Dimensions In Rhythm

Polymeter in a 4/4 Groove ?

Excerpt from, “The Harmonic Perspective of Rhythm”

Binary Structure and the Proximate Three

First, it will be noticed that all the examples cited above (adding-metric-multi-dimensionality-to-your-perception-concept-and-playing) were in ternary-based metres. That is, the main beats were divided primarily into threes. The two and three foundational relationship is clear and easy to derive in such a base. Two beats of triplets, or six subdivisions, with every other subdivision articulated, gives three against those two main beats. This can be seen in various guises in the previous examples of Burns, Novotney, Peñalosa and Kubik, among many others. Alternatively, in a metre of three, as is common in Spanish Flamenco music, the inverse relationship exists, where the two is easily derived by halving the three ‘main’ beats.[1]

But what about music that is clearly in a binary metre with mainly binary subdivisions of the beats? In this regard, Toro is in agreement with Peñalosa and others that the near ubiquitous figure rendered by three subdivisions, three subdivisions, and then two subdivisions, shown here,

Figure 1. The Binary Three Correlate.

Audio Example 1. The Binary Three Correlate (CD Track 14).

is a sort of binary substitution for the three component of the harmonic foundation (Toro 2012b). This figure is found not only in African Diaspora folk and popular music, but is quite common in Indian, middle eastern and other musics as well. In fact, given its modern dispersion by popular styles of dance music, from Michael Jackson to Hip-Hop, House, and Bollywood, it would be a difficult but worthy task to assess the dispersion of this figure through the world’s musical traditions before the advent of mass media, and whether the phenomena does indeed represent some kind of dispersion, or, more to Toro’s estimation, a parallel genesis based on the naturalness of the relationship it represents[2]

In any case, as has been documented by Peñalosa (Peñalosa and Greenwood 2009) Acquista (Acquista 2009) and described to me by Toro, the aforementioned figure, also known as tresillo cubano, ‘mambo bass’ (Toro 1993b) or just tresillo (Mauleón 2011, 1999) can be shown quite clearly, from an African Diaspora perspective, to relate to the ternary relationships described previously. Any suggestion of a chronological relationship indicating a parent/child or evolutionary ordering is, to me, less clear cut. This is a relationship that was shown to me back around 1989 or 1990 by Yoruba Dundun (‘talking drum’ of the Yoruba people of Nigeria) master Francis Awe. Although the tresillo rhythm is common in many parts of the globe, its use in the African context, where simultaneous multiple metric references are the rule rather than the exception, suggests its function as a proximate three metre quite clearly, as explained below.

The Proximate Three: Clarity from Africa

To begin with, a review of a few basic African rhythmic concepts will be helpful. One of the most common organizational, foundational, or timeline patterns (Novotney 1998; Hernandez 2000; Peñalosa and Greenwood 2009; Kubik 2010; Locke 1998; Arom et al. 2004; Locke 2010; Burns 2010; Agawu 1995; Nketia 1974; Chernoff 1981, et. al.) is shown below:

Figure 2. Ternary ‘Standard Pattern’ or ‘Key Pattern’.

Audio Example 2. The Standard Pattern (CD Track 15).

Written in this way, it is perhaps clear that it is comprised of the three metre derived previously by articulating every other ternary subdivision (there are two threes in succession, to make six). In this case, however, the first three are ‘on’ the beat, that is, they line up with the one, and the last four are offbeat, in relation to the three metre (again, there are two in a row, six altogether). Seen in this way, it is perhaps clearer to hear them as, ‘on – on – on off – off – off – off (on),’ such as this:

Figure 3. Ternary Standard Pattern from Six.

Audio Example 3. The Standard Pattern from Six (CD Track 16).

But now we must consider the two component, essentially built into the 12/8 metric format; it is the reason 12/8 is much preferable to 6/4, which would have been acceptable in the previous graphic. 12/8 implies the metre of four, subdivided as triplets. When these triplets are articulated in groups of two (or every other triplet, as above) we get the naturally occurring two/three cross-rhythm (Peñalosa and Greenwood 2009; Novotney 1998), polyrhythm, or polymetre, twice in succession:

Figure 4. Two and Three in Compound Quadruple Time, or Four and Six.

We saw and counted the same in box notation in Adding Metric Multi-dimensionality to your Perception, Concept, and Playing. So, our timeline pattern with the four metre looks like this:

Figure 5. Standard Pattern with Four.

And finally, the same thing, grouped in four, with all the subdivisions implied, which should be easier to read[3]:

Figure 6. Standard Pattern with Four and Implied Subdivisions.

Audio Example 4. Three and Two to Standard Pattern with Two (CD Track 17).

