The symmetry of tonality

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:

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;
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.