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. This form can
be seen as a linear mapping of an interval that is (L+s)/(L+(L+s)), so
in a 9L&2s mapping for example, the first mediant would give a 9/20,
and in the form [LLLLsLLLLLs] this indexing template would be (where
"e" is just a variable that indicates any EDO):

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)

So in a 5L&2s mapping in the form [LLsLLLs], 16e for example will have
a periodic table index of [44-2444-2]:

(5) 3 1 6 4 2 0
12 (10) 8 13 11 9 7
19 17 (15)(20) 18 16 14
26 24 22 27 (25) 23 21
33 31 29 34 32 (30) 28
40 38 36 41 39 37 (35)
47 45 43 48 46 44 42
(54) 52 50 55 53 51 49
59 57 62 60 58 56
(66) 64 69 67 65 63
71 76 74 72 70
(78) 83 81 79 77
(90) 88 86 84
95 93 91
(102)100 98
107 105
(114)112
119
(126)

and an index of [3222331] when taken:

round (e/12*0)
round (e/12*2)
round (e/12*4)
round (e/12*5)
round (e/12*7)
round (e/12*9)
round (e/12*11)
round (e/12*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 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
(I'm using "m" here as a variable to generically indicate the first
mediant, i.e., [L+(L+s)]):

round (e/m*2)
round (e/m*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

I wrote,

>This form can be seen as a linear mapping of an interval that is
(L+s)/(L+(L+s)), so in a 9L&2s mapping for example, the first mediant
would give a 9/20, and in the form [LLLLsLLLLLs]

9/20 should have read 11/20.