The successive addition of short durations in (a) resembles previous observations about relationships among common periodic rhythms in other Balkan repertories, such as the variable numbers of short durations in Singer’s (1974, 387–89) generative rules for Macedonian folk dance rhythms. By contrast, the three longer rhythms in (b) are 2 For instance, Kirilov (2007, 350–60) transcribes a bootleg recording of Trakiya performing the otkrivane with a pravo horo with a time signature of 6/8 instead of a kopanitsa, and an analysis of this recording might emphasize the three measures of 6/8 that do not participate in my chains of inclusion relations, as well as other differences between the two versions. Goldberg 25 apparently rare in Balkan music from before the latter part of the twentieth century. Their use on the recording also differs from that of the other recurring rhythms, in that these three patterns occur only fleetingly, contributing to a technically virtuosic conclusion to the composed portion of the piece and arguably standing in a less direct relationship to meter. The fact that both groups of rhythms can be organized into inclusion relations suggests the possible relevance of additivity for constraints on the periodic rhythms of wedding band music, while the separation of the inclusion relations into two chains that are transformed by adding adjacent short durations, in the first case, and adjacent long durations, in the second case, reflects the stylistic distinction between the two types of periodic rhythms. Note that this understanding of additivity as a property of a repertory rather than a procedure in a single piece is still much more limited than claiming that Balkan rhythm is inherently additive: additivity is only one of the principles involved in the generation of a certain type of periodic rhythm that underlies some of the music, not a general characteristic pervading all rhythmic thinking or processes. The reformulation of additive rhythm thus shows some potential as an analytical concept with respect to individual stylistic choices, the experience of listening to particular recordings, or the properties of the periodic rhythms in a repertory. Of course, establishing the relevance of additive rhythm for folk- Goldberg 26 inflected Balkan music of the past few decades would require a survey of many more performing ensembles. Moreover, I would not argue that additive rhythm is essential to an understanding of rhythm even in recordings by Bistrik Orchestra and Trakiya: within the purview of each of the three analytical applications I have suggested, additive rhythm describes only a small part of the music’s rhythmic organization, and these applications themselves might well be ideologically suspect for reasons similar to those that prompted my attempt to redefine additive rhythm in the first place. Thus, I hope only to have recuperated additive rhythm as an optional concept in the context of Balkan music, and thereby to have drawn attention to the importance of a careful evaluation of the goals and assumptions that underlie our analytical decisions. Goldberg 27 Discography Krstić, Bilja, and Bistrik Orchestra. 2002. Zapisi. Produced by Dušan Ševarlić. Hi-Fi Centar CD 10264. ———. 2007. Tarpoš. Produced by Voja Aralica. Intuition INT 3406 2. Papasov, Ivo, and His Bulgarian Wedding Band. 1989. Orpheus Ascending. Produced by Joe Boyd and Rumyana Tzintzarska. Hannibal Records HNCD 1346. Papasov, Ivo, and His Orchestra. 1991. Balkanology. Produced by Joe Boyd. Hannibal Records HNCD 1363. Works Cited Agawu, Kofi. 2003. Representing African Music: Postcolonial Notes, Queries, Positions. New York: Routledge. Arom, Simha. 2004. “L’aksak: Principes et typologie.” Cahiers de musiques traditionnelles 17: 11–48. Bar-Yosef, Amatzia. 2009. “Comparative Musicology Revisited: The Problem of Cross- Cultural Comparison as Reflected in Sachs’ Theory of Additive vs. Divisive Rhythm.” Muzyka 54: 29–35. Bra iloiu, Constantin. 1984. “Aksak Rhythm.” In Problems of Ethnomusicology, edited by A.L. Lloyd, 133–67. Cambridge: Cambridge University Press. Buchanan, Donna A. 1996. “Wedding Musicians, Political Transition, and National Consciousness in Bulgaria.” In Retuning Culture: Musical Changes in Central and Eastern Europe, edited by Mark Slobin, 200–30. Durham: Duke University Press. ———. 2006. Performing Democracy: Bulgarian Music and Musicians in Transition. Chicago: University of Chicago Press. Goldberg 28 Clayton, Martin. 2000. Time in Indian Music: Rhythm, Metre, and Form in North Indian Ra g Performance. Oxford: Oxford University Press. Cler, Jérôme. 1994. “Pour une théorie de l’aksak.” Revue de Musicologie 80, no. 2: 181– 210. Hasty, Christopher. 1997. Meter as Rhythm. New York: Oxford University Press. Huron, David. 2006. Sweet Anticipation: Music and the Psychology of Expectation. Cambridge, MA: MIT Press. Kirilov, Kalin. 2007. “Harmony in Bulgarian Music.” Ph.D. diss., University of Oregon. Krstić, Bilja, and Bistrik Orchestra. 2012. “About Us: Bilja Krstić.” Bilja Krstić and Bistrik Orchestra website. Accessed March 10. http://bilja.rs/about.php?aboutStranica=193. London, Justin. 2001. “Rhythm.” In The New Grove Dictionary of Music and Musicians, 2nd edition, edited by Stanley Sadie and John Tyrrell, vol. 21. London: Macmillan Publishers Limited. ———. 2004. Hearing in Time: Psychological Aspects of Musical Meter. New York: Oxford University Press. Morris, Robert. 2004. “The Survival of Music: Musical Citizenship in South India.” Perspectives of New Music 42, no. 2: 66–87. Nketia, J.H. Kwabena. 1974. The Music of Africa. New York: W.W. Norton and Company. Rasmussen, Ljerka V. 2002. Newly Composed Folk Music of Yugoslavia. New York: Routledge. Rice, Timothy. 1994. May It Fill Your Soul: Experiencing Bulgarian Music. Chicago: University of Chicago Press. Pressing, Jeff. 1983. “Cognitive Isomorphisms between Pitch and Time in World Musics: West Africa, the Balkans, and Western Tonality.” Studies in Music (Australia) 17: 38–61. Goldberg 29 Sachs, Curt. 1953. Rhythm and Tempo: A Study in Music History. New York: W.W. Norton and Company. ———. 1960. “Primitive and Medieval Music: A Parallel.” Journal of the American Musicological Society 13, nos. 1–3: 43–49. Silverman, Carol. 2007. “Bulgarian Wedding Music between Folk and Chalga: Politics, Markets, and Current Directions.” Muzikologija: C asopis Muzikološkog Instituta Srpske Akademije Nauka i Umetnosti 7: 69–98. Singer, Alice. 1974. “The Metrical Structure of Macedonian Dance.” Ethnomusicology 18, no. 3: 379–404. Tomlinson, Gary. 2007. Music and Historical Critique: Selected Essays. Aldershot, UK: Ashgate. Widdess, D. Richard. 1980–81. “Rhythm and Time-Measurement in South Asian Art- Music: Some Observations on Ta la.” Proceedings of the Royal Musical Association 107: 132–38.