Understanding Musical Intervals
Introduction
In Chapter 3 we will be looking more deeply at how musical intervals are derived from harmonic ratios.
For the moment, I just want to introduce the basic terms known to all practicing musicians, so we have the vocabulary that will allow us to go more deeply into the topic.
Beginning with the major pentatonic scale, which as you learned in the last section could be derived from a basic human singing pattern, we can use middle C as a starting note and label the intervals that are formed above it:
We can also begin on the high tone and look at the intervals below the high C:
Listen to the playback of these intervals and be certain you can match the shape to the sound.
Notice that, when looking at the notes of the major pentatonic, the type of interval switches between major and minor, depending on whether the interval is ascending or descending. From C up to D is a major second, but from C down to D is a minor seventh. From C up to E is a major third, but from C down to E is a minor sixth.
This brings us to the concept of perfect intervals, major intervals, minor intervals and inversions.
Perfect Intervals and Inversions
Perfect intervals are those that have the musical ratios of 1:1, 2:1, 3:2 or 4:3. We will learn more about the actual ratios in Chapter 3. For the moment, suffice it to say that these intervals form the unison, octave, perfect fifth and perfect fourth, as you see above.
We can then gather that:
- A perfect fifth is formed from C up to G
- A perfect fourth is formed from C down G.
- A perfect fifth plus a perfect fourth from the interval of an octave.
- A perfect fifth is then the inversion of the perfect fourth.
- A perfect fourth is then the inversion of the perfect fifth.
We can therefore define the inversion of an interval as a change of position. To invert the perfect fifth from C to G, keep the G as the lowest note and place the C above it. This results in a shift of interval from perfect fifth to perfect fourth. The ratio changes from 3:2 to 4:3. Similarly (and somewhat obviously), the perfect unison inverts to a perfect octave. The ratio changes from 1:1 to 2:1:
Major/Minor Intervals and Inversions
Major and minor intervals are those that have the musical ratios of 5:4 and 6:5, respectively. Again, we will explore these ratios more in Chapter 3. As we did with the perfect intervals, we can draw the following conclusions with the major and minor intervals:
- Major and minor intervals are found between the root note and the D, E and A of the pentatonic scale.
- When we measure up from the bottom note, these intervals are considered to be major.
- When we measure down from the top note, these intervals are considered to be minor.
- A major interval inverts to a minor interval.
- A minor interval inverts to a major interval.
- Thus a minor seventh is the inversion of a major second
- A minor sixth is the inversion of a major third.
Listen to the playback of the above intervals. You will developing your ear to hear these intervals using the Practica Musica software.
The Full Major Scale and the Semitone (half step)
Up til now, we have only formally discussed the pentatonic scale. In chapter 3 we will look more deeply at the true genesis of the pentatonic and the full seven note (septatonic) scale. For now, by way of basic orientation, let's add the other two notes to the basic major pentatonic scale and look at what is created by this addition. It is assumed in this class that you have already played or sung the basic scale sometime in your life, or that you are learning it in the Rudiments portion.
The reason I introduce the major pentatonic scale first, followed by the full major scale, is to emphasize the intervals that are formed when the two additional notes, the F and the B, are added to the pentatonic scale.
When F and B are added to the scale, we create semitones between E and F and between B and C, and we also form the interval of the tritone between F and B:
ALERT: It is incredibly important that you understand these two concepts, as all of interval reading and musical literacy and notation is based on few simple facts:
- The interval between any two adjacent notes is a second.
- Major seconds, also known as whole tones or whole steps, are formed between all natural notes, except...
- ...except between the steps of E and F and between B and C. These are the minor seconds, also known as semitones or half steps.
- The interval between any two notes formed by two line
notes with a line between, or two space notes with a space between is
fifth.
- Perfect fifths are formed between all natural notes except...
- ...except the fifth formed between B and F. This is considered a diminished fifth, also known as a tritone. It is diminished because it a semitone smaller than a perfect fifth. It is called a tritone because it can be seen as comprising three whole steps.
- The interval between any two notes formed by a third plus a step (line
to space with a skip or space to line with a skip) is a fourth.
- Perfect fourths are formed between all natural notes except...
- ...except the fourth between F and B. This is considered an augmented fourth, and also forms the interval of the tritone.
Thus:
Listen to the playback and try to hear the difference between the major/minor seconds and between the perfect fourths/fifths and the tritone. You will practice this in your ear training software.
Summarizing the Intervals and their Inversions
From the above information, we can now better define all the intervals of the natural notes. These are also called the diatonic intervals.
