The system of notation that has developed in conjunction with European--and subsequently, American--music over approximately the last 1000 years is unique among the musics of the world. While avoiding claims of superiority of one style or type of music over another, it is nevertheless possible to assert that no other culture in the world has a system of music notation that is as highly developed and versatile as the "Western" system. Some cultures, such as China for example, have long used a system of notation for their music, but it is a cumbersome notation, used primarily for preserving a piece of music rather than reading from it. The Western system, as it has developed into the twenty-first century, is capable of indicating virtually all parameters of music simultaneously--pitch and rhythm (duration), to be sure, but also, in varying degrees of sophistication, dynamics, tempo, and articulation, and phrasing.

Let us first study the notation of the duration of sound, or rhythm. It is interesting to note that it is largely because Western music developed polyphony (music in more than one part simultaneously) to a higher degree than any other culture that a precise system of rhythmic notation was necessary from a rather early date (at least by the late twelfth century). If music is monophonic (i.e., only one voice at a time), rhythmic notation is not so necessary, even if several people perform the monophonic line simultaneously. But polyphonic music that follows rather strict rules of consonance and dissonance--as did medieval polyphony--requires precise coordination of the element of time.

That our system of rhythmic notation is a clearly proportional one is at least partially due to the fact that historically, music was considered a "mathematical art." Figure I-1 shows the standard rhythmic symbols and their corresponding rests, arranged in descending order of value. Each note symbol has a corresponding rest that indicates a silence of exactly the same duration as the note. The basis of the system is the simple proportion of 2:1: in the table, adjacent types of notes and rests are related according to that ratio. Thus, for example, a whole note equals two half notes, a half note equals two quarter notes, and so on. This relationship is illustrated in Figure I-2.

Figure 1-2 gives relative rather than absolute values for notes and rests. The exact value of a note or rest--that is, the number of beats or fractional part of a beat that it receives--can be determined only by reference to a meter signature, discussed below.

Continuing our proportional terminology, a dot following a note increases the value of that note by half. Thus a dotted half note is worth one-and-one-half times as much as a "regular" half note, and also worth three times the value of a quarter note (see Figure 1-3). In the most common time signature, 4/4 (or C), a dotted half note in this time signature receives three beats; this will be explained further below.

A note may be followed by two dots. In this case, the second dot receives one-half the value of the first dot. Again using the most common time signature, 4/4 (or C), a double-dotted half note in this time signature receives 3 ½ beats (see Figure 1-4).

When we deal with meter signatures (also called "time signatures"), we are once again dealing with mathematical proportions. Most of our modern time signatures, in fact, originally were called "proportions," because of their proportional relationship to one of the four original time signatures, called prolations. Modern meter signatures can be grouped into three categories: simple, compound, and asymmetrical.

Simple meters are aptly named, for they are the easiest category to understand. All meter signatures or time signatures consist of two numbers, placed one above the other in the manner of a fraction. Following mathematical terminology merely for the sake of convenience, we may call the top number the "numerator" and the bottom number the "denominator." Almost all simple meters have the number 2, 3, or 4 for the numerator. The number 1 as a numerator is quite rare, but it may also be considered "simple." All simple meters share one characteristic in common: the first level of subdivision below the level of the beat is binary; that is, when a note that receives one beat is subdivided, its most common subdivision is by halves. Proportions again!

The signatures for simple meters are relatively easy to interpret: the numerator indicates the number of beats in each measure, while the denominator indicates the type of note that receives one beat. In 4/4 (C, or "common") time, for example, the denominator of the meter signature is 4. This means that a "one-fourth" note, or quarter note, receives one beat. If the denominator is 2, then a half note receives one beat. Since the denominator must represent a common note value, typically only 2, 4, 8, and 16 are used as denominators--1 and 32 are occasionally used as denominators; 64 is theoretically possible, but in fact quite rare. Figure I-5 offers some illustrations.

Notice that in Figure I-5, the "initial subdivision of the beat" is indicated for each meter given. Notice that in each case, two notes of the next common value below the level of the beat are used. This is an important point to remember as we proceed to examine compound meters.

Simple meters are subdivided into three generic classifications--simple duple, simple triple, and simple quadruple. All simple duple meters have 2 for their denominator, while all simple triple meters have 3, and all simple quadruple meters, 4. Figure I-6 shows all the common (and some not-so-common) simple meters, arranged according to these three categories. The most commonly used meter signs are marked with asterisks (*).

