Formalization of direct diatonic modulations by means of seventh-chords

 

Seventh-chords: (4,3,4) - major seventh (formally "major/major seventh", denoted as "maj/maj"), (4,3,3) - dominant seventh (denoted as "dom"), (3,4,4) - major seventh with minor third (formally "minor/major seventh", denoted as "min/maj"), (3,4,3) - minor seventh (formally "minor/minor seventh", denoted as "min/min"), (3,3,4) - half diminished seventh (formally "diminished/minor seventh", denoted as "dim/min"), (4,3,3) - augmented seventh (formally äugmented/major seventh", denoted as äug/maj"), and (3,3,3) - diminished seventh (formally "diminished/diminished seventh", denoted as "dim/dim") may also serve as a basis for modulation. Starting from an equivalent initial table we can derive corresponding tables of direct diatonic modulations by means of seventh-chords which are common for the initial and the final tonality.

major

minor

h. minor

m. minor

I

0

maj/maj

0

min/min

0

min/maj

0

min/maj

II

2

min/min

2

dim/min

2

dim/min

2

min/min

III

4

min/min

3

maj/maj

3

aug/maj

3

aug/maj

IV

5

maj/maj

5

min/min

5

min/min

5

dom

V

7

dom

7

min/min

7

dom

7

dom

VI

9

min/min

8

maj/maj

8

maj/maj

9

dim/min

VII

11

dim/min

10

dom

11

dim/dim

11

dim/min

i=2,5,7

 

major

®

major

i=5

maj/maj

IV-I

i=2

min/min

III-II

i=5

min/min

VI-III

i=7

min/min

VI-II

i=2,5,7

 

natural minor

®

natural minor

i=5

maj/maj

VI-III

i=2

min/min

V-IV

i=5

min/min

IV-I

i=7

min/min

V-I

i=2

 

melodic minor

®

melodic minor

i=2

dom

V-IV

i=2

dim/min

VII-VI

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

 

major

®

natural minor

i=4

maj/maj

I-VI

i=9

maj/maj

I-III

i=9

dom

V-VII

i=2

min/min

II-I

VI-V

i=4

min/min

III-I

VI-IV

i=7

min/min

II-V

i=9

min/min

II-IV

III-V

VI-I

i=11

min/min

III-IV

i=9

dim/min

VII-II

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

 

natural minor

®

major

i=3

maj/maj

III-I

i=8

maj/maj

VI-I

i=3

dom

VII-V

i=1

min/min

VI-III

i=3

min/min

I-VI

IV-II

V-III

i=5

min/min

V-II

i=8

min/min

I-III

IV-VI

i=10

min/min

I-II

V-VI

i=3

dim/min

II-VII

i=0,4,9,11 (9)

 

major

®

harmonic minor

i=4

maj/maj

I-VI

i=0

dom

V-V

i=9

min/min

II-IV

i=11

min/min

III-IV

i=9

dim/min

VII-II

i=0,1,3,8 (3)

 

harmonic minor

®

major

i=8

maj/maj

VI-I

i=0

dom

V-V

i=1

min/min

IV-III

i=3

min/min

IV-II

i=3

dim/min

II-VII

i=0,2,7 (0,2)

 

major

®

melodic minor

i=0

dom

V-V

i=2

dom

V-IV

i=0

min/min

II-II

i=2

min/min

III-II

i=7

min/min

VI-II

i=0

dim/min

VII-VII

i=2

dim/min

VII-VI


i=0,5,10 (0,10)

 

melodic minor

®

major

i=0

dom

V-V

i=10

dom

IV-V

i=0

min/min

II-II

i=5

min/min

II-VI

i=10

min/min

II-III

i=0

dim/min

VII-VII

i=10

dim/min

VI-VII


i=0,2,3,7 (0)

 

natural minor

®

harmonic minor

i=0

maj/maj

VI-VI

i=7

maj/maj

III-VI

i=3

dom

VII-V

i=0

min/min

IV-IV

i=2

min/min

V-IV

i=7

min/min

I-IV

i=0

dim/min

II-II


i=0,5,9,10 (0)

