The Distance Between Pitches
The interval between two notes is the distance between the two pitches – in other words, how much higher or lower one note is than the other. This concept is so important that it is almost impossible to talk about scales, chords, harmonic progression, cadence, or dissonance without referring to intervals. So if you want to learn music theory, it would be a good idea to spend some time getting comfortable with the concepts below and practicing identifying intervals.
Scientists usually describe the distance between two pitches in terms of the difference between their frequencies. Musicians find it more useful to talk about interval. Intervals can be described using half steps and whole steps. For example, you can say “B natural is a half step below C natural”, or “E flat is a step and a half above C natural”. But when we talk about larger intervals in the major/minor system, there is a more convenient and descriptive way to name them.
The first step in naming the interval is to find the distance between the notes as they are written on the staff. Count every line and every space in between the notes, as well as the lines or spaces that the notes are on. This gives you the number for the interval.
Figure 4.33. Counting Intervals
To find the interval, count the lines or spaces that the two notes are on as well as all the lines or spaces in between. The interval between B and D is a third. The interval between A and F is a sixth. Note that, at this stage, key signature, clef, and accidentals do not matter at all.
The simple intervals are one octave or smaller.
Figure 4.34. Simple Intervals
If you like you can listen to each interval as written in Figure 4.34: prime, second, third, fourth, fifth, sixth, seventh, octave.
Compound intervals are larger than an octave.
Figure 4.35. Compound Intervals
Listen to the compound intervals in Figure 4.35: ninth, tenth, eleventh.
Exercise 4.5.1. (Go to Solution)
Name the intervals.
Exercise 4.5.2. (Go to Solution)
Write a note that will give the named interval.
So far, the actual distance, in half-steps, between the two notes has not mattered. But a third made up of three half-steps sounds different from a third made up of four half-steps. And a fifth made up of seven half-steps sounds very different from one of only six half-steps. So in the second step of identifying an interval, clef, key signature, and accidentals become important.
A to C natural and A to C sharp are both thirds, but A to C sharp is a larger interval, with a different sound. The difference between the intervals A to E natural and A to E flat is even more noticeable.
Listen to the differences in the thirds and the fifths in Figure 4.38.
So the second step to naming an interval is to classify it based on the number of half steps in the interval. Familiarity with the chromatic scale is necessary to do this accurately.
Primes, octaves, fourths, and fifths can be perfect intervals.Note
These intervals are never classified as major or minor, although they can be augmented or diminished (see below).
What makes these particular intervals perfect? The physics of sound waves (acoustics) shows us that the notes of a perfect interval are very closely related to each other. (For more information on this, see Frequency, Wavelength, and Pitch and Harmonic Series.) Because they are so closely related, they sound particularly good together, a fact that has been noticed since at least the times of classical Greece, and probably even longer. (Both the octave and the perfect fifth have prominent positions in most of the world’s musical traditions.) Because they sound so closely related to each other, they have been given the name “perfect” intervals.Note
Actually, modern equal temperament tuning does not give the harmonic-series-based pure perfect fourths and fifths. For the music-theory purpose of identifying intervals, this does not matter. To learn more about how tuning affects intervals as they are actually played, see Tuning Systems.
A perfect prime is also called a unison. It is two notes that are the same pitch. A perfect octave is the “same” note an octave – 12 half-steps – higher or lower. A perfect 5th is 7 half-steps. A perfect fourth is 5 half-steps.
Figure 4.39. Perfect Intervals
Listen to the octave, perfect fourth, and perfect fifth.
Major and Minor Intervals
Seconds, thirds, sixths, and sevenths can be major intervals or minor intervals. The minor interval is always a half-step smaller than the major interval.
Major and Minor Intervals
- 1 half-step = minor second (m2)
- 2 half-steps = major second (M2)
- 3 half-steps = minor third (m3)
- 4 half-steps = major third (M3)
- 8 half-steps = minor sixth (m6)
- 9 half-steps = major sixth (M6)
- 10 half-steps = minor seventh (m7)
- 11 half-steps = major seventh (M7)
Figure 4.40. Major and Minor Intervals
Listen to the minor second, major second, minor third, major third, minor sixth, major sixth, minor seventh, and major seventh.
Exercise 4.5.3. (Go to Solution)
Give the complete name for each interval.
Exercise 4.5.4. (Go to Solution)
Fill in the second note of the interval given.
