Dimensions In Rhythm

Crossing the Lines: In Pursuit of (Linear) Freedom, Part 3

Excerpt from, “The Harmonic Perspective of Rhythm.” Part 1. Part 2.

Helpful Signposts and Observations

Before leaving the linear perspective, however, I would like to discuss a few more points. In developing the capacity to diverge from the predominate metric cycle through a subdivision regrouping exercise, such as the dotted note, I found it interesting and helpful to realize that several distinct perspectives emerge, independent of the actual combinations. This was touched on above. To take a common example, if we work with groupings of four and three subdivisions at the same time, we can conceive of them this way:

Figure 1. Subdivisional Groups of Four and Three.

The top line can be counted while played, without subdivisions, as:

ONE, two, three, FOUR, one, two, THREE, four, one, TWO, three, four, ONE, etc.

Audio Example 1. Groups of Four and Three Subdivisions, Counting on Four (CD Track 43).

The numbers keep the metric cycle, the emphasis, shown in all capitals, the smaller repetitive cycle, or, where the two lines meet in rhythmic unison. To make things more consistent, I will call this the cycle of relation, as this is where the relationship between the two lines repeats, regardless of their positions in one metre or the other. This cycle of relation is a three with four polyrhythm/polymetre. With subdivisions, we’d hear:

ONE, e, and, A, two, e, AND, a, three, E, and, a, FOUR, e, and, A, etc.,

nested inside the previous count. This time the capitals emphasize the bottom line articulations, or the dotted note.

Audio Example 2. Counting Subdivisional Groups of Four and Three (CD Track 44).

From, ‘the other side,’ with the dotted note in our foreground, we could perceive it as this:

or, simply this:

Figures 2 and 3. The Dotted Perspective: Subdivisional Groups of Four and Three, Repeating in One Bar, or, The Cycle of Relation.

This can be counted almost the same as above, but repeating after three beats:

ONE, e, and, A, two, e, AND, a, three, E, and, a, ONE, etc.,

or, looking from the bottom line and emphasizing the top:

ONE, trip, let, two, TRIP, let, three, trip, LET, four, trip, let.

Audio Example 3. Subdivisional Groups of Four and Three, Cycle of Relation from Both Perspectives (CD Track 45).

This is, of course, the very same cycle of relation. It repeats four times before the larger, cycle of repetition has run its course. This happens when both lines have beat one on the downbeat of both ‘metres’, i.e., when the cycle of relation meets the original, 4/4 metric cycle.

In summary, we can hear this dotted note example or other similar ‘crossing’ pattern: 1) As the original metric cycle with a dotted note complement; 2) As the cycle that results from the subdivision regrouping exercise (here, the dotted note); this now becomes the foreground (here, triplets), with the other line in a ‘crossing’ relationship; 3) As the cycle of relation, which will be a smaller poly-rhythmic cycle, which repeats x number of times within the larger cycle, x being the original metre.

More Than the Dot? Perspective…

So what if, as we have seen in the examples drawn from Indian Classical rhythmic practice in Kippen and Montfort, we use a higher order regrouping factor than three? (Kippen 1988; Montfort 1985) Say, five and four? Does the dotted note still somehow apply? I would say it does, in at least two ways: One has to do with perspective and the other is a creative, technical trick (though far from original). In terms of perspective, the dotted note, like the two and three relationship at its essence, is a bridge to multi-dimensional, rhythmic perception. It is a bridge because on the one hand it is easy to derive and execute. As in simpler examples, it is based on downbeats and upbeats only. As mentioned in the section, “Two and Three: The Foundation” (page 95, or in this blog, here), we have only to divide the beats in half. On the other hand, to play it and stay with it until its full resolution while keeping track of the original metric cycle or pattern, requires—and develops—the ability to perceive both at once, and to feel the motion of each, coexisting within you; to realize that they are both related and independent.

ET: This confronts you with that problem that these must be two different…(gesticulates)…they must be two different things within you.

JD: So that’s what you’re training: To have a split awareness.

ET: Not a split awareness! To have perspective. And to be able to understand that there’s pulse and that there’s motions; there’s different waves, that are part of this thing. But they’re their own, even though they’re generated by this (hands move to show pulse). The linear way is to subdivide this…doesn’t work! The harmonic way is the way. Where things…the three…there’s no three in any waves. Waves are just downbeats and upbeats. So the three comes from the third frequency. And that third frequency is not a subdivision of this. It’s its own monster. And that’s a concept, that must be understood. But what it is (is) that every wave is part of a harmonic system. It’s its own monster. Because it has its own downbeat and upbeat. And I noticed that by playing. I said ‘Ahh. Fantastic.’ You know because you get the feeling, you know? (Plays For Your Hands Only page 18: a triplet ostinato with the last line, which is offbeat sixteenths, i.e., even numbered 32^nd notes, four to the beat.) Those are those two frequencies. And when I played that, I went, ‘Bata music.’ Ok? And then I heard all the music that has that, that I know. I go, ‘Of course.’ But to get here, the first pages were grueling. (Toro 2012a)

