Indirect modulations by means of fifth-chords and their optimization

When direct diatonic modulation by means of a common fifth-chord is not possible, the need for an indirect diatonic modulation arises. For example, a direct diatonic modulation from C major to E-flat major is not possible, since their initial tonal pitches form the interval i=3 which does not belong to the set of interval values (i=2,5,7,10) that allow direct major-to-major modulation. An indirect modulation can be carried perfected by means of different tonalities.

This indirect modulation may be carried out by means of a major tonality. If we wish to employ a major as our indirect tonality, we must choose the interval i1 which allows the modulation from the initial major tonality (C major) to the intermediary major tonality to occur, and interval i2 which allows the modulation from the intermediary major tonality to the final major tonality (E-flat major), so that i1+i2=1 mod 12. Since the possible values for i1 and i2 are 2,5,7,10, and i=3, the following pairs only are eligible: (i1,i2)=(5,10), or (10,5) (because i=i1+i2=5+10=10+5=3 mod 12).

We can read the indirect modulations for (i1,i2)=(5,10) directly from the table for major-to-major direct modulations. As the result, we obtain following indirect modulations:

1) 2) i1=5 maj IV-I I-V, i2=10 maj IV-V;

3) 4) i1=5 maj IV-I I-V, i2=10 min II-III;

5) 6) i1=5 min VI-II III-VI, i2=10 maj IV-V;

7) 8) i1=5 min VI-II III-VI, i2=10 min II-II

Since i1=5, indirect modulation occurs in all eight cases by means of F major. As a result we get the following direct modulations from C major to E-flat major by means of F major:

1. C major to F major modulation, by means of major fifth-chord 5,9,12 (F1-A1-C2) positioned on the IV degree of C major and the I degree of F major, then the modulation from F major to E-flat major by means of major fifth-chord 10,14,17 (B-flat1-D2-F2) positioned on the IV degree of F major and the V degree of E-flat major;

2. C major to F major modulation by means of the major fifth-chord 12,16,19 (C2-E2-G2) positioned on the I degree of C major and the V degree of F major, then the modulation from F major to E-flat major by means of the major fifth-chord 10,14,17 (B-flat1-D2-F2) positioned on the IV degree of F major and the V degree of E-flat major;

3. C major to F major modulation by means of the major fifth-chord 5,9,12 (F1-A1-C2) positioned on the IV degree of C major and the I degree of F major, then the modulation from F major to E-flat major by means of the minor fifth-chord 7,10,14 (G1-B-flat 1-D2) positioned on the II degree of F major and the III degree of E-flat major, etc. (Fig. 6.1)

For the values of (i1,i2)=(10,5) we get the next eight modulations from C major to E-flat major by means of B-flat major.

Figure 6.1. Three indirect modulations from C major to E-flat major by means of F major.

If we wish to use a natural minor as the intermediary tonality, we must choose interval i1 which will allow the modulation from a major key into a natural minor and interval i2, which will allow the modulation from the natural minor key into the major, so that i1+i2=1 mod 12. Since the possible values for i1 are 2,4,7,9,11 (9), and for i2 they are 1,3,5,8,10 (3), while i=3, the following pairs are eligible: (i1,i2)=(2,1),(7,8) (because i=i1+i2=2+1=7+8=3 mod 12).

We can read the indirect modulations for (i1,i2)=(2,1) from the tables for direct modulations from major to natural minor and from natural minor to major:

1) 2) 3) 4) i1=2 maj I-VII IV-III, i1=2 min II-I VI-V;

5) 6) 7) 8) i2=1 maj VI-V, i2=1 min IV-III.

