5.1 Introduction

Up until now in this book we have concerned ourselves primarily with the duration and patterning of musical events (rhythm) and the way they are organized in time (meter). But there are many other ways to describe musical sounds—many other factors, in other words, that contribute to their acoustic identity and the way they relate to one another. We may, for example, speak of the loudness or softness of sound—what musicians refer to as dynamics. We might also speak of the various properties that allow a listener to distinguish between two notes played similarly on, say, a violin and a flute—one of a group of characteristics that fall under the category of musical timbre. In tonal Western art music, however, the most important of these factors is arguably pitch: the “highness” or “lowness” of musical tones.

We will begin this chapter by defining pitch in terms of musical acoustics. We will then outline a widely used system for naming and classifying pitches according to the way they sound. The chapter will conclude with a discussion of the different ways pitches may be written in staff notation as well as a more accurate method of identifying them by name.

5.2 Frequency

As mentioned above, the term pitch refers to the “highness” or “lowness” of a particular tone. The shrill whistling of a tea kettle is an example of a high pitch. The deep, resonating horn of an enormous freighter ship is an example of a low pitch. It is important to remember, though, that highness and lowness are relative—this is particularly important when describing musical tones, where subtle changes in pitch can have dramatic effects on a listener’s experience. A high-pitched musical tone, in other words, may be described as low when compared to another, even higher pitched tone.

The terms “highness” and “lowness” are quite common in discussions of pitch. The image they suggest—of pitches placed along a vertical axis in physical space—is, however, just an analogy. When we talk about the high speed of a train, we are not referring to the elevation of the tracks and in music there is nothing inherent to a high pitch that places it physically above any other. Nonetheless, the vertical imagery is helpful, particularly when it comes to the way pitches are written in staff notation, as we will see momentarily.

Although a detailed discussion of musical acoustics is beyond the scope of this book, we may define pitch more accurately by considering the physical phenomena that produce sound. When an object vibrates, it sets the air around it into motion. The air molecules are compressed and decompressed in correspondence with the motion of the vibrating object. These tiny waves of pressure then emanate outward, away from their source. Human ears are capable of perceiving these vibrations in the air as sounds. If the pulses of compression happen regularly, they will be perceived as having pitch.

Pitch corresponds with the frequency of these vibrations: objects producing high pitches vibrate very quickly, objects producing low pitches less so. Pitch is measured in hertz (Hz), a unit indicating the number of vibrations happening over a time span of one second. Example 5–1 presents a 440 Hz tone, a pitch produced by vibrations happening 440 times every second:

Changing the frequency of the vibrations changes the pitch. When the vibrations happen more frequently, we perceive a higher pitch. Example 5–2 presents a 493.88 Hz tone. It sounds slightly “higher”—more urgent or energetic—than the tone in Example 5–1.

There are an infinite number of pitches. It follows, then, that there are also an infinite number of pitches between any two pitches. Some pitches are so close in frequency that it is impossible to discern the difference between them. Furthermore, some pitches are either so high or so low in frequency that they are imperceptible to the human ear. Generally speaking, humans are capable of hearing pitches in the 20 Hz to 20,000 Hz range. In tonal Western art music, though, the pitches one encounters tend to be much more limited in range and in number.

5.3 Pitch names

A piano keyboard is a useful tool when it comes to conceptualizing and visualizing pitch in tonal Western art music. Each key on the keyboard, black and white, corresponds with a different pitch. The lowest pitch is located on the far left end of the keyboard and each key to the right plays a successively higher pitch. The 440 Hz pitch heard in Example 5–1 corresponds with one of the white keys around the middle of the keyboard, as shown by the arrow in Example 5–3:

Pitches are named using the letters. The indicated key/pitch in Example 5–3 is called “A.”

Moving from left to right along the keyboard, each of the white keys corresponds with a new letter from A up to G, after which the names cycle back to A. This pattern of letters continues in both directions indefinitely (including pitches above and below the range of the piano):

All of the white keys on a piano keyboard have a letter name, A through G. In addition to showing the letter names for all of the white keys on the keyboard, Example 5–4 identifies one key/pitch in particular. The C that lies five white keys to the left of A440 is known as “middle C,” so named because of its location in the middle of the keyboard. Middle C plays an important role in Western music theory and it is important that you be able to identify it on a piano. (On a standard 88-key piano, it is the fourth C from the left.)

