# Additive meter

Our initial discussion of bars and beat division was based on music in which time is divided into units of equal length, which may themselves be subdivided into smaller units again of equal length, and so on to any level of division. This principle of “divisive meter” is common to most Western music along with many other traditions around the world.

However, some music is based, not on dividing time into equal units, but on adding together units of time, some of which may be longer than others. The resulting sequences of unequal time units may themselves repeat in a regular way, like the “bars” we are already familiar with. But because the bars are formed by addition rather than division, the number of beats in a bar may be a prime number such as 5, 7, or 11. If it is a divisible number, it may nevertheless be made up of unequal units, as in an 8-beat bar consisting of 3+3+2 beats.

In itself, an unusual number of beats per bar does not present a particular problem for global notation: the number of beats and the beat division can simply be specified at the beginning in the usual way. The opening riff of Pink Floyd’s “Money,” for instance, has 7 beats per bar with an implicit beat division by 3. (This should be clear without worrying about the pitch aspect of the notation, but if you require an explanation of that, see Specified pitch.)

Things become a little more complicated if we want to indicate how the beats are grouped into unequal units within each bar. The Pink Floyd riff doesn’t convey a very strong sense of beat grouping, although it’s probably easiest to hear as 4+3. In contrast, the North Indian rhythmic cycle rupak tal also has 7 beats per bar (or cycle), but they are always explicitly grouped as 3+2+2.

When the beat grouping is explicit and consistent in every bar, it can be indicated as part of the meter/tempo marking at the beginning of the score. The beat lines at the beginning of each group can be distinguished by extending them upwards only, while keeping them the same thickness as other beat lines, so that they will still be less prominent than the bar lines. The numbers that indicate the beat grouping, 3+2+2, are placed within square brackets to emphasize that, taken together, they comprise one bar.

The above example doesn’t specify any tempo or beat division, because rupak tal can be played at any tempo and the beat division tends to vary with the tempo. If it were played at 100bpm with beat division by four, the meter/tempo marking would be [3+2+2]/4@100. The “/4” indicates that any of the beats within the brackets may be divided into four pulses. (In this way, additive and divisive principles are often found at different levels within the same music; in fact, the additive principle rarely operates at more than one level in a given piece.)

Different considerations come into play when the tempo is so fast that the beats can’t be divided at all—that is, when what we have been calling “beats” are really pulses (see Beats). Such music can be simply notated with very fast beats and no beat division, and that is perhaps the right approach for North Indian classical music, where there is usually a gradual acceleration of tempo and no clear point at which the beats become pulses. But in a lot of other music with additive meter, particularly dance music from Eastern Europe, it seems clear that the unequal units are groups of pulses rather than of beats, and that what they form are in effect “long beats” and “short beats.”

Most often, the long beats consist of 3 pulses, the short beats of 2. The meter of the Bulgarian kopanitsa dance has 2+2+3+2+2 pulses, but the only way to dance to this is to feel it as short-short-long-short-short beats.

One way in which global notation tries to avoid a cluttered appearance is by not marking individual pulses in the score. With music using unequal groups of pulses, this results in beat lines having unequal amounts of space between them, which captures the difference between long and short beats quite intuitively. For example, a bar with one long beat followed by two short beats will appear as follows.

The remaining challenge is how to express the meter and tempo at the beginning. To indicate a number of beats per minute, some recurring time unit must be taken as “the beat.” When the beats are not all the same length, one solution might be to specify the tempo in pulses per minute and the meter in pulses per bar. That is in effect how staff notation usually handles additive meter. But this solution loses the distinction between beats and pulses that remains experientially important even when the beats vary in length.

Global notation therefore favors a different solution. The basic unit—1 beat—is taken to be the “short” beat. The “long” beat is therefore equal to 1.5 beats. A bar need not contain an exact number of beats: the preceding example for instance has 3.5 beats per bar. Its meter could simply be specified as 3.5/2 (since the underlying pulses are twice as fast as “1 beat”). But if the sequence of short and long beats is consistent in every bar, it can also be specified with the numbers 1 and 1.5: in this case, [1.5+1+1]/2. All of this can then be followed by the tempo in [short] beats per minute.

For a simple example: the Russian composer Aleksandr Borodin notated the second movement of his Symphony No. 3 in a meter of 5/8, but the music is far too fast to feel (or to conduct) 5 beats per bar. In reality, it has 5 pulses per bar, consistently grouped into a short and a long beat. This is apparent from the way each of these beats (except the first of the phrase) has a “tonic accent”: a note that is higher in pitch than the notes before and after it. The contrast between short and long beats is further highlighted by the fact that the pulses within the short beats are distinctly articulated (by tonguing) whereas those within the long beats are played legato and articulated only by changes of pitch.

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The same principle can be applied to any additive meter. Here is an example in the 3.5 beat meter discussed above, with the steps of the Macedonian lesnoto dance indicated with foot symbols. (The sequence of steps occupies three bars. As always, the left-hand edge of the symbol represents the onset timing.)

Note that if the first beat of the meter is a long one, the hurdle that indicates the tempo-defining beat goes over the first short beat.

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Source of audio:

Alexandr Borodin, Symphony No. 3 in A minor, second movement, performed by L’Orchestre de la Suisse Romande conducted by Ernest Ansermet, The Essential Borodin, Decca CD, ASIN: B0000261KO (1998), disk 2 track 11.

### 2 comments on Additive meter

1. Han says:

If I want to divide a bar in two and two thirds beats, with three pulses per beat, I could write down the division as say:

4/3+1/3+1/3

This is a problem, because it could also be interpreted as a bar consisting of two beats. I suggest using brackets like this:

(4/3+1/3+1)/3

I think brackets will be more clear for any composite beats, even if you don’t use any fractures.

1. Originally I didn’t use brackets, but this comment has led me to incorporate them and update the page.
One reason I didn’t originally use brackets was that, while they make the meter/tempo indication look more like a mathematical formula, they don’t actually the mean the same as they would in mathematics. For instance, [3+2]/2 doesn’t mean that the sum of 3+2 is to be divided by 2, but rather that each of the five beats within the bracket can be divided by 2.
Another reason was that the brackets would actually be redundant, not conveying any information that wasn’t already there. That’s because the beat division figure “/2” (or whatever) is already understood to apply to all the digits that come before it, even without brackets.
However, I agree that brackets will make it more obvious that the number of beats in the bar is indicated by more than one digit and that the beat division applies to all of them. Brackets would also enable us to notate a meter in which different beats were divided differently, such as [2/2+1/3]@120 (i.e. three beats to a bar, the first two divided by two and the last by three, all at a tempo of 120bpm).
If I were to notate a meter of two-and-two-thirds beats, with each beat divided into three pulses, I would write 2.67/3. (The .67 is a decimal approximation of two-thirds, but you could write a two-thirds fraction if you want to take the trouble). If I wanted to specify where the long and short beats were, I could write, say, [1+1.67]/3.