Dimensions In Rhythm

The Harmonic Perspective of Rhythm Revisited

Excerpt from, “The Harmonic Perspective of Rhythm”

Bolstering Expressive Capacity: A New Rhythmic Paradigm

The hypothesis herein is that the side that needs conceptual bolstering, in the schooled musician, at least, is that which is largely left to fate in traditional (western) music lessons, namely, the expressive capacity of the student. Only one component of this capacity will be considered here, but it is arguably the most significant: Rhythm. This is not to say that traditional music teachers do not have many effective ways of teaching rhythm, but instead that the perceptual grid that is taught and, significantly, that is conceptually and perceptually engrained, is limited. It is hypothesized that by working with a new, more extensive model such as that proposed by Toro, the possibility of both conceptual (explicit) and perceptual/implicit understanding will be available at a much finer, multi-dimensional resolution, thereby decreasing the gap in rhythmic command, flexibility, and grounded-ness that tends to exist between the experts of AT/Musics (Aurally Transmitted, see Traditional-vs-modern-apprenticeship) and those of more theoretically concerned pedagogical systems. Such a new model offers the AT/Music proponent, including those from outside the original cultural milieu, a more comprehensive theoretical position from which to understand and master their craft. It also offers the theoretically minded a broader rhythmic model from which to work, thereby deepening expressive capacity and/or facilitating the understanding and potential influence from the arguably more rhythmically sophisticated traditions of the world.

One more analogy: With some pitched instruments, such as fretless string instruments, one must learn to ‘tune’ every note. Many others only require periodic tuning. Some like the piano, are usually tuned by someone else! In rhythm, however, everyone must ‘tune’ every note, every time it is played. As tuning systems have been developed to follow Pythagorean tuning, even-tempered tuning, tunings using five, seven, twelve, twenty-two, twenty-four or some other number of notes to the octave, in various intervallic arrangements, so the parameters with which rhythm is ‘tuned’ can be seen from different perspectives. Some of these perspectives appear to better represent what we hear from the musics of the world than the (western) system in standard use. That said, it is not suggested the current system be scrapped, as some theorists have (Koetting 1970; Arom et al. 2004), only that it be expanded and re-conceptualized.

The Harmonic Perspective of Rhythm Revisited

As the name suggests, this new perspective allows for the consideration of two or more concurrent conceptions of time measurement. This contrasts with a linear perspective of rhythm, in which all ideas—even from distinct metric systems like two and three—are conceived within one line of subdivisions equal to the smallest division of the beat (the ‘lowest common denominator’) necessary to represent all. Theoretically, this linear way of thinking can of course account for anything possibly articulated to within a margin of error beyond the range of human perception—think of digital technology in which sounds are represented by discreet and very tiny bits of information that, when interpreted by the playback device, give us a relatively perfect picture of any aural phenomenon we wish to preserve. Conceptually and practically, however, this linear way of thinking seems to fall short of some of the needs of the rhythmically sophisticated human music-maker.

Polymetre

Before moving on I would to clarify an important term and its use in this document: polymetre. I am also going take a controversial stance on its use, but one that I feel is important in the aim of semantic efficacy and necessary for the position argued herein vis-a-vis the harmonic perspective of rhythm. Peñalosa, Agawu and Novotney, in particular, make their case for the term cross-rhythm, applicable to many of the examples to be dealt with in upcoming discussions. (Peñalosa and Greenwood 2009; Agawu 2003; Novotney 1998) Novotney in particular goes to lengths to distinguish cross-rhythm as a subset of the more inclusive term polyrhythm, which he says is, “a general and non-specific term for the simultaneous occurrence of two or more conflicting rhythms, of which, the term cross-rhythm is a specific and definable subset.” (Novotney 1998, 13)

Peñalosa says: “The secondary beat cycle is the next tier of the rhythmic matrix. This secondary cycle is composed of cross-beats (or counter-beats), beats that regularly and systematically contradict the primary beats. Cross-beats create excitement, rhythmic tension and a sense of forward momentum.” (Peñalosa and Greenwood 2009, 21)

