Introduction

Metre is a pervasive presence in most music.  In Western music, particularly that of the common-practice period, metre is often reduced to a simple concept of “beats in the bar” or “time signatures.”  This has a limited practical value—it should not be ignored but it only provides a simplistic template for processing musical time.^[1]

Therefore, before we discuss examples of standard metre designation (aka “time signatures”) in Western classical music, it is essential to consider some of the more fundamental concepts around metre.^[2]

Periodicity and Entrainment

Periodicity is the quality of something recurring at a constant rate—that is, each occurrence is separated by the same time interval. Entrainment (sometimes called attunement) is a process of human cognition. It may be described as the ability to mentally “tune in” to a periodic stimulus.

Imagine a the sound of a tap dripping steadily. Each drip is a stimulus, and it is not too long (maybe about three or four drips) before we can “lock on” to the recurrent pattern.  At this point we have entrained to the sound stimulus—the series of drips.  If someone tightens the tap and the dripping stops, we can still project the periodic pattern into the future.

There are limits to our capacity for entrainment.  If the tap is dripping very slowly, for instance, then beyond a certain time interval our capacity to fix onto the pattern and project its recurrence (predicting accurately when the next drip will occur) is lost.

There is also a limit for us to entrain at faster rates beyond a certain threshold. Instead of a tap, imagine a spinning a bicycle wheel while holding a flexible object, like a plastic ruler, in the spokes.  The ruler will make a clacking sound as it hits each spoke.  Spin the wheel slowly, and you will hear the individual clacks and be able to locate a pattern of recurrence, but at a certain speed, the clacks will start to become a stream rather than a series of individual sounds.

Note also, that we do not entrain to recurrent stimuli that are not periodic. Imagine a dog barking persistently, but somewhat randomly.  We will not form any mental pattern that could enable projection of the next bark with any precision. (This lack of predictability might explain why we often find a barking dog such an intrusive sound.)

Measuring periodic events

For the time interval range of recurrent stimuli that we entrain to, it seems likely that this is connected to everyday things that form recurrent patterns with which we are familiar—regular movements such as walking or running, bodily phenomena such as heart rate and breathing.  We can assign values to these kinds of recurrent phenomena in two different ways:

1. by measuring the frequency of periodic events over a certain time span (e.g. a minute) or,
2. by measuring how much time elapses between each event—this is called the “inter onset interval” (IOI).

Imagine walking at an average pace. For most of us this is somewhere around 100 steps per minute.^[3] The time between each step (IOI) at this pace would be 0.6 s (or 600 ms).

beats and Pulse

Let’s imagine a new periodic pattern of stimuli—this could be anything, but it is most practical to imagine that this is a pattern of sound stimuli.  The example we will use below is striking a bass drum.  Any individual sound will have a certain profile or “envelope,” including its characteristics of attack (how fast and clear the initial onset is) and decay (how the sound sustains and abates over time).  Percussion instruments generally produce individual sounds with strong attack and rapid decay, as shown in Example 1.  This is the wave form for two successive hits on a bass drum with an IOI of 0.5 s. To hear it, use the player below.

For any sound stimulus in our consideration of metre, it is the onsets and the time between successive onsets (IOI) that we are primarily concerned with. Note that:

• Each onset occurs at a time point (without duration).
• We call these time points “beats”
• A periodic series of beats forms a pulse.
• The quality of being regularly spaced by the same time span (or IOI) is called isochrony.

We can represent a pulse visually by using a series of regularly spaced dots—see the line of dots marked T in Example 2— and aurally by a regular series of hits on our drum.  We call each time point a “beat” (as in “beats per minute”—see above).

There are a couple of things to note about the representation of a pulse Example 2:

• It should be understood as continuous, we have we have stopped it arbitrarily after 20 beats, but it extends indefinitely
•  A pulse does not have to be aurally present at all times in a piece of music for us to entrain to it.  Certain conditions do have to met for it to be felt, but a literal series of sounds on every beat is not one of them.  The sounds of the drum and other percussion instruments we will use in the following examples are to be understood as representations of our inner entrainment, not literal musical elements.

On its own, a pulse presents one basic requirement for metre—a single level of periodic events that we can entrain to and project into the indefinite future.  However, on its own, it does not create a metre.