Notice that the white keys and black keys on the keyboard follow a very specific pattern. Black keys are placed between some of the white keys in alternating groups of two and three. This pattern allows the player to quickly identify a particular pitch. (Imagine a keyboard with 88 keys that looked exactly the same!) Notice, too, that pitches/keys with the same name always appear in the same place relative to the black/white key pattern. It is important that you be able to identify pitches on the keyboard based solely on their position within this pattern.

Activity 5-2

Activity 5–2

Identify the note letter name associated with each of the indicated piano keys.

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Exercise 5–2a:

Question

What note letter name is associated with the dotted key on the keyboard below?

Answer

C

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Exercise 5–2b:

Question

What note letter name is associated with the dotted key on the keyboard below?

Answer

C

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Exercise 5–2c:

Question

What note letter name is associated with the dotted key on the keyboard below?

Answer

F

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Exercise 5–2d:

Question

What note letter name is associated with the dotted key on the keyboard below?

Answer

B

An interval is the perceived distance between two pitches. The size of an interval depends on the frequencies of the pitches in question. We will discuss intervals at greater length in Chapter 11, but there are two intervals in particular that will be useful here: octaves and semitones.

An octave is the interval between a pitch and the next pitch above or below it sharing the same name. The two As in the following example are an octave apart. (The audio file accompanying Example 5–5 plays the two As in succession, first the lower A, then the A an octave above.)

Pitches an octave apart share more than just a name: they also sound very similar. This is due in part to the ratio found between their respective frequencies. The lower of the two As in Example 5–5 is a pitch with a frequency of 440 Hz. The higher of the two As is 880 Hz, exactly double the frequency of the lower A. The next A above 880 Hz would have double the frequency again (1760 Hz) and so on. Try playing all of the As on a keyboard and notice how similar they sound. Try playing some other notes and notice the difference. The term pitch class refers to the complete group of all pitches related by one or more octaves. Pitch class A, for example, is a set of pitches that includes all of the As.

A semitone is the interval between adjacent keys on the keyboard. The pattern of black and white keys on the keyboard repeats every octave. If one were to play each successively higher key above the low A in Example 5–5, they would hear twelve pitches before arriving at the A an octave above it. An octave, in other words, is equal to twelve semitones.

Semitones appear on the keyboard between every B and the C above it, as well as every E and the F above it. But semitones also appear between adjacent pairs of white and black keys:

To hear the sound of a semitone, listen again to the pitches heard in Example 5–1 and Example 5–2. These two pitches are a semitone apart.

Activity 5-3

Activity 5–3

Identify each of the following intervals as either a semitone or an octave.

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Exercise 5–3a:

Question

What type of interval appears between the two keys with dots?

Hint

Remember, a semitone is the interval between two adjacent keys and an octave is the interval between one key and the next one above or below it sharing the same name.

Answer

semitone

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Exercise 5–3b:

Question

What type of interval appears between the two keys with dots?

Hint

Remember, a semitone is the interval between two adjacent keys and an octave is the interval between one key and the next one above or below it sharing the same name.

Answer

octave

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Exercise 5–3c:

Question

What type of interval appears between the two keys with dots?

Hint

Remember, a semitone is the interval between two adjacent keys and an octave is the interval between one key and the next one above or below it sharing the same name.

Answer

octave

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Exercise 5–3d:

Question

What type of interval appears between the two keys with dots?

Hint

Remember, a semitone is the interval between two adjacent keys and an octave is the interval between one key and the next one above or below it sharing the same name.

Answer

semitone

The black keys on the keyboard—and their corresponding pitches—are named according to one of the adjacent white keys. Symbols called accidentals are used to indicate this kind of adjacency. A sharp symbol (#) indicates that the pitch in question is a semitone higher than a reference pitch. A flat symbol (b) indicates that the pitch in question is a semitone lower than a reference pitch. The following example shows the location of a Bb and an F# on the piano keyboard:

A natural symbol (§) is used to clarify when a pitch is unaffected by sharps or flats. A C, in other words, may also be referred to as C§. When distinguishing C and C#, for example, one might use the label C§ to help avoid confusion. (Note that black keys can never be labeled with a natural sign.)