All of the meters within a given category share certain characteristics in common--apart from the fact that they have the same number of beats per measure. In simple duple, for example, the two beats are differentiated by their relative strengths. Typically the first beat of each measure is accented, the second, unaccented. Using the letters "S" for strong (accented) and "w" for weak (unaccented), the accentual structure of a typical measure in 2/4 time--S w S w, etc.--is illustrated in Figure I-6. (Notice that the letters "S" and "w" are assigned only to the beats. The eighth notes that fall in between the beats (subdivisions of the beat) in the second measure of the example are of course still weaker than the "weak" second beat of the measure.) The accentual pattern illustrated here is of course the same in all simple duple meters; only the level of the beat changes.

In simple triple meters, typically the first beat of each measure is accented (strong), while the second and third beats are unaccented (weak). The accentual structure is thus S w w S w w, etc. (See Figure I-7.)

In simple quadruaple meters, typically the first and third beat of each measure is accented (strong), while the second and fourth beats are unaccented (weak). Theretically, though both the first and third beats are strong, the first is even stronger than the third. Thus in Figure I-8, capital S is used for the strongest beat (first beat) in each measure, while lower-case s is used for the subsidiary strong beat, on beat 3.

Compare Figures I-6 and I-8 and you will notice that in terms of sound, simple quadruple meter differs from simple duple meter only in that there are two different levels of "strong" beats (designated S and s, respectively, in Figure I-8). This subtle difference in theory is not always discernible in practice. When listening to an unfamiliar piece of music, it may be difficult even for the trained listener to distinguish between simple duple and simple quadruple.

In Figures I-6, I-7, and I-8, while all the illustrations are given in meters with 4 as the denominator, the accentual structure of a meter within a given classification remains the same, regardless of the denominator. One might reasonably ask, then, why each generic classification contains several different meter signatures (each with a different denominator). Or, put in a different way, why would a composer choose 3/2 rather than ¾? In general, the smaller the denominator, the slower the implied tempo. Trained musicians generally expect that a piece in 3/2 will be slower than a piece in ¾ (though this is not always true), and similarly, a piece in 3/8 will be faster than a piece in ¾. Larger note values tend to suggest a slower tempo.

Compound meter signatures generally have 6, 9, or 12 for the numerator. 4, 8 and 16 are the most common denominators in compound meter; 1, 2, and 32 are possible, but rare. We cannot interpret denominators in compound time in the same way we did for simple meters. In compound meters the denominator does not denote the note that receives one beat, as in simple meters, but rather the note that receives one pulse. For all compound meters, three pulses equal one beat. The pulse, then, is the initial subdivision of the beat.

We noted above that in simple meters, the initial subdivision of the beat is duple; that is, in 4/4 time, for example, the initial subdivision of the quarter note, which receives one beat in this meter, is into two eighth notes. In compound meters, however, the initial subdivision of the beat is triple--i.e., in 6/8 time, there are two beats in a measure, with the dotted quarter receiving one beat. The initial subdivision of this dotted quarter note is into three eighth notes. As we noted above, each eighth note receives one pulse. This is illustrated in Figure I-9, with regard to 6/8 time.

All common compound meters (and some not-so-common) are illustrated in Figure I-10, subdivided, as with simple meters, according to three types--compound duple, compound triple, and compound quadruple. The most common triple meter signatures are marked with an asterisk.

Remember that in each type of compound meter, three pulses equal one beat. Remember also that the note receiving one beat in any compound meter is always a dotted note. Beats are classified as strong and weak in exactly the same way as in simple meters: in any compound duple meter, the pattern of beats is S w; in compound triple, S w w; in compound quadruple, S w s w. Notes that represent the pulse rather than the beat (i.e., q in 6/4) are of course even weaker in accentuation than those note that represent weak beats.

What we noted above concerning the relationship between simple duple and simple quadruple meters is also true of the relationship between compound duple and compound quadruple: the distinction between the two is often more theoretical than real. Listening to an unfamiliar piece, it is frequently difficult to distinguish compound quadruple from compound duple, because the two different levels of "strong" beats (represented by S and s respectively in Figure 1-11) are not well differentiated.

Anyone who has performed music extensively, and has read from notation, will be aware that there are certain conventions for rhythmic notation. Anyone notating a piece of music, whether an original composition or an arrangement, must be aware of these conventions. Performers today frequently read music at sight, or perform it with a relatively small amount of rehearsal. Under such conditions, it is imperative that the music is notated in a way that will be easy to read. Particularly in a sight-reading session, sloppy or non-standard notation can waste a lot of precious time.

