i i i i Chapter 3 Some Music Theory The musical language that made the classical style possible is that of tonality, which was not a massive, immobile system but a living, gradually changing language from its beginning. —Charles Rosen In this chapter we explore some music theory, concentrating on pitch and harmony. This will help us to appreciate some of the structural patterns in musical compositions. We begin by discussing musical intervals and chords, which describe how different pitches are related and combined. We then describe the basic guidelines that have been used for centuries to create music within the diatonic keys (the major and minor keys described in the previous chapter). Following those guidelines produces music that we shall refer to as diatonic music.1 A commonly used technique for composing diatonic music is tonal sequencing. Tonal sequencing is a prime example of a mathematical procedure in musical composition. We conclude the chapter by examining sequencing when the chromatic scale is used, which is known as real sequencing. 3.1 Intervals and Chords Most of Western music is composed from notes on the chromatic scale, often restricted to a specific diatonic scale (or key).2 When comparing two notes, we can consider the change on a score that separates one note from another. This change is called an interval. There are two types of intervals, melodic intervals and harmonic intervals. A melodic interval is the change on the score between two successive notes. A harmonic interval is the change on the score between two notes that are played simultaneously, as in a chord. Diatonic music uses intervals and chords for melody and harmony in a characteristic way, which we describe in the next section. 3.1.1 Melodic and Harmonic Intervals When two notes occur successively, as in a melody, then the interval between them is called a melodic interval. To compute a melodic interval, you count the number of spaces and lines on the score between the two notes, starting with 1 for the first note. For example, the note E4 lies on the lowest staff line of the treble clef staff. If the next note is G4, then it lies on the next staff line above. Counting all the lines and spaces between these two notes, inclusively, we get a count of 3. So the interval between E4 and G4 is called a third. Another example would be the notes C4 and B3, for these notes we count 1 ledger line and 1 space, inclusively. So the interval between C4 and B3 is a second. In Figure 3.1 we show some melodic intervals. Notice that the intervals are not dependent on whether the pitches are going up or down. To specify pitch direction we would say, in this example, 1It is also called tonal music, but we prefer to call it diatonic music. 2Although there has been a tendency within the last hundred years to avoid emphasis on diatonic scales, focusing exclusively on the chromatic scale (sometimes called atonal music or post-tonal music). 51 i i i i i i i i 52 3. Some Music Theory Third Second Fourth Fifth Sixth Figure 3.1. Some melodic intervals. that the first two notes go up a third, while the last two notes go down a sixth. There are two special cases of intervals. When two notes are exactly the same, then we say their interval is a unison. When two notes are separated by an octave, then we say their interval is an octave (rather than an eighth). Major and Minor Intervals If we consider the number of half steps between two pitches, then most intervals fall into two cate- gories, called major and minor. For example, the interval between G4 and B4 is a third, and so is the interval between D4 and F4. However, the number of half steps in the former case is 4, while in the latter case it is 3. The interval between G4 and B4 is called a major third, while the interval between D4 and F4 is called a minor third. In Figure 3.2, we show the classification of intervals according to whether they are major or minor. Notice that the intervals of unison, fourth, fifth, and octave do not Unison Min. 2nd Maj. 2nd Min. 3rd Maj. 3rd Per. 4th Aug. 4th Per. 5th Min. 6th Maj. 6th Min. 7th Maj. 7th Octave 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 Unison Min. 2nd Maj. 2nd Min. 3rd Maj. 3rd Per. 4th Aug. 4th Per. 5th Min. 6th Maj. 6th Min. 7th Maj. 7th Octave 0 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10 −11 −12 Figure 3.2. Top: Some ascending melodic intervals. Minor is abbreviated as Min., and Major is abbreviated as Maj. Perfect is abbreviated as Per., and Augmented is abbreviated as Aug. Bottom: Some descending melodic intervals. Beneath the label for each interval are the number of half steps from the first note to the second note. have major or minor forms. The interval of a fifth is called a perfect fifth. An interval of a fourth can either be a perfect fourth (5 half steps), or an augmented fourth (6 half steps). The adjective perfect stems from ancient Greek theories of music. The augmented fourth is also referred to as a tritone.3 This categorizing of most intervals as major or minor is musically important. Because the number of half steps is related to the magnitude of frequency ratios between pitches, we will hear two notes that are a minor third apart as closer in pitch than two notes a major third apart. We have described the basic forms for melodic intervals. There are some additional classifications for intervals which occur less frequently, but we shall not discuss them.4 Harmonic Intervals When two pitches are played simultaneously then the interval between the notes is called a harmonic interval. The classification of harmonic intervals is essentially the same as for melodic intervals. The 3The precise definition of a tritone is an interval where the notes are 6 half steps apart on the chromatic clock. So, the notes ♯ ♭ D4 and G4 are an augmented fourth, a tritone. However, the notes C4 and G4 are a diminished fifth, also a tritone. In both cases, the notes are separated by 6 half steps. 4Any book on music theory will describe the complete classification of intervals. See, for example, S. Kostka and D. Payne, Tonal Harmony, with an introduction to twentieth century music, Sixth Edition, McGraw-Hill, 2009. i i i i i i i i 3.1 Intervals and Chords 53 counting of spaces and lines, and half steps, is always done from the lower pitch note to the higher pitch note. We summarize the main types of harmonic intervals in Figure 3.3. Min. 2nd Maj. 2nd Min. 3rd Maj. 3rd Per. 4th Aug. 4th Per. 5th Min. 6th Maj. 6th Min. 7th Maj. 7th Octave +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 Figure 3.3. Some harmonic intervals. In Figure 3.3, we did not include a unison interval because only one staff (for one instrument) was displayed. In scores where multiple instruments are employed, it is possible for two instruments to play the same notes simultaneously. The interval between these identical notes is a unison. 3.1.2 Chords When multiple pitches are played simultaneously, we say thata chord is being played. While the two notes in a harmonic interval can be thought of as a chord (called a dyadic or 2-note chord), the most important chords in diatonic music are the triadic chords. Triadic chords are 3-note chords. The most basic triadic chords in diatonic music are created by combining two harmonic intervals that are thirds. If the lowest pitch note and the middle pitch note form a major third, and the middle pitch note and the highest pitch note form a minor third, then the chord is called a major chord. If the lowest pitch note and the middle pitch note form a minor third, and the middle pitch note and the highest pitch note form a major third, then the chord is calleda minor chord. For example, in Figure 3.4 there are two chords with their notes and the half steps separating them, and a diagram illustrating how they are diagrammed on the chromatic clock. The first chord has a G A +3 ↑ +4 ↑ E F +4 ↑ +3 ↑ C D Figure 3.4. C-major chord and D-minor chords in root position, and their diagrams on the chromatic clock. major third between its lowest pitch note and middle pitch note, and a minor third between its middle pitch note and its highest pitch note. Therefore it is a major chord. It is called a C-major chord. The second chord has a minor third between its lowest pitch note and middle pitch note, and a major third between its middle pitch note and its highest pitch note. So it is a minor chord. It is called a D-minor chord. This diagram illustrates the following principle: Reading clockwise on the chromatic clock, a major chord always has 4 hours, followed by 3 hours, separating its notes. While a minor chord always has 3 hours, followed by 4 hours, separating its notes. Using this principle, we can identify underlying chords in musical passages by diagramming them on the chromatic clock. Examples of i i i i i i i i 54 3.