Although some of the fixed pitch 7 from decimal scales have low

efficiency, there still good enough to be in Carl's list by the looks of

it.

The best candidate is probably

0 1 3v 4v 6 7 9v 0

3 5 3 7 3 5 5 /31

4 7 4 9 4 7 6 /41

It's strictly proper. 6 out of 7 thirds are either 5:4 or 6:5. 4 out of

7 fifths are 3:2. For 31-equal, Lumma stability is 35.5% and Rothenberg

efficiency is 52.1%. In 41=, efficiency goes down to 46.6%. Here's a

lattice:

0-----6

/ \ / \

/ \ / \

1-----7-----3v----9v

\ /

\ /

4v

Another one is

0 1 3^ 4^ 6 7 9^ 0

3 7 3 5 3 7 3 /31

4 9 4 7 4 9 4 /41

Improper this time. 6 thirds either 6:5 or 5:4 and 4 fifths 3:2.

Efficiency 50.6% for either tuning, Lumma stability 31.7% in 41=, 29% in

31=. Lattice

3^----9^

/ \ /

/ \ /

4^----0-----6

/ \ /

/ \ /

1-----7

I've also been looking at the 6 from 10 scale 1 2 2 2 1 2. It's a subset

of the 7 from 10 MOS. I don't like it so much, but it works well with the

criteria. The grid is

1 2 2 2 1 2

3 4 4 3 3 3

5 6 5 5 4 5

7 7 7 6 6 7

8 9 8 8 8 9

It'd be nice if the second row could be 5:4 thirds and 4:3 fourths, but

fixed scales don't work out like that. Still, the middle row can be 4:3,

5:7, 7:10 and 3:2 whereas the first row has lots of 9:8 or 8:7 or 7:6s.

There are a couple of fixed scales that work like this.

4 6 6 6 4 5 /31

5 8 8 8 4 7 /41

and

3 6 6 6 3 7 /31

4 8 8 8 4 9 /41

They both depend on two consonances (8:7 and either 9:8 or 7:6) counting

as the same pitch class. Then, they fulfil Carl's original rules.

Rothenberg efficiency is always 49.4%.

It also happens that the scale works as 6 from 26,

3 5 5 5 3 5

8 10 10 8 8 8

13 13 13 13 11 13

18 18 18 16 16 18

21 23 21 21 21 23

5 steps are 8:7, 11 steps are 4:3 and 13 steps are 7:5. It's strictly

proper, Lumma stability of 53.8% and 66.1% efficiency. I thing it should

go on the list, although it'll have trouble with tetrachordality.

Graham