Music publishers of course must deal with this question each time they print a piece of music. And while every publishing house has its own standards for rhythmic notation, these standards are not absolutely uniform from one publishing house to the next. Moreover, different types of music sometime necessitate a different style of notation. An example of this is vocal music. While the practice of notation of vocal music has undergone some changes in recent years, it was once common practice for music with text to beam separately every note that carries a syllable. All notes of the value of the eighth note or smaller were beamed together--up to the maximum number of notes customarily beamed together in instrumental notation (see below). This style of notation is illustrated in Figure I-12. But some publishers would beam together all the eighth notes in XXX.

The three main principles of rhythmic notation are clarity of the economy, clarity of the beat, and convention. They are not presented in priority order; in any given situation, one may take precedence over the other, and questions of priority are resolved differently by different music publishers.

The principle of economy holds that the fewer notational symbols, the better. To take an extreme example, notating four beats of rest as shown in m. 1 of Figure I-13a would be ludicrous; similarly, one would notate a single note of four beats duration as in m.2 of the same figure. Figure I-13b presents the standard notation in both instances.

Similary, dots should be used, rather than ties, wherever in possible (in general). Thus Figure I-14a is prefereable to I-14b.

Clarity of the beat is very important, particularly in music that will be sight-read. The most important aspect of sight-reading is knowing exactly where to place notes in relation to the beat. Examine Figure I-15a, which shows an incorrect form of notation, with the beats very difficult to see. Figure I-15b, however, shows a more correct form of notation, In Figure I-15b, the beats are easy to distinguish; but notice that clarity of the beat has taken precedence over economy here: Figure I-15a uses fewer symbols.

Notice that while Figure I-15a uses few symbols that I-15b--i.e., it is more economical, the position of the beats is not at all clear. As we learned above, 9/8 time is a form of compound triple meter. Each beat is divided into three pulses. If we are sight-reading this music, we want to see the beats, which coincide with the first, fourth, and seventh pulses. I-15b shows the same pattern, re-notated so that the beats are clearly visible.

The third principle of rhythmic notation, convention, dictates that some rhythmic patterns are notated in a certain way just by tradition. Rests in particular are subject to "conventions" of notation. For example, the whole rest may be used to indicate a silent measure, regardless of the time signature (with some exceptions). This is true even in those meters for which the whole rest literally represents more beats than than there should be in a single measure. Figure I-16 illustrates this phenomenon.

In Figure I-16a-b, it would be logical to assume that since a dotted half-note fills a complete measure, a dotted rest would as well. But convention dictates that an entire measure of rest in ¾ is indicated by a whole rest, even though, technically speaking, with a denominator of 4, the whole rest technically should receive four beats in a measure. A similar situation holds for I-16c-d: since a half note fills and entire measure in 2/4 time, it would be logical to assume that a silent measure would be indicated by a half rest. But by convention, a whole rest customarily is used instead.

The principle of clarity of the beat should be taken in a relative sense. In Figure I-17, for example, it does not make sense to notate the pattern as in I-17b or I-17c; I-17a is better. In notes of longer values--especially those that are relatively simple--the performer is not overly concerned about where a note ends, particularly if it terminates at the end of a measure or even at the end of a beat. The performer is more concerned about where the note begins.

Notice that the pattern of notation in I-17c shows clearly where every beat of the measure falls. But no experienced musician would prefer this pattern to I-17a. The rhythmic pattern is after all a relatively simple one, and there is nothing at all confusing about I-17a. Similarly, I-17b has the virtue of showing where the second "strong" beat of the four-beat measure falls--on beat 3. But as has already been said, it is a simple pattern, and will confuse no experienced musician. In the case presented here, economy may be said to take precedence over clarity of the beat. But convention plays a role as well, because it is simply "conventional" to notate this simple pattern in this way.

Now consider Figure I-18. The situation here is slightly more complicated.

