Z-relations

For rank-1 Z-relations, there are many species (perhaps infinite, because new kinds get created the higher up one goes in -tET.....)

However, I seem to be getting a handle on classification. The first thing I have noticed is that Z-relations based on complementation are a breed of their own (such as 12,6 hexads)
however, even these can be classified (there are 3 flavors of Z-relations here:

1. 1-tritone sets, where you reflect one pitch.
2. 1-tritone sets, where you reflect two pitches.
3. 2-tritone sets, where you reflect one pitch (same as 1)
4. 2-tritone sets, with a more complicated algorithm, (however, same for remaining sets)

So of the 15 Z-related pairs here, there are 4 in class 1, 4 in class 2, 1 in class 3 and 6 in class 4. For class 1 to 3, you reflect an element or two by a tritone, across a wedge of symmetry, with 1 or 2 tritones held fixed. For class 4, it's more complicated, with one element going down 2 and one going up 2. More later...

However, for the simpler noncomplemental type of Z-relation, you get better patterns.
in 12,5 there are two pairs that merely do the tritone reflect thing. There is one more unusual one which reflects by 2 pi / 3 (4 pitches) across a point with a triune fixed. (0 4 8)

I thought this might be of the augmented (128/125) variety but it is just plain old meantone like all the others (tritone motion). in 12,4 there is just one, with tritone motion and intvec <111111> (A FLID)

This can be extended to rank-2 and higher, by considering Z instead of just Z/nZ. I'll leave that to a colleague.

I have found that by spelling sides of a Z-relation in terms of difference sets one can relate the Z-relation itself by a comma, say meantone. Here is the 12,5 case mentioned above:

0 4 8 1 3 -> 0 4 8 3 5. Notice how 3 acts as a fixed mirror (hyperplane) and 1 reflects through it to 5, a motion of 4, or 2 pi / 3 with 0 4 8 fixed (a triune). Now a good spelling is
1/1, 5/4, 25/16, 25/24, 225/192 -> 1/1, 5/4, 25/16, 225/192, 125/92.

Long story short, is that you merely get 9/8 -> 10/9 for elements of the "odd plane"
so that is clearly a meantone Z-relation!!!

More on odd and even planes later, they can be manipulated by D4 X S3, but I digress.

pgh

PS I have been adumbrated by Mike with respect to rank-2 and so forth, his development.