Augmented and Diminished Intervals
If an interval is a half-step larger than a perfect or a major interval, it is called augmented. An interval that is a half-step smaller than a perfect or a minor interval is called diminished. A double sharp or double flat is sometimes needed to write an augmented or diminished interval correctly. Always remember, though, that it is the actual distance in half steps between the notes that determines the type of interval, not whether the notes are written as natural, sharp, or double-sharp.
Figure 4.43. Some Diminished and Augmented Intervals
Listen to the augmented prime, diminished second, augmented third, diminished sixth, augmented seventh, diminished octave, augmented fourth, and diminished fifth. Are you surprised that the augmented fourth and diminished fifth sound the same?
Exercise 4.5.5. (Go to Solution)
Write a note that will give the named interval.
As mentioned above, the diminished fifth and augmented fourth sound the same. Both are six half-steps, or three whole tones, so another term for this interval is a tritone. In Western Music, this unique interval, which cannot be spelled as a major, minor, or perfect interval, is considered unusually dissonant and unstable (tending to want to resolve to another interval).
You have probably noticed by now that the tritone is not the only interval that can be “spelled” in more than one way. In fact, because of enharmonic spellings, the interval for any two pitches can be written in various ways. A major third could be written as a diminished fourth, for example, or a minor second as an augmented prime. Always classify the interval as it is written; the composer had a reason for writing it that way. That reason sometimes has to do with subtle differences in the way different written notes will be interpreted by performers, but it is mostly a matter of placing the notes correctly in the context of the key, the chord, and the evolving harmony. (Please see Beginning Harmonic Analysis for more on that subject.)
Figure 4.45. Enharmonic Intervals
Any interval can be written in a variety of ways using enharmonic spelling. Always classify the interval as it is written.
To invert any interval, simply imagine that one of the notes has moved one octave, so that the higher note has become the lower and vice-versa. Because inverting an interval only involves moving one note by an octave (it is still essentially the “same” note in the tonal system), intervals that are inversions of each other have a very close relationship in the tonal system.
Figure 4.46. Inverting Intervals
To find the inversion of an interval
- To name the new interval, subtract the name of the old interval from 9.
- The inversion of a perfect interval is still perfect.
- The inversion of a major interval is minor, and of a minor interval is major.
- The inversion of an augmented interval is diminished and of a diminished interval is augmented.
Exercise 4.5.6. (Go to Solution)
What are the inversions of the following intervals?
- Augmented third
- Perfect fifth
- Diminished fifth
- Major seventh
- Minor sixth
Here is a quick summary of the above information, for reference.
|Number of half steps||Common Spelling||Example, from C||Alternate Spelling||Example, from C||Inversion|
|0||Perfect Unison (P1)||C||Diminished Second||D double flat||Octave (P8)|
|1||Minor Second (m2)||D flat||Augmented Unison||C sharp||Major Seventh (M7)|
|2||Major Second (M2)||D||Diminished Third||E double flat||Minor Seventh (m7)|
|3||Minor Third (m3)||E flat||Augmented Second||D sharp||Major Sixth (M6)|
|4||Major Third (M3)||E||Diminished Fourth||F flat||Minor Sixth (m6)|
|5||Perfect Fourth (P4)||F||Augmented Third||E sharp||Perfect Fifth (P5)|
|6||Tritone (TT)||F sharp or G flat||Augmented Fourth or Diminished Fifth||F sharp or G flat||Tritone (TT)|
|7||Perfect Fifth (P5)||G||Diminished Sixth||A double flat||Perfect Fourth (P4)|
|8||Minor Sixth (m6)||A flat||Augmented Fifth||G sharp||Major Third (M3)|
|9||Major Sixth (M6)||A||Diminished Seventh||B double flat||Minor Third (m3)|
|10||Minor Seventh (m7)||B flat||Augmented Sixth||A sharp||Major Second (M2)|
|11||Major Seventh (M7)||B||Diminished Octave||C’ flat||Minor Second (m2)|
|12||Perfect Octave (P8)||C’||Augmented Seventh||B sharp||Perfect Unison (P1)|
Summary Notes: Perfect Intervals
- A perfect prime is often called a unison. It is two notes of the same pitch.
- A perfect octave is often simply called an octave. It is the next “note with the same name”.
- Perfect intervals – unison, fourth, fifth, and octave – are never called major or minor
Summary Notes: Augmented and Diminished Intervals
- An augmented interval is one half step larger than the perfect or major interval.
- A diminished interval is one half step smaller than the perfect or minor interval.
