Wondering how to get to grips with additive metres?
Awed by the complexities of flamenco palmas?
Despair not, help is at hand!

As a prelude to aksak “limping” metres, we might start with quintuple metres, which go far back, even in WAM. By the baroque period there are niche examples by composers such as Schmelzer, and they feature in 19th-century Russian music—a most popular instance being the “limping waltz” of Tchaikovsky’s Pathétique symphony (2+3).

This reminds me of two other subversions of the waltz, albeit in regular triple metre: the 2nd movement of Berlioz’s Symphonie fantastique (cf. here and here), and Ravel’s La valse, a “surreal nightmarish vision of a decaying society through a broken kaleidoscope”, as I said in my Ravel page. Elsewhere, perhaps inspired by the Basque zortzico dance, Ravel was partial to additive metres—such as in his piano trio, and the quintuple-time Bacchanale finale of Daphnis and Chloé (both also featured on that page, the latter from 45.29). And Messiaen was most partial to complex yet catchy additive rhythms.

Quintuplets, of course, are something else altogether; as are the creative use of additive rhythms in minimalism (see also examples from Reich and Meredith).

From Tchaikovsky we might graduate to

• the Pearl and Dean theme (which we may hear as two groups each of 3+3+2; or for a Bulgarian, perhaps two fast groups of 3+3+3+3+2+2)
• Un homme et une femme (after the three upbeats, 3+3+2+2: the first two 3s, in the original “Dabadabada, badabadaba”, were later remembered as “Chabadabada“, a word that entered the language to denote alternating male and female candidates in electoral lists!)
• and Lalo Schifrin’s theme to Mission impossible (5/4, with a duplet over the first 2 beats).

If you can hum along to such easy examples, then that’s a good start in mastering the intricacies of so-called aksak metres around east Europe and the Middle East…

Indeed, Take five was inspired by hearing Turkish musicians. Rather more challenging is the opening section of Blue Rondo à la Turk (2+2+2+3):

Note the helpful BTL comment there (only without the punctuation!):

Taco, taco, taco, burrito. Taco, taco, taco, burrito. Taco, taco, taco, burrito.
[SJ: not to be confused with potato, potato]

Still, that’s a rather crude, mechanical usage, the melody merely marking out the metre in regular quavers—whereas further east, melodic rhythms are infinitely varied within the basic metre.

Admittedly, the additive patterns of the Rite of spring have been transcribed in 4/4—was it really Boulez who had this drôle idea?! Cf. Slonimsky‘s help for Koussevitzky, here). Indeed, the scores for both the Pearl and Dean and Un homme et une femme tunes were written in duple metres.

And in the first movement of Winter, Max Richter’s welcome recomposition of the Four Seasons mixes in some great limping 7/8 bars (2+2+3—just the two tacos before the burrito today, thanks waiter) (from 1.18):

An intriguing instance is I say a little prayer, with its quirky insertion of a triple-time bar in the chorus—which no-one apparently even has to think about. And do listen to Herbie Hancock‘s addictive Ostinato.

* * *

But all this is mere child’s play compared to folk music. Though such metres are quite widespread, Bartók, Brailiou et al. coined the term “Bulgarian rhythm”.

A classic essay is

• Constantin Brăiloiu, “Aksak rhythm” (in Brăiloiu, Problems of ethnomusicology, 133–67, based on a 1951 lecture),

which contains far more detailed schemata. His work followed that of

• Béla Bartók, “The so-called Bulgarian rhythm” (1938).

A transcription by Bartók of a Turkish zurna–davul shawm band shows how, over the basic metre, melodic and percussion rhythms seriously thicken the plot:

Jerome Cler’s splendid website on the yayla has a thoughtful analysis of an aksak (here).

The whole repertoire of players like Ivo Papazov is based on aksak metres:

I don’t think I’m quite ready for Sedi donka (Plovdivsko horo), a 25-beat pattern divided

    7             7                 11
3+2+2 | 3+2+2  | 2+2+3+2+2

For more on the diverse musical cultures of Bulgaria and environs, see here. And for a wide-ranging discussion, see

• John Blacking, “Irregular rhythms: movement, dance, music, and ritual”, ch.3 of A common-sense view of all music (1987).

Back with Merrie England, there’s a fascinating article:

• Vic Gammon and Emily Portman, “Five Time in English Traditional Song”, Folk music journal 10.3 (2013).

* * *

Further east, an example from the muqam of the beleaguered Uyghurs of Xinjiang is sadly topical. A common metre consists of one long beat divided into two equal stresses, followed by two regular beats—which we might notate cumbersomely as

with the initial duplet over a notional 3/8 unit:

Some sections add another duple unit, like this dastan from Chebiyat muqam (actually a duplet over 3/8,  followed by 3/4):

And some muqam have still more metrically complex segments to explore.

As with many world genres, the Uyghurs have no tradition of notation, and seem to have no terminology for such metres (though see Rachel Harris’s chapter in Harris and Stokes (eds.), Theory and Practice in the Music of the Islamic World). As with flamenco, this kind of thing is only an issue for those (like me) hampered by a visual classical education. The trick is to internalise it in the body—and to dispense with notation. Let’s remember that much of this music accompanies dance.

Uyghur musical traditions are part of a rich culture that is currently being systematically erased in Xinjiang (see roundup here).