One potentially confusing aspect of this system is that it allows for pitches to be named in more than one way. The black key between A and B, for example, may be referred to as A# or Bb. When two names refer to the same pitch, those names are said to be enharmonically equivalent. They may sound the same, but they are spelled differently. The following example shows the common names given to all of the pitches corresponding with keys on the keyboard:

Complicating matters further, sharps and flats are not exclusive to black keys. A C, for example, is enharmonically equivalent to a B# and an E is enharmonically equivalent to Fb. Enharmonically equivalent names might seem redundant and therefore superfluous—particularly when we think of them abstractly, as we are doing here. In a musical context, however, these names become much more meaningfully valuable. The way a pitch is spelled—that is, the way it is labeled with a pitch-letter name—indicates how it sounds and functions in relation to the notes around it. D# and an Eb might refer to the same pitch, but in a musical context these labels might mean something very different.

5.4 Staff notation

Pitch can also be represented using staff notation. A staff is a set of horizontal lines upon which notes are written. (The plural of “staff” is “staves.”) The number of lines in a staff is variable, but the standard is five. The distance between adjacent lines is exactly the same as the height of a notehead. Noteheads may be placed directly on one of the lines or in one of the spaces, completely filling the gap between two lines:

The location of a notehead on a staff corresponds with the pitch of the note: higher notes appear toward the top and above the staff, lower notes at the bottom and below. Each successive line and space corresponds with a pitch letter name. Starting on the lower of the two notes Example 5–9, count the number of lines and spaces it takes to get to the higher note. You will find that the higher note is eight steps above the lower. The pitches represented by these two notes, then, would be an octave apart.

The staff may be extended up and down with one or more small lines called ledger lines. (The spacing of the ledger lines continues the spacing of the staff lines.)

A single staff, however, is incapable of showing all 88 of the pitches that a piano can play—at least, not without an unmanageable number of ledger lines! Instead, a staff is designed to be versatile. It shows pitch in a relative way. We can see, for example, that the two pitches in Example 5–9 are an octave apart, but we do not know what pitches they are. In order to notate specific pitches, we need an extra symbol.

A clef is a large symbol that appears at the very beginning of a staff. The purpose of a clef is to assign one of the lines or spaces on a staff to a specific pitch. One of most commonly encountered clefs is the treble clef. The treble clef symbol includes a little swirl around the second line from the bottom of the staff and specifies that this line corresponds with the G just above middle C. (The symbol is derived from a stylized cursive “G” and is sometimes known as a “G clef.”)

With one of the lines assigned to a specific pitch, all of the remaining lines and spaces become similarly linked. The following example shows four notes on a staff with a treble clef and the location of the corresponding keys on a piano keyboard. Each note, in other words, now indicates a specific pitch:

The bass clef occurs almost as frequently as the treble clef. The bass clef symbol includes two dots centered around the second line from the top of the staff and specifies that this line corresponds with the F just below middle C. (The symbol is derived from a stylized cursive “F” and is sometimes known as an “F clef.”)

Again, with one of the lines assigned to a specific pitch, all of the remaining lines and spaces become similarly linked. The following example shows the same four noteheads as Example 5–13, now with a bass clef at the beginning of the staff:

Notice that with a bass clef instead of a treble clef, these same four noteheads refer to very different pitches.

Treble and bass clefs often appear together on a pair of staves called a grand staff:

Notice that with ledger lines there are multiple ways of writing the same pitch on a grand staff. Both of the noteheads in Example 5–15 refer to middle C. Next to a treble clef, middle C is on the first ledger line below the staff. Next to a bass clef, middle C is on the first ledger line above the staff.

A third type of clef—the C clef, which is derived from a stylized “C”—specifies the location of middle C as one of the lines on a staff. Unlike the G- and F- clefs, which generally appear in the same position on a staff, C clefs tend to appear in different places. When the C clef is centered on the middle line of the staff it is referred to as an “alto clef”; when it is centered on the second line from the top of the staff it is referred to as a “tenor clef”

Accidentals may be written on the staff as well. In the following example, a sharp symbol turns an F into F# and a flat symbol turns a D into Db:

Notice how the accidentals are lined up with the noteheads. The small rectangle in the middle of the sharp sign and the little loop at the bottom of the flat sign are lined up either directly on a line or between two staff lines, just like noteheads. Notice, too, that accidentals in staff notation are placed to the left of the noteheads they affect—unlike with pitch letter names, where accidentals are placed after the letter (e.g., “A#“).

