In the previous chapter, we dealt with only one element of music — duration. But music has other characteristics which are also easily measured. One such characteristic is that which we describe as being “higher” or “lower”: a difference in PITCH.
Determining the Relative Heights of Pitches
In Western music, we use a metaphor to describe pitch. We say that certain pitches are “higher” or “lower” than others. We incorporate this metaphor into our system of Western music notation, which maps pitches onto higher and lower positions on the page (see Unit 5).
Some musicians even refer to “pitch space.” So, our investigation of pitch will begin with the ability to discriminate between higher and lower pitches, and to recognize notes of identical pitch.
When we hear a siren, we think of it as starting “low,” reaching a “high” point, and then heading “lower” again. This is an example of a continuous change in pitch. Between any two pitches, there is an infinite number of other possible pitches. However, most Western music (like most musics around the world) doesn’t make use of continuous sliding between pitches. Instead, we sing, play, hear, and think of discrete pitches — pitches separated from one another by specific distances in pitch space.
Let’s examine the relative heights of the discrete pitches in the folk tune “Frère Jacques.” Sing the first fourteen notes of the tune (“Frè-re Jac-ques, Frè-re Jac- ques, dor-mez vous, dor-mez vous?”) and think about the height (in pitch) of each note compared to the others. Which notes are the same in pitch? Which are higher or lower? Sing the tune and trace the height of each pitch with your hand in the air.
This relative up-and-down motion of pitches is called MELODIC CONTOUR. We can show melodic contour on the page by drawing horizontal lines for each note:
Take note of where the pitches ascend, descend, and stay the same. Which pitches are the same even though they’re separated by different intervening pitches? How many unique pitches are in this part of “Frère Jacques”?
There are, in fact, five different pitches among these fourteen notes. In order from the lowest to the highest, they are the pitches for the syllables: (1) “Frè–” and “–que”; (2) “–re”; (3) “Jac–” and “dor–”; (4) “–mez”; (5) “vous.” If you label each syllable with its above number you get the following:
Sing the tune for “Frère Jacques,” but sing the above numbers instead of the words.
The Major Scale & Scale Degrees
Now think of the tune that goes with these words:
Try drawing horizontal lines which show the direction of those pitches. Here’s what you should arrive at:
The pitches move steadily downward for each and every word, so that their arrangement is even less complicated than that in “Frère Jacques.” But how do these pitches correspond to those in “Frère Jacques”? In other words, where do the five unique pitches in “Frère Jacques” fit into the eight pitches in “Joy to the World”?
You may experiment with this, stopping on various pitches in one song and trying to start the other song from that point. The result of any such experiment will lead you to see (or, more appropriately, hear) that the last pitch in “Joy to the World” corresponds to the first pitch in “Frère Jacques.” This means that the five unique pitches in the beginning of “Frère Jacques” correspond to the lowest five pitches in “Joy to the World.” (Test this out: sing “Frère Jacques” and stop on the word “vous,” then start in the middle of “Joy to the World” with “world” on that same pitch.)
Let’s now examine the relationship between the highest and lowest pitches in the first line of “Joy to the World.” Sing all the way through this first line, emphasizing the last, lowest pitch (“come”). Then sing the first, highest pitch again (“Joy”). Isolate the two; sing one after the other. It’s not the same in pitch — it’s obviously higher — but it is the same in character; you can even start singing “Frère Jacques” with it, way up there. Although there is a great distance between these two pitches, they are very much alike — so much alike that musicians use the same label to refer to both. So, if we use the label “1” to refer to the lowest pitch in “Joy to the World,” we should also label its highest pitch “1” as well.
So now we may write out number-labels that correspond to the pitches in “Joy to the World”:
Like “Joy to the World,” much Western music makes use of seven unique pitches. If we arrange these seven pitches in ascending and/or descending order, we produce what musicians call a SCALE.
We can number each of the seven unique pitches in a scale. Each of these numbered pitches is called a SCALE DEGREE.
We can refer to a scale degree by number (for example, “scale degree one” or “the first scale degree”). When writing about scale degrees, musicians often abbreviate “scale degree” by placing a caret above the numeral — for example, 1 stands for “scale degree one.”