Notice that I-18b shows the location of every beat of the measure--1, 2, 3, and 4--quite clearly. It is, however, less economical than I-18a. The latter uses eight symbols, while the former uses six. Once again, economy wins out over clarity of the beat, and convention also plays a role. It is "traditional" to notate such a pattern as in I-18a. But another, subsidiary principle is at work here, and it has to do with strong beats versus weak beats. Notice that in the illustration above, while I-18a does not show where beats 2 and 4 fall at all, it clearly shows where the strong beats--1 and 3--fall. So we can make a value judgment here: the principle of clarity of the beat is more concerned with strong beats than with weak beats. It is difficult to make a categorical statement concerning such issues, but the subsidiary principle certainly works well with such a simple, commonly used pattern as the one illustrated here.

Another situation in which economy wins out over clarity of the beat is illustrated in Figure I-19. In I-19ba even the strong third beat of the measure is obscured by the syncopated position of the half note. Yet probably no publisher would notate the pattern as in I-19b, even though this pattern clearly shows the location of all four beats--and particularly the strong third beat of the measure. Simplicity is the key here--the pattern is quite simple to perform.

The beaming of notes of the value of the eighth note and smaller follows certain conventions. For the moment we will ignore the issue of vocal notation--in which notes that carry a single syllable are customarily flagged separately, regardless of their value or position in the measure--and discuss beaming as it applies to instrumental notation. In general, where notes of the value of the eighth note or smaller appear together, and are not in syncopated position, they should be beamed together (see Figure I-20).

Some principles to follow in beaming include:

• No more than four notes should be beamed together (Figure I-21a).
• A beam should not obscure the beat (Figure I-21b-c)
• All notes within a single beat that can be beamed together should be beamed together (Figure I-21d).
• If six eighth notes appear in a measure of ¾ time, the beam may be broken either between beats 2 and 3, or between beats 3 and 4 (Figure I-21e).
• A rest may note intervene between two notes that are beamed together (Figure I-21f).
• Notes of different values may be beamed together, as long as there are no more than four beats under one beam. Such patterns are best used where the beam covers one beat (Figure I-21g).

Conventions concerning notation have evolved over the years, as musical styles change. The common use of syncopation in jazz, for example, has rendered certain rhythmic patterns so common to musicians familiar with the style that clarity of the beat may be sacrificed in favor of economy. Figure I-22 is such an example. In I-22a, even the strong third beat of the measure is obscured by the notation; but the pattern is so common, particularly in jazz styles, that it has come to be acceptable to notate the pattern in this way. Both I-22a and I-22b may be considered "correct," though musicians familiar with jazz styles will find I-22a more comprehensible that those who are not.

The notation of rests follows different conventions from those governing notes in some instances, although music publishers differ considerably in their policies in this regard. Traditionally, dotted rests have been avoided, except when they represent values below the level of the beat. According to George Heussenstamm (The Norton Manual of Music Notation [New York: Norton, 1987], pp. 34-39), modern practice now leans somewhat more to the use of dotted rests, particularly in compound meters. Realizing that there is room for variation in this matter, I offer the following guidelines for notating rests, which represents a relatively "modern" approach.

• Dotted rests should be used only in compound meters (Figure I-23a), though two rest symbols may be used in place of the dotted rest--a more "traditional" pratice (Figure I-23b)
• A dotted rest should be used, rather than two separate symbols, when the value it represents is less than a beat (Figure I-23c).
• Rests should never obscure the strong third beat in quadruple meter (Figure I-23d).
• The whole rest can be used to indicate a whole measure of rest in any meter except 4/2 (Figure I-23e).
• In simple triple meter, use two separate rest symbols to indicate two beats of rest (Figure I-23f).
• In compound meters, use two separate rest symbols to indicate two pulses of rest (Figure I-22g).

In music of the "common-practice period," as we learned earlier in this chapter, we expect to find certain numbers as numerators in meter signs, and not others. For simple meters, the numbers 2, 3, and 4 are common; for compound meters, 6, 9, and 12. Other numerators are rare in music before 1900. But in the twentieth century composers began to use such numerators as 5, 7, 11, and others, fairly frequently. For want of a better term, these meter signs are often known as "asymmetrical meters."

The asymmetrical meter signatures are generally interpreted as with simple meters. While asymmetrical meters are more common in "classical" music of the twentieth century, Paul Desmond’s Take Five, written for the Dave Brubeck Quartet, provides an interesting and relaitvely well-known example of quintuple meter, in this case 5/4 time (see Figure I-12; recordings are available). Notice that in this piece the five beats are uniformly divided 3 + 2, in every measure. Most compositions in quintuple meter--whether in the "classical" or jazz idiom--may be subdivided into regularly recurring patterns of 3 + 2, or 2 + 3.