Now, in 12/8, the two/three (four/six) relationship is inherent in the timeline structure. One component, the three, is in the timeline and the other, the two, in the beat structure. In practice, these two metres and derivative patterns using only some of the articulations, on and off the beat, and/or ornamenting their respective metre, will be distributed to various instrumental, vocal, and dance patterns. (Chernoff 1981; Agawu 1995; Burns 2010; Anku 1997; Locke 1998; Scherzinger 2010, et. al.) In this way the dual metric implication is always implied. Short of long term experience performing and dancing to this kind of music, common in most of sub-Saharan Africa, it is recommended to learn to play the aforementioned two rhythmic combinations, and then to keep the top line while alternating between four and six to the bar on the bottom, like this:

Figure 7. Standard Pattern with Four then Six.

Note that this is a simplified, ‘bare bones,’ version of an African polyrhythmic/polymetric experience, and even traditionally derived variations to these exercises are extensive. Nevertheless, the basic concepts should now be sufficient for the discussion at hand. With this key pattern and its dual metric implications understood, we can now turn to the binary correlate.

As the previous timeline pattern is fundamentally multi-dimensional, its relationship to the binary correlate can be seen from different perspectives. If we consider the rhythmic three element above and its correlate, tresillo, then, instead of,

Figure 8. Three with Two (Four with Six) in Ternary Structure

we have,

Figure 9. Three with Two (Four with Six), Binary Correlate.

which might look clearer as this:

Figure 10. Binary Correlate as Dotted, Dotted, Non-Dotted.

A similar quantization of the standard pattern into common time then changes this,

Figure 11. Standard Pattern, Ternary.

into this:

Figure 12. Standard Pattern, Binary.

As mentioned, this very pattern and its clear similarity to the ternary version were first shown to me in Yoruba music lessons by Francis Awe. Both these patterns are common in that music. The binary version is also quite common in Congolese popular music such as Rumba and Soukous. Moreover, in teaching I sometimes call this pattern a ‘composite clave’, as it contains all the elements that make up both the Rumba Clave and Son Clave, so central to the Cuban/Caribbean/’Salsa’ rhythmic concept.

Figure 13. Son Clave (top). Rumba Clave (bottom).

Some scholars even suggest that all African Diaspora music is founded on the clave concept (Peñalosa and Greenwood 2009; Spiro and Ryan 2006), and the present discussion follows a similar line of thinking, though we are concerned here primarily with the implication of polymetre. I mention the clave concept to indicate once again how ubiquitous is the aforementioned rhythmic pattern—call it the binary standard pattern—especially if we consider the various claves as its relatives or derivatives.

Lest the discussion get overly theoretical, the experiential perspective is again in order, and thus, more suggested exercises.

First, changing from the ternary to binary feel and back again, while maintaining the same four background. I’ll write it in common time with triplets to avoid changing time signatures:

 Figure 14. Three and Two Ternary to Tresillo with Two Binary.

Audio Example 5. Three and Two, Ternary to Binary (CD Track 18).

Now, the same idea with the standard pattern:

 Figure 15. Standard Pattern Ternary to Binary Feel.

Audio Example 6. Standard Pattern,Ternary to Binary (CD Track 19).

And finally, the binary version with the switch from four to ‘six‘ underneath:

 Figure 16. Binary Standard Pattern with Four and with Binary Six Correlate/Tresillo.

Audio Example 7. Binary Standard Pattern with Four and ‘Six‘ (CD Track 20).

It essential to note, however, that these are not just exercises but actual foundational patterns found throughout the African Diaspora. It is the commonality and transmutability of these feels that constitutes the strongest evidence for the proposal that the tresillo pattern is actually a proximate three, maintaining the polymetric foundation even in rhythmic structures that appear to be outside a strict 12/8 interpretation.

[1]   It should be clear by now that, depending on the aesthetic predilections of the piece or style in question, there need not be a ‘main’ beat or metre, but that several can co-exist and take perceptual predominance in turns according to the performance, the soloist, the dance, and of course, the listeners’ awareness.

[2]   As alluded to elsewhere, Toro is interested in Jung’s theory of the collective unconscious, as well as Joseph Campbell’s concept of the monomyth, both ideas based on the underlying assumption that humans are predisposed to certain universal attributes of mind; this is as opposed to personal attributes, which are based on an individual’s experience. Thus, patterns of behaviour, myth, creative archetypes, dreams and such allegedly show similar themes across the world.

[3]   In certain applications, I prefer this type of notation, similar to that used by Billmeier and Keïta. (Billmeier and Keïta 2004) It not only helps the reader with subdivisions and rhythmic groupings, but is more accurate. In percussion music, strictly speaking, all note values are short and similar, thus the mixing of quarter notes, eighth notes and half notes for bell and drum strikes is less than correct but often employed as it is easier on the eye than a great number of small rests. The notation system used here partially avoids that circumstance though would also be a bit cumbersome for long orchestrations. It is best suited to writing ostinatos. The eighth notes here are also not accurate enough to represent a real performance, as compared to duration values from a live performance into a midi device, for example. This discussion will be taken up when we come to feel. Thus introduced I will use this system when appropriate.

Bibliography

As you’ve seen in the text, I have included links to most of these works. If you are interested in purchasing any of them, please consider clicking through and helping to support this labor of love. Thank you! YIR (Yours In Rhythm), John