I do not have a problem with this terminology, for the limited context in which they are working: African and Diaspora accompaniment patterns with a predominant main (primary) metre and other, secondary cycles related by the regrouping of subdivisions in twos and threes. Toro, as will be seen shortly, uses this process as a substantial part of his method for increasing rhythmic perception and catalysing interdependence in performance. However, though African music was said to be polymetric by Chernoff (and earlier scholars, Jones, King, Brandel), the general state of scholarship now shuns the term. (Chernoff 1981; Jones 1959; King 1960; Brandel 1959)

However, the polymetre that is being thrown out with the proverbial bathwater is that which requires us to think in one metre after another, in additive fashion (such as using sequential 5/8 and 7/8 bars to notate a 12 pulse pattern), as in the transcriptions of A.M. Jones and Rose Brandel. (Jones 1959; Brandel 1959) I agree that their perspectives were skewed, difficult to manipulate, and mostly unrelated to the perspective used by performers of the musics in question. The problem, as I see it, is that the scholarly world thinks immediately and uniquely of polymetre as tactus-preserving polymetre, where the underlying tactus continues, uninterrupted, while the sequentially applied metres change its (the tactus’) metric structure. This is a linear, one-dimensional application. Measure-preserving polymetre, on the other hand, allows for different metres to unfold in the same space of time, as discussed throughout this document. This is usually referred to as polyrhythm. (“Meter (Music)” 2014)

Nor do I reject the term Polyrhythm, but, as Novotney noted, it is very general, relating equally to an outrageously complex combination like eight and nine, a simple one like three and two, or even to two rhythms that are obviously of the same metre but emphasizing different parts of the beat structure so as to ‘conflict’ in the listeners’ mind. (Novotney 1998, 13)

And, though I think cross-rhythm also has its appropriate usage in certain contexts—like the ones discussed by the aforementioned authors—it is also limited. I do agree that most if not all musics are fundamentally based on a primary metric cycle, but other cycles do exist; in complex rhythmic systems like those from Africa and India, they must exist for the music to be aesthetically whole. I propose a return to the implication of the component words, poly and metre. Two metres, in many cases, are occurring; whether they can each be derived with the other’s commonly used subdivisional unit is irrelevant. This is a key point where the harmonic perspective of rhythm differs from conventional viewpoints. All polymetres can be derived as cross-rhythms (by the regrouping of a common subdivision) but at a certain point it becomes difficult, as suggested several times in this thesis. Do we arbitrarily respect some line of complexity as an absolute line, either side of which we change terms, or do we allow the consideration of one measure cross-rhythms to imply different perspectives, and, different metres?

Put another way, certain metres, those with twelve pulses (or subdivisions) in particular, can be divided into several co-existing cycles, or perspectives, at the same time: two, three, four and six, to be more specific. They all repeat in one metric cycle, or measure. And yes, I might refer to them as cross-rhythms. However, a cycle of five, eight or even nine can also be used within this measure. Eight is quite common in an African Diaspora schema, even as an accompaniment part, but somewhat difficult to derive as a cross-rhythm, requiring as it would, 24 subdivisions; it is much easier to hear it as double the four. Five or nine with twelve in the space of one measure would be virtually impossible to think of cross rhythmically. Theoretically, this poses a problem. Realistically it insists that we open our ears and our perspectives. As argued here and by Toro, these other metric perspectives are in the music, “All the time.” (Toro 2012f, 2012j, 2012m) This is essentially another perspective on the ‘off-beat five’ discussion above.

If, however, we were to regroup the twelve subdivisions into fives—which would last five measures until resolving (beat ‘one’ with regroup cycle ‘one’)—I would certainly refer to this as a cross-rhythm. Its accents also imply, or derive, if you will, a five with twelve ‘polymetre’, but that is not the usual way I prefer to use the term, as it is happening over the course of five metric cycles. Cross-rhythm is a much better term, in this case. This phenomenon, incidentally, will be covered in much greater detail in the section, “Linear Modalities” (page 153).

Furthermore, as will be detailed in the section, “Into the Dot: Playing with Perspective” (page 159), the method discussed herein is meant to engender a malleable perspective. If we speak strictly of primary and secondary, foreground and background, it can be very helpful for the student, but ultimately are we not cheating the music of yet another dimensional referent? As these structures—both harmonic (in one bar) and linear (regroupings, cross-rhythms)—become more clear and stable in the body, it becomes possible to hear one from the other and vice versa; that is, to hear three from two or two from three. There is still a primary beat structure, yes; in music, this would be the four main beats in most African music or the tal in India, but in rhythm, perhaps just the one.