By convention, accidentals apply to all other noteheads appearing on that line or space for the remainder of the measure in which it appears unless otherwise indicated. Consider the accidentals in the fourth measure of the violin part in the following example (the note-letter names are written above the staff):

There are three noteheads in the lowest staff space in m. 4 of the violin part. The note on the downbeat is an F#, as indicated by the accidental. The third note in the measure is also an F# because the accidental applies to all following notes in the same space for the remainder of the measure. This effect may be canceled out, however, by another accidental. The last note of the measure, then, is once again F§, as specified by the accidental.

In some cases, a composer or publisher will include redundant accidentals as helpful reminders to performers. Consider the five notes in the vocal part, mm. 17–18 in the following example:

In m. 17, all four notes in the voice part are to be sung as Ab. The natural symbol in m. 18 is not entirely necessary, since the flat symbol in m. 17 applies only until the end of the bar. It is included here as a helpful reminder to the singer that the note should not be Ab like in the previous measure. Accidentals such as these are called “courtesy accidentals” and are sometimes—though not always—written in parentheses as in Example 5–22.

There are two other accidentals besides sharps, flats, and naturals that you are likely to encounter. A double sharp symbol (x) indicates that the pitch in question is two semitones higher than a reference pitch. And finally, a double flat symbol (bb) indicates that the pitch in question is two semitones lower than a reference pitch. Sharp, flat, and natural symbols occur quite frequently, double sharps and double flats less so.

5.5 Scientific pitch notation

Before concluding this chapter, there is one more system for naming pitches that is worth describing. In the discussion above, we used unique names for two specific pitches: middle C and A440. By itself, though, the name “C” specifies a pitch class. It does not specify any one C in particular. “C” might refer to middle C or it might refer to any other “C.” As you can imagine, it would be helpful to have a system that would allow us to quickly name or identify any specific pitch.

Scientific pitch notation does just that by combining pitch letter names with numbers. In this system, middle C is labeled “C4.” The C an octave below middle C is labeled “C3,” the C an octave above middle C is labeled “C5,” and so on:

All other pitches are numbered according to the number of the next lower C. The E on the bottom line of the treble clef, for example, would be E4 since it lies above C4 (and below C5). The G at the top of the bass staff would be G3 for the same reason. This system allows quick and accurate naming of all available pitches.

5.6 Summary

The term pitch refers to the frequency of sound or the perceived highness or lowness of a musical tone. An interval is the perceived distance between two pitches. An octave is the interval between one pitch and another pitch that doubles the frequency of the first. In tonal Western art music, the octave is broken up into twelve smaller intervals called semitones.

The pitches used in tonal Western art music are easily visualized using a piano keyboard. White keys and their corresponding frequencies are labeled with letters, A through G. Black keys and their corresponding frequencies are labeled according to their proximity to white keys using accidentals: sharp symbols (#) or flat symbols (b). Natural symbols (§) are sometimes used for clarity when referring to pitches that are neither sharp nor flat. The pattern of white and black keys—and, therefore, the pattern of pitch-letter names—repeats every octave.

A staff is a set of five lines upon which notes are written. A staff by itself can show the relative distance between two pitches, but cannot indicate specific pitches. A clef—a symbol appearing at the beginning of a staff—links one line of the staff to a specific pitch, thereby linking all of the other lines and spaces to specific pitches. Treble and bass clefs are very common while C clefs (alto and tenor) are slightly less common. A grand staff is a pair of staves, the top with a treble clef and the bottom with a bass clef.

There are multiple, redundant ways to write pitches, both in letter names and in staff notation. Different spellings that refer to the same sound are said to be enharmonically equivalent. Double sharp symbols (x) and double flat symbols (bb) increase the number of possibilities for enharmonic equivalence. Scientific pitch notation extends the pitch-letter naming system with numbers to indicate specific pitches.