THE SUPERIMPOSITON OF SYMMETRICAL FIGURES
I will now illustrate how one can utilize symmetrical figures superimposed over an existing symmetry. Say for example that we have a pentatonic mode built around the tritone D-#G. The mode has the formula I-II-#IV-#V-VI.
By superimposing a symmetrical figure over D and then #G we have the possibilitiy to extend the mode with related notes. These notes can be freely chosen however must utilize the established symmetry and must be symmetrically constructed. An example can be as little as two notes per symmetric member or as much as one wishes, even up to an endless string of notes. Nicolas Slonimsky had written an entire Thesaurus of such figures.
The mode in lettered notes reads as such: D-E-#G-#A-B. A superimposed symmetrical figure could be as simple as D-bE-#G-A.
This is a simple and direct superimposition. One can as well take the figure and invert the intervallic relationship from #G giving us a variant for example. D-bE-#G –G. Both of these figures are for me superimposed symmetrical extensions.
One other variant would be to substitute an augmented triad because of its symmetrical relationship to the tritone and build an elaborate figure based on graduated major thirds. An example in letters would be D-F-E-bA-bD-C and now I transpose and invert the figure on the major third #F-#D-E-C-G-bA and then I chose to transpose the figure in its original form once again a major third higher bB-bD-C-E-A-bA.
Now the cycle is complete. We can use this method vertically or horizontally.
CANTUS SYMMETRICUS AND THE TWELVE TONE ROW
As one can read with my third scored example, I have a chain of modes lineally connected through their symmetrical relationships. These modes form a structured matrix upon one can write freely a superimposed twelve-tone technique.
Interesting enough one has the perfect equality of notes as practiced in the twelve tone technique and for example, a bass voice utilizing the chaining together of modes connected with the basic principle of controlled transposition as stated in the beginning of the theory.
The twelve-tone technique can of course be transposed as wished and built out of the notes from the given motives of a given piece. Therewith one can be assured of a motivic and harmonic unity. We can as well take the modal structures and build harmonies and superimpose the existing twelve tone rows.
POLYMODAL EXTENSION AS PRACTICED IN MODERN JAZZ
Many times I truly wondered where McCoy Tyner’s 70ties music derived its notes from. A very clear example would be looking at the undertoning of fourths where the usual blues scale would be for example in Cminor.
An example looks as such. C with a fourth underneath it, G and once again then D. If we have a blues melody C-bE-F-bG-F then the combination will be triads built in fourths playing three modes at once.
D-G-C; F-bB-bE; G-C-F;
bA-bD-bG; G-C-F.
One can as well make quadrads with this technique. bD-E-A-C; E-G- C-bE; bG-A-D-F; G-bB-bE-bG; bG-A-D-F. This polymodal technique bases itself on given triads that may appear even in complex structures. An example could be a C7b9-#11-6 chord. The three notes at the top of the chord form an F#minor chord. One can use exactly the same technique with polymodal extension over F#minor and therefore over the C7b9-#11-6 chord.
A very simple example is the CMAJ7 chord. It has as well inclusive with the third, fifth and seventh, the triad Eminor in its basic structure. It is practiced often that a jazz musician improvises Eminor blues over the CMAJ7 chord.