This study and the theories therein, though largely examined using African music as a reference, are also meant to examine and generate more universal perspectives. Still, though it may be a small point to the strictly African music scholar, the thinking and terminology in current use—i.e., primary cycle and cross-rhythm—seems to hierarchize the whole show, making it one-dimensional; that is, there is one main beat and the others are linear relations. The reader will understand, I hope, that this is a subtle point, as I still prefer and agree to start from one main metric reference; but the capacity to hold several in mind is, I believe, an essential aesthetic for the astute participant. I also hope that the reader will not object to the conceptual use of multiple metric references in the course of this analysis…and that it will become clear as to why.

Finally on this subject, I will attempt to deal with the comment of Justin London that there is, “no such thing as polymeter.” (London 2004, 50) In the preparation stages of this research, I was taken aback by this rather blunt statement. Now, with a few allowances added, I can consider it as a possibility. The allowance I refer to in particular is this: I would prefer if London would have included a qualifying term such as, ‘psychologically,’ or ‘consciously,’ before his bold statement. This would imply that the part of our mind we normally think of as waking consciousness (and which, in the course of this study, has appeared to me to be a significantly linear apparatus) is not in fact able to attend to two metres at the same time, but rather jumps quickly from one to the other. Even with a fair bit of experience in this arena, I cannot say that such a statement would be categorically wrong. However, I would argue, that we certainly do manage to play polymetres. And, moreover, music with multiple metric references seems to exhibit a great power to entertain, and even to overwhelm the mind, in rare cases to the point of trance. This is a controversial subject, outside the bounds of this discussion. However, regardless of whether people enjoy, are moved by, and/or potentially overwhelmed by multi-dimensional rhythmic phenomena because they manage to perceive several of these dimensions at once, or because their minds are quickly jumping back and forth between them, the fact remains that we are able to play them. We play them in group settings, but also as individuals. Toro’s methods push that envelope further than any other I know. My view is that if one person can maintain polymetre, some part of his or her consciousness must therefore be able to perceive it. From a linear (regrouping) perspective, it is easy to play three with two and several more polymetres of a slightly more difficult nature. But eventually, it is argued herein, that method fails. Even with difficult examples like seven and five, perhaps the mind does indeed jump rapidly back and forth from one metric reference to another (and/or, hears them as one composite melody), but the body manages to comprehend and execute them. Indeed, the body seems to be able to manage many movement tasks on its own, even some, like driving a car while adjusting the stereo or using a phone that should probably be more consciously attended to! But, if the body can perform two complexly related metres at the same time (not to mention maintaining one, or two, while manipulating the other) is it not implied that the mind is involved at some level? The nervous system, after all, extends throughout the body, and the mind/body division is, I believe, mostly a discredited notion[1]. It is suggested elsewhere that Toro’s multi-dimensional perspective and exercises and even the general process of playing music in a very engaged way, is akin to meditation, which, working well, is contingent on the cultivation of a wide, less acutely focused awareness (Distinct from the ‘normal,’ linear, conscious thought process). I posit that this is one way the mind/body system deals with the consideration of multi-dimensional rhythmic phenomena. In my experience, polymetre certainly does exist, but I invite London or others to explain a statement to the contrary.

The Linear Approach

Conceptually, we saw in the example of the offbeat five figure that humans seem to be quite capable of dealing with simultaneous or near simultaneous consideration of what would be, in a linear conception, quite distantly related time reference systems (metres). Practically, in order to conceive of such an example, one would have to work with a subdivision of twenty (four beats by five beats) and then, in order to hear or perform the two together, would have to group the twenty into fives (four groups) to articulate the metre of four, and into fours (five groups) to articulate the metre of five.

Figure 1. The Linear Derivation of Four with Five.

Audio Example 1. Four and Five Linear Derivation (CD Track 9).

While it is true that at slow tempos this might be possible for a fairly clever musician, at faster tempos, not to mention in the heat of improvisational use, this method quickly becomes cumbersome and distracting—the epitome of explicit knowledge in its potential to distract from the implicit experience. Moreover, this sort of mental juggling seems to require simultaneous processing of multiple concepts—becoming cognitively ‘harmonic’—anyway. And, remember that the example in question used the offbeat of five, meaning we would actually need to count and doubly group not twenty but forty divisions per bar. Finally, we have established, albeit in a generalized sense, that in the world at large many of the people who regularly use such devices are not theoretically trained. This points to the possibility that there is another way to approach the problem.