Summary Notes: Inversions of Intervals
- To find the inversion’s number name, subtract the interval number name from 9.
- Inversions of perfect intervals are perfect.
- Inversions of major intervals are minor, and inversions of minor intervals are major.
- Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.
Solutions to Exercises
Solution to Exercise 4.5.1. (Return to Exercise)
Solution to Exercise 4.5.2. (Return to Exercise)
Solution to Exercise 4.5.3. (Return to Exercise)
Solution to Exercise 4.5.4. (Return to Exercise)
Solution to Exercise 4.5.5. (Return to Exercise)
Solution to Exercise 4.5.6. (Return to Exercise)
- Diminished sixth
- Perfect fourth
- Augmented fourth
- Minor second
- Major third
4.6. Harmonic Series II: Harmonics, Intervals, and Instruments*
Frequency and Interval
The names of the various intervals, and the way they are written on the staff, are mostly the result of a long history of evolving musical notation and theory. But the actual intervals – the way the notes sound – are not arbitrary accidents of history. Like octaves, the other intervals are also produced by the harmonic series. Recall that the frequencies of any two pitches that are one octave apart have a 2:1 ratio. (See Harmonic Series I to review this.) Every other interval that musicians talk about can also be described as having a particular frequency ratio. To find those ratios, look at a harmonic series written in common notation.
Figure 4.53. A Harmonic Series Written as Notes
Look at the third harmonic in Figure 4.53. Its frequency is three times the frequency of the first harmonic (ratio 3:1). Remember, the frequency of the second harmonic is two times that of the first harmonic (ratio 2:1). In other words, there are two waves of the higher C for every one wave of the lower C, and three waves of the third-harmonic G for every one wave of the fundamental. So the ratio of the frequencies of the second to the third harmonics is 2:3. (In other words, two waves of the C for every three of the G.) From the harmonic series shown above, you can see that the interval between these two notes is a perfect fifth. The ratio of the frequencies of all perfect fifths is 2:3.
Exercise 4.6.1. (Go to Solution)
- The interval between the fourth and sixth harmonics (frequency ratio 4:6) is also a fifth. Can you explain this?
- What other harmonics have an interval of a fifth?
- Which harmonics have an interval of a fourth?
- What is the frequency ratio for the interval of a fourth?
If you have been looking at the harmonic series above closely, you may have noticed that some notes that are written to give the same interval have different frequency ratios. For example, the interval between the seventh and eighth harmonics is a major second, but so are the intervals between 8 and 9, between 9 and 10, and between 10 and 11. But 7:8, 8:9, 9:10, and 10:11, although they are pretty close, are not exactly the same. In fact, modern Western music uses the equal temperament tuning system, which divides the octave into twelve notes that are equally far apart. (They do have the same frequency ratios, unlike the half steps in the harmonic series.) The positive aspect of equal temperament (and the reason it is used) is that an instrument will be equally in tune in all keys. The negative aspect is that it means that all intervals except for octaves are slightly out of tune with regard to the actual harmonic series. For more about equal temperament, see Tuning Systems. Interestingly, musicians have a tendency to revert to true harmonics when they can (in other words, when it is easy to fine-tune each note). For example, an a capella choral group, or a brass ensemble, may find themselves singing or playing perfect fourths and fifths, “contracted” major thirds and “expanded” minor thirds, and half and whole steps of slightly varying sizes.
The harmonic series is particularly important for brass instruments. A pianist or xylophone player only gets one note from each key. A string player who wants a different note from a string holds the string tightly in a different place. This basically makes a vibrating string of a new length, with a new fundamental.
But a brass player, without changing the length of the instrument, gets different notes by actually playing the harmonics of the instrument. Woodwinds also do this, although not as much. Most woodwinds can get two different octaves with essentially the same fingering; the lower octave is the fundamental of the column of air inside the instrument at that fingering. The upper octave is the first harmonic.Note
In some woodwinds, such as the clarinet, the upper “octave” may actually be the third harmonic rather than the second, which complicates the fingering patterns of these instruments. Please see Standing Waves and Wind Instruments for an explanation of this phenomenon.
It is the brass instruments that excel in getting different notes from the same length of tubing. The sound of a brass instruments starts with vibrations of the player’s lips. By vibrating the lips at different speeds, the player can cause a harmonic of the air column to sound instead of the fundamental. Thus a bugle player can play any note in the harmonic series of the instrument that falls within the player’s range. Compare these well-known bugle calls to the harmonic series above.
