Beats of unequal length

A time signature of 9/8 in a Western classical score means three beats of three. A time signature of 9/8 in a Bulgarian dance means four beats — short, short, short, long — and the two are not variants of one another. They are different structures that share a numerator, in the way two ragas can share a pitch set and be different ragas.

The Balkan version is called aksak, a Turkish word meaning limping, and the limp is the point.

What a metre normally is #

Ordinary Western metre is built by repeated division. A bar divides into equal beats; each beat divides into two or three equal parts; those parts divide again. Every level is uniform, and a time signature names two of those levels — how many of the second, and what size they are.

That construction is why 4/4 and 6/8 are described as simple and compound rather than as different numbers. Both have beats; they differ only in whether a beat splits in two or three. The tree in the figure is the whole system, and everything in it is regular.

Additive metre is not built that way. A bar of aksak nine is assembled from unequal groups — 2+2+2+3 — and the assembly is the primary structure. There is no equal-beat level to find, because there is not one.

Which level the inequality is at #

The distinction that matters, and that the notation hides, is where in the tree the irregularity sits.

Take a bar of nine quavers. It can be read as:

Three equal beats of three. The quaver is the subdivision and the dotted crotchet is the beat. Regular at both levels.

Four unequal beats. Three beats of two quavers and one of three. The quaver is still the subdivision, but there is no single beat length — three of them are two units long and one is three.

The first is a division; the second is an addition. In the first, the number nine is a consequence of $3 \times 3$; in the second it is a consequence of $2 + 2 + 2 + 3$, and the two facts happen to produce the same total.

A dancer knows which reading is in force immediately, because the two produce different steps. A reader of the notation may not, because 9/8 says nothing about it, and a great deal of Balkan music is printed with the grouping indicated by beaming alone.

Which computation produced the number #

The trees are drawn from the grouping, not from the time signature.

The generator is given a list of group sizes and lays each beat out at a horizontal position proportional to where it starts within the bar: a group of size $g$ starting after $a$ units occupies the span from $a/N$ to $(a+g)/N$, where $N$ is the sum. So a beat’s width on the page is its duration, and an aksak bar comes out visibly lopsided while a compound bar comes out even.

The sound buttons work from the same list. A pattern is built with a one at the start of each group and zeros elsewhere, played on a grid of $N$ equal units, so the clicks fall exactly where the drawing puts them.

That equivalence is the useful part. The picture and the sound are the same list of group sizes, so a caption cannot claim a limp the audio does not have.

Two, three, and nothing else #

Look across the aksak repertoire and one regularity holds almost everywhere: the groups are twos and threes, and nothing else.

That is not a stylistic preference. Any whole number greater than three can be written as a sum of twos and threes, so twos and threes are a complete toolkit; and a group of four is heard as two twos, while a group of five is heard as a two and a three. There is no such thing as an indivisible group of five at these speeds, because the tempo is fast enough that five units is long enough to subdivide.

So an additive metre is a word in an alphabet of two letters, and the metres in use are the words that recur: 2+3 for the Bulgarian paidushko, 2+2+3 for rachenitsa, 2+2+2+3 for daichovo, 3+2+2+3+2 for the ten-beat patterns, and considerably longer words in the Macedonian repertoire.

The connection to maximal evenness is direct and worth stating carefully. Bjorklund’s algorithm, asked for four onsets in nine steps, produces $2+2+2+3$ — the daichovo grouping — because that is the most even arrangement nine allows. Asked for three in seven it produces $2+2+3$. The aksak groupings are, in a large number of cases, exactly what an evenness criterion would generate.

That is not a claim that anybody derived them that way. It is a claim that the same constraint — divide a number that does not divide, as fairly as it will go — produces the same answers whether a dancer or an algorithm is applying it.

It is also weaker than it sounds, and the section after next says how much weaker by counting what else the constraint left available. The short version is that on most of these numbers it left nothing, so the agreement is a fact about arithmetic rather than about either the dancer or the algorithm, and the two cases that carry any information at all are split one each way.

How it is counted #

The quickest way to establish which reading is in force is to listen to how people count it, and the counting is unambiguous.

