The symmetry of harmony

In the previous chapters we have introduced the concepts of intervals and chords. In the course of their study we will apply two types of reduction:

1. vertical analytic reduction (reduction of doubling and reduction of span);

2. horizontal analytic reduction (condensing of arpeggiated chords) (Despic, 1987) 1.

As a result of this procedure we get chords with identified same-name tones whose span does not exceed an octave (12 semitones). In the further course of our presentation it will be implied that all intervals and chords are given in their reduced form.

A double-toned five-three chord, shown in an ascending sequence of tonal pitches y1, y2, y3, y4 is located in the octave position if i14=y4-y1=0 mod 12, in the third position if i14=3,4 mod 12, and in the fifth position if i14=6,7 mod 12.

Our next goal is to determine the chord position and its basic tone and type. In the case of the five-three chord and its inversions, by looking at intervals i12=y2-y1 and i23=y3-y2, we can very easily come to the following conclusions:

1. if i12 < 4 and i23 < 4, the five-three chord is in its basic position, thus its basic tone is y1;

2. if 5 < i12 < 6, we are talking about the first inversion of the five-three chord (the six-three chord) and, consequently, its basic tone is y3;

3. if 5 < i23 < 6, this is the second inversion of the five-three chord (six-four chord) and its basic tone is y2.

We can determine the type of five-three chord very easily by bringing it down to its basic position and by reducing its tonal pitches y1, y2, y3 modulo 12. As a result we get the following chords: major triad (0,4,7) (maj), minor triad (0,3,7) (min), diminished triad (0,3,6) (dim), and augmented triad (0,4,8) (aug) which can also be expressed in the form (i12,i23) respectively as (4,3), (3,4), (3,3) and (4,4). Both forms will be used equally in our thesis, without special remarks.

Similarly, in the case of a seventh-chord given by an ascending sequence of tonal pitches y1, y2, y3, y4 which determines intervals i12, i23 and i34=y4-y3, we conclude that:

1. if all three intervals i12, i23, and i34 are greater than 2, then the seventh-chord is in its basic position and its basic tone is y1;

2. if i34=1,2, then we are dealing with its first inversion, the six-five chord. Consequently, its basic tone is y4;

3. if i23=1,2, then we are dealing with its second inversion, the four-three chord. Consequently, its basic tone is y3;

4. if i12=1,2, then we are dealing with its third inversion, the four-two chord. Consequently, its basic tone is y2.

After this we bring it down to its basic position, reduce the tonal pitches modulo 12 and determine the type of seventh-chord: major/major seventh (4,3,4) (maj/maj), major seventh with minor third (3,4,4) (min/maj), dominant seventh (4,3,3) (dom), minor/minor seventh (3,4,3) (min/min), diminished/minor seventh (3,3,4) (dim/min), diminished/diminished seventh (3,3,3) (dim/dim), and augmented/major seventh (4,4,3) (aug/maj). A similar procedure can be applied to ninth-chords of which there are 12 types.

Because we are using numeric notation for chords, we will not distinguish diatonic chords and altered chords of diatonic type.

1As in previous chapters, the terminology has been taken from D. Despi's book, Harmonic Analysis (1987). The musical and theoretical precepts used in this chapter are based on the works of the following authors: D. Despic (1971, 1981, 1989), O.L. Shrebkova and S.S. Shrebkov (1952), Yu. Holopov (1974), V. Berkov (1980), L.A. Mazel and V.A. Zuckerman (1967), L. Mazel (1979) and N. Cook (1987).