Formalization of direct diatonic modulations by means of
seventhchords
Seventhchords: (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 seventhchords 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 

IVI 
i=2 
min/min 

IIIII 
i=5 
min/min 

VIIII 
i=7 
min/min 

VIII 
i=2,5,7
natural minor


® 
natural minor

i=5 
maj/maj 

VIIII 
i=2 
min/min 

VIV

i=5 
min/min 

IVI 
i=7 
min/min 

VI 
i=2
melodic minor


® 
melodic minor

i=2 
dom 

VIV 
i=2 
dim/min 

VIIVI

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

® 
natural minor



i=4 
maj/maj 

IVI 


i=9 
maj/maj 

IIII 


i=9 
dom 

VVII 


i=2 
min/min 

III 
VIV 

i=4 
min/min 

IIII 
VIIV 

i=7 
min/min 

IIV 


i=9 
min/min 

IIIV 
IIIV 
VII 
i=11 
min/min 

IIIIV 


i=9 
dim/min 

VIIII 


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


® 
major 


i=3 
maj/maj 

IIII 


i=8 
maj/maj 

VII 


i=3 
dom 

VIIV 


i=1 
min/min 

VIIII 


i=3 
min/min 

IVI 
IVII

VIII 
i=5 
min/min 

VII 


i=8 
min/min 

IIII 
IVVI 

i=10 
min/min 

III 
VVI 

i=3 
dim/min 

IIVII 


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

® 
harmonic minor

i=4 
maj/maj 

IVI 
i=0 
dom 

VV 
i=9 
min/min 

IIIV 
i=11 
min/min 

IIIIV

i=9 
dim/min 

VIIII 
i=0,1,3,8 (3)
harmonic minor


® 
major 
i=8 
maj/maj 

VII 
i=0 
dom 

VV 
i=1 
min/min 

IVIII 
i=3 
min/min 

IVII 
i=3 
dim/min 

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

® 
melodic minor

i=0 
dom 

VV 
i=2 
dom 

VIV 
i=0 
min/min 

IIII 
i=2 
min/min 

IIIII

i=7 
min/min 

VIII 
i=0 
dim/min 

VIIVII 
i=2 
dim/min 

VIIVI 
i=0,5,10 (0,10)
melodic minor


® 
major 
i=0 
dom 

VV 
i=10 
dom 

IVV 
i=0 
min/min 

IIII 
i=5 
min/min 

IIVI 
i=10 
min/min 

IIIII 
i=0 
dim/min 

VIIVII 
i=10 
dim/min 

VIVII 
i=0,2,3,7 (0)
natural minor


® 
harmonic minor

i=0 
maj/maj 

VIVI 
i=7 
maj/maj 

IIIVI 
i=3 
dom 

VIIV 
i=0 
min/min 

IVIV

i=2 
min/min 

VIV 
i=7 
min/min 

IIV 
i=0 
dim/min 

IIII 
i=0,5,9,10 (0)
harmonic minor


® 
natural minor

i=0 
maj/maj 

VIVI 
i=5 
maj/maj 

VIIII

i=9 
dom 

VVII 
i=0 
min/min 

IVIV 
i=5 
min/min 

IVI 
i=10 
min/min 

IVV 
i=0 
dim/min 

IIII 
i=3,5,8 (3,5)
natural minor


® 
melodic minor

i=3 
dom 

VIIV 
i=5 
dom 

VIIIV 
i=3 
min/min 

IVII 
i=5 
min/min 

VII 
i=8 
min/min 

III 
i=3 
dim/min 

IIVII

i=5 
dim/min 

IIVI 
i=4,7,9 (7,9)
melodic minor


® 
natural minor

i=7 
dom 

IVVII 
i=9 
dom 

VVII 
i=4 
min/min 

III 
i=7 
min/min 

IIV 
i=9 
min/min 

IIIV 
i=7 
dim/min 

VIII 
i=9 
dim/min 

VIIII 
i=0,3,5 (0,3,5)
harmonic minor


® 
melodic minor

i=0 
dom 

VV 
i=0 
min/maj 

II 
i=3 
min/min 

IVII 
i=3 
dim/min 

IIVII 
i=5 
dim/min 

IIVI 
i=0 
aug/maj 

IIIIII 
i=0,7,9 (0,7,9)
melodic minor


® 
harmonic minor

i=0 
dom 

VV 
i=0 
min/maj 

II 
i=9 
min/min 

IIIV 
i=7 
dim/min 

VIII 
i=9 
dim/min 

VIIII 
i=0 
aug/maj 

IIIIII 
By comparing the results obtained for fifthchords and
seventhchords we can identify both their similarities and their
specific differences. Where major and minor fifthchords 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 seventhchord dominates among all the
seventhchords, occurring eight times. In regards to symmetry, the
results for the fifthchords and the seventhchords also differ. The
antisymmetric arrangement of the major and minor fifthchords 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 seventhchord structure
(Fig. 5.15) has caused a lower level of symmetry in the
corresponding tables of modulation by means of seventhchords. They
satisfy only the (anti)symmetry that occurs in the inverted tables
in regards to the inversion of intervals (complementarity of
intervals i and 12i).
As to their application, the tables of direct diatonic modulations
by means of seventhchords 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
seventhchord) 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 seventhchord
from harmonic C minor to harmonic E minor. Since i=4, the
following modulations are eligible:
1) modulation (i_{1},i_{2})=(0,4) by means of C major;
2) modulation (i_{1},i_{2})=(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
seventhchords) from harmonic C minor to harmonic E minor. It
renders two modulations:
1) i_{1}=9 dom VVII, i_{2}=7 maj/maj IIIVI;
2) i_{1}=9 dom VVII, i_{2}=7 min/min IIV.
These modulations are carried out in the following manner:

by means of the dominant seventhchord 19,23,26,29 (G^{2}B^{2}D^{3}F^{3}) 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 seventhchord 12,16,19,23 (C^{2}E^{2}G^{2}B^{2}),
positioned on the III degree of natural A minor and on the VI
degree of harmonic E minor;

by means of the minor/minor seventhchord 9,12,16,19
(A^{1}C^{2}E^{2}G^{2}), 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 fifthchords
and seventhchords 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
ninthchords is very simple, but offers few possibilities because of
the limited choice of possible intervals for direct and,
consequently, for indirect diatonic modulations.
