Formalization of direct diatonic modulations by means of fifth-chords

Diatonic modulations are based on the invariance of the fifth-chord - the preserving of the fifth-chord (as a sequence of tonal pitches) during the transition from one tonality to another. We will first examine all possibilities for a direct diatonic modulation. After that, we may carry out each indirect modulation as a combination of two or more direct ones.

We will analyze direct diatonic modulations on the example of major-major modulations. In this case there are six transformations of major-to-major fifth-chords:

1. by transposing a scale for the interval i=2, the major fifth-chord on the V degree is pronounced the major fifth-chord on the IV degree, which can symbolically be denoted as i=2 maj V-IV;

2. by transposing the scale for the interval i=5, the major fifth-chord on the IV degree is pronounced the major fifth-chord on the I degree (i=5 maj IV-I);

3. by the same transposition of a scale for the interval i=5, the major fifth-chord on the I degree is pronounced the major fifth-chord on the V degree (i=5 maj I-V);

4. by transposing the scale for the interval i=7, the major fifth-chord on the V degree is pronounced the major fifth-chord on the I degree (i=7 maj V-I);

5. by the same transposition of the scale for the interval i=7, the major fifth-chord on the I degree is pronounced the major fifth-chord on the IV degree (i=7 maj I-IV);

6. by transposing the scale for the interval i=10, the major fifth-chord on the IV degree is pronounced the major fifth-chord on the V degree (i=10 maj IV-V).

Possible values of the transposition interval are i=2,5,7,10. This means that by means of a major fifth-chord a modulation can be carried out from one major tonality (with an initial tone pitch of Y0) to another tonality (with an initial tonal pitch of Y1), if the interval i=Y1-Y0 equals 2,5,7, or 10. The mentioned direct modulation is carried out by using major fifth-chords which correspond to the values of interval i. For example, a possible direct modulation is from C major to G major by using the major fifth-chord, since their initial tonal pitches Y0=C1=0 and Y1=G1=7 make the interval i=Y1-Y0=7. This direct modulation can be carried out by means of the major fifth-chord 7,11,14 (G1-B1-D2) which is positioned on the V degree of C major and on the I degree of G major, or by means of the fifth-chord 12,16,19 (C2-E2-G2) positioned on the I degree of C major and on the IV degree of G major.

In the same manner we can find all diatonic modulations from one major tone into another by means of a minor fifth-chord. In this case, too, there are six transformations of minor fifth-chords into minor ones:

1. by transposing the scale for the interval i=2, the minor fifth-chord on the III degree is pronounced the minor fifth-chord on the II degree (i=2 min III-II);

2. by transposing the scale for the interval of i=5, the minor fifth-chord on the VI degree is pronounced the minor fifth-chord on the III degree (i=5 min VI-III);

3. by the same transposition of the scale for the interval i=5, the minor fifth-chord on the II degree is pronounced the minor fifth-chord on the VI degree (i=5 min II-VI);

4. by transposing the scale for the interval i=7, the minor fifth-chord on the VI degree is pronounced the minor fifth-chord on the II degree (i=7 min VI-II);

5. by the same transposition of the scale for interval i=7, the minor fifth-chord on the V degree is pronounced as the minor fifth-chord on the VI degree (i=7 min III-VI);

6. by transposing the scale for the interval i=10, the minor fifth-chord on the II degree is pronounced the minor fifth-chord on the III degree (i=10 min II-III).

We note that the values of the transposition intervals i=2,5,7,10 are the same as in the previous case. Also, (anti)symmetry in respect to interval inversion (the complementarity of intervals i and 12-i) holds as well. Both properties are the results of an antisymmetric arrangement of the major and minor fifth-chords within major and minor scales (Fig. 5.3). The result thus obtained means that, by means of a minor fifth-chord, a modulation from one major tonality (with an initial tonal pitch of Y0) to another major tonality (with an initial tone of Y1) can be carried out if interval i=Y1-Y0 equals 2,5,7, or 10.

