LEARNING MORE ABOUT THE MUSIC WE SING [N.B.: The following statements about the sizes of intervals are based on arithmetic—on the science of acoustics—which will be explained later.] Lesson 1, for March 23 Scales and Just Intonation: Let’s start with the C-major scale. Its notes in ascending order are C D E F G A B C In C major the note B is the leading tone, which “leads” both melodically and harmonically into the note above it, C, the tonic of the scale. C major is known as a diatonic scale, consisting a series of whole-steps and half-steps. These same whole-steps and half-steps, and in the same order, are also used for the key of A-minor, which starts a minor third lower. (A minor is also a diatonic scale.) A B C D E F G A The only difference between C major and A minor is that the minor key has its own leading-tone: G#. Thus, more accurately, the notes of A-minor are: A B C D E F G# A We will only be concerned at the moment with C major. The C-major scale consists of the following ascending intervals: whole-step (C-D whole-step (D-E) half-step (E-F) whole-step (F-G) whole-step (G-A) whole-step (A-B half-step (B-C) Thus, five whole-steps and two half-steps. If we were to sing in piano tuning, this is all you would need to know about the major scale. But we sing as much as possible in just intonation. In order to have pure, cleaner-sounding intervals, we sing with two different sizes of whole-step and two different sizes of half-step. Let us examine why two different sizes of whole-step and half-step are necessary in just intonation. To begin with, notice that the diatonic major scale has three major thirds. In C major, these are C-E, F-A, and G-B. You already know that the pure major third is a little narrower than the major third of the piano [this will be demonstrated later, when we look at the arithmetic of musical acoustics]. In order to create this slightly narrower major third, we can think of the bottom note pulling up slightly (I like to think of it as a “high” note relative to the third above it) while the upper note pulls down slightly (I like to think of it as a “low” note relative to the third below it). In order to achieve pure major thirds as we ascend up the scale, each one will consist of one “large” whole-step plus one “small” whole step. Although all pure major thirds consist of one large and one small whole-step, the two types of whole-steps are not always in the same ascending order. Thus, the major third from C-E consists of a large whole-step C-D plus a small whole-step D-E; the major third from F-A consists of a large whole-step F-G plus a small whole-step G-A; but the major third from G-B consists of a small whole-step G-A plus a large whole step A-B. The two notes of the half-steps in the diatonic scale have different letter names: E-F and B-C in the key of C major. These are diatonic half-steps, as opposed to half-steps with the same letter name, such as F-F#, which are chromatic half-steps. In just intonation, diatonic half-steps are wider than their corresponding half-steps on the piano, and chromatic half-steps are narrower than their corresponding half-steps on the piano. Let’s look briefly at the F-major scale, which has a key signature of one flat. Its notes in ascending order are F G A Bb C D E F The three major thirds in F major are F-A, Bb-D and C-E. We can also look briefly at the G-major scale, which has a key signature of one sharp. Its notes in ascending order are G A B C D E F# G The three major thirds in G major are G-B, C-E, and D-F#. Here is a little rule of thumb to determine whether notes are high or low. If a note has a flat in front of it, it is high (we want the Bb to pull up toward the D in the key of F). If a note has a sharp in front of it, it is low (we want the F# to pull down toward the D in the key of G). This rule applies to key signatures with one, two, or three flats and to key signatures with one, two, or three sharps. These are the keys we sing in most often. End of Lesson 1. Click here to continue to Lesson 2, for March 25. Click here to return to the Lesson List.