Inverting Cage's Sqare Root Form/Wilson's Geometries

It has taking me a while to figure this out. The answer finally came when I was working on another problem. My brain is just not that linear.

Some time back while looking at Takemitsu’s application of Cage’s square root idea to Pitch, I stumbled upon its relationship to certain geometric structures of Erv Wilson. Pitch or duration doesn’t matter in that the structure is the same. I posted this to the Silence list that I was on at the time. I repeat the jest of that below.

This is the simplest example using only 4 durations. This gives us a good visual representation as to what happens with more complex examples. Here is the resulting square using A B C D to be what ever you want.

AA AB AC AD

BA BB BC BD

CA CB CC CD

DA DB DC DD

Now compare this with this excerpt from one of Wilson’s letters to John Chalmers found here
http://anaphoria.com/CPStoC-pt1.PDF shown above

One has only to replace the number by the letters.
What interests me though is the second diagram on this page.
Which shows a possible variation of Cage’s square root idea in reverse with each successive element having the reverse relationship above. To make this square, one uses the complementary elements not included in the original. This is what i had missed despite being right there in front of me.

BCD CD BD BC

DC ACD AD AC

DB DA ABD AB

BC CA BA ABC

It is interesting to observe where the two intersect which he does on the next page.






















More often than not such a form is used with more elements. With 6 elements [here using the first 6 odd numbers] we can see below what happens as a like wise inter structure and its complements.




The thing about the set made by the intersection forms a cycle and makes it easy to proceed to the next element in a smooth fashion. same with the other ones i mention later.
There is no reason at this point no to consider the combinations of 3 out of 6 and if we look at those points where it interests with it complement we get the Eikosany. One of Erv’s Favorite structures. Which you can find it the article linked above.

I had a dream two nights ago where i had my Meru bars in a pentagram which i post next. Perhaps my mind is more linear than i thought.