Minimal Metre

If you listen to the pulse in Example 2 above, or take any periodic sound series (e.g. the ticking of a clock) and hear it for long enough, your brain is likely to start grouping the beats into regular, larger cycles—typically in cycles of two, three or four beats—even if there is no differentiation between the regular sounds.  In other words, you might find your self counting, for example, “1, 2, 1, 2” or “1, 2, 3, 1, 2, 3” or “1, 2, 3, 4, 1, 2, 3, 4.” This human tendency is usually called “subjective rhythmisation.” However, given we are talking about metre and not rhythm a better term would be “subjective metricisation.”^[4]

When you perform this spontaneous subjective grouping, you are mentally imposing a metre over the basic pulse.

To make this more explicit, consider Example 3. This represents two separate pulses: the original one (T) and a new one (labelled S) whose frequency is half that of T (60-bpm).  These two pulses combined represent a metre—a set of two pulses {S, T}.

Note that an essential component of this metre is that the slower pulse (S) is included in the faster pulse (T).  That is, for every beat in pulse S, there is a simultaneous beat in T.

^[5]

In the case above, the metre {S, T} is a minimal duple metre.  “Minimal” because there are only two levels, and “duple” because for every beat in S there are two in T.

Note that the beats that are included in both pulses are called “down beats.”

In metrical music, two other minimal metres are commonly able to be perceived: minimal triple and minimal quadruple metres.  These are represented in the now familiar way below.  In Example 4, each beat in the triple metre’s S pulse is included with every third beat in its T pulse while, clearly, in Example 5 each beat in the quadruple metre’s S pulse is included in every fourth beat in T.

Beyond these parameters (duple, triple and quadruple), groupings of the faster pulse with respect to the slower pulse are either non-divisible (e.g. prime numbers such as 5, or 7) or so readily divisible into smaller units that we will impose this division mentally through subjective metricisation (e.g. 6 might divide into 2 or 3, 9 will divide into 3, and so on), creating a a more complex metre.

Deep metre

Subjective metricisation, the act of spontaneously subdividing larger collections of beats (e.g. 9 into 3 groups of  3, and so on), brings us to the next consideration, which is “deep metre.”^[6] Deep metre is inherent in the minimal quadruple metre (as in Example 5, above) to the extent that we will often mentally impose a intermediary division into two of the slower S pulse through subjective metricisation. If this is concretised as in Example 6, then the original S pulse is displaced by a new form of S and moved up the hierarchy to a new level, which is below labelled R (now heard on the triangle in our sonic representation).

There are two minimal duple metres here: {S, T} and {R, S} which combine to form a deep metre {R, S, T}. Note that both R and S are included in T

Let’s take another example: here we introduce a triple metre as one of the minimal metres (Example 7).

In this case {S, T} is a triple metre, {R, S} is a duple metre. We might call {R, S, T} a “deep duple-triple metre.” R and S are still both included in T. 

In Example 8, we reverse the situation, where by {S, T} is duple and {R, S} triple, making {R, S, T} a “deep triple-duple metre.”Again, (R) and (S) are still both included in (T).

To complete all the permutations, let’s finally consider a “deep triple metre” {R, S, T} where both {R, S} and {S, T} are minimal triple metres (Example ).  Again, R and S are still included in T.

Other ways of representing metre

The graphics of pulses as dots used above are quite explicit, but not very efficient ways of representing metre.  We could describe them verbally using a kind of top-down approach. E.g. where both minimal metres were duple, we might say something like “duple-duple” metre, or regarding the other two, “duple-triple,” “triple-duple” or “triple-triple.” However, when you consider that we can have more than three layers of deep metre, this is also a rather inelegant description.

Because the deep meter is an ordered set of two or more adjacent and overlapping minimal metres, we could use a kind of set notation.  If 2 stands for a duple metre and 3 stands for a triple meter, then each of the metres above (Examples 6 to 9), in order of appearance cold be represented as <2 2>, <3 2>, <2 3> and <3 3> respectively.

