Harmonic Playing and Perception: Stacking Meters.
Excerpt from, “The Harmonic Perspective of Rhythm”
Polymetre
It should be clear by now that motional ‘down’ beats and ‘up’ beats occur on many levels of the rhythmic hierarchy. An illustration of this phenomenon was presented in figures 1 and 2, Crossing the Lines in Pursuit of Linear Freedom, Part 1, the preliminary exercises of Efrain Toro’s book For Your Hands Only. The first line of ‘downs’ is followed by the second line of ‘ups’. Those two are then combined to make ‘downs’ at double the pace, which are followed by their respective ‘ups’, and so on, until practically unfeasible. Key to rendering this sort of exercise valuable, once again, are aural and motional awareness. Down strokes can be played on the ‘down’ beats at a particular hierarchical level of the rhythmic structure; they can be played this way with up strokes in between, so that motional and aural/structural awareness coincide; or, they can be played on the ‘up’ beats, in which case motion and hearing are opposed. With all these capacities firmly embedded in our approach, we can consider their value in approaching polymetre.
Always the Wave
At the primary metric level, where there is just one downbeat (beat one), there is also just one upbeat. This perspective can be applied to any metre. It is thus, in Toro’s view, the key to combining metres, in lieu of the linearly conceived counting methods covered in the last section (Crossing the Lines: In Pursuit of (Linear) Freedom, Parts 1, 2, and 3). If we take the cycle of relation from any of those exercises, we will see that the accented subdivisions representing each of the two grouping structures repeat in a symmetrical pattern around the primary upbeat of that cycle. In the case of three and two, the second articulation of two IS the primary upbeat, marking this as a primordial polymetric pattern, at the foundation of multi-dimensionality.
However, the symmetry exists, no matter the combinatory second metre.
Figures 1 and 2. Symmetry About the Primary Upbeat.
Audio Example 1. Five, Six and Seven with Two (CD Track 53).
Note the new time signature. Although not functionally different from common time (4/4 or 2/2), it nevertheless carries the implication that neither the five, six, seven or two are the ‘main’ unit of counting. They all relate to one, and its upbeat.
This symmetry can thus be used as a reference when combining metres, or superimposing them on the same one cycle.
Figure 3. Downbeat/Upbeat Symmetry without the Upbeat Reference.
Audio Example 2. Four and Three, Five and Three, Seven and Five (CD Track 54).
It is as if their durations about the primary upbeat, and therefore each other, are in rhythmic retrograde from one half to the other. This perspective helps to simplify the relationship even in complex combinations such as the five and seven above or, here, five and six:
Figure 4. Symmetry in Five with Six.
Audio Example 3. Six and Five (CD Track 55).
It as if we hear, ‘together – – – – – six five – – – six – five – – six – – five – six – – – five six – – – – ; or, after hitting together on the downbeat, they strike very near, with the faster one (six) preceding, less near, then six in the middle, surrounded by equal space, then the reverse: Five preceding six, less near, then very near.
Here is seven and five again. I have spread the empty, upbeat area, indicated with an up arrow, in order to make the downbeat and upbeat halves stand out.
Figure 5. Symmetry in Five with Seven, Notes Spread About Upbeat Mark.
Jogging the Memory
Symmetry is one of several perspective enhancing shortcuts I use to hear or remember the interactive melody between the two metres. Other ideas I find helpful are:
- Deriving the order of interaction from the metric relationship: Using the difference between the metre factors, we can get a quick picture of their interaction pattern. For example, three from four leaves one extra articulation to be distributed in the four line. It will occur in the first interaction and the rest will alternate—four, four, three, four, three, four (also see six and five, above).
Figure 6. Four and Three. Interaction Pattern: 2:1, 1:1, 1:1.
If the difference is more than one, the next instance of two successive articulations of one metre will occur in the middle, around the upbeat (also see five and three above).
Figure 7. Seven and Five. Interaction Pattern: 2:1, 1:1, 2:1, 1:1, 1:1.
As the difference gets greater, the interaction pattern will proceed toward a simpler ratio; in seven and four we see a 2:1 ratio for all but the last interaction:
Figure 8. Seven and Four. Interaction Pattern: 2:1, 2:1, 2:1, 1:1.
And with eight and four everything is evened out:
Figure 9. Eight and Four.
Proceeding to a still greater difference, the process begins again, adding to the first interaction to make 3:1, 2:1, 2:1, 2:1:
Figure 10. Nine and Four.
Then 3:1, 2:1, 3:1, 2:1, which can be conceived of as five and two twice:
Figure 11. Ten and Four.
And 3:1, 3:1, 3:1, 2:1:
Figure 12. Eleven and Four.
Audio Example 4. Seven, Eight, Nine, Ten and Eleven with Four (CD Track 56).
