Flying through the expanded Magic Squares.

Etienne Deleflie is a composer working at the University of Wollongong on compositions dealing with the perception and simulation of spatial orientation of sound. He originally asked for help in how to map Patch’s Scale onto a matrix that lead to a deeper and deeper joint exploration of some other intonational possibilities.


It quickly became apparent that he envisioned a just intonational matrix in which one could 'fly' through, as a space where nearby harmonies would have some relation to each other yet still retain a special identity in the overall field to aid in the sense of movement. This reawakened my own interest in parallel ideas of unifying space and pitch.


Those familiar with lambdomas [of which Partch’s diamond is an example] would recognize this as a logical choice, but these result with everything higher on one end. Thus the results are rather unbalanced and also lack a harmonic variety we sought for.


The situation instead seemed ripe for other types of intonational possibilities and I thought that some of the simpler recurrent sequences would work well.


So with a bit of experimenting I found that 'magic squares' placed in the middle provided good seeds for making different series running in four directions . Nearby tones would retain various difference tone reinforcements yet over ‘space’ one would move away or toward where one started harmonically in a significant yet varied way. Yet there was something musically appealing to having certain harmonics stand out and even be repeated yet then diverge in different directions from there.


After going into the anechoic chamber where Deleflie has his speakers set up and listening to some of the results, we found it warranted pursuing an actual installation with small speakers in a space where people can walk through. We are working toward this at the moment.


While this idea continues to develop further, this diagram shows some of the ways I am constructing my matrices. This one serves as a good illustration in that it includes more than one series embedded into it. [Click to enlarge.]

The numbers refer to harmonics (that in turn have to be multiplied to get them in the hearing range). First I started with a magic square made of numbers 4-12 that one can see in the outlined box in the near middle. From here I construct recurrent sequences for each direction of two varieties.

I illustrate.

Take a number, say 5, in the box and call this A with B and C being the numbers above it. Now if we create a sequence such as A + C = D we take 5 + 9 =14 which is the number above 9. Now if we move the sequence up we add 10 + 14 = 24 and we continue this pattern. Next the same formula is applied moving to the right where 5 + 7 = 12 and 12 + 12 = 24 etc. For the other two directions I used a different recurrent sequence. This one is A + B = D. So in this case we add 7 + 12 = 19 and 12 + 5 = 17 in the case of the bottom row moving left or in the case of moving down 9 +10 = 19 and 10 + 5 =15 etc. These two sequences I learned from Erv Wilson who uses them to create his Meta-Pelog and Meta-Slendro scales. He might be the first to have found them embedded in Pascal's Triangle or Meru Prastara, as it was known centuries earlier . His papers on this can be seen here.

You will notice that being a magic square the number 24 comes out being the sum of the row or columns in the box, and if one follows any of the lines in either direction where one has three 24s in a row you might notice we get simple harmonics of this 24. Quickly each quadrant deverges in its own unique way on either side of these rows. The corners become the highest notes with the center the lowest, which is a useful balanced arc. Yet it is possible to treat this whole series as a subharmonic series where the corners become the lowest and the center the highest.

Etienne has realized this using SuperCollider. The piece is now available here https://anaphoria.bandcamp.com/album/second-flight-over-an-island-interior