Symmetry of fifthchord coefficients of stability
The instability of a fifthchord can be measured according to its similarity with the tritone triangle (Despi\' c, 1987). Based on this, we define the coefficient of instability n of a fifthchord as the number of common tones it shares with a tritone triangle (i.e., as a coefficient of similarity of a fifthchord in regard to tritone triangle). Vice versa, the coefficient of stability of a fifthchord can be expressed by the formula s=3n ^{ 1}. Thus the stability coefficient of the tonic fifthchord is 3, for the subdominant it is 2, and for a dominant it is 1. Consequently, the harmonic crescendo in the case of a major key has the form TSD and in the case of a minor key, the tritone occurs on the II degree, whose secondary function is s (stritone), which produces a harmonic crescendo TDS. The corresponding stability coefficient s can be found for any fifthchord on any degree. For example, in the case of a major key, s(\textII)=1, as the fifthchord which has 2 common tones with a tritone triangle is on the II degree (its instability coefficient is n=2, so s=1). In this manner we get the following table of stability coefficients s of the fifth chords that are positioned on different degrees of the major and minor key.
By using an analogous procedure it is possible to study the seventhchord structure of natural scales. Of course, the transition to seventhchords will not have any impact on harmonic functions. The table of the stability coefficient S=4N of the seventhchords is as follows:
Besides dtritone on the VII degree, stritone appears on the II degree in the harmonic minor (Fig. 5.4)
(a)
(b) Figure 5.4 A diagram of the interval structure of the harmonic minor (a) and melodic minor (b). In the case of harmonic minor, by using the proposed procedure we get a diagram of the symmetry of interval structure. In this diagram an identification of complementary intervals has been made, i.e., the principle of minimal samename intervals has been applied. In this diagram we can directly read the fifthchord structure of the harmonic minor. The appearance of the tritone rectangle IIIVVIVII is evident at first sight. The instability of the fifthchord on certain degrees is expressible in terms of their similarity with the tritone rectangle. As a result we get the following table of the stability coefficients of the fifthchords:
In this table we note the complete stability of the tonic fifthchord and the uniformity of the kinetic role of the subdominant and dominant function. As for the stability of the fifthchord, the function of the VI degree is equally close to the tonic and subdominant function. However, the distinctive stability of the seventhchord on the VI degree (S=3) prevails in favor of the tonic function. The stability coefficients of seventhchord S are as follows:
Based on this table we can deduce that the harmonic crescendo in the case of the harmonic minor will have the form TSD. As opposed to the asymmetric interval structure of the harmonic minor (Fig. 5.4.a), the melodic minor has a symmetric interval structure (Fig. 5.4.b). There appears the axis of symmetry which contains the V degree and passes between the I and II degree. In regards to the symmetry of the fifthchord structure, in the melodic minor the ambivalent symmetric function of the III degree stands out. Both the axis of symmetry for the VI and VII degree (dimdim) and the axis of antisymmetry for the V and I, and the IV and II degree (majmin) cross it simultaneously (Fig. 5.5). In the melodic minor the dtritone on the VII degree represents a source of instability. The ttritone in the diminished fifthchord on the VI degree has an inferior role, so that when calculating the coefficient of stability it occurs only as a correctional factor on the VI degree. Accordingly, the coefficients of stability of fifthchords s and the coefficients of stability of the seventhchords S have the following values:
Based on this, the form of the harmonic crescendo is TSD.
Figure 5.5 The ambivalent symmetry function of the III degree of the melodic minor. ^{ 1}For the tritone fifthchord n equals 3. Consequently, its stability coefficient is s=0.