And Additive Rhythms

The other way I have found the dotted note to be helpful, especially when the subdivision or regrouping unit is of a higher order like five or seven, is its use as a building block for those larger structures. This has been mentioned several times throughout this thesis; all rhythms can be broken down into two and threes. It was seen above in part 2, figures 5 and 9 (groups of five and seven with sixteenths and tresillo, respectively). This additive perspective is inherent in Konakol, the Indian, syllabic, rhythmic counting method as well. (Montfort 1985; “Konnakol” 2014; “Konnakol-The Art of South Indian Vocal Percussion”) Three subdivisions are counted, ‘takita’, four, ‘takadimi’, five, ‘taka-takita (two and three), seven, takadimi-takita (four and three), nine, ‘takadimi-takatakita’, (four and five), and so forth; odd numbers are always reconciled with the ‘takita’ three grouping unit[1].

To look at the linear, regrouping principle with a few familiar and a few more complicated examples, I put some of the key elements in table form as an aid to grasp the various analytical perspectives. The most important rows are shaded.

Table 1. Some Helpful Perspectives on Subdivision Regrouping.

The first three shaded rows are the choices that determine the form of the regrouping exercise. The next shaded line (7) tells us how long the cycle of relation is in beats. Note it is always the same as the number of subdivisions in the regrouping. The next shaded line (9) shows how many bars the whole cycle will take. Note that this is also equal to the number of subdivisions in the regrouping unit, except in examples C and D, where the regrouping unit has the same number of subdivisions as the bar has beats. In these cases, the cycles of relation and repetition are the same length. A dotted note does not bestow linear, bar crossing dimensionality in three, but one bar of polymetre. Finally, the last two are perhaps the most useful, especially for the longer, more complex examples. These show how, as suggested previously, we can think of the whole cycle as one or several repetitions of the polyrhythmic cycle of relation, whose ratios are shown in the last row. These ratios, as indicated in parenthesis, show how many main beats occur per each regrouping unit; this is the inverse of the main subdivision value to the regrouping value. In other words, if we take 5/4 time—five beats with four sixteenths each—and regroup in three, we get a three:four polyrhythm (regroup:subdivisions) that repeats every three beats (regroup). Five of these cycles (starting on each of the five beats) will happen in the three bars it takes to complete the cycle.

The following are illustrations of the examples presented in the table.

Example A: Two beats of two subdivisions grouped in threes; 3:2 polyrhythm happens twice in three bars:

Figure 4.

Audio Example 4. Linear Example A (CD Track 46).

Example B: Two beats of four subdivisions grouped in threes; 3:4 polyrhythm happens twice in three bars:

Figure 5.

Audio Example 5. Linear Example B (CD Track 47).

Example C: Three beats of four subdivisions grouped in threes; 3:4 polyrhythm happens once (resolves in one bar):

Figure 6.

Audio Example 6. Linear Example C (CD Track 48).

Example D: Three beats of five subdivisions grouped in threes; 3:5 polyrhythm happens once:

Figure 7.

Audio Example 7. Linear Example D (CD Track 49).

Example E: Four beats of four subdivisions grouped in fives; 5:4 polyrhythm happens four times in five bars; note additive 3+2 grouping:

Figure 8.

Audio Example 8. Linear Example E (CD Track 50).

Example F: Four beats of five subdivisions grouped in sevens; 7:5 polyrhythm happens four times in seven bars; additive grouping is 2+2+3

Figure 9.

Audio Example 9. Linear Example F (CD Track 51).

Example G: Five beats of three subdivisions grouped in fours; 4:3 polyrhythm happens five times in four bars

Figure 10.

Audio Example 10. Linear Example G (CD Track 52).

Once again, the experiential perspective comes into play and some of these examples, especially example F, are quite difficult to play. The chart and the examples, in providing a few structural signposts, are meant as an aid for someone willing to put in the effort to work through these types of rhythmic challenges. They are also intended as a partial codification of the largely intuitive process that might go on in the mind of someone who is skilled at these types of manipulations, perhaps an Indian Classical rhythmic specialist, though in that case the rhythmic structure would likely be rendered and manipulated in konakol syllables or bols representing the sounds on tabla, mrdingam or other instruments.

The foregoing has been about the method of deriving polymetric relationships based on subdivision, here as linear grouping patterns within another metre. The cycle of relation, by itself, with its accents considered as beats in themselves, is polymetre. And, as warned, it can become quite difficult to derive and manipulate complementary metric structures from this perspective. Fortunately, there is another way, which leads us to back to The Odd in You. Part 1. Part 2.

[1]   Toro would take issue with some of these, stating that the proper, ‘natural’ way to render them is with the larger number first, i.e., takatakita-takadimi. This way the symmetry about the two/global upbeat remains intact. Pillay did teach me a tabla phrase of nine subdivisions this way: na ki na num ki–na ki num ki. (Pillay 2011)

Bibliography

As you’ve seen in the text, I have included links to some 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