If you sing or play the notes of the scale, you’ll notice that ending with 7 sounds incomplete. Because 7 sounds so incomplete, musicians usually perform scales by repeating 1 at the top and bottom of the scale. We can write out the notes of a scale as follows:
Sing this scale using numbers. The scale you’re singing is called the MAJOR SCALE. (For the moment, take the scale you’ve been singing as an axiom. For now, we’ll simply label it as a “major” scale to distinguish it from others we’ll encounter later. We’ll define the major scale more rigorously later in this unit, and we’ll compare the major scale with other scale types in later units.)
Names for Scale Degrees
Given its important position as the first and last note in the scale, we will pay special attention to !. This pitch has important properties as a resting tone — the note to which others in a composition seem to want to resolve. This pitch is called the TONIC.
Every scale degree has a name (in addition to a number). Here is a list of those names:
Vibrations and Pitch
We learned in Unit 1 that all sound is vibration. Various things can vibrate and produce a pitch: a string (as on a guitar or violin, for example), a column of air (as in a trumpet or flute, for example), and so on. There is a correspondence between the perceived pitch of a sound and its rate of vibration. The faster something vibrates, the higher we perceive its pitch.
We can use modern instruments (like oscilloscopes and tuners) to investigate the exact frequency of pitches we hear, but for at least 2,500 years, Western musicians have investigated pitch using a simple device known as a monochord. The monochord consists of a string capable of being plucked or bowed to produce pitches. The string is stretched between two fixed points, and one or more movable bridges may be placed under the string to adjust the lengths of its vibrating parts. Here is a diagram of a monochord, in which the red line represents the string, the black triangles represent the fixed end points, and the red triangle represents the movable bridge:
You can find instructions on building your own inexpensive monochord here
A video showing how to make a more elaborate (and expensive) one appears here
When we divide a string in half we also double its frequency of vibration. When we divide a string in thirds, we triple its vibration rate, and so on. Thus, there is an inverse relationship between the length of a string and its rate of vibration.
Dividing a vibrating string (or column of air) into halves, thirds, quarters, and so on, produces a series of higher and higher pitches. This series is called the OVERTONE SERIES. All the pitches in the series are called PARTIALS. The lowest pitch in the series (the string vibrating in its entire length) is called the FUNDAMENTAL. All the higher pitches (the whole-number divisions of the string) are called OVERTONES (sometimes also called harmonics).
A characteristic of most vibrating bodies is that they vibrate not only in their entirety, but also in halves, thirds, quarters, and so on at the same time. This means that most pitches you hear consist not only of the fundamental, but of a combination of overtones as well. Although this combination of overtones strikes our ears along with the fundamental, we tend to perceive the pitch of only the fundamental (in part because it’s so much louder than the others). Most people are blissfully unaware of the pitches of the overtones that strike their ears all day long.
We’ll see the specific pitches of the overtone series when we learn about notating pitches in Unit 5, and in Unit 16, we’ll learn how the combination of overtones produces timbre.
The 2:1 string-length proportion, which corresponds to a 1:2 proportion in frequency (in cps or Hz), spans the exact same distance in pitch space covered by the lowest and highest pitches in “Joy to the World” — in other words, from 1 up to the next highest 1. Because this covers all the possible pitches in between, the ancient Greeks called this distance the diapason (which literally refers to “across all” the pitches). If we refer back to the major scale, we can see (and hear) that this distance spans eight pitches — 1 through 7, plus the next higher 1. Because it spans eight pitches in most Western scales, Western musicians have come to call this distance an OCTAVE (sometimes abbreviated 8va, for the Italian ottava).
We noted earlier that the highest and lowest pitches in “Joy to the World” share a certain identity, to the point that musicians use the same label to refer to both. In the same way, any pitches an octave (or multiple octaves) apart share an identity. We call this identity OCTAVE EQUIVALENCE, which means that pitches an octave apart are somehow equivalent to one another. Because of this, we give similar labels to any pitches separated by one or more octaves.