A compound nine is counted “one two three, two two three, three two three” — three groups, each with its own beat number, subdivisions numbered within. An aksak nine is counted “one two, two two, three two, four two three”, or more usually as a string of names: quick quick quick slow.

The second is a list of beats of two kinds. The first is a list of beats of one kind. No amount of staring at a time signature distinguishes them, and everyone inside the tradition does it instantly, because the counting is taught with the steps.

The dance is the reason. In rachenitsa the long beat carries a weight transfer that the short beats do not; in daichovo it carries a hop. The unequal group is not a rhythmic decoration applied to a regular substrate — it is the moment in the step pattern that takes longer, and the metre is a description of a body.

That also explains why the groupings are stable across enormous repertoires. A dance is a fixed sequence of movements, so its metre cannot drift, and a village repertoire of dozens of tunes in one metre is a repertoire of tunes for one dance.

The words that are actually used #

Listing the common metres with their groupings makes the alphabet visible, and lets each be checked against what an evenness criterion would produce.

Five, as 2+3. Paidushko. The algorithm’s E(2,5) is 2+3.

Seven, as 2+2+3. Rachenitsa, the most widespread of all. E(3,7) is 2+2+3.

Nine, as 2+2+2+3. Daichovo. E(4,9) is 2+2+2+3.

Eleven, as 2+2+3+2+2. Kopanitsa. E(5,11) is 2+2+2+2+3 — the same cyclic sequence rotated to a different starting point, which on a circle rather than a line is the same necklace.

Ten, as 3+2+2+3. A Macedonian grouping. E(4,10) is 3+2+3+2, which is not a rotation of it. Here the algorithm and the tradition genuinely disagree, and the tradition is not being more even than it needs to be — it is doing something else.

Four agreements and one disagreement out of five looks like a real pattern and is not one. The next section counts how many other answers were available.

How many groupings the alphabet even allows #

The agreements are worth nothing unless the tradition could have chosen otherwise, and mostly it could not. Fix the alphabet at twos and threes — which the section above argues for on independent grounds — fix the number of beats, and ask how many distinct groupings exist at all, counting rotations of one arrangement as one thing because the bar line is arbitrary on a circle.

Seven of the nine had one possible answer. Two threes will not fit inside an odd number alongside a maximal count of twos, so an odd bar taken at its fastest beat level is forced: one three and as many twos as will go, in the only cyclic arrangement there is. The algorithm and the tradition agree at nine and eleven and thirteen and fifteen because nothing else exists to disagree with, and an agreement that could not have failed is not evidence about anything.

Which leaves the two cases where a choice was genuinely available. At twelve the tradition takes the arrangement the algorithm takes. At ten it takes the other one. One agreement and one disagreement, which is a coin landing once each way, and the resemblance that made this section worth writing turns out to be an arithmetic of small numbers almost in its entirety.

That is a stronger version of the caution the closing section gives, and it points the same way. The place to look for evidence is not the odd metres, where the alphabet decides; it is the even ones — ten, twelve, fourteen, sixteen — where the choices multiply. Fourteen at six beats has three groupings and fifteen at six has four; seventeen at seven has five. If a maximal-evenness criterion is doing real work anywhere in this repertoire it will show there, on the bars where a tradition had somewhere else to go and did not.

Is the long beat exactly one and a half #

The notation says the long beat is three units and the short beat two, which makes their ratio exactly 3:2. Measurements of performance say otherwise, and the direction of the discrepancy is consistent.

In recorded Bulgarian and Macedonian dance music the long beat is typically shorter than 1.5 short beats — often closer to 1.4, and sometimes as low as 1.3 at fast tempi. The pattern is the same one swung quavers show: a proportion notated as a simple ratio, played as something else, with the discrepancy varying with tempo, and with the same floor on how short an event can be doing the work.

The explanation is likely to be the same as well. At a fast dance tempo a unit is 130 or 140 milliseconds, and the long group’s third unit is the one under pressure — there is a floor on how short an articulated event can be, and it is the short groups that hit it first, so the long one compresses relatively.