The Harmonic Approach

In a harmonic conception of metre, on the other hand, the two time reference systems (metres) are only related to one. That is, they take place in the same space of time; they begin and end in that same space of time; they count beat one together, and so on. One is one long beat that lasts the whole metric cycle.

In this conception, these simultaneously conceived metres also relate to the offbeat of one, which is essentially the same as two. As in a wave, each metre ‘checks in’ at the offbeat; its second half is either a repetition of the first, in the case of even metres, or, in the case of odd metres, is a symmetrical mirror image about the offbeat. This is why our offbeat five figure met up with the second beat (or the upbeat of the entire metric cycle, the one.). The five figure could also be thought of or derived as the even numbered beats (beats two, four, six, eight and ten) of a cycle of ten, which, being even, repeats on the two/global upbeat (beat six of ten).

The figure below shows this idea in graphic form and in music notation.

Figure 2. The Harmonic Perspective: One through Nine.

Audio Example 2. The Harmonic Series (CD Track 10).

The Harmonic Series: Pitch or Rhythm

The figure above is, of course, a representation of the harmonic series, using musical notes (here in common, 4/4 or 2/2 time) to represent rhythmic values. The harmonic series is more commonly used as the model for our understanding of the physical relationship between vibrating bodies that produce various, musically related pitches; we hear them as related because they display varying degrees of consonance based on the relationship of their frequencies. When describing musical pitches, these frequencies are expressed as Hertz, or vibrations per second. This same model can serve as the template for rhythmic pedagogy and analysis, but at a different speed; rhythmic frequencies are expressed rather as beats per minute (bpm)[2].

The analogy extends beyond the mere use of frequency as a measure of pitch or rhythmic speed, however. If we double a frequency expressed as rhythm, say 100 bpm, we get its most closely related frequency, one that is twice as fast, double time, that has more ‘energy,’ and so on. Depending on the speed and the instruments involved (snare drum or tuba?) this may present technical difficulties to play or to attend to, but there is no real change in conceptual complexity. Likewise, when we double a frequency at the level of musical pitch, we get the most fundamentally related note, the octave. Here again, we might feel more ‘energy’ (this is more of an emotional concept as the physical energy required to produce a lower note is often greater; think of the low strings on a cello or piano, or the mallet and stroke required for the bass drum versus the tenor or snare drums), and the technique might be more demanding, as for the higher notes of the brass, but, conceptually, harmonically, there is little added. The doubling of the frequency, as demonstrated in the nomenclature, produces essentially the same note, and thus carries the same name.

To produce a different note, rather than the same note in a different octave, the frequency ratios between the first and second notes have to exist in a more complex ratio than 1:1 or 1:2. The next option in the harmonic series is a tripling of the first frequency, or a ratio of 1:3. It is at this point that the discussion can become muddled, but with careful attention, I hope that it will be clear that the next steps produce the fundamental model that we see in both the pitch and rhythmic arenas. The frequency ratio of 1:3 gives us the musical interval of a fifth.[3] That is, if 220 Hz produces the note ‘A,’ 660 Hz will produce the note ‘E.’ As the second frequency (represented by the ‘3’) will be higher than the octave already generated by the ratio of 1:2, however, the actual note will be a twelfth higher than the first; the ratio that generates the fifth—the next most common, most powerful intervallic relationship after the octave, and the first to offer a fundamentally distinct note and rhythmic relationship—is 2:3. It might help to visualize this discussion on a piano keyboard. The fundamental, or the ‘1’ of the 1:2:3 ratio, might be the note ‘C’. The ‘2’ would be the note ‘C’ an octave higher. The ‘3’ would be the note ‘G’ above that.

