Musical Set Theory and M12

I have found that any set of 12 pentad transpositions in the M12 group (produced by Steiner(5,6,12) will use exactly the 12 different pitches of 12-tET in its "supernote" (the sixth note, which is determined by the first five.

I am working on a proof, although intuitively it makes complete sense.
What it comes down to is this: Since you cannot have a fixed pentad,
and two different "supernotes" (Because the pentad would be in two
different Steiner sets, which is against the definition), you also
cannot have two different pentad transpositions against the same
fixed note (kinda like set theory "relativity"). I misplaced the proof
(which is not an earth-shattering one by any means)

This is helping to make sense of some of my other theories (expounded on in tuning under Musical Set Theory and 12-tET and some here too), but
as a side bar, what this also shows is that any pentad type (there are 66) can be used to construct a 12-tET Twelve-tone row. The order of
the row could merely be that shown on my spreadsheet (Files Paul Hj Stuff 110209.xls, 022310 tab (I am just dating everything now) with
my ABCD nomenclature, in a kind of canonical ordering.

You could also match the transposes of a given pentad one to one
with the permuted sixth-notes of the Steiner system like (0,5), (1,4),
(2,3), (3,6), etc. and use that for something. Of course the choice
of 0 in the pentads is completely arbitrary but the choice in the sixth
note completely literal. One way would be to use the root of the
(0,6) tritone in the first position as 0 ("C"). This is also indicated
by "C" in my nomenclature for a tritone in a (0-6)-slot.

PGH

PGH

--- In tuning-math@yahoogroups.com, "paulhjelmstad" <phjelmstad@...> wrote:
>
> I have found that any set of 12 pentad transpositions in the M12 group (produced by Steiner(5,6,12) will use exactly the 12 different pitches of 12-tET in its "supernote" (the sixth note, which is determined by the first five.
>
> I am working on a proof, although intuitively it makes complete sense.
> What it comes down to is this: Since you cannot have a fixed pentad,
> and two different "supernotes" (Because the pentad would be in two
> different Steiner sets, which is against the definition), you also
> cannot have two different pentad transpositions against the same
> fixed note (kinda like set theory "relativity"). I misplaced the proof
> (which is not an earth-shattering one by any means)
>
> This is helping to make sense of some of my other theories (expounded on in tuning under Musical Set Theory and 12-tET and some here too), but
> as a side bar, what this also shows is that any pentad type (there are 66) can be used to construct a 12-tET Twelve-tone row. The order of
> the row could merely be that shown on my spreadsheet (Files Paul Hj Stuff 110209.xls, 022310 tab (I am just dating everything now) with
> my ABCD nomenclature, in a kind of canonical ordering.
>
> You could also match the transposes of a given pentad one to one
> with the permuted sixth-notes of the Steiner system like (0,5), (1,4),
> (2,3), (3,6), etc. and use that for something. Of course the choice
> of 0 in the pentads is completely arbitrary but the choice in the sixth
> note completely literal. One way would be to use the root of the
> (0,6) tritone in the first position as 0 ("C"). This is also indicated
> by "C" in my nomenclature for a tritone in a (0-6)-slot.
>
> PGH

I guess this isn't always true. I found the use of
0,0,1,1,2,4,6,6,7,7,8,10 Where there are three hexad types:
R1 (1,7) J1 (0,1,4,6,7,10) Z (0,2,6,8) (Set and sixth note)
Curiously, if you move R1 to (2,8) and Z to (3,5,9,11)it works.

PGH