Another graphic way of representing metre that is more compact and efficient than the stacked lines of dots is the “ski hill graph.”  In this each node represents a pulse, with slower pulses placed higher (up the “ski slope”) faster ones. There are two diagonal paths down from one node to the next, one leading left the other leading right. These connecting paths represent a minimal metre.  “Skiing” down to the left represents a duple metre, to the right a triple metre.  Here is a graphic representation (Example 10) of the deep metres discussed above (Examples 6 to 9).

The Tactus

The of the tactus is an important concept in relation to understanding the perception (and to an extent, the notation) of metre.  While metres can involve a range of interrelated periodicities, as shown above, human perception and cognition prioritises for special attention, pulses within a certain range.  This range includes rates at which we might naturally tap our feet, dance, or count, or at which a conductor might make their main gestures.  Historically, this pulse is known as the tactus.  Within a  deep metrical complex, the tactus might sit somewhere in the middle, between faster subdivisions where the speed works against natural movement or counting, and slower multiples that assume increasingly structural or even abstract meaning.

We can illustrate this with reference to the opening four bars of J. S. Bach’s Prelude in C major, from volume 1 of his Well-Tempered Clavier.  We will do this without reference, for the time being, to the notated score. Below is a performance and a graphic representation of the complex metre of this excerpt.

This particular piece is a famous (perhaps the most famous) example of a figuration prelude.  It takes a short musical “figure,” in this case a series of eight notes of equal duration, and recycles this figure through various harmonic (pitch) changes over the course of the whole composition.  The individual notes of the figure are heard at the fastest pulse level (v). We call this the musical “surface.”  At the tempo adopted in the performance (which is a fairly common tempo for this piece) this surface pulse is heard at 320bpm.  While discernible and even countable, this is not a comfortable or natural rate for cognitive processing as a pulse.

A representation of the “pitch profile” of the eight-note figure is shown at the bottom of the graph, with each note numbered and pitch represented approximately by height. This confirms what we hear: note 1 is the lowest, note 5 the highest (echoed again at note 8).  As the figure repeats, the position of these notes — 1 low and 5 high — create a second pulse, four times slower than the fastest pulse (labelled T).The alternation of the low and high pitch of the figure at regular intervals reinforces this pulse, which runs at 80bpm.  This is a frequency which falls into a much more natural (for humans) counting/moving/gesturing range.  This pulse is the most likely position of what we would identify as the tactus, the counting level at which we might think “1, 2, 3, 4.”^[7]

A minimal quadruple metre {T, v} is identified with the slower frequency at the tactus level, and the faster frequency at the surface level.  Note that subjective metricisation leads us to infer a mediating subdivision of the tactus (labelled u), thus creating a deep metre {T, u, v}. Because this particular prelude does not mark this pulse level for attention in the same way that it does T (with low and high notes) and v (with every note), it is not as strong. For this reason, we represent its dots in grey.  None the less, it is present in our cognitive metrical processing of this musical surface.

Altogether, we can think of these faster pulse levels below the tactus as subdivisions of the tactus (and, to reinforce this idea, we have used lowercase letters to represent them.)

If we turn our attention to the pulses above the tactus, we now have multiples rather than subdivisions.  These levels also have qualities that mark them for attention.  (S) is marked by return to the low note of each eight-note figure and, slightly more abstractly, marks an immediate repetition of the figure. (R) is marked by a change of pitch in certain notes of the figure.

Interestingly, our way of attending to pulse layers above and below the tactus pulse is different.  The faster levels (subdivision) fit into the “groove” of the tactus somewhat automatically.  The slower layers might entrain in a more “abstract way” and more slowly.  For instance, we need to hear the figure’s repetition to grasp the overall meaning of S and we need to appreciate the even slower unfolding of pitch change to grasp its meaning.  The subdivisions are more visceral, the slower pulse levels are more “structural,” or “architectural.”

Below we represent the metre of this prelude in the ski-hill graph format.  On the left, showing the pulses as labelled in the dot matrix above, and on the right using the notational values which come from the score, which is shown below.

Elasticity

The examples we have considered so fart have shown a very uniform and regular motion through time.  Human listeners and performers are not machines, however.  In most music, there is room for give and take, or push and pull in the time dimension.  Our attention and capacity for entrainment can accomodate a degree of flexibility in tempo within certain thresholds and still recognise the phenomena of pulse and metre.