In summarizing our results we may conclude the following: in order to find out whether direct major-to-major modulation is possible, it is sufficient to verify if the initial tones Y0 and Y1 of these tonalities make up the interval i=2,5,7, or 10. For example, a direct modulation from E major to F-sharp major is possible, because their initial tonal pitches Y0=E1=4 and Y1=F-sharp1=6 together make up the interval i=Y1-Y0=2. This direct modulation may be carried out by means of a major fifth-chord 11,15,18 (B1-D-sharp 2-F-sharp 2), which is positioned on the V degree of E major and on the IV degree of F-sharp major, or by means of a minor fifth-chord 8,11,15 (G-sharp 1-B1-D-sharp 2) which is positioned on the III degree of E major and on the II degree of F-sharp major.

As a final result we get the table of direct major-major modulations which can be realized only for the values i=2,5,7,10.

 major ® major i=2 maj V-IV i=5 maj IV-I I-V i=7 maj V-I I-IV i=10 maj IV-V i=2 min III-II i=5 min VI-III II-VI i=7 min VI-II III-VI i=10 min II-III

A table of direct natural minor to natural minor modulations is obtained by an analogous procedure. In this case the direct modulations can be carried out for the values of i=2,5,7,10.

 natural minor ® natural minor i=2 maj VII-VI i=5 maj VI-III III-VII i=7 maj VII-III III-VI i=10 maj VI-VII i=2 min V-IV i=5 min IV-I I-V i=7 min V-I I-IV i=10 min IV-V

Direct diatonic modulations from harmonic minor to harmonic minor or from melodic minor to melodic minor allow the use of the diminished (dim) fifth-chord as a means of modulation. The result is a table of direct harmonic minor to harmonic minor modulations which are possible only for the values i=1,5,7,11 (3,9). The values of interval i for diminished chords are given in parentheses. They allow direct modulations by means of a diminished fifth-chord.

 harmonic minor ® harmonic minor i=1 maj VI-V i=11 maj V-VI i=5 min IV-I i=7 min I-IV i=3 dim II-VII i=9 dim VII-II

An analogous table of direct melodic minor to melodic minor modulations for the values of i=2,10 (2,10) is:

 melodic minor ® melodic minor i=2 maj V-IV i=10 maj IV-V i=2 min II-I i=10 min I-II i=2 dim VII-VI i=10 dim VI-VII

Since all previous tables refer to diatonic modulation from one tonic material to the same tonic material (major
® major, natural minor ® natural minor, etc.), in all cases (anti)symmetry occurs in regards to the inversion of intervals (the complementarity of intervals i and 12-i). This is not true in the case of diatonic modulations to a tonic material different from the initial one (e.g., major ® natural minor, etc.). However, antisymmetry in respect to major and minor fifth-chords will still occur.

The following is a table of direct major to natural minor modulations for i=2,4,7,9,11 (9):

 major ® natural minor i=2 maj I-VII IV-III i=4 maj I-VI V-III i=7 maj IV-VII i=9 maj I-III IV-VI V-VII i=11 maj V-VI i=2 min II-I VI-V i=4 min III-I VI-IV i=7 min II-V i=9 min II-IV III-V VI-I i=11 min III-IV i=9 dim VII-II

Owing to the fact that all major and natural minor scales consist of the same interval material, an inverted table of direct modulations from natural minor to major is antisymetric with the previous one in regard to the inversion of intervals (i.e. the complementary intervals i and 12-i). For the values of i=1,3,5,8,10 (3) the table is as follows:

 natural minor ® major i=1 maj VI-V i=3 maj III-I VI-IV VII-V i=5 maj VII-IV i=8 maj III-V VI-I i=10 maj III-IV VII-I i=1 min IV-III i=3 min I-VI IV-II V-III i=5 min V-II i=8 min I-III IV-VI i=10 min I-II V-VI i=3 dim II-VII

By looking at this table we may deduce that, for example, a direct diatonic modulation from C-sharp minor to A major is possible, since Y0=C-sharp 1=1, Y1=A1=9, i=Y1-Y0=8. This modulation can be derived in the following ways:

1. by means of major fifth-chord 16,20,23 (E2-G-sharp2-B2) positioned on the III degree of C-sharp minor and on the V degree of A major;

2. by means of major fifth-chord 9,13,16 (A1-C-sharp 2-E2) positioned on the VI degree of C-sharp minor and on the I degree of A major;

3. by means of minor fifth-chord 13,16,20 (C-sharp 2-E2-G-sharp 2) positioned on the I degree of C-sharp minor and on the III degree of A major and

4. by means of minor fifth-chord 18,21,25 (F-sharp 2-A2-C-sharp 3) positioned on the IV degree of C-sharp minor and on the VI degree of A major.

