stability

I did some stability calculations for ET MOS this morning similar to my
efficiency calculations. The results were very interesting. _Both_ of our
conjectures, Paul, regarding the Fibonacci series MOS's were incorrect:
not only does it have a lower efficiency than the 12-t diatonic, but it
also has a lower stability! In fact the only stability higher than the
diatonic for it's scale-tree layer is the pentatonic 2 2 2 2 1 (all MOS's
with only one small step are strictly proper). Let me plot these
stabilities in the form of the scale-tree. First, note these two facts:
all ET MOS with L:s < 2 and their MOS subsets are strictly proper, and all
ET MOS with s = 1 and L:s > 2 are improper except when they have only one
small step. Therefore, only the stabilities of the 2:1 MOS are
interesting, so I've only listed these. Generators are in parenthesis next
to stability values.

(1/3)1.0

(1/4)0.667 (2/5)1.0

(1/5)0.5 (2/7)1.0 (3/7)0.7 (3/8)0.9

(1/6) (2/9) (3/10) (3/11) (4/9) (5/12) (4/11) (5/13)
0.4 1.0 0.714 0.857 0.524 0.952 0.643 0.893

1/7)2/11)3/13)3/14)4/13)5/17)4/15)5/18)5/11)7/16)7/17)8/19)5/14)7/19)7/18
.333 1.0 .722 .833 .533 .933 .618 .891 .417 .972 .618 .848 .491 .945 .641
(8/21).872

Notice that the same families I posited for the tree of efficiencies are
relevant here; take diagonals across the tree from down-left to up-right
and you get families of similar stabilities:

(2/5)1.0 (2/7)1.0 (2/9)1.0 2/11(1.0),
(3/13)0.722 (3/10)0.714 (3/7)0.7,
(3/14)0.844 (3/11)0.857 (3/8)0.9,

That first family is the one-small-step family. The family containing the
12-t diatonic is second highest in stability to it until a new family is
formed on the last layer listed by generator 7/16. The step pattern for
the 7/16 2:1 MOS is 222212221. That is, its the next MOS with only two
small steps after the diatonic. This relates to the the two correlations
we found for efficiency. In fact, the very same correlations determine
stability, except that a higher number of gaps for a given cardinality
_increases_ stability whereas it _decreases_ efficiency. Higher
cardinality decreases both values, but as you can see in the following
table, the effect is fairly weak for stability if we regard the proportion
cardinality:gaps as the quantity strongly correlated with stability,

Generator______Step Pattern_card:gaps__stability
2/11 222221 6:5 1.0
2/9 22221 5:4 1.0
2/7 2221 4:3 1.0
2/5 221 3:2 1.0
1/3 21 2:1 1.0
5/12 2221221 7:5 0.952
3/8 22121 5:3 0.9
5/13 22122121 8:5 0.893
3/11 2212121 7:4 0.857
3/10 2121211 7:3 0.714
3/7 21211 5:2 0.7
1/4 211 3:1 0.667
4/11 21211211 8:3 0.643
4/9 2112111 7:2 0.524
1/5 2111 4:1 0.5
1/6 21111 5:1 0.4

jason

--- In tuning@egroups.com, Jason_Yust <jason_yust@b...> wrote:

_Both_ of our
> conjectures, Paul, regarding the Fibonacci series MOS's were
incorrect:
> not only does it have a lower efficiency than the 12-t diatonic,
but it
> also has a lower stability!

I doubt I ever had enough of a grasp of efficiency and stability to
make a conjecture about Fibonacci series MOS's, nor did I express the
Fibonacci series MOS's in terms of ETs as you're doing here.

> That first family is the one-small-step family. The family
containing the
> 12-t diatonic is second highest in stability to it until a new
family is
> formed on the last layer listed by generator 7/16. The step
pattern for
> the 7/16 2:1 MOS is 222212221.

FWIW, that's a mode of Goldstein's scale . . .

--- In tuning@egroups.com, Jason_Yust <jason_yust@b...> wrote:

http://www.egroups.com/message/tuning/14149

> (1/3)1.0
>
> (1/4)0.667 (2/5)1.0
>
> (1/5)0.5 (2/7)1.0 (3/7)0.7
(3/8)0.9
>
> (1/6) (2/9) (3/10) (3/11) (4/9) (5/12) (4/11)
(5/13)
> 0.4 1.0 0.714 0.857 0.524 0.952 0.643
0.893
>
>
1/7)2/11)3/13)3/14)4/13)5/17)4/15)5/18)5/11)7/16)7/17)8/19)5/14)7/19)7
/18
> .333 1.0 .722 .833 .533 .933 .618 .891 .417 .972 .618 .848 .491
.945
.641
>
(8/21).872
>

This looked a bit better in the original (!!)

Sorry to be so bothersome, but I have a question about this
diagram... It is clearly the Stern-Brocot tree, since the sum of the
numerators and denominators are the ratios in the next row below.

However, are these ratios actually PITCHES (??) What are they,
OTONAL sets (??) Are the "generators" for these "noble" scales
actually PITCHES (??)

Sorry about this... I was swimming, but the water is getting deep,
and my head is slowly going under...
__________ ___ __ _
Joseph Pehrson

Joseph wrote,

>Sorry to be so bothersome, but I have a question about this
>diagram... It is clearly the Stern-Brocot tree, since the sum of the
>numerators and denominators are the ratios in the next row below.

>However, are these ratios actually PITCHES (??) What are they,
>OTONAL sets (??) Are the "generators" for these "noble" scales
>actually PITCHES (??)

Joseph, these are not frequency ratios, they are fractions of an octave (in,
say, cents). However, we keep bringing up the fact that the Stern-Brocot
tree is useful (for different things) under _both_ interpretations -- the
fractions being frequency ratios, or the fractions being fractions of an
octave. Jason Yust was using it in the latter sense here, because it is the
generator's size as a fraction of an octave, not as a frequency ratio, which
is relevant to Rothenberg issues (propriety, efficiency, stability). You
could think of these as non-acoustical properties of a scale.