The Building Blocks of Music: The Interval

The Building Blocks of Music: The Interval

When we have two notes to consider, we describe the relationship between the two as an interval. We can look at the interval in two ways: the vertical (when the notes sound simultaneously or at least overlap considerably) and the horizontal (when one note follows another). Because it is easier to hear horizontal intervals, we will begin with them.

Now for a personal aside: identifying intervals by ear was a stumbling block for me as a student of music. At the University of California, Santa Cruz, we had a grueling 64 Interval Test, of which we had to identify 58 correctly. I finally passed it, and have had little trouble with identifying intervals since then, but it did not come easily. I actually ended up with a rough perfect pitch as a result of working on hearing those intervals (you can wake me up in the middle of the night with a gun to my head and ask for A440, and I can give you A440 (although as a baroque geek, this is disturbing. Proper A is 415Hz)).

The first division of intervals is into consonance and dissonance. We will first look at the consonant intervals.

If you are sitting at a piano, please play C and then G, preferably around Middle C, but it can be any octave. What you have just heard is the Perfect Fifth. What this means is that the interval between the notes is a certain ratio (I will not confuse you with the exact ratios), which remains constant, no matter what the lower note is, so a Perfect Fifth above D is A, above E it is B, etc. By calling it Perfect we are recognizing two things: first, that this interval is found early in the harmonic series, which we discussed in Building Blocks: Tone, and, second, that the first note is found in the Major scale of the second and vice versa. So, in our example, one encounters a G in the key of C, and a C in the key of G. The Perfect consonances are dangerous to the composer, because they imply such a strong harmonic affinity. When we get into counterpoint, we will see the care that composers use in handling the Perfect Fifth (from here on referred to as P5), the Octave (P8) and the Unison (P1).

The P8 interval is simply a doubling of pitch, so an octave is C to C. The P1 is an identity: play the same note twice and you have it.

The other Perfect interval is the Perfect Fourth (P4). It is a tricky one, because it can sound incredibly dissonant. In fact, in the study of counterpoint, the P4 is considered a dissonance to be avoided (or treated with the care that one would treat any dissonance). Part of the reason is that the P4 is an INVERSION of the P5, so if we play C and then F, the P4, we know that F to C is a P5 (hint, inversions add up to 9).

If we play C and then E, we get an imperfect consonance, called a Major Third (M3). It is called Major, because E is in the key of C Major, but C is not in the key of E Maj. If we play C and then E flat, it is a minor third (m3), because the E flat is not in C Maj, but the C is in the Key of E flat Major (the Major key with three flats: B, E, and A). The inversion of the third is the sixth, so if we start on E our Major sixth (M6) is C sharp, and the minor sixth (m6) is C.

As you play these intervals, notice the associations with songs that come out of them. An ascending P5 (C up to G) may bring the Star Wars theme to mind. A descending P5 (G up to C) might call to mind Born Free. Use these clues to help you learn your intervals. If you know of handy guides to the intervals please feel free to post them in the comments box, to help others learn these. My examples are often a little esoteric, so many of you who are more attuned to popular music might be more useful here.

Now we have the dissonant intervals. We will start with the second. From C to D we have a Major Second (M2), or a whole step. If we move from C to D flat we have a minor second (m2), or a half-step. If we invert the second (2+X=9, and to add to the joy, an inversion of a Major interval is a minor interval and vice versa) we get the seventh. C up to B is our Major seventh (M7), while C up to B flat is our minor seventh (m7).

These are the natural intervals. If we augment an interval, it means that we add a half-step (so an augmented fifth (+5) is the same, for our purposes, as m6). Likewise to diminish an interval means to deduct a half-step (so a diminished fourth (-4, actually the degree sign is preferable, but I have no idea how to render that from the computer, so we will have to use the less preferable -sign) is the same as M3. All of this leads to the problem child of intervals: the tritone (also known as Diabolus in Musicam – the devil in the music), a thoroughly unstable interval. To hear a tritone, play C and then F sharp. Think, “Maria” from West Side Story. Avoiding the tritone is a serious business in writing counterpoint. Not only is it dissonant, but it gravely damages the melodic flow, and cannot even be outlined in three notes (C, E, F#). In tonal harmony the shifting nature of the tritone makes it incredibly useful in the V7 chord, but that is way down the road, so no need to worry about it now.

So the intervals are, in ascending order from the root:

P1
m2
M2
m3
M3
P4
Tritone (+4/-5)
P5
m6
M6
m7
M7
P8
From here on, the intervals are the same, but added to the octave. No one really talks about anything larger than P12, or P5 an octave up.

The Vertical Interval. In some ways a note with a complex wave form can be described in the same language as a chord, or a stack of notes played simultaneously. The way to separate these two concepts is to declare that in looking at the vertical interval, we are concerned with the tones that have roughly equal sonic energy. Certainly each one will come with a host of overtones, but each of these overtones will have diminished sonic energy, as we saw in the Building Block: The Tone.

For this level of conversation we will only concern ourselves with two notes played simultaneously, so no need to think of the more complex harmonies of triads yet.

Basically, the language of dissonance and consonance is the same, but our ear has a little more work to do to tell which interval is which. Play the examples from above, but play them at the same time. Notice whether the resultant combination has a settled feel or whether it sounds like something needs to come after it. You will get an idea of the strength of consonance and dissonance: m7 is less dissonant than M7, for instance, P5 is more consonant than M6, etc.

When you contemplate these relationships and how they work together, you get a good idea of how marvelous the Cosmos is. All of these relationships are built into the very nature of sound! Almost. We cannot talk about the interval without talking about tuning.

For a variety of reasons, the ratios that make up the circle of fifths, that wonderful acoustic phenomenon that generates the 12 tones of the scale by moving up in fifths (C,G,D,A,E,B,F#,C#,G#,Eb,Bb,F,C) does not add up all the way, so if we keep the ratios of the fifths pure, the C will not be the octave C we started with. The gap is the Pythagorean comma. Resolving that gap has resulted in many tuning systems that either abandoned the use of some keys (most of the just tempered scales do this), or compromised the intervals so that the octaves work out, and the intervals are usable.

What you are used to hearing is the Equal 12 system, in which the ratio between each tone in the chromatic scale is identical as you go from one note to the other. To do this, the fifths, thirds, and sixths are compromised, so they are not as consonant as they should be. But temperament is a topic for another Building Block, so I will leave it at this introduction!

So, read this, plunk out the intervals, think about it, ask questions, digest it, and we will get to the analysis of Victimae paschali laudes.