Analogous tables have been made for all possible direct diatonic modulations:

i=0,2,4,5,9,10,11 (0,9)

 major ® harmonic minor i=0 maj V-V i=4 maj I-VI i=5 maj I-V i=9 maj IV-VI i=10 maj IV-V i=11 maj V-VI i=2 min II-I i=4 min III-I VI-IV i=9 min II-IV VI-I i=11 min III-IV i=0 dim VII-VII i=9 dim VII-II

i=0,1,2,3,7,8,10 (0,3)

 harmonic minor ® major i=0 maj V-V i=1 maj VI-V i=2 maj V-IV i=3 maj VI-IV i=7 maj V-I i=8 maj VI-I i=1 min I-II i=3 min IV-II I-VI i=8 min I-III IV-VI i=10 min I-II i=0 dim VII-VII i=3 dim II-VII

i=0,2,4,5,7,9,10 (0,2)

 major ® melodic minor i=0 maj IV-IV V-V i=2 maj V-IV i=5 maj I-V i=7 maj I-IV i=10 maj IV-V i=0 min II-II i=2 min II-I i=4 min III-I i=7 min VI-II i=9 min VI-I i=0 dim VII-VII i=2 dim VII-VI

i=0,2,3,5,7,8,10 (0,10)

 melodic minor ® major i=0 maj IV-IV V-V i=2 maj V-IV i=5 maj IV-I i=7 maj V-I i=10 maj IV-V i=0 min II-II i=3 min I-VI i=5 min II-VI i=8 min I-III i=10 min I-II i=0 dim VII-VII i=10 dim VI-VII

i=0,1,2,3,5,7,8 (0,3)

 natural minor ® harmonic minor i=0 maj VI-VI i=1 maj VI-V i=2 maj VII-VI i=3 maj VII-V i=7 maj III-VI i=8 maj III-V i=0 min I-I IV-IV i=2 min V-IV i=5 min IV-I i=7 min I-IV V-I i=0 dim II-II i=3 dim II-VII

i=0,4,5,7,9,10,11 (0,9)

 harmonic minor ® natural minor i=0 maj VI-VI i=4 maj V-III i=5 maj VI-III i=9 maj V-VII i=10 maj VI-VII i=11 maj V-VI i=0 min I-I IV-IV i=5 min IV-I I-V i=7 min I-IV i=10 min IV-V i=0 dim II-II i=9 dim VII-II

i=0,1,3,5,7,8,10 (3,5)

 natural minor ® melodic minor i=1 maj VI-V i=3 maj VI-IV VII-V i=5 maj VII-IV i=8 maj III-V i=10 maj III-IV i=0 min I-I i=3 min IV-II i=5 min IV-I V-II i=7 min V-I i=3 dim II-VII i=5 dim II-VI

i=0,2,4,5,7,9,11 (7,9)

 melodic minor ® natural minor i=2 maj IV-III i=4 maj V-III i=7 maj IV-VII i=9 maj IV-VI V-VII i=11 maj V-VI i=0 min I-I i=5 min I-V i=7 min I-IV II-V i=9 min II-IV i=7 dim VI-II i=9 dim VII-II

i=0,1,2,3,5,10 (0,2,3,5)

 harmonic minor ® melodic minor i=0 maj V-V i=1 maj VI-V i=2 maj V-IV i=3 maj VI-IV i=0 min I-I i=3 min IV-II i=5 min IV-I i=10 min I-II i=0 dim VII-VII i=2 dim VII-VI i=3 dim II-VII i=5 dim II-VI i=0 aug III-III

i=0,2,7,9,10,11 (0,7,9,10)

 melodic minor ® harmonic minor i=0 maj V-V i=1 maj VI-V i=0 maj V-V i=9 maj IV-VI i=10 maj IV-V i=11 maj V-VI i=0 min I-I i=2 min II-I i=7 min I-IV i=9 min II-IV i=0 dim VII-VII i=7 dim VI-II i=9 dim VII-II i=10 dim VI-VII i=0 aug III-III