Measuring Intervals Using Ordinal Numbers
We have developed a basic understanding that pitches are high or low in relation to one another, but we haven’t yet investigated the specific distances in pitch space between them. In comparing the pitches of two notes, the distance between them is called an INTERVAL.
One way to measure the interval between two pitches is to count the number of scale degrees spanned by those pitches. Thus, for example, the interval from 1 up to 3 is called a 3rd (since it spans three scale degrees — 1, 2, and 3). Using this method, musicians use ordinal numbers (“2nd…3rd…4th…”) to name intervals, with several exceptions: (1) the interval from one scale degree to itself (no higher, no lower) is not called a “1st,” but rather a “unison”; (2) the interval that spans eight scale degrees is not called an “8th,” but rather an “octave” (as we noted above); (3) all intervals larger than an octave may be referred to as a combination of an octave and some smaller interval, called a compound interval (for example, “an 8va and a 5th”); (4) musicians almost always use compound-interval names to refer to intervals that span more than twelve or thirteen scale degrees (for example, you might hear of either a “10th” or “an 8va and a 3rd” but you will probably never hear of a “14th” — it would be called “an 8va and a seventh”)
Steps & Skips
The 2nds formed by adjacent scale degrees are also called STEPS. Thus, the interval from 1 to 2 is a step, from 2 to 3 is a step, and so on. Any interval larger than a step is called a SKIP or a LEAP.
Whole Steps and Half Steps & the Structure of the Major Scale
The generic idea “step” is a useful one, but not very precise. Are all steps of equal distance? As yet, we have no absolute means of measuring these distances.
How can we compare the sizes of the various steps in our major scale? One way of measuring the intervals between pitches involves shortening the length of a vibrating string and measuring the physical shortening of that string. You may use a monochord for this purpose, but if you don’t build or have access to one you may use any unfretted string instrument with a fingerboard, such as a violin, viola, cello, or double bass. (The instructions below are for an unfretted string instrument. If you use a monochord, don’t stop the string with your finger, but use a movable bridge instead.)
Working with a single string, pluck it or bow it to produce a pitch, without stopping the string at any spot. Sing that pitch and call it 1. Sing the beginning of a major scale, starting with that pitch as 1. Return to the 1 produced by the unstopped string, and sing 1 and 2. Make the string slightly shorter by stopping it with your finger. Now pluck or bow the string, and adjust your finger to make the string match the sound of your 2. Measure the distance from the end of the fingerboard (where the string stops vibrating) to the spot where you stopped the string to produce 2. Measure it with a ruler. You may find it helpful to place thin strips of tape on the fingerboard to mark these spots. Now repeat the process to find 3. Measure the distance from 2 to 3. Compare the distance from 2 to 3 with the distance you measured from 1 to 2. You’ll notice that they’re about the same.
Now continue this process to find 4. If you compare the distance from 3 to 4, you’ll see that it’s much smaller than the earlier distances we traveled. In fact, it’s about half the distance from 1 to 2 or 2 to 3. Not all steps are equal to the others. We are dealing with two sizes of steps. Musicians call the first a WHOLE STEP, and the other a HALF STEP. (In England and many areas strongly influenced by England’s special musical vocabulary, the whole step is called a “tone” and the half step is called a “semitone.”)
You can also try this with a fretted instrument, such as a guitar. The frets of the guitar show, physically, just how much the string has been shortened.
Moving through the rest of the scale, you’ll find the rest of the intervals to be these: 4 to 5 spans a whole step; 5 to 6 also covers a whole step; 6 to 7 is also a whole step; but 7 up to 1 requires a move of only one half step. Here, then, are the intervals between the notes of the major scale:
- Laid out from left to right…
- Laid out from bottom to top…
This pattern of whole and half steps can serve as a more rigorous definition of the MAJOR SCALE, which we accepted axiomatically earlier in this unit.
You should also try to feel the difference between a whole step and a half step with your voice. Try singing both whole and half steps above and below a single note.
In addition, try to hear the difference between whole and half steps. Have someone sing or play various ascending and descending whole and half steps while you practice identifying them.
This material corresponds to UMass OWL Homework 2