BACKGROUND
On the 3^3, many people use variants of a 3-cycle of edges,
more-or-less as follows:

let Rs = Right slice = R'L + roll cube so front is up

T  (for tri-cycle?) = Rs U2 Rs' U2  = (UF,DF,UB)  
                            to cycle 3 edges on a slice

Some people use it in the form
      R2 D' - T - D R2 = (UF,UR,UB) to get 3 on a face, without

flips, and some use

Rs' U - T - U' Rs = Rs' U Rs U2 Rs' U Rs = (UF,LU,RU)

for 3 on a face, with flips, since it saves a move due to

       U2 U' = U.
END BACKGROUND

Now, this move, when transferred to the 4^3, seems to be the basis
of both Richard Pavelle's "S" move of 16-May, and Bernie Cosell's "M"
move of 17-May (Bernie using a left slice version of the third form).
The move Roger Frye describes on 7-June is also the same as this
(re-oriented to a different face), and so is the "quite long" tool I
use. To move only edges requires S3, and to move only centers
requires S5!!! I still would like a "nice" (preferably short) center
pair move.

SPOILER          SPOILER             SPOILER

We have now several 3-cycles:

(UBl,UFr,LDf)  =  U2 f' D f - U2 f' D' f      (from Minh Thai's book)

(UFl,BUl,RUb)  =  l' - R U R' U' - l - U R U' R' (Minow, 6-June)

(RDf,URf,UFl)  =  R' d R - U' - R' d R - U       (Frye, 7-June)

Note that all of these have corresponding 3-cycles of centers, by
simply substituting a slice for its' corresponding edge in each of the
moves. For instance, in Thai, substitute d for D; in Minow, r for R,
etc. There is also a version which cycles an edge an center together,
by rotating the appropriate face and slice together.
In fact, it looks as though any simple move on the 4^3 will have a
left and right version, a forward and back version, and a slice and
edge version or two, along with combinations of these. Is there any
way to put these into a canonical form so we can recognize related
moves? What about canonical form on the 3^3?
One more spoiler: to switch opposite centers (all 4 cubies):

(F,B)  = (r2 U2 l2 U2)3.

END SPOILER          END SPOILER        END SPOILER

Can the notation be usefully extended so as to talk about UB as
the pair of upper back edges, F as the front 4 center cubies, and Fu
as the upper two of F, etc. It might make notating the permutations
easier. Is there any notation to make various repeats easier; for
instance, some expansion of "squared" to indicate "primed"
(commutator) type repeat, or a repeat with all quarter twists in the
opposite direction. Maybe ( )2 means repeat; ( )i2 means repeat
with sub-parts "primed", and ( )-2 repeats with qtws opposite. The
purpose is to try to make it more obvious what each move is really
doing, and to be able to compare moves easier.
Has anybody given any thought to notation the 5^3?
By the way, if Bill Mann is listening, would you describe your
transform (mentioned by Edmond on 23-May)?
--- Stan
-------