a counterpart to Yasser's Fibonacci scales

I think an interesting counterpart to Yasser's m-out-of-n Fibonacci
scales would be the m-out-of-n scales based on adjacent fraction (or
"Ls index") series.

Here's the two "2s5L" series:

0/1 -1/2, 1/2 2/5, 2/3 5/8, 3/4 8/11, 4/5 11/14, ...

0/1 1/3, 1/2 3/5, 2/3 5/7, 3/4 7/9, 4/5 9/11, ...

The first creates this m-out-of-n series:

3-out-of-5
7-out-of-12
11-out-of-19
15-out-of-26
19-out-of-33, ...

And the second creates this m-out-of-n series:

4-out-of-7
7-out-of-12
10-out-of-17
13-out-of-22
16-out-of-27, ...

The following Ls index series would give Yasser's 12-out-of-19:

-1/1 -2/1, 1/3 1/4, 3/5 4/7, 5/7 7/10, 7/9 10/13, ...

2-out-of-3
7-out-of-11
12-out-of-19
17-out-of-27
22-out-of-35, ...

The "compliment" of this would be:

-2/1 -1/1, 1/4 1/3, 4/7 3/5, 7/10 5/7, 10/13 7/9, ...

2-out-of-3
7-out-of-12
12-out-of-17
17-out-of-24
22-out-of-31, ...

(Note the expansion of faux-tetrachordal like motifs based on the size
of the generator in these scales.)

--Dan Stearns

Dan,

I don't understand this but can you show the 13-out-of-22 and 22-out-of-31
scales? (and 7-out-of-12 and 12-out-of-19 if they're different from
Yasser's)

-Paul

Paul H. Erlich wrote,

> can you show the 13-out-of-22 and 22-out-of-31 scales? (and
7-out-of-12 and 12-out-of-19 if they're different from Yasser's)

The 7 and 12-tone scales are the same as Yasser's. The 13 and 22-tone
ones are the 7:12 like Mos scales; the 1:2^(17/22), and the
1:2^(24/31).

Here's the Golden 13s9L:

0 70 114 184 228 298 342 385 456 499 570 613 657 727 771 841 885 928
999 1042 1113 1156

Here's the Golden 4s9L:

0 105 209 314 378 483 588 652 757 862 926 1031 1135

I'll also include the Silver 4s9L as it works the fifth and the major
third closer to just (and as it's a slightly warped subset of the
Golden 13s9L):

0 113 225 338 384 497 610 656 769 882 928 1041 1153

--Dan Stearns

Dan wrote,

>The 13 and 22-tone
>ones are the 7:12 like Mos scales; the 1:2^(17/22), and the
>1:2^(24/31).

So you're saying they're generated by an approximate 12/7, or 7/6. A chain
of subminor thirds. Weird!

Paul!
This happens to be how my 22 is mapped out (chains of "7/6") which is a subset of
dallesandro as a 31 tone scale.

"Paul H. Erlich" wrote:

> So you're saying they're generated by an approximate 12/7, or 7/6. A chain
> of subminor thirds. Weird!

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Kraig wrote,

>This happens to be how my 22 is mapped out (chains of "7/6") which is a
subset of
>dallesandro as a 31 tone scale.

Thanks, Kraig! Would you mind elaborating on this?

Paul!
As you know one pictures is worth a 1024 words so This weekend I will just put up graphic!
too busy till then!

"Paul H. Erlich" wrote:

> Kraig wrote,
>
> >This happens to be how my 22 is mapped out (chains of "7/6") which is a
> subset of
> >dallesandro as a 31 tone scale.
>
> Thanks, Kraig! Would you mind elaborating on this?

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Paul H. Erlich wrote,

> So you're saying they're generated by an approximate 12/7, or 7/6. A
chain of subminor thirds. Weird!

Right. But here's what I was getting at in the "Yasser sense". If you
take an adjacent fraction series you get a sequence where it's easy to
see each preceding m-out-of-n inside of each successive one. Take this
series for example:

-1/0 -1/1, 0/1 1/3, 1/2 3/5, 2/3 5/7, 3/4 7/9, ...

The first one here, -1/0 -1/1, is a 1-out-of-2. Now if you look at
this as the "extra" step of the tetrachord, L here, you can see how
each new m-out-of-n adds a tetrachordal LLs, and how each new
incarnation always contains all its previous incarnations:

|----|
|-------------|
|------------------------|
|------------------------------------|
|------------------------------------------------|
0 2 4 6 7 9 11 12 14 16 17 19 21 22
L L L s L L s L L s L L s

1-out-of-2 = L
4-out-of-7 = L LLs
7-out-of-12 = L LLs LLs
10-out-of-17 = L LLs LLs LLs
13-out-of-22 = L LLs LLs LLs LLs

-- Dan Stearns

Here's the adjacent fraction (or Ls index) series that gives the
Yasser 12-out-of-19:

-1/1 -2/1, 1/3 1/4, 3/5 4/7, 5/7 7/10, 7/9 10/13, ...

This would give a neutral generator based tetrachord of sLsLL, and a
tetrachordal "leftover" of sL. Here's this taken to the 17-out-of-27
m-out-of-n.

|---|
|--------------|
|-----------------------------|
|--------------------------------------------|
s L s L s L L s L s L L s L s L L
0 1 3 4 6 7 9 11 12 14 15 17 19 20 22 23 25 27

2-out-of-3 = sL
7-out-of-11 = sL sLsLL
12-out-of-19 = sL sLsLL sLsLL
17-out-of-27 = sL sLsLL sLsLL sLsLL

Note that when you rotate "m" so that the generator occupies the tonic
you get a sort of generalized major scale arrangement.

-- Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> Paul H. Erlich wrote,
>
> > So you're saying they're generated by an approximate 12/7, or
7/6. A
> chain of subminor thirds. Weird!
>
> Right. But here's what I was getting at in the "Yasser sense".

> 1-out-of-2 = L
> 4-out-of-7 = L LLs
> 7-out-of-12 = L LLs LLs
> 10-out-of-17 = L LLs LLs LLs
> 13-out-of-22 = L LLs LLs LLs LLs

Wow. That's quite different from how Yasser's scales would look --
looks like you've discovered something new here.

Paul Erlich wrote,

> Wow. That's quite different from how Yasser's scales would look --

Right, Yasser's m-out-of-n series expand towards ever closer
approximation of Golden Meantone. These adjacent fraction series
expand by adding a tetrachord to the "disjunction" (is that what the
tetrachordal "leftover" is called?), and m is always working the two
stepsizes down towards the subcommatic range while n is working the
generator towards P.

I think it's pretty easy to see a generalized tetrachord "rule" in
here somewhere, but as you've pointed out in the past, any such "rule"
will loose its meaning if the generalized tetrachord doesn't occupy a
sizable enough portion of the scale...

I'm really just starting to look at this a bit. Any ideas?

--Dan Stearns