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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
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
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
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.
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.
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,
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
(a) (b)
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
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
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
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
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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.
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