Scales

4.1 Definitions

The word "scale" comes from the Italian, where scala is used to describe a ladder.  In a sense, a scale is like a ladder in that there are distinct rungs (notes) that a melody can use to move up or down in the pitch space.  It is easy to move from one rung to the next, but harder to jump three or four rungs (up or down!), and the same is true for melodies, which tend to move primarily in a step-wise motion.

In music we describe a scale as being a collection of pitches that have a specific relationship with one another.  There usually are eight notes in a scale (e.g. the major and minor scales) but scales of five, six, ten, and twelve notes are commonly found.  For purposes of this discussion, however, we will only discuss major and minor scales.

4.2 Introduction to the Major Scale

Rather than talk about what a major scale does (or why it is named "major") we will instead talk about how one is built.

Recall that intervals have a quantity (size) as well as a quality (number of half steps).  Remember also the definitions of "whole step" and "half step," (where a half step is the distance from a key to its nearest neighbor; the whole step is another half step again).

With this in mind, the rule for building a major scale is:

A major scale is built entirely of whole steps, except between the third and fourth degrees of the scale, and the seventh and eighth degrees.

The diagram below illustrates, graphically, where the half steps fall in a major pattern:

With this paradigm in mind, we can build a major scale, shown below on a piano keyboard:

Play this example several times..  Listen and notice how the half steps on the keyboard correspond to the half steps in our model (above), placing the half steps between 3-4 and 7-8. 

Make some mistakes with your scale, and hear how even a slight adjustment of half step and whole step arrangement alters the sound of the scale.

4.3 Building major scales in various keys

The special collection of notes known as the major scale can be built using many combinations of notes (twelve, in fact, one for each white or black key on the keyboard).  To differentiate between these collections, we refer to each as being a specific "key." The scale above, for example, since it is built on C is referred to as "the key of C major." 

To build collections on other notes, simply follow the formula above (half steps between 3-4 and 7-8) and generate each scale starting on the desired note.  For example, let us build a scale on G major, using our formula:

The notes in the G major scale then, are G A B C D E (F#) G.  The F# is necessary because a half step is needed between the 7^th and 8^th degrees of the scale and the F must be raised a half step to meet this criteria.

We say the key of G major has one sharp, F sharp, to reflect this.  This is often called a key signature, in that the only major key to have one F sharp alteration is G major.

4.4 The minor scale

Minor scales simply employ a different arrangement of whole and half steps to make a different type of sound.  The general rules we discussed above still apply, the only change is the order of whole and half steps, which is:

As you can see, whole steps are employed throughout, but as in the major scale, there are two points where half steps are employed, between 2-3 and 5-6.  Using this formula as a guide, let us build a minor scale on C:

We see, then, that a C minor scale employs flats to achieve the desired relationship: half steps between 2-3, and 5-6.

4.5 Forms of the minor scale

As if knowing the major and minor scale formulas weren't enough, we often find music using variations of the minor scale.  It is almost as if no one can decide exactly how it works so composers will adjust the scale in certain situations to fit their needs.  You need to be aware of some of these modifications.

a) Natural Form

The natural minor is the formula we've discussed above, with half steps between 2-3 and 5-6.  It is "natural" because no modifications have been made to the scale; it appears in its pristine state.

b) Harmonic Form

As you'll discover later in the semester, harmonic concerns cause the minor scale to become changed from time to time.  A minor scale that uses the harmonic form undergoes a change based on harmonic concerns. 

The harmonic form is exactly like the natural form (above), with the exception that the seventh scale degree is raised a half step.  Below are some examples:

For each of the examples above, you see two versions of each scalethe natural and the harmonic.  Note how the harmonic form uses the same pitches except changes the seventh by raising it by a half step.

a) Melodic Form

The third possible condition of the minor scale is known as "melodic minor." It is exactly like the harmonic minor except the 6^th and 7^th degrees are both raised a half step.  In fact, at this point, the harmonic minor almost begins to resemble a major scale

You can see, then, that by raising the sixth and seventh degrees, the melodic minor scale resembles the major scale except for the location of the first half step (in major, it appears between 3-4 and in minor between 2-3).

In that sense, you may think of the melodic minor scale as being a major scale with a lowered third degree.

4.6 The order of sharps

Although it is critical to understand the theory behind generating major and minor scales,  it is much easier to memorize the associated keys with their key signatures.

One interesting phenomenon in our system is that if we keep track of the number of sharps in a key signature, and make a list like so:

we can begin to see a pattern emerge: as we add sharps to the key signature (that is, as we read down this list), the key associated with a key signature is a fifth higher than the previous key!  

In other words, if we know that D major has two sharps, and we are asked what the key signature for E major is, we simply move up by fifths until we reach E, and we add a sharp to the key signature in the process to obtain four sharps.  (D up a fifth is A, which has three sharps, then another fifth up yields E, which has four sharps).

4.7 The order of flats

If we compile a similar list of flat keys and arrange them in order of the number of flats in the key signature, we get a list that looks like:

Once again, we see a similar phenomenon, except each key is now a fifth below the previous one.  For example, the key of Bb major has two flats, and if we descend a fifth, we reach the key of Eb, which has three flats.

4.8 The circle of fifths

Using the idea that all of these keys are related by fifths to one another (that is, if we go up or down a fifth, we end up adding or taking away an accidental), it is possible to arrange the keys into a circle, which is a very helpful way to remember the order of sharps and flats.  Memorize it well.  The circle below illustrates the circle of fifths with major and minor keys (which exhibit the same phenomenon).  Major keys are in capital letters on the outside of the circle, and minor keys are on the inside in lowercase.

4.9  Advanced intervals

You will recall that intervals have a quantity and a quality. As we demonstrated above, not all thirds are created equal!

There are names for the qualities of all intervals. One of the easiest ways to remember the interval names is to memorize your scales first, then derive the interval names from the scale.

One helpful hint: some intervals are always known by the terms "perfect," while others use the terms "minor" and "major" to describe them. This is very consistent, so be on the lookout for patterns.

Consider the example below, which uses our major scale formula (see above) and illustrates how intervals are named within a scale:

In this example, we can see that all intervals of the quantity 1, 4, 5, and 8 are all named "perfect." The remaining intervals (2, 3, 6, 7) are all named "major."

In the natural minor system, note which intervals have their names changed and which ones remain the same:

> Notice how the intervals that were "perfect" in the first example (1, 4, 5, and 8) are still perfect. The intervals that were major are all now minor, with the exception of the distance from 1-2.

4.10  A final comment about intervals

A point of some confusion is what happens when intervals are modified through some accidental or other modification. For example, consider the distance from C to F#. What is the name of that interval?

Well, using our knowledge of intervals, we know that is a fourth (C, D, E, F) but it is not a perfect fourth, because F# represents a note not found in the C major scale.

The identification of these "odd intervals" can best be described through the table below: