November 29, 2007

Arom on polyrhythms

Here's a sort of prequel to my post on "Rock syncopation: stress shifts or polyrhythms?" (11/26/2007). It consists mostly of some notes and quotes from Simha Arom's African Polyphony & Polyrhythm, Cambridge University Press, 1994 (originally published as Polyphonies et Polyrhymies Instrumentales d'Afrique Centrale, SELAF, 1985). The point is to help those of us trained in European metrical traditions to understand some other ways to think about meter and rhythm. My own interest comes from attempts to understand the ways that words fit into American popular and folk music of the past century or so.

The stuff in red is quoted from Arom's book. The rest is my commentary, summary and exemplification.

Some of Arom's terminology (p. 230):

The period provides a temporal framework for rhythmic events. It is invariably composed of whole numbers. These numbers are usually even, i.e. divisible by two ... This means that the structure of the period is symmetric. The constituents of this structure are pulsations.

The pulsation is an isochronous reference unit used by a given culture for the measurement of time. It consists of a regular sequence of reference points in relation to which rhythmic events are ordered. Moreover, in polyrhythmic music, the pulsation is the common denominator, from the standpoint of musical organization, for all the parts in a piece. ...

The pulsation is, however only rarely given material existence in this region of Central Africa. While it can always be given the material form of handclaps, it is nevertheless usually implicit.

The pulsation can be subdivided in three different ways: binary when it is slit into two of four equal parts; ternary when it is split into three or, rarely, six equal values; composite when it is split, in a combination of the two preceding ways, into five equal values. ...

The minimal operational value is the smallest relevant duration obtained after subdivision; all other durations are multiple of this value.The period is thus equal to the total number of these values. A period based on twelve pulsations will then contain thirty-six operational values in the case of ternary, and twenty-four (or forty-eight) in the case of binary, subdivision of the pulsation. ...

The preceding sections have dealt with the metric organization of the period as a temporal framework for rhythmic events. In African music, however, several rhythmic events are usually found to occur simultaneously. This is what we call polyrhythmics  ... [T]he superposed rhythmic figures in a polyrhythmic context are of varying lengths, yet always stand in simple ratios, such as 2:1, 3:1, 3:2, 4:2, and multiples thereof.

Here's a discussion of one particular class of rhythmic figure (p. 246):

One particular form of asymmetry which is very frequently found in Central Africa may be called rhythmic oddity. When the number of pulsations in the periods involved is divided by two, the result is an even number. The figures contained in this period are nevertheless so arranged that the segmentation closest to the middle will invariably yield two parts, each composed of an odd number of minimal vaues, wherever the dividing line in placed. These figures are always constructed by the irregular juxtaposition of binary and ternary quantities. The resulting rhythmic combinations are remarkable for both their complexity and their subtlety. They follow a rule which may be expressed as 'half - 1/half + 1'. [...]

The figure with eight minimal values (i.e., containing two binary pulsations) ... is articulated as follows: 3/3.2 = 3/5 = 4-1/4+1.

This, of course, is nothing other than the familiar habanera rhythm, discussed in my earlier post:

 x - - - x - - -|x - - - x - - -
|1 2 3 4 5 6 7 8|1 2 3 4 5 6 7 8|
 o     o     o   o     o     o
 o     o         o     o

Note that the two binary "pulsations" (indicated with x's in my schematic above) are implicit in this figure, though they are where a musician from this tradition would mark time with hand-claps -- and they are positions that might well wind up marked by notes in a different figure performed simultaneously. (See my earlier post for an example.)

A figure with a period containing twelve operational values arranged into four ternary pulsations ... may be segmented into 3.2/3.2.2 = 5/7 = 6-1/6+1.

 x - - x - - x - - x - -|x - - x - - x - - x - -
|1 2 3 4 5 6 7 8 9 A B C|1 2 3 4 5 6 7 8 9 A B C|
 o     o   o     o   o   o     o   o     o   o
 o         o             o         o

Here's a midi implementation of this schema -- for clarity, I've put the "minimal operational values" in the background. In Arom's examples, both the "minimal operational values" and the "pulsation" would be implicit, and the whole thing would be much faster -- typically the "pulsations" are around 150 per minute (2.5 per second, 400 msec. each), and thus in this case, the minimal operational values would be around 450 per minute (7.5 per second, 133 msec. each):

A figure with four pulsations subdivided into sixteen minimal values ... has the following arrangement: 3.2.2/2.3.2.2 = 7/9 = 8-1/8+1.

 x - - - x - - - x - - - x - - -|x - - - x - - - x - - - x - - -
|1 2 3 4 5 6 7 8 9 A B C D E F G|1 2 3 4 5 6 7 8 9 A B C D E F G|
 o     o   o   o   o     o   o   o     o   o   o   o     o   o
 o             o                 o             o

And so on for some longer figures.

