mapping and indexing two step size (L&s) cardinality

I've been having an off-list correspondence with Joe Monzo dealing
with some of the posts I've done recently about mapping and indexing
two step size (L&s) cardinality, and I've decided to go ahead and post
some of this as it might better explain some particular aspect of what
I've previously posted... Before I joined the TD last January I just
sort of groped through my own ways doing and (ahem...) "explaining"
things, and while there's little doubt that for the most part I still
do; I'd also hope that a year of being exposed to people laying things
out in a coherent and intelligible manner has rubbed off at least a
little bit! (And if this isn't the case - which it probably isn't,
well I still feel as though I've really benefited from the exposure
anyway...)

[Joe:]
> Your original explanation to me of how the periodic table
> works said that it might be clearer if I relate it to the
> table which appears later in your post, showing the
> 1/35-'octave' space. But it wasn't. That only mystified
> me further, as I saw no connection whatever between the
> periodic table and the 1/35 diagram.

Ah ha! Think of the two fifths, 3/5 (at 720�) and 4/7 at (~686�). Now
I'm sure that this diagram will make sense with that in mind... as
each fraction represents the fifth size over the EDO, or put another
way the linear interval capable of creating the scale (which in this
case is seven note scale with five large and two small step sizes) --
or that falls into this 1/35 space between 3/5 (at 720�) and 4/7 at
(~686�) -- and the EDO that contains it.

(4/7) (3/5)
\ /
7/12
/ \
10/17 11/19
/ \ / \
13/22 14/24 15/26
/ \ / \ / \
16/27 17/29 18/31 19/33
/ \ / \ / \ / \
19/32 20/34 21/36 22/38 17/40
...

Another way to look at this (without the linear interval clearly
illuminated) is:

(5) (7)
(10) 12 (14)
(15) 17 19 (21)
(20) 22 24 26 (28)
(25) 27 29 31 33 (35)
(30) 32 34 36 38 40 42
(35) 37 39 41 43 45 47 49
40 42 44 46 48 50 52 54 56
45 47 49 51 53 55 57 59 61 63
50 52 54 56 58 60 62 64 66 68 70
...

This also shows how the "periodic table" expands towards "L" without
the truncation... The vertical and horizontal expansions could in a
sense be seen as the cases of maximum ambiguity in a given mapping,
as s=0 and s=L&L=s are what the horizontal (or diagonal) and vertical
expansions are working towards, i.e., L (equal) and L+s (equal). And
while the truncation -- where the table is seen as a horizontal
sequence of -s (mod L+s) truncated by letting 0=L+s -- addresses the
horizontal (or diagonal) expansion, I'm somewhat inclined to think
that the vertical expansion is probably best addressed by commonsense
and ones personal inclination to work (or not work as the case may be)
with larger and larger EDO's... If I were to show the previous example
with the truncation it would look like this:

(5) (7)
(10) 12 (14)
(15) 17 19 (21)
(20) 22 24 26 (28)
(25) 27 29 31 33 (35)
(30) 32 34 36 38 40
(35) 37 39 41 43 45 47
42 44 46 48 50 52 54
49 51 53 55 57 59 61
56 58 60 62 64 66 68
63 65 67 69 71 73 75
...

Compare that with this:

(5) 3 1 6 4 2 7
12 (10) 8 13 11 9 14
19 17 (15)(20) 18 16 21
26 24 22 27 (25) 23 28
33 31 29 34 32 (30)(35)
40 38 36 41 39 37 42
47 45 43 48 46 44 49
54 52 50 55 53 51 56
61 59 57 62 60 58 63
...

Now just as the mediants, or "scale tree" like example I gave of this
before:

(3/5) (4/7)
7/12
10/17 11/19
13/22 17/29 18/31 15/26
16/27 23/39 27/46 24/41 25/43 29/50 26/45 19/33
...

affords a very efficient view of what's going on (i.e., only the two
borders between which the linear interval must fall and then the EDOs
which don't represent duplicate mappings, in other words 12 & 24e at
2212221 and 4424442, etc.), the "periodic table" allows one to see
everything of a given mapping at a glance... i.e., the EDOs that won't
work (which could be seen as both a handy convenience as well as a
possible stable framework from which to index two step size
cardinality in a given mapping), the borders (which are the expansions
"L" and "L"+"s"), and the EDOs that do "work."

And while we've been pretty consistently sticking to the seven note
scale with five (identical) large and two (identical) small step
sizes, (i.e., a 5L&2s mapping) as an example, L&s are just variables
here, and the 5L&2s just one example...

Dan