Because i1=2, the modulation is carried out by means of D minor. This means that for the values of (i1,i2)=(2,1) we can carry out eight different modulations from C major to E-flat major by means of the natural D minor:

1) i1=2 maj I-VII, i2=1 maj VI-V;

2) i1=2 maj I-VII, i2=1 maj IV-III;

3) i1=2 maj IV-III, i2=1 maj VI-V;

4) i1=2 maj IV-III, i2=1 min IV-III;

5) i1=2 min II-I, i2=1 maj VI-V;

6) i1=2 min II-I, i2=1 min IV-III;

7) i1=2 min VI-V, i2=1 maj VI-V;

8) i1=2 min VI-V, i2=1 min IV-III.

The modulations are carried out in the following order:

1. by means of the major fifth-chord 12,16,19 (C2-E2-G2) on the I degree of C major and the VII degree of D minor, then by means of the major fifth-chord 10,14,17 (B-flat 1-D2-F2) positioned on the VI degree of D minor and the V degree of E-flat major;

2. by means of the major fifth-chord 12,16,19 (C2-E2-G2) positioned on the I degree of C major and the VII degree of D minor, then by means of major fifth-chord 7,11,14 (G1-B1-D2) positioned on the IV degree of D minor and on the III degree of E-flat major;

3. by means of major fifth-chord 5,9,12 (F1-A1-C2) positioned on the IV degree of C major and the III degree of D minor, then by means of major fifth-chord 10,14,17 (B-flat 1-D2-F2) positioned on the VI degree of D minor and the V degree of E-flat major, etc. (Fig. 6.1).

Another sequence of indirect modulations from C major to E-flat major is carried out for the values of (i1,i2)=(7,8) by means of the natural G minor.

The application of the given tables of direct diatonic modulations helps us find all possible indirect modulations from one tonality to another. For example, in the case of C major and E-flat major (i=3), these are the following possibilities:

1) modulations by means of the major key (i1,i2)=(5,10),(10,5);

2) modulations by means of the natural minor (i1,i2)=(2,1),(7,8);

3) modulations by means of the harmonic minor (i1,i2)=(0,3),(2,1),(5,10);

4) Modulations by means of the melodic minor (i1,i2)=(0,3),(5,10),(7,8),(10,5);

or:

1) modulations by means of F major and B-flat major;

2) modulations by means of natural D minor and G minor;

3) modulations by means of harmonic C minor, D minor and F minor;

4) Modulations by means of melodic C minor, F minor, G minor and B-flat minor.

We may also want to exactly determine the optimal indirect diatonic modulation, i.e., the indirect modulation by means of a tonality which is maximally related (similar) to the initial and final tonality. It is necessary in this case to determine the coefficient of similarity (p. 55) of the initial and the intermediary tonality and the coefficient of similarity between the intermediary and final tonality, and then choose the direct modulations with the maximum sum total of the said coefficients. In the previous example we get the following result:

1) in the case of modulations by means of a major (i1,i2)=(5,10), (10,5), the sum of the coefficients of similarity is 11;

2) in the case of modulations by means of a natural minor (i1,i2)=(2,1),(7,8), the sum of the coefficients of similarity is 11;

3) in the case of modulations by means of a harmonic minor (i1,i2)=(0,3),(2,1),(5,10), the sum of the coefficients of similarity is 8,6,7, respectively;

4) in the case of modulations by means of a melodic minor (i1,i2)=(0,3),(5,10),(7,8),(10,5) the sum of the coefficients of similarity is 9,8,8,11, respectively.

Thus, there are five optimal direct modulations from C major to E-flat major, by means of F major, B-flat major, natural D minor, natural G minor and melodic B-flat minor.

The construction of diatonic modulation sequences is based on the same principles as those of direct diatonic modulation. Here we are dealing with a sequence of modulations (i1,...,in), in which case i1+...+in=i mod 12, where i1,...,in is a sequence of intervals determined by the initial tonal pitches of consecutive intermediate tonalities, and interval i is determined by the initial tonal pitches of the initial and final tonality. We may also speak of optimal modulation sequences. These are modulation sequences with a maximum sum total of corresponding similarity coefficients.