Read this article’s companion on Time Signatures.

Meter is the structure of beats. Bars contain beats; but unlike a box of unbuilt Legos the beats are ordered in a hierarchical structure within the bar. Time signatures count the number of Legos, but also tell you something about how they are put together to fill the bar. They’re hardly ever just chucked in willy-nilly.

Metrical structure exists between beats, and within beats. Between beats meter can be duple (in twos) or triple (in threes). It can also be quadruple, but that’s really a sub-species of duple (two duples). This parameter specifies the hierarchical structure of the beats in the bar, and is sometimes called the tempus or the “time”:

  • Duple meter contains two beats in the bar. The first beat is stronger than the second.
  • Triple meter contains three beats in the bar. The first beat is strongest, the other two are weaker. (Very often the third beat is stronger than the second beat, but this need not be the case.)
  • Quadruple meter is the grouping of two duple groups within a duple structure. Thus the first duple group is stronger than the second duple group; and the first element of each duple group is stronger than the second element in the same group. This means that the first and third beats are stronger than the second and fourth, but also that the first beat is stronger than the third (and the fourth weaker than the second).

Meter can be simple or compound. This parameter specifies the number of subdivisions within a single beat, and by extension, the hierarchy of those subdivisions; it is sometimes called the prolatio or “prolation,” from the Latin for “bringing forth,” which makes reference to how the individual beat is brought forth or realized:

  • Simple meters have duple beat subdivisions. They are counted “1 and 2 and 3 and…”
  • Compound meters have triple beat subdivisions. They are counted “1 and a 2 and a 3 and a…”

Note that there is no special term for meters with quadruple beat subdivisions; this is because quadruple is just a special variety of duple. Meters with quadruple subdivisions are also classified as simple meters. They are counted “1 ee and a 2 ee and a 3 ee and a…”.

The numerator of the time signature unambiguously specifies the metrical structure. Behold:

NUMERATORS Duple Triple Quadruple
Simple 2 3 4
Compound 6 9 12

Thus any time signature where the top number is 2 is a “simple duple” meter; any time signature where the top number is 9 is a “compound triple” meter; and so on.

It is useful to note that the values in the second row of the above table are exactly three times that of the first row. The reason for this stems from the design of the notational proportions. I’m referring here to the fact that—in any and all time signatures—whole notes are divided into two half notes, which are divided into two quarter notes, which are divided into two eighth notes, and so on. In other words, the notational system is metrically simple by default, which is to say every note value is a duple subdivision of some other note value. Simple meters can express the subdivision of their beats naturally using note values of the next smaller denomination: quarter note beats can have duple eighth note subdivisions. In compound meters, the numerator is multiplied by 3 to override this simple (duple) default: a numerator of 6 can be divided into two groups of three, such that each of two beats can have a triple subdivision. The multiplying of the numerator of a simple meter by 3 to derive its compound meter equivalent is related in an important way to the denominators of compound meters, discussed below.

The denominator of the time signature does not specify the metrical structure; it specifies at what level the metrical structure signified by the numerator applies. Denominators must be powers of 2 because of the built-in duple-ness of the notational proportions—wholes have two halfs, which have two quarters, etc. Thus denominators can be 1 (for whole notes), 2 (for half notes), 4 (for quarter notes), 8 (for eighth notes), 16 (for sixteenth notes), etc. Independent of the numerator, the denominator specifies at what level the meter given by the numerator is operating.

The numerator alone specifies the meter; theoretically the denominator can be any power of 2. However, certain tendencies arise in the common practice:

DENOMINATORS Duple/Quadruple Triple
Simple 2, 4 (rarely 8&#41 2, 4 (often 8&#41
Compound 4, 8 (rarely 16) 4, 8 (often 16)

The first thing to note is that the table shows the most common denominators of time signatures to be 4 and 8. Roughly speaking, this just means that most music (again, in the common practice) is written in quarter notes and eighth notes. There is no rational reason for this except for convention; but intuitively we can appreciate that music notated at the whole-note or double-whole-note level leaves little room for notating longer durations without the cumbersome use of ties, and music written in very short durations, say thirty-second notes or sixty-fourth notes, becomes very cumbersome to read because the subdivisions of these notes will be 128th and 256th notes. Specifying meter at the half-, quarter-, and eighth-note levels therefore is a nice middle area that leaves room for longer and shorter durations on either side of the beat-level.

The second thing to notice is that simple meters use a denominator of 8 relatively infrequently, and compound meters do not typically use 2 as their denominator. One will also readily notice that the denominators of compound meters (second row) are double that of their simple meter counterparts (first row). This arises for the same reason that the numerators of compound meters are three times that of simple meters, as described above. It is a way of working around the fact that there is no intrinsic way to notate a subdivision of three in the Western notational system, where note values are by definition twice or half as long as other basic note values.

Taking a simple meter, if we multiply its numerator by 3 and its denominator by 2, we derive its compound meter equivalent. Thus 68is the compound meter equivalent of 24, not 34. Time signatures do not reduce like fractions (because they aren’t fractions); to convert from simple to compound meter, one must multiply by the ratio 3:2.

  1. Multiplying the denominator by 2 changes the beat level to the next smaller (quicker) note, effectively halving the length of the beats. For example (and most commonly), quarter-note beats in X4become eighth-note beats in X8time.
  2. To keep the bars the same length, we would have to multiply the numerator by 2 also. This would double the number of beats after we halved the length of each beat (in Step 1). Twice as many half-beats is the same as the original number of whole beats.
  3. Instead, we multiply the numerator by 3, which crams in an extra half-beat into each original whole beat.
  4. This extra half-beat allows us to notate a triple subdivision of the original whole beat. For example, what was originally a quarter-note beat now has an artificial third half in it, which allows us to notate its subdivision as three eighth notes (each of which is half a quarter note).
  5. Of course, there still are only two halves, not three, in any whole. So to make it all add up, we add a dot such that what was originally, for example, a quarter note in simple meter becomes a dotted-quarter note in compound meter.

One bar of 68, which contains 6 eighth notes is still longer than one bar of its “equivalent,” 24, which contains only 4 eighth notes. This is not necessarily a problem: a composer may wish to use this discrepancy for musical effect. For example the Mission Impossible theme song can be notated as a constant alternation of unequal bars of compound and simple duple meters, thus: “DUN dun dun DUN dun dun DUN dun DUN dun.” The first two groups of triple “DUN dun dun”s comprise the 68bar, the second two groups of “DUN dun”s comprise the 24bar. But at other times, a composer may wish to make these “metrical equivalents” really add up to be “temporal equivalents,” such that each bar of 68has exactly the same duration as each bar of 24. How is this possible, given that 68has two extra eighth notes? Simple: by equating the dotted-quarter note of 68with the (undotted-)quarter note of 24, each bar has two equal beats, made equal by arbitrary definition. In 68each beat has a triple subdivision, whereas in 24each beat has a duple subdivision. Thus simple and compound metrical equivalents may be proportional or nonproportional.

In this way we can see that time signatures, first of all, simply describe the number of a certain kind of beat in a bar. But secondarily they very strongly imply—and in the common practice, necessarily imply—a certain metrical structure.

27. April 2012 by Matthew Hall
Categories: Crash Courses | Tags: , | 2 comments

Comments (2)

  1. Pingback: Time Signatures | Chamber Music of Brahms

  2. Matthew..Many Thanks. You are efficient ! I hope to absorb all of this, in the aftermath of an ancient theory class, by next week .