Which means the aksak nine notated as 2+2+2+3 is, in performance, something closer to 2+2+2+2.8, and the notation is a categorical label rather than a measurement. As with swing, that is a good enough label to teach from and a poor one to sequence from.

What the notation could have said #

Western notation has three ways of indicating an additive grouping, and all of them are workarounds.

The first is beaming: write 9/8 and beam the quavers 2+2+2+3, leaving the reader to infer the beats. It is the commonest solution and it is silent on the page — a reader who is not looking for it will not see it.

The second is a compound time signature: write the numerator as $2{+}2{+}2{+}3$ over 8, which some twentieth-century editions do. It is unambiguous and it is ugly, and it has never been standard.

The third is changing the metre bar by bar, alternating 4/8 and 5/8, which is what a good deal of Stravinsky looks like. It carries the grouping exactly and destroys the sense that the bar is a repeating unit — which for a dance is precisely the wrong thing to destroy.

None of the three is bad, and the fact that there are three of them is the informative part. A notation designed around division has no natural slot for an addition, so the addition has to be expressed by a beaming convention, an extended signature or a stream of metre changes. The music was there first, and the notation is being asked to hold something it was not built for.

Where the model stops #

The grouping is not always fixed. Some aksak repertoire allows the long group to move within the bar between phrases, and some pieces alternate groupings. Drawing a metre as one tree implies a permanence that not every dance has.

The ratio measurements are patchy. The published work on Balkan performance timing is much thinner than the equivalent jazz literature, covers few performers, and does not always separate tempo from region. The 1.4 figure quoted above should be read as a direction rather than a value.

“Additive versus divisive” is a modern analytical distinction. The terms come from twentieth-century theory — Curt Sachs popularised the pair — and they are a useful description rather than something practitioners of either kind would recognise as a choice. A great deal of music is not cleanly one or the other.

Nothing here concerns what the melody does. The metre is the grouping of a pulse. Whether melodic phrases align with the groups, cross them, or ignore them is a separate question, and Balkan melodic practice frequently does the last of those.

The evenness connection is barely a resemblance. The counting above finds seven of nine agreements forced by the alphabet, one genuine agreement and one genuine disagreement, so there is very little here for a mechanism to explain. What is left is a hypothesis about the even-numbered metres and no measurement of them.

Whose music, and when #

Aksak metres are concentrated in a band running from Bulgaria and Macedonia through Greek Thrace and into Turkey, and they are overwhelmingly dance metres — which is the most important thing about them, because the grouping is a sequence of steps before it is anything else.

Bartók collected and transcribed a great deal of this material in the 1910s and 1930s, coined the phrase “Bulgarian rhythm” for it, and wrote six of the Mikrokosmos pieces in such metres. His transcriptions are the reason the repertoire entered Western art music, and also the reason it entered it in a form that reads as an exotic complication rather than as an ordinary way of building a bar.

The Turkish usul system is a related and much older tradition of long rhythmic cycles built from named strokes, in which cycles of nine, ten, thirteen and considerably larger numbers are standard, alongside a theory of pitch built on fifty-three commas, and in which the structure is a specified sequence rather than a division.

Beyond that band, additive construction turns up in Indian tala — where cycles such as the ten-beat jhaptal, grouped 2+3+2+3, are structural — and in a good deal of twentieth-century Western art music that borrowed the device deliberately. Stravinsky’s Rite of Spring is the famous case, and it is a genuinely different use: the groupings there change constantly, which makes them a compositional device rather than a metre a dancer could hold.

The distinction is worth preserving. A metre is something a body can be in. A bar-by-bar sequence of unequal groups is something a score can specify, and the two produce different music even where they produce identical notation.

The ladder from here #

Later rungs on this anchor: the longer aksak words and why some of them recur while others do not. Measured performance timing in Balkan dance, and how the ratios move with tempo. The Turkish usul cycles and the relationship between a stroke sequence and a metre. Indian tala, where the additive structure carries a great deal more information than a grouping. And the general question this raises about Euclidean generation: how much of the world’s traditional rhythm is what an evenness criterion produces, and how much of the resemblance is a coincidence of small numbers.

Nine quavers, four beats, and no level at which anything is the same length as anything else.