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:
As a result of this procedure we get chords with identified samename 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 doubletoned fivethree chord, shown in an ascending sequence of tonal pitches y_{1}, y_{2}, y_{3}, y_{4} is located in the octave position if i_{14}=y_{4}y_{1}=0 mod 12, in the third position if i_{14}=3,4 mod 12, and in the fifth position if i_{14}=6,7 mod 12. Our next goal is to determine the chord position and its basic tone and type. In the case of the fivethree chord and its inversions, by looking at intervals i_{12}=y_{2}y_{1} and i_{23}=y_{3}y_{2}, we can very easily come to the following conclusions:
We can determine the type of fivethree chord very easily by bringing it down to its basic position and by reducing its tonal pitches y_{1}, y_{2}, y_{3} 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 (i_{12},i_{23}) 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 seventhchord given by an ascending sequence of tonal pitches y_{1}, y_{2}, y_{3}, y_{4} which determines intervals i_{12}, i_{23} and i_{34}=y_{4}y_{3}, we conclude that:
After this we bring it down to its basic position, reduce the tonal pitches modulo 12 and determine the type of seventhchord: 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 ninthchords 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.
^{
1}As 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).
