L & s -- some thoughts on indexing two step size cardinality (revisited)

L & s -- INDEXING TWO STEP SIZE CARDINALITY:

In mapping scales with two step size cardinality, I use the first
mediant of L&s where L=2 & s=1 as the indexing template, you could
look at this as the left-hand side of Erv Wilson's Scale-Tree (or
Pierce sequence):

1/0
1/1
2/1

which could also be seen more broadly in this generalized L & s
expression:

L s
L+s
2L+s L+2s
3L+s 3L+2s 2L+3s L+3s
...

This indexing template can be seen as a linear mapping of an interval
(the variable "i" in the following) that is i/(x+y) where I'm letting
"x" be L, and "y" be L+s, so in a 9L & 2s mapping for example, i=11,
and i/(x+y) is 11/20, and in an [LLLLsLLLLLs] configuration you have a
linear form of -1 0 1 2 3 4 5 6 7 8 9:

9---0---11---2---13---4---15---6---17---8---19

It's also interesting to note that this could also be illustrated as:

4
/|
/ |
13-19
/| /
/ |/
2--8
/| /
/ |/
11-17
/| /
/ |/
0--6
/| /
/ |/
9--15
| /
|/
4

Where i/(x+y), i.e. 11/20, is synonymous with the "3/2," and round
(11/20)*11 is synonymous with the "5/4."

This 9L & 2s indexing template would be:

round (e/20)*0
round (e/20)*2
round (e/20)*4
round (e/20)*6
round (e/20)*8
round (e/20)*9
round (e/20)*11
round (e/20)*13
round (e/20)*15
round (e/20)*17
round (e/20)*19
round (e/20)*20

Where "e" is just a variable that indicates any EDO, and the
expressions I've written are synonymous with the general expression of
(LOG(2^(n/E))-LOG(1))*(e/LOG(2)) were "n" is any 0,1,2,...E number,
and "E" indicates the EDO of the indexing template (i.e., x+y).

So in a 5L & 2s mapping where i=7, an [LLsLLLs] configuration gives
a -1 0 1 2 3 4 5 linear form of:

5---0---7---2---9---4---11

And using 16e as an example, you would have a 5L&2s periodic table
index of [44-2444-2] -- this can be seen in the following L/s periodic
table example:

(1/0) 1/-1 1/-2 2/-2 2/-3 2/-4 2/-5
2/1 (2/0) 2/-1 3/-1 3/-2 3/-3 3/-4
3/2 3/1 (3/0) (4/0) 4/-1 4/-2 4/-3
4/3 4/2 4/1 5/1 (5/0) 5/-1 5/-2
5/4 5/3 5/2 6/2 6/1 (6/0) 6/-1
6/5 6/4 6/3 7/3 7/2 7/1 (7/0)
7/6 7/5 7/4 8/4 8/3 8/2 8/1
(8/7) 8/6 8/5 9/5 9/4 9/3 9/2
9/7 9/6 10/6 10/5 10/4 10/3
(10/8) 10/7 11/7 11/6 11/5 11/4
11/8 12/8 12/7 12/6 12/5
(12/9) 13/9 13/8 13/7 13/6
(14/10) 14/9 14/8 14/7
15/10 15/9 15/8
(16/11) 16/10 16/9
17/11 17/10
(18/12) 18/11
19/12
(20/13)

However, 16e would also have an index of [3222331] when taken:

round (e/12)* 0, 2, 4, 5, 7, 9, 11, 12

By making the first index match the second you get an index of
[4,4,,-2''''4,,4,4,-2'''], where a ['] is always an indication to
raise the index by 1, and a [,] is always an indication to lower an
index by 1. And as L & s are always synonymous with the first mediant
form of:

round (e/E)*2
round (e/E)*1

This in turn could then be reduced to a final index of two step size
cardinality:

[33,1'3,331]

These would be the complete indexes for the EDOs in the 5L & 2s
mapping that do not have a generating interval that falls between 4/7
and 3/5 and are neither of the ambiguous expansions of L and L+s (5
and 7e in this case, as the horizontal and vertical expansions of the
periodic table could be seen as the cases of maximum ambiguity in a
given mapping, as s=0 and s=L & L=s are what the horizontal and
vertical expansions are working towards, i.e., L and L+s):

35e L=6 & s=3 [6636,663]
30e L=5 & s=3 [5535553,]
28e L=5 & s=2 [55,2'5,552]
25e L=4 & s=2 [4424'442]
23e L=4 & s=2 [4424,442]
21e L=4 & s=2 [44,24,44,2]
20e L=3 & s=2 [33'2,3'332]
18e L=3 & s=2 [3323332,]
16e L=3 & s=1 [33,1'3,331]
15e L=3 & s=1 [33,133,31]
14e L=2 & s=1 [22'122'21]
13e L=2 & s=1 [2212'221]
11e L=2 & s=1 [2212,221]
10e L=2 & s=1 [22,1222,1]
9e L=2 & s=1 [22,12,22,1]
8e L=1 & s=1 [11'1,1'111]
6e L=1 & s=1 [1111111,]
4e L=1 & s=0 [11,0'1,110]
3e L=1 & s=0 [11,011,10]
2e L=0 & s=0 [00'000'00]
1e L=0 & s=0 [0000'000]

Dan