Recently, I have been so busy that I overlooked the blog a bit too much. Even if I had no time to post anything, I didn’t put my ideas on the back burner and now I am ready to share a few of them. Let’s start considering the first three hexachord of the mirrored palindromic series that I showed in this post: <5,8,9,7,6,3>, <2,5,6,4,3,0>, and <1,4,5,3,2,11>. If we consider the pitch class integers (C=0, C#=1 … A#=10, B=11) of a hexachord as units of linear distance (with C=0 defined as 12 units) and align the resulting three distance sequences one after the other in a proportional way, we obtain a multiple alignment like the following.
Note that every unit (the space between two blocks) is meant to describe the temporal extent of some event (e.g., a sound, or a silence). Thus the very last three blocks of such a multiple alignment are actually the starting points of the first three units of a new alignment. Moreover, the number of units between two positions of the alignment depends on the metric choice. For instance, considering a quarter note as unit, the distance between the sixth block of the first sequence of distances (position 160 in the alignment) and the third block of the second sequence of distances (position 76 in the alignment) is a half note and 5 thirty-second notes (2 + 5/8) according to the metric of the first sequence of distances (i.e., 5 units between position 1 a 160). The distance between the same two blocks is approximately 2 + 1/5 according to the metric of the second sequence of distances, and approximately 1 + 2/5 + 1/7 according to the metric of the third sequence of distances.
Of course it is possible to measure the linear distances between all points of the alignment but it is also necessary to introduce some kind of approximation to avoid complex musical notation. We can approximate those distances using the closest option available on a preordered table of note values and derive three different patterns starting from the original multiple alignment.
After converting all the linear distances into note values, we can decide which kind of event (sound, silence, dynamics, etc.) must be bound to the fundamental elements of each pattern. In the following example, I used the third pattern as template and assigned dynamics, articulations and rests respectively to the elements of the distance sequences derived from the third (blue blocks), the second (red blocks), and the first (black blocks) hexachord. By way of information, the serialization of these three parameters were obtained from the diagonals of the matrix of permutations of the generating row.