These are not actual ratios, of course, but patterns of interaction. Some help with the timing of the interactions follows.
- Subdivision size: The differences between the adjacent beats, from the perspective of one metre, is some factor of subdivisions equal to the other metre. This is especially helpful in smaller metric combinations such as three and five, where we might think, ‘ONE, two, (trip), LET, three, four, TRIP, (let), five.’ This is a combination between linear regrouping and harmonic hearing; the interaction pattern of the articulations from the respective metres being known, the timing is taken from knowledge of their mathematical relationship. This is obviously less helpful in combining two higher order metric structures with correspondingly smaller subdivisions, such as seven and five or higher.
- Mnemonics: Although less than precise and with their own potential for bias or rigid thinking, verbal phrases can nevertheless help to jog the memory. I first learned this technique from Frank Bennet, a South Indian and jazz drummer, in a class he taught at UCLA, entitled ‘Cross-Cultural Perspectives in Jazz.’ Examples are: ‘Pass the butter,’ for three and two; ‘Pass the Gosh Darn Butter,’ (expletives tamed by me) for four and three (for me, the four comes out more), or, alternatively, ‘Hot chillies hit the spot,’ (in which I hear emphasis on the three); ‘Hot chillies sure do hit the spot,’ for five and four, and so on.
Wave Interactions, Tricky Surf
The preceding ideas were some of my observations gathered during the process of learning and remembering various polymetric combinations. The next idea is Toro’s approach to this process. For him this is the essential concept that sheds light on these relationships of contrasting metres. It is the direct application of the downbeat/upbeat concept to the combination of metres. As mentioned previously, when odd metres are seen about the upbeat, they occur part before, part after[1]. Even numbered metres repeat on the upbeat. When dividing odd metres in two then, the first half will have one more count than the second half[2] (though the extra count is only one half long, see footnotes, below). Thus, three will divide 2 + 1 (on-beat articulations). Five will divide 3 + 2. Seven is 4 + 3, and so on. Therefore, when combining metres, the process is greatly simplified by considering how their halves combine. Four and three is like two and two, then, two and one. Seven and four and is like four and two, then, three and two. Seven and five is like four and three, then, three and two.
The reader will note the word ‘like’, in the above statements. The timing is not exactly those ratios. If it was, the cycle would start over in the middle and would effectively be the doubling of a reduced ratio; for example, we saw previously that ten and four is really just five and two twice. Alternatively, the cycle would consist of two ratios in a row with one of the component metres kept steady (isochronous) and the other adjusted as two, independent halves with different timings (e.g., seven and five as four:three then three:two; if one metre is isochronous the other is dependent and non-isochronous). Nevertheless, the concept, with a keen ear, helps to reduce the task into more sizeable chunks. A keen ear is of paramount importance all through this discussion, as is the awareness of steady, interdependent motions as all but the ponderous method of subdivision regrouping gives but approximations of the timings involved. Thus, once the relative interaction pattern is well established by some combination of the above methods, the ear and the motion of the body must take over to refine the different patterns into more or less even (isochronous) renditions of their respective counting structures. I say, ‘more or less,’ because as we’ve seen in the discussion of feel, multi-dimensional music, as produced by human bodies in motion, is not calculatedly ‘perfect.’ If it was, it would not sound right to most of us. It is the ‘perfection of the imperfection’ (See, Harmonic Feel, Part 1) which we seem to find more profound.
A Walk Through
As an example process from my experience, to play seven and five, I might do the following, not necessarily in this order:
- Having played this combination before, I would think of the two half, downbeat/upbeat interaction pattern. This is four with three then three with two.
- Knowing that the second half does not start together, I would play the four and three then the three and two, but on this second ratio I would use a slightly staggered start, the articulation of the seven limb in front of (before) the other, five limb.
- I would now try to listen to the two, count each to make sure they are right and attempt to adjust them to be more or less even.
- Remembering that the timing of the interactions in the second half is a mirror image to those in the first half, I would try to hear and adjust my strikes to hear this quality.
- If and when it feels pretty good, I might check with a metronome[3].
At this point, I imagine an observer might have several nagging objections, for example:
- What is the use of all this? Is this just rhythmic ‘gymnastics’ or is this relevant to music?
- Why not use the permutation regrouping method? I prefer to be organized, didactic and accurate in my studies.
[1] Only the central offbeat of an odd metre coincides with the primary upbeat of the one cycle, but to consider it is akin to doubling the odd metre in question thus making it even.
[2] The offbeats are the reverse, making the two halves even in length, e.g., 2 ½ + ½ 2 = 5.
[3] There are also polymetric capable, computer based metronomes such as the Bounce Metronome. (“Check Out the Astonishing Bounce Metronome Pro”)
Bibliography
I don’t really cite books or articles in this section, but here are some related works:
Leave a Reply