 

harmonic minor

®

natural minor

i=0

maj/maj

VI-VI

i=5

maj/maj

VI-III

i=9

dom

V-VII

i=0

min/min

IV-IV

i=5

min/min

IV-I

i=10

min/min

IV-V

i=0

dim/min

II-II


i=3,5,8 (3,5)

 

natural minor

®

melodic minor

i=3

dom

VII-V

i=5

dom

VII-IV

i=3

min/min

IV-II

i=5

min/min

V-II

i=8

min/min

I-II

i=3

dim/min

II-VII

i=5

dim/min

II-VI


i=4,7,9 (7,9)

 

melodic minor

®

natural minor

i=7

dom

IV-VII

i=9

dom

V-VII

i=4

min/min

II-I

i=7

min/min

II-V

i=9

min/min

II-IV

i=7

dim/min

VI-II

i=9

dim/min

VII-II


i=0,3,5 (0,3,5)

 

harmonic minor

®

melodic minor

i=0

dom

V-V

i=0

min/maj

I-I

i=3

min/min

IV-II

i=3

dim/min

II-VII

i=5

dim/min

II-VI

i=0

aug/maj

III-III


i=0,7,9 (0,7,9)

 

melodic minor

®

harmonic minor

i=0

dom

V-V

i=0

min/maj

I-I

i=9

min/min

II-IV

i=7

dim/min

VI-II

i=9

dim/min

VII-II

i=0

aug/maj

III-III


By comparing the results obtained for fifth-chords and seventh-chords we can identify both their similarities and their specific differences. Where major and minor fifth-chords are concerned there is a balance as to their frequency: each of them occurs in brief in all scales ten times. As for the frequency of their occurrence, the minor seventh-chord dominates among all the seventh-chords, occurring eight times. In regards to symmetry, the results for the fifth-chords and the seventh-chords also differ. The antisymmetric arrangement of the major and minor fifth-chords within major and natural minor scales (Fig. 5.3) has caused a special kind of (anti)symmetry in the corresponding tables of direct diatonic modulations. A lower level of symmetry of seventh-chord structure (Fig. 5.15) has caused a lower level of symmetry in the corresponding tables of modulation by means of seventh-chords. They satisfy only the (anti)symmetry that occurs in the inverted tables in regards to the inversion of intervals (complementarity of intervals i and 12-i).

As to their application, the tables of direct diatonic modulations by means of seventh-chords offer the same possibilities as the previous ones. The difference between them consists of the fact that because of the limited possibilities of direct diatonic modulations, the more frequent use of indirect diatonic modulations is necessary. For example, there is no direct diatonic modulation (by means of a seventh-chord) from harmonic minor to harmonic minor, etc.

As an illustration of the application of the obtained results we will show indirect diatonic modulations by means of a seventh-chord from harmonic C minor to harmonic E minor. Since i=4, the following modulations are eligible:

1) modulation (i1,i2)=(0,4) by means of C major;

2) modulation (i1,i2)=(9,7) by means of natural E minor.

In the first case the sum total of the coefficients of similarity equals 8, and in the second, 10. Thus, the modulation by means of natural A minor is the optimal modulation (by means of a seventh-chords) from harmonic C minor to harmonic E minor. It renders two modulations:

1) i1=9 dom V-VII, i2=7 maj/maj III-VI;

2) i1=9 dom V-VII, i2=7 min/min I-IV.

These modulations are carried out in the following manner:

  1. by means of the dominant seventh-chord 19,23,26,29 (G2-B2-D3-F3) positioned on the V degree of the harmonic D minor and on the VII degree of natural A minor, then by means of the major/major seventh-chord 12,16,19,23 (C2-E2-G2-B2), positioned on the III degree of natural A minor and on the VI degree of harmonic E minor;

  2. by means of the minor/minor seventh-chord 9,12,16,19 (A1-C2-E2-G2), positioned on the I degree of natural A minor and on the IV degree of harmonic E minor.

The combinations of diatonic modulations by means of fifth-chords and seventh-chords are also possible, as are the constructions of various modulation sequences, etc. The expansion of the proposed system for the construction of diatonic modulations to ninth-chords is very simple, but offers few possibilities because of the limited choice of possible intervals for direct and, consequently, for indirect diatonic modulations.