THE CURIOUS CASE OF CYCLIC SYMMETRIC OCTATONIC SCALES

If the interval set the symmetric scale also repeats itself in an exact manner over the entire sound spectrum then we can describe these scales as both cyclists and symmetric. For practical scales there are two closely related octatonic scales that would satisfy this definition. These are generated by the interval sets O1=[12211221] and O2=[21122112]. In the investigation of the properties of these two scales we are going to define S as the "tonic" of a scale which is simply the starting tone. S will take on the values 0, to 11 which corresponds to the 12 tones of the tempered scale with C=0. The only operation were going to use is transposition represented by T number of semitones. We can define any octatonic scale by the 4-member set X of the tones that are not present in the scale.

If we investigate O1 generated scales, we observed that if S is even then the members of the set X are also even and if S is odd then the members of X are also odd. Suppose the scale is transposed by T; if S is odd and T is even then X is odd. Similarly, if both are odd or even then X is even and if one is odd the other is even then X is odd. Using this information we can generate a table of X sets for each starting tone.

The generation of such a table has surprising results mixed with the obvious:

a) We need only 6 X sets to cover the entire 12 tones.

b)  Any tone transposed by six semitones has the identical set X.

c) No special techniques would be required to use the results.

                        Table 1. The generating sets for O1 scales:

                                        S                       X

                                  0   and   6         2   4   8   10

                                  2   and   8         0   4   6   10

                                  4   and  10        0   2   6     8

                                  1   and    7        3   5   9   11

                                  3   and    9        1   5   7   11

                                  5   and  11        1   3   7     9

 Suppose we would like to generate the CSS for F#. F# =  6  thus we have:

            0 -  1  -  2 – 3    –   4 – 5 – 6   -  7 – 8 – 9 – 10 – 11

            C    C#        D#            F    F#    G        A           B   è  F# G  A B C C# D# F F#

It is as simple as that.

A close examination of the table suggests that with the use of few accidentals we can manoeuvre over the entire set of harmonic relationships with relative ease and smooth transition from one region (tonality) to another. The reason for the ease in the transitions is the additional shared members of some of the X sets. For example, 1 and 9 share 5 and 11. Therefore transposition from 1 to 9 would require moving C# to D# and G to A, the rest of the notes stay the same. Here we note that if he transformation is within the same class i,e, odd to odd and even to even, the transformation would not be noticeable except for uncommonly attentive listeners, If transformation is between classes, the contrast would be strong and it would be very pronounced to be recognisable except for uncommonly tone deaf.

If we apply the same methods of calculating X sets for O2 we find out that the two formulations are nearly the same except for O2 giving opposite results for the odd even values of S. Therefore the transformations and going from one to the other for the two scales could be reduced to the table given below:

Table 1. The generating sets for O1 and O2 scales:

            O1              O2                           X

       0   and   6        3   and   9        2   4   8   10

       2   and   8        5   and  11       0   4   6   10

       4   and  10       1   and    7       0   2   6     8

       1   and   7        4   and   10      3   5   9   11

       3   and   9         0   and    6       1   5   7  11

       5   and 11        2   and    8       1   3   7    9

One might be tempted to say that O2 is obtained from O1 by minor 3^rd transposition. But, we will resist the temptation and discuss a composition application.

An example of the use of both cyclic symmetric scales is available here:

https://soundcloud.com/naesmen/enigmatic-symmetry-op-90

This composition process did not require any special adjustments for my normal routine except that I paid closer attention to the rhythmic structure. The harmonic structure more or less fell into to the rhythmic frame work. Initially, selecting a harmonic flow required by the flow of the music called for considerable amount of piano time but eventually it was possible to compose as I usually do directly to the paper. As a demonstration, the piano piece was necessarily kept simple in many aspects. However having used the scales once, I feel that it has considerable dramatic potential and it is intuitive  in the sense that it covers by and large, a well-travelled  ground by many 20^th/21^st  century musicians. The outcome, as you will hear in the example piece, can be as traditional as one chooses to make it so. On the other hand there is no limitation in generating very interesting harmonic structures – of course this depends on the composer's choices.

As always we would be more than happy to answer any questions on the above.