Scales are one of the primary building blocks of tonal music. The Mozart sonata example in the previous lesson demonstrates how powerful the scale can be in shaping an entire phrase, introducing a motif, and creating tension that propels the phrase forward to its cadence. Now, you might be surprised if I claim that a major scale is, in fact, derived from the overtone series. However, Schoenberg already demonstrated this in his masterwork, Theory of Harmony. To explore this further, let us look again at the first through fifth overtones of C2 (Figure 1a):

For our current purpose, the fundamental, second, and fourth overtones are the most essential because they give us unique pitch materials: C, G, and E, respectively (Figure 1b). (In contrast, the first, third, and fifth overtones are duplicates.) Overtones beyond the fifth are either duplicates or dissonances and have very low amplitudes, making them inconsequential to our current discussion.

Between G and E, the former is more prominent simply because it is louder. If we treat G as a new fundamental note, its overtone series will introduce two new pitch materials to our collection—D and B (Figure 2).

You are probably thinking that the same line of thought could be applied to E, whose overtone series would provide B and G# as new pitch materials. But as we discussed earlier, the amplitude of E4 is so low that its overtones would be negligible.

But what happens if we look at this the other way around? G3 came into being because it is the second overtone of C2. So, C2 itself must be the second overtone of a hypothetical F0. Figure 3 shows the overtone series of F, which produces F and A as new pitch materials.

To summarize, when C is played, G and E also occur as overtones. G, being the second overtone and the stronger of the two, can be treated as a new fundamental tone that provides D and B as its own overtones. The C that we played, in turn, is the second overtone of F, whose fourth overtone is A. And, surprise! Altogether, these notes form a complete C major scale (Figure 4).

You should already be familiar with the structure of the major scale. I’ll just briefly remind you again, for the sake of formality, that all consecutive scale degrees are a major second apart, except between the mediant and subdominant and between the leading tone and the tonic, which are only a minor second apart. We’ll discuss the voice-leading and other implications of these intervallic relationships later. For now, it is important to note that the seven notes are not equal. In the C major scale, C is the most important because it is the only note that we actually played. Next comes G, the strongest overtone of C, which is also treated as a fundamental. F is the next most important. You may think that, since C is an overtone of F, F should be the most important note. But in reality, F is only a hypothetical note—conceptually significant but not actually being played. In any case, these three notes are more important than the other members of the scale. These three fundamentals—C, G, and F, one played, another an overtone, and the last hypothetical—form the primary tones of the C major scale: the tonic, dominant, and subdominant, respectively.

Since, both physically and conceptually, all the other members of a major scale are derived from a single note—the tonic—there should never have been any doubt about which note “owns” the scale. However, our predecessors did not have the benefit of hindsight that we do. Historically, arriving at the “correct” tonic involved many detours, and the struggle for supremacy among the seven tones lasted more than half a millennium. Musicians in earlier times did recognize the existence of the seven tones; it was simply not clear which one was the master. From this perspective, it is plausible to view the church modes, and the entire modal system of early music, as an artistic experiment in trial and error on a grand scale. After all, why not let the seven tones take turns serving as the tonic, allow things to play out for a while (i.e., half a millennium!), and see which tone is the last one standing? Thus, each of the seven tones was assigned a tonic function (Figure 5):

Of these seven modes, theoretically, the B Locrian is the weakest: it is the only mode in which the distance between the first and fifth scale degrees is a diminished fifth (B–F). The other modes all contain the perfect fifth, which reflects the important relationship between the fundamental and second overtone, and between the tonic and dominant. If your reasoning follows this line, congratulations! History agrees with you. The Locrian was historically the least used mode and was the first to fall out of fashion.

With our knowledge of tonality (which was not yet fully formed during the era of modal music), let us categorize the other six modes as either major or minor. A mode is major if its first and third scale degrees form a major third. This applies to C Ionian, F Lydian, and G Mixolydian. These three major modes were eventually consolidated into the C Ionian mode, later known as the C major scale. And this is as it should be. After all, of these three tonic notes, C is the only one we actually play!

The Dorian, Phrygian, and Aeolian, on the other hand, have their first and third scale degrees forming a minor third. Therefore, we categorize them as “minor” modes. In the next lesson, we will discuss these more complicated minor modes. But before we move on, why don’t you show me that you really know how to construct a major scale? Try the following exercises.

Exercises

I’m sure you can construct any major scale, one way or another. But if you are thinking about key signatures, let’s not do that. Why? Imagine a student trying to recite an E major scale. He first thinks of the key signature. Then, to be absolutely sure, he compares each note of the scale against the key signature. Peeking into his head, you’d hear this:

“OK, E major. That’s four sharps: F#, C#, G#, D#. All right, E major scale: E; F#; G, uh—F#, C#, G#—yes, G#! A, hmm—F#, C#, G#, D#—nope! Just A. Just B too, I remember! C, well, now—F#, C#—oh, C#! OK, D, let’s see—F#, C#, G#, D#—ah, D#!”

Now, if you tell him to recite it again, he’d probably go through a similar routine one more time. How terrible was that? You’re thinking, of course, that you can do it faster than this student. Sure, I don’t doubt it. I exaggerated a bit there. But do you really think you can recite a scale by checking it against the key signature faster than you can play that scale on your main musical instrument? No way. The fastest way to recognize anything musically is to imagine playing it on your instrument, in your mind. If you cannot do that, it simply means that you cannot play that particular scale on your instrument, which is no excuse at all.

“But I’m a singer!” When my voice-major students complained—and they usually did every year in my freshman harmony class—I always asked them to feel the vibration around their larynx for pitch reference . . . No! If you are a singer, then it’s time to make the piano your secondary instrument. A physical instrument that you can easily observe will greatly help you grasp abstract musical concepts. Keyboard, string, and mallet instruments are the most convenient for this purpose.

So, recite ALL the theoretical major scales, which I have redundantly listed below in a random order just to make it a bit more difficult. Each scale should take no more than 10 seconds to complete—that’s almost 1.5 seconds per note! Speak all the pitches clearly, from the tonic up to the octave, including the sharps and flats. For Ab major, you’d say, “Ab, Bb, C, Db, Eb, F, G, Ab” quickly, for example. Ready? Go!

Db major

Eb major

F# major

Bb major

E major

C major

A major

B major

Gb major

Ab major

C# major

F major

D major

G major

Cb major