First of all, let's complete the discussion of inversions.
- We've seen that the major second and the major third inverts to the minor seventh and the minor sixth.
- We've seen that the perfect fourth inverts to the perfect fifth, and vice versa.
- It follows then that the minor second will invert to the major seventh
- It also follows, since both the augmented fourth and the diminished fifth are considered to be a tritone (three whole tones), and that they are same pitches, that the diminished fifth inverts to the augmented fourth, and the augmented fourth inverts to the diminished fifth. Note that this is the only inteval where the inversion is the same interval, with the same number of steps.
With all our diatonic intervals defined, we can draw the following conclusions:
Seconds:
- A major second is considered the basic whole tone, which exists between the majority of intervals. (See above)
- A minor second occurs less frequently in natural scales. It does not exist at all in pentatonic scales. In normal diatonic scales, such as the scale of C major, it occurs twice. (See above).
- The succession of major and minor thirds in the diatonic scale is in the graphic above. Again, just remember that E to F and B to C are semitones, and the rest are whole tones.
Thirds:
- A major third is a combination of two whole tones, for example C to E is a combination of C to D and D to E.
- A minor third is a combination of a whole tone and a half tone, for example C down to A is a semitone from C to B and a whole tone from B to A.
- The succession of major and minor thirds along the diatonic scale are as follows. Notice that the third formed above the notes C, F and G are major thirds, and the third above the notes D, E, C and B are minor thirds. This is essential information!
Fourths and Fifths:
- A perfect fourth is a major third plus a semitone.
- A perfect fifth is a major third plus a minor third.
- A tritone (diminished fifth or augmented fourth) is a combination of three whole tones (F to G, G to A, A to B). To find the three whole tones from B to F (the diminished fifth), you need to use chromatic notes:
The perfect fourths and fifths along the diatonic scale are in the graphic above.
Sixths:
- A major sixth is a perfect fifth plus a whole tone, for example C up to G, then G to A.
- A minor sixth is a perfect fifth plus a semitone, for example C down to F, then a semitone to E.
Sevenths
- A major seventh is perfect fifth plus a major third, for example C up to G, then G up to B.
- A minor seventh is a perfect fifth plus a minor third, fo rexample C down to F, then F down to D.
Know these rules will allow you to form any interval above or below a given note. This will, in fact, be your assignment for this page.
ALL INVERSIONS:
As to inversions, we can now state the essential rules:
- All major intervals invert to minor intervals.
- All minor intervals invert to major intervals.
- All perfect intervals invert to perfect intervals.
- All diminished intervals invert to augment intervals. All augmented
intervals invert to diminished intervals.
- Thus, the tritone, the diminished fifth/augmented fourth, inverts to itself.
NOTE WELL: As to the interval type, it is very easy. The two intervals should add up to the number 9:
- Seconds invert to sevenths (2+7 = 9)
- Thirds invert to sixths (3+6 = 9)
- Fourths invert to fifths (4+5 = 9)
- Fifths invert to fourths (4+5 = 9)
- Sixths invert to thirds (3+6 = 9)
- Sevenths invert to seconds (7+2 = 9)
Even though arithmetically intervals add up to the number 9, musically these are actually harmonic ratios, and thus the interval plus its inversion actually always equals an octave. This is very important to remember!
Chromatic Intervals
The final step in interval recognition is an extension of all the above materials. Once the existence of the minor second between E and F and between B and C is clear in your mind, you can use the above rules to determine intervals involving chromatic tones.
For seconds: major seconds equal two minor seconds. Thus major seconds will include a sharp note to a sharp note or a flat note to a flat note except when they involve in E-F or a B-C combination.
For thirds: major and minor thirds can be determined by the formula of combining two whole steps (= major third) or a whole step plus a half step (= minor third).
For fourths: Most fourths are perfect -- they combine a major third with a half step. They are augmented if they combine a major third plus a whole step.
For fifths: Most fifths are perfect -- they combine a major third with a minor third (or you can think of it as a perfect fourth plus a whole step). Fifths are diminished if they combine two minor thirds or they combine a perfect fourth plus a minor second.
For sixths: find the perfect fifth interval from the lowest note, then determine whether the interval above the fifth is minor second (= minor sixth) or a major second (= major sixth).
For sevenths: find the perfect fifth interval from the lowest note, then determine whether the interval above the fifth is minor third (= minor seventh) or a major third (= major seventh).
Study the following chart and recognize these patterns of intervals. Be clear why they express the interval in question.
