Indirect modulations by means of fifthchords and their optimization
When direct diatonic modulation by means of a common fifthchord is not possible, the need for an indirect diatonic modulation arises. For example, a direct diatonic modulation from C major to Eflat 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 majortomajor 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 i_{1} which allows the modulation from the initial major tonality (C major) to the intermediary major tonality to occur, and interval i_{2} which allows the modulation from the intermediary major tonality to the final major tonality (Eflat major), so that i_{1}+i_{2}=1 mod 12. Since the possible values for i_{1} and i_{2} are 2,5,7,10, and i=3, the following pairs only are eligible: (i_{1},i_{2})=(5,10), or (10,5) (because i=i_{1}+i_{2}=5+10=10+5=3 mod 12). We can read the indirect modulations for (i_{1},i_{2})=(5,10) directly from the table for majortomajor direct modulations. As the result, we obtain following indirect modulations: 1) 2) i_{1}=5 maj IVI IV, i_{2}=10 maj IVV; 3) 4) i_{1}=5 maj IVI IV, i_{2}=10 min IIIII; 5) 6) i_{1}=5 min VIII IIIVI, i_{2}=10 maj IVV; 7) 8) i_{1}=5 min VIII IIIVI, i_{2}=10 min IIII Since i_{1}=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 Eflat major by means of F major:
For the values of (i_{1},i_{2})=(10,5) we get the next eight modulations from C major to Eflat major by means of Bflat major.
Figure 6.1. Three indirect modulations from C major to Eflat major by means of F major. If we wish to use a natural minor as the intermediary tonality, we must choose interval i_{1} which will allow the modulation from a major key into a natural minor and interval i_{2}, which will allow the modulation from the natural minor key into the major, so that i_{1}+i_{2}=1 mod 12. Since the possible values for i_{1} are 2,4,7,9,11 (9), and for i_{2} they are 1,3,5,8,10 (3), while i=3, the following pairs are eligible: (i_{1},i_{2})=(2,1),(7,8) (because i=i_{1}+i_{2}=2+1=7+8=3 mod 12). We can read the indirect modulations for (i_{1},i_{2})=(2,1) from the tables for direct modulations from major to natural minor and from natural minor to major: 1) 2) 3) 4) i_{1}=2 maj IVII IVIII, i_{1}=2 min III VIV; 5) 6) 7) 8) i_{2}=1 maj VIV, i_{2}=1 min IVIII. Because i_{1}=2, the modulation is carried out by means of D minor. This means that for the values of (i_{1},i_{2})=(2,1) we can carry out eight different modulations from C major to Eflat major by means of the natural D minor: 1) i_{1}=2 maj IVII, i_{2}=1 maj VIV; 2) i_{1}=2 maj IVII, i_{2}=1 maj IVIII; 3) i_{1}=2 maj IVIII, i_{2}=1 maj VIV; 4) i_{1}=2 maj IVIII, i_{2}=1 min IVIII; 5) i_{1}=2 min III, i_{2}=1 maj VIV; 6) i_{1}=2 min III, i_{2}=1 min IVIII; 7) i_{1}=2 min VIV, i_{2}=1 maj VIV; 8) i_{1}=2 min VIV, i_{2}=1 min IVIII. The modulations are carried out in the following order:
Another sequence of indirect modulations from C major to Eflat major is carried out for the values of (i_{1},i_{2})=(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 Eflat major (i=3), these are the following possibilities: 1) modulations by means of the major key (i_{1},i_{2})=(5,10),(10,5); 2) modulations by means of the natural minor (i_{1},i_{2})=(2,1),(7,8); 3) modulations by means of the harmonic minor (i_{1},i_{2})=(0,3),(2,1),(5,10); 4) Modulations by means of the melodic minor (i_{1},i_{2})=(0,3),(5,10),(7,8),(10,5); or: 1) modulations by means of F major and Bflat 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 Bflat 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 (i_{1},i_{2})=(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 (i_{1},i_{2})=(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 (i_{1},i_{2})=(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 (i_{1},i_{2})=(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 Eflat major, by means of F major, Bflat major, natural D minor, natural G minor and melodic Bflat 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 (i_{1},...,i_{n}), in which case i_{1}+...+i_{n}=i mod 12, where i_{1},...,i_{n} 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.
