A simple example is where a four note motif is transformed using a structure called the hexany. The hexany basically take the combination of 2 points out of 4 and condenses each into a single location that functions as both. I relate the method here to those cubist drawings that use a single line can represent both the edge of a face seen looked at forward or to the side. There are many examples of multiple meaning in line and forms in this style. Wilson's paper above working in 19 ET shows how the hexany can be used as the basic of generating a diamond or lambdoma of itself.
|Template and Example of Melodic Hexany|
Since in melody one deals with the relative position of notes to each other, it is possible to have a single note have more than one relative positions to others. Starting with a four note motif, I will show one way it can be transformed in a simple form of which the reader can go into as much depth as one desires. each tone in the six tone figure can function as two different relative points in the melody. but will function as the two opposite points in regard to the inversion. without getting bogged down in any more technicalities than needed is the point that the hexany will exhibit properties of 4 note motif while decentering it position and location. in other words it properties become spread out over larger pitch terrain.
The same can be applied to a scale here shown in terms of a pentatonic. In 12 ET one will more often than not will have notes duplicated which in a sense undermines the purity of the structure but still with preserve an unfolding of the scale as if it is turned inside out. Here are example below starting with a pentatonic A B C# E F also showing it in rotation and with a chart accenting some 8 tone cycles within these 10 note dekanies i happen to have on hand
|rotations of the A B C# E F type pentatonic with examples of some 8 tone patterns found within|
The process can be pushed even to heptatonic scales but many of the Combination product sets will involve up to 35 tones with 3 out of 7 set. Included is an example of one with the 2 out of 7 set which encompasses 21 tones. Interesting even though notes are repeated, where they are will more often than not will include those relationships not used elsewhere, so even if one uses the same note, one will be using it in a different context. The 3 out of 7 template and blank are included for ones own use and exploration. Such structures can also be used for rhythmic relationships but beyond of what we will cover in this blog.
|Example of the 2 out of 7 set seeded with a heptatonic scale|
|Blank 2 out of 7 set lattice|
|Lattice of 3 out of 7 set from Wilson Hebdomekontany paper.|
|Blank 3 out of 7 lattice|