Since the "pulsations" subdividing the periods in these examples are all isochronous -- 4+4 or 3+3+3+3 or 4+4+4+4 -- should we think of these rhythmic patterns as dislocations of underlyingly evenly-spaced beats, in the style of Temperley's theory of "rock syncopation? Arom suggests a different sort of generating process for this case:

The technique of rhythmic oddity is based on the principle of progressively inserting binary quantities into configurations bounded by ternary quantities. This is clear from the paradigmatic representation of how this principle applies:

Cycle of 8 minimal values    3.   3. 2
Cycle of 12 minimal values    3. 2 3. 2.2
Cycle of 16 minimal values    3. 2.2 3. 2.2.2
Cycle of 24 minimal values    3. 2.2.2  3. 2.2.2.2

The figure shown in ex. 36, however, applies an inversion of this principle, by inserting ternary quantities in configurations bounded by binary values. The twenty-four constituent minimal values are articulated thus:

2.3.3.3  2.3.3  2.3

Note that these patterns are not yet polyrhythmic -- they're examples of some of the basic figures out of which polyrhythmic music can be made.

The simultaneous performance of different rhythmic figures engenders a polyrhythmic block or formula. Since each figure has its own period, but all the periods stand in simple ratios, the period of a polyrhythmic formula will always be the period of the longest figure. (p. 277)

One example of such a polyrhythmic formula, provided by Arom, is the scheme for the mò.kóndí dance of the Aka people, "the music for a ritual dance used to consecrate a new campsite" (p. 289 ff.).

We may remark in passing that the Aka musicians generally perform three different rhythmic figures on a single two-headed skin drum lying on the ground: the è.ndòmbà ['child'] part is played on the narrower end, and the ngúé ('mother') par on the wider end. The two drummers straddle the drum, sitting with their backs turned. The dì.kpàkpà [hard wooden sticks] player crouches between them facing the drum, and strikes the barrel in the middle.

The instrument known as dì.kétò consists of striking together two iron strips.

The four parts of mò.kóndí are interwoven in a way schematized below (in a simplified form in which only the accented notes of each part are shown).

            1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
pulsation:  x  -  -  x  -  -  x  -  -  x  -  -  x  -  -  x  -  -  x  -  -  x  -  -
è.ndòmbà:   o        o     o  o     o  o        o        o     o  o     o  o
dì.kpàkpà:  +     +        +     +        +        +        +     +        +
ngúé:       ^        ^  ^        ^  ^  ^        ^        ^  ^        ^  ^  ^
dì.kétò:    *     *  *     *     *     *     *     *     *  *     *     *     *

The dì.kétò part is one of those "rhythmic oddity" patterns, configured as 3.2.2.2.2.2 3.2.2.2.2 = 13/11 (where the 3s are further divided as 2.1).

Note that there is no direct connection between this culture and the polyrhythmic aspects of American popular music, aside from some very recent effects. African musical traditions are extraordinarily diverse. The Aka people are hunter-gatherers of the Central African Republic, whereas the musical and the traditions that influenced American music would have come mainly from agricultural and city-based cultures in West Africa and Angola. (And perhaps especially through Lagos, via the 19th-century Lagos/Bahía/Havana/New Orleans pathways discussed by J. Lorand Matory in "The English Professors of Brazil: On the Diasporic Roots of the Yorùbá Nation", Comparative Studies in Society and History, 1999.)

However, the patterns and formulae from Central Africa illustrate some general principles of African (and perhaps American) polyrhythmics. Quoting Arom again (p. 211):

In most traditional African music, time is organized according to the following principles:

(1) A strictly periodic structure (isoperiodicity) is set up the repetition of identical or similar musical material, i.e. with or without variations.
(2) The isochronous pulsation is the basic structural element of the period. Whether the figures it contains are binary or ternary or a combination of these, the period is defined by the invariant number of pulsations which constitutes its temporal framework.
(3) There are no regular accentual matrices The pulsations or beats on which the period is based all have the same status. There is thus no intermediate level (measure "strong" beats") between the pulsation and the period.
(4) The pulsation is not necessarily materialized.

It is easy to confirm that these principles exist. It will suffice to ask a member of a given culture to superpose hand-claps on a piece of music from that culture. Whether the experiment involves one or more informants, at brief or longer intervals, we have found that the results are always the same:

- the beat is isochronous and perfectly regular: the beats always fall at the same points in the melodic and/or rhythmic material contained in the period.
- concomitantly, however this material may be distributed, it always reappears after exactly the same number of beats.

These essential principles of all measured Central African music, both monodic and polyphonic, also seem to govern most measured music throughout sub-Saharan Africa.

Perhaps these principles also apply to a certain part of American popular music since 1890 or so, or at least to one available way of conceptualizing some of it. When we think about the patterns of tune-text alignment in that music, we should keep this in mind.

Posted by Mark Liberman at November 29, 2007 06:50 AM