The symmetry of tonality

In the chapters The Symmetry of Melodic Structure and The Symmetry of Harmony we have looked at the versatility of fifth-chords - the appearance of the same fifth-chord within different tonalities. The versatility of fifth-chords is the basis of diatonic modulation - the transition from one tonality to another within a certain piece which takes place with the help of a common fifth-chord.

We will show a table of the distribution of major (4,3) (maj), minor (3,4) (min), diminished (3,3) (dim) and augmented (4,4) (aug) fifth-chords within the major, natural minor, harmonic minor and melodic minor keys 1. We will include the interval that builds the initial tone of the chord with the initial tone of the scale and the type of chord:

 major minor h. minor m. minor I 0 maj 0 min 0 min 0 min II 2 min 2 dim 2 dim 2 min III 4 min 3 maj 3 aug 3 aug IV 5 maj 5 min 5 min 5 maj V 7 maj 7 min 7 maj 7 maj VI 9 min 8 maj 8 maj 9 dim VII 11 dim 10 maj 11 dim 11 dim

Each of the scales is defined by its initial tonal pitch y0 (y0=0,1,...,11) and a corresponding sequence of intervals (0,2,4,5,7,9,11), (0,2,3,5,7,8,10), (0,2,3,5,7,8,11) or (0,2,3,5,7,9,11).

All diatonic modulations can be classified into two basic groups:

1. direct, and

2. indirect.

In the case of direct modulations the transition is carried out directly from one tonality to another with the help of a common fifth-chord. If direct modulation is not possible, i.e., if the initial and final tonality do not have a common fifth-chord, we resort to indirect modulation. This is done by using a tonality which connects the initial with the final tonality (has common fifth-chords with both the initial and the final tonality). Another criterion for the classification of direct modulations is a type of fifth-chord (maj, min, dim, aug) which is a means for completing the modulation. Since every indirect modulation represents the result of two or more direct modulations, we may speak of combined types of modulation (e.g., maj-min, min-min,...). In accordance with the harmonic function of individual degrees of the scale of modulation we can classify direct modulations according their type T-T, T-D,... from the aspect of their tonal functions, as well as indirect modulations of the type T-D/T-T, etc. Finally, it is also possible to study modulation sequences.

1As in the previous chapters, the terminology and the denotation has been taken from the works of D. Despic (1971, 1981, 1987, 1989).