goldberg p

8/12/2019 Goldberg p

http://slidepdf.com/reader/full/goldberg-p 1/29

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-short-

short-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 withDurations in a Ratio of 2:3

long-short-short

short-short-short-long

short -short -long-short-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 value-

laden 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 objectionssimilar 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

23 2

32 22 3 2

3 2 2

+ + = 7

+ = 7+

Commutativity amongRhythms of Equal Length

32 2+ + = 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 13

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

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-short-

short 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 amongRhythms of Different Lengths

3 2 2+ +

2+3 2 2+ + 2+3 2 2+ +

2+3 2 2+ + 2+

3 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 theoreticallyviable 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.

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

4

5

6

8

1

2

3

4

7

(a) (b)



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 goal-

oriented 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 ofTrakiya performing the otkrivane with a pravo horo with a time signature of 6/8instead of a kopanitsa, and an analysis of this recording might emphasize the threemeasures of 6/8 that do not participate in my chains of inclusion relations, as wellas 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 RecordsHNCD 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 musiquestraditionnelles 17: 11–48.

Bar-Yosef, Amatzia. 2009. “Comparative Musicology Revisited: The Problem ofCross- 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 inTransition. 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 ofOregon.

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 andMusicians, 2nd edition, edited by Stanley Sadie and John Tyrrell, vol. 21.London: Macmillan Publishers Limited.

———. 2004. Hearing in Time: Psychological Aspects of Musical Meter . NewYork: Oxford University Press.

Morris, Robert. 2004. “The Survival of Music: Musical Citizenship in SouthIndia.” Perspectives of New Music 42, no. 2: 66–87.

Nketia, J.H. Kwabena. 1974. The Music of Africa. New York: W.W. Norton andCompany.

Rasmussen, Ljerka V. 2002. Newly Composed Folk Music of Yugoslavia. NewYork: 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 WorldMusics: 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 AsianArt- Music: Some Observations on Ta la.” Proceedings of the Royal

Musical Association 107: 132–38.

goldberg p

Documents