Figure 4.54. Bugle Calls
Although limited by the fact that it can only play one harmonic series, the bugle can still play many well-known tunes.
For centuries, all brass instruments were valveless. A brass instrument could play only the notes of one harmonic series. (An important exception was the trombone and its relatives, which can easily change their length and harmonic series using a slide.) The upper octaves of the series, where the notes are close enough together to play an interesting melody, were often difficult to play, and some of the harmonics sound quite out of tune to ears that expect equal temperament. The solution to these problems, once brass valves were perfected, was to add a few valves to the instrument; three is usually enough. Each valve opens an extra length of tube, making the instrument a little longer, and making available a whole new harmonic series. Usually one valve gives the harmonic series one half step lower than the valveless intrument; another, one whole step lower; and the third, one and a half steps lower. The valves can be used in combination, too, making even more harmonic series available. So a valved brass instrument can find, in the comfortable middle of its range (its middle register), a valve combination that will give a reasonably in-tune version for every note of the chromatic scale. (For more on the history of valved brass, see History of the French Horn. For more on how and why harmonics are produced in wind instruments, please see Standing Waves and Wind Instruments)Note
Trombones still use a slide instead of valves to make their instrument longer. But the basic principle is still the same. At each slide “position”, the instrument gets a new harmonic series. The notes in between the positions aren’t part of the chromatic scale, so they are usually only used for special effects like glissandos (sliding notes).
Figure 4.55. Overlapping Harmonic Series in Brass Instruments
These harmonic series are for a brass instrument that has a “C” fundamental when no valves are being used – for example, a C trumpet. Remember, there is an entire harmonic series for every fundamental, and any note can be a fundamental. You just have to find the brass tube with the right length. So a trumpet or tuba can get one harmonic series using no valves, another one a half step lower using one valve, another one a whole step lower using another valve, and so on. By the time all the combinations of valves are used, there is some way to get an in-tune version of every note they need.
Exercise 4.6.2. (Go to Solution)
Write the harmonic series for the instrument above when both the first and second valves are open. (You can use this PDF file if you need staff paper.) What new notes are added in the instrument’s middle range? Are any notes still missing?
The French horn has a reputation for being a “difficult” instrument to play. This is also because of the harmonic series. Most brass instruments play in the first few octaves of the harmonic series, where the notes are farther apart and it takes a pretty big difference in the mouth and lips (the embouchure, pronounced AHM-buh-sher) to get a different note. The range of the French horn is higher in the harmonic series, where the notes are closer together. So very small differences in the mouth and lips can mean the wrong harmonic comes out.
Playing Harmonics on Strings
String players also use harmonics, although not as much as brass players. Harmonics on strings have a very different timbre from ordinary string sounds. They give a quieter, thinner, more bell-like tone, and are usually used as a kind of ear-catching special-effect.
Normally a string player holds a string down very tightly. This shortens the length of the vibrating part of the string, in effect making a (temporarily) shorter vibrating string, which has its own full set of harmonics.
To “play a harmonic”, the string is touched very, very lightly instead. The length of the string does not change. Instead, the light touch interferes with all of the vibrations that don’t have a node at that spot.
Figure 4.56. String Harmonics
The thinner, quieter sound of “playing harmonics” is caused by the fact that much of the harmonic series is missing from the sound, which will of course affect the timbre. Lightly touching the string in most places will result in no sound at all. This technique only works well at places on the string where a main harmonic (one of the longer, louder lower-numbered harmonics) has a node. Some string players can get more harmonics by both holding the string down in one spot and touching it lightly in another spot, but this is an advanced technique.
Solutions to Exercises
Solution to Exercise 4.6.1. (Return to Exercise)
- The ratio 4:6 reduced to lowest terms is 2:3. (In other words, they are two ways of writing the same mathematical relationship. If you are more comfortable with fractions than with ratios, think of all the ratios as fractions instead. 2:3 is just two-thirds, and 4:6 is four-sixths. Four-sixths reduces to two-thirds.)
- Six and nine (6:9 also reduces to 2:3); eight and twelve; ten and fifteen; and any other combination that can be reduced to 2:3 (12:18, 14:21 and so on).
- Harmonics three and four; six and eight; nine and twelve; twelve and sixteen; and so on.
Solution to Exercise 4.6.2. (Return to Exercise)
Opening both first and second valves gives the harmonic series one-and-a-half steps lower than “no valves”.