Octaves Do Not Harmony Make

With the interval of a fifth, it is clear that a harmonic conception of pitch is possible, for now we have two different notes to work with. Likewise in rhythm, a harmonic concept springs into possibility not with a doubling of frequency—the equivalent of adding an octave—but rather with this foundational ratio of 1:2:3. Essentially, this means the foundation of a harmonic perspective of rhythm is the ratio of 2:3; But the one is always a given. The one is the unbroken time that the cycle takes to repeat itself. It is the fundamental frequency which generates the others. In rhythm, this is a conceptual relationship to be learned, although in music it is felt quite naturally without any verbal description; one is the beginning and end of the cycle to which all variations eventually refer. Thus music can be understood as a cultural representation of the physical world. In pitch, this ultimate reference to one is a physical property of vibrating bodies. The basic physics of acoustics tells us that a vibrating body such as a stretched string or the air in the tube of a wind instrument, once set in motion by an energetic force, will vibrate as its longest, fundamental length, and simultaneously at ratios of the harmonic series: integer ratios of that fundamental, as represented in figure 2. In his book, All of Rhythm, Toro gives this visual reference:

When looking at light from the sun or a lightbulb you will see mainly one color and if a prism is located between you and the light a series of colors that are the components of light will be seen. The same thing happens when you hear a sound from the piano or guitar. For example the note C on the piano is composed of a series of waves vibrating at the same time. You hear not only the fundamental pitch but all of the softer tones that give the sound its quality. (Toro 1995, 1)

In rhythm, the equivalent of the first three frequencies which produce the fundamental, the octave and a fifth above that would be a simultaneous conception of whole notes, half notes, and half note triplets, in a bar of common time (though it should now be clear that four beats to the bar is one of many options), as we saw at the top of figure 2.

There are of course many books that go into these relationships in much greater detail than is necessary here. To clarify a few more of these ratios however, suffice it to say that further up the harmonic series, the third harmonic, which vibrates at four times the fundamental frequency[4], produces a note two octaves higher (two times the first octave, which is itself two times the fundamental). In rhythm, we get a stream of notes four times as fast as the fundamental, which would be quarter notes in the previous conception and in figure 2.

The fourth harmonic, which vibrates at five times the fundamental gives the note that is approximately (in tempered tuning) a major third above this second octave. The equivalent rhythm, continuing in our standard notation example, would be quarter note quintuplets. The fifth harmonic would be six times the fundamental, a fifth above that second octave (an octave higher than second harmonic, which we remember is three times the fundamental). This would be in our example quarter note sextuplets (which, oddly or not is the same as quarter note triplets). The potential for confusion is sometimes higher in prosaic descriptions, and sometimes the other way around. As this example seems to follow the former condition, a summary table is in order; I would also strongly recommend visualization and aural training at the keyboard for the pitch examples and visualization and aural training using figure one for the rhythm examples.

Table 1. Harmonic Nomenclature, Pitch and Rhythmic Relationships

I have now taken the perhaps perilous course of giving the reader homework. Nevertheless, as experiential learning is ‘fundamental’ to this study, it is also fundamental to the transmission of the perspective under scrutiny.

Pitch and Rhythm are Analogous

Music scholars may find some difficulty in finding equivalence in pitch and rhythm in this way. Although some ((Jay; Toro 1995) come close to claiming there are in fact connections between these cognitively separate worlds of frequency perception, I am not advocating such a position here. I do, however suggest a strong, functional analogy, in that a world of octaves is a comparatively sparse world, whether in pitch or rhythm. Such a world cannot really be said to have harmony, nor to require simultaneity of perception and awareness.

[1]   “It is not only the separation between mind and brain that is mythical: the separation between mind and body is probably just as fictional. The mind is embodied, in the full sense of the term, not just embrained.” (Damasio 2008, 118)

[2]   Their relationship is expressed by the simple formulae, bpm / 60 = Hz, and Hz x 60 = bpm, but only in octave equivalents would any two values work with our sensory abilities.

[3]   The ‘natural’ tuning we are hypothetically generating differs from other, human adjusted tunings, in particular the even-tempered tuning in use in the Western world since approximately the time of Johan Sebastian Bach (who made the flexibility of this now ubiquitous tuning to operate in multiple, changing key centres famous in his Well-Tempered Clavier). As such, I left out the usual designation of ‘perfect’ before the ‘fifth’ as not to dishonestly represent the micro-tonal variations between the naturally derived tuning being discussed and the even-tempered or other tunings familiar to the reader.

[4]   This system of nomenclature can also cause initial confusion. Nevertheless, it is the system in common usage.

[5]   Suffice it to say that we start counting with the fundamental, but that the harmonics are counted starting at double the fundamental, hence their number is always one less than the total number the ratio has been applied, e.g. the fundamental=1x, first harmonic=2x, etc.

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