Music Theory Tutorial: Scales and Intervals

I have found that many music educators, especially those of us who spend most or all of our time at the elementary level, have lost a good deal of what we once knew of music theory. We seldom teach much of what we learned in undergraduate music theory classes, never much more than note names and diatonic intervals. Neapolitan chords, and especially augmented 6th chords are but a faint memory, if that. Now don’t worry, I am not going to be writing about those things here. What I am going to explain are diatonic scales, and the diatonic and non-diatonic intervals that are derived from those scales.

To begin, diatonic means that only notes from a given scale are used. If a melody is in the key of C minor, then the melody is diatonic if it stays in C minor, and it is entirely diatonic if every note of it is a note from the C minor scale. Any tone that appears that is not in the C minor scale is a non-diatonic tone. The course of music history is one of gradually adding and assimilating more and more non-diatonic notes into the norm.

Intervals are distances between two tones, with number of scale tones being the unit of measure. A fifth describes two tones that are five scale tones apart, inclusive. Inclusive means that you count the two tones in your reckoning rather than counting from them. So with an F major scale, the distance between G and D is a fifth. We calculate this by counting, G, A, B, C, D. We named 5 tones from G to D, inclusive, so the distance between G and D is a fifth. With one exception, fifths between any two notes in a major scale are classified as perfect. The interval G to D is a perfect fifth. So is F to C, A to E, and so forth. There is one fifth in a major scale that is not perfect; that is the one between the seventh scale tone and the fourth (in F major that would be E to Bb). This fifth is called diminished because it is smaller and much more dissonant than all the other fifths. Another name for this interval is the tri-tone. Other intervals in the perfect category are fourths, octaves, and unisons. Unisons are two tones of the same pitch.

The smallest interval in Western tonal music is the half-step, or minor second. Half steps are most easily explained by looking at a piano keyboard. From a white key, such as C to the adjacent black key, C#, is a half step. There are no keys in between these two. Going down from C, there is no adjacent black key; the closest key to C going downward is B, so B is a half step from C. When we discussed the tri tone above, we said E to Bb was a diminished fifth. Diminished means made smaller, but made smaller from what? From a perfect fifth. E to B is a perfect fifth. E to Bb is a half step smaller, that is Bb is a half step closer to E than is B natural, so the Bb makes the interval smaller, or diminished. We can also make an interval bigger by a half step. When this is done, the resulting interval is called augmented. If E to B is a perfect fifth, then Eb to B is an augmented fifth, because Eb is a half step further away from B than E. We now know of three kinds of fifths–perfect, diminished, and augmented. The same three kinds of intervals apply to fourths, octaves, and unisons; they can all be perfect, diminished, or augmented.

Now what of intervals that are not fifths, fourths, octaves, or unisons? For seconds, thirds, sixths, and sevenths, the same designations of diminished and augmented apply, but there are two different ones that replace perfect. Seconds, thirds, sixths, and sevenths that are made of two diatonic tones are either major or minor, depending on how many half steps there are within the interval. Here is a table of these intervals and the number of half steps in each.

MajorMinorDiminishedAugmented
Second210 (same as
unison)
3 (same as
minor 3rd)
Third432 (same as
minor 3rd)
5 (same as
perfect 4th)
Sixth987 (same as
augmented 5th)
10 (same as
minor 7th)
Seventh11109 (same as
major 6th)
12 (same as
perfect octave)

Intervals for which I have indicated “same as” sound the same but are notated differently. For example, C to C# is an augmented unison, but C to Db is a minor second, though as can be seen by looking at a piano keyboard, C# and Db are the same. Because most intervals that are diatonic are major, minor, or perfect, these tend to sound more consonant or pleasant. That is because we are most accustomed to these intervals compared to the non-diatonic ones. As more diminished and augmented intervals came into more common use, people became familiar with the sounds and regarded them as less dissonant that formerly. It is this process of enculturation that explains why Le Sacre du Printemps was so scandalous when it was premiered, but is now is reliably heard without fisticuffs.

Most non-diatonic tones were first introduced into music as controlled dissonances; that is, they were meant to sound “wrong” or unpleasant in order to create musical tension, an then were immediately followed by a diatonic tone which restored pleasantness or consonance to the music. These non-diatonic tones were most often classified as passing tones (approached by step and left by step in the same direction) neighbor tones (approached by step and left by step in different directions), suspensions (approached by unison with the dissonance created in another voice, then left by descending step), though there were others as well. Gradually, more of the tones became dissonant, and many of them were no longer followed by diatonic tones so that the music overall sounded more dissonant. Because listeners had learned to make sense of music using consonance as their guide, when that consonance was fragmented or removed, music written this way became confusing. This kind of dissonant music, called non-tonal, must be listened to in a different way if it is to be understood.

When intervals are “stacked” so that two or more of them are heard simultaneously, harmony is formed and the structure so made is called a chord. Chords have the same designations as intervals. There are major, minor, augmented and diminished chords. The notes of the chord are named based on their interval from the root, which is the tone the chord is named after. So for a C major chord, the root is C, the third is the tone an interval of a third above C, which is E, and the fifth is the tone an interval of a fifth above C, which is G. When the third is a major third above the root, the chord is a major chord; when the third is a minor third above the root, the chord is minor. When the fifth is an augmented fifth above the root and the third is a major third above the root, the chord is augmented, and when the fifth is a diminished fifth above the root and the third is a minor third above the root, the chord is diminished.

Scales cannot be augmented, diminished, or perfect, but they can be major or minor. When a series of eight adjacent tones is arranged in seconds and in order from lowest to highest, a scale is formed. When the seconds occur in the order of major, major, minor, major, major, major, minor, then that scale will be major, and will span a perfect octave. When the seconds occur in the order of major, minor, major, major, minor, major, major, then that scale will be minor. Composers have usually raised the seventh tone a half step to strengthen the tonality, forming a minor second between the last two notes, and an augmented second between the sixth and seventh tones. Other alterations have also been used to improve the flow of the music. So when a musical work is said to be in D major, for example, that means that the composer has utilized tones that comprise the D major scale, the one where the starting and lowest tone is D. Canon over a ground bass by Pachelbel, so popular at weddings, is in D major. So is Tchaikovsky’s Violin concerto. Pieces written in a minor key are constructed from notes of a minor scale. Beethoven’s overly famous Symphony no. 5 is in C minor.

Intervals and scales are but the beginning of music theory. I hope to explore more aspects of the topic in future posts. If you have a particular topic within music theory you would like me to write on, please let me know by using the contact page. If you enjoyed this post or would like to see more like, please like this article and leave a comment.

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