11-out-of-31

Hi,

can anyone come up with a rational "interpretation" of the following
11-note subset of 31-equal--or even any rotation of it?

0, 3, 6, 10, 13, 14, 17, 20, 24, 27, 30, (31)

3 3 4 3 1 3 3 4 3 3 1

I'm looking for a "nice" (though it'll have to be non-standard) set of
ratios that are best approximated by this scale or one of its modes.

(By the way, this could be an example for the poster looking for scales
with 3 sizes of step a couple of days ago, too)

thanks, jon

Jon Wild wrote,

>can anyone come up with a rational "interpretation" of the following
>11-note subset of 31-equal--or even any rotation of it?

>0, 3, 6, 10, 13, 14, 17, 20, 24, 27, 30, (31)

If you are using it polyphonically or homophonically, it is best to look at
a lattice diagram:

-0 / `. /,' / 3
`. / 20 / /
24 / / `. / /
,'/ `. / / 13--------0 /
30--/-----17 / / `. ,' `. /
| / `. / / 6--------24
|/ 10 / / \`. ,'/ `.
14 / `. / / \ 30--/-----17
/ 3 / \/| /
/ / `. / /\|/
/ / 27--------14
13---------0 / / `. /,'
`. ,' `. / / 20
6--------24 / / `.
/ \`. ,'/ `. / / 13---------0
/ \ 30--/-----17 / / `. .'
/ \/| / `. / / 6
/ /\|/ 10 / / \`.
27--------14 / `. / / \ 30
`. /,' / 3 / \/|
20 / / `. / /\|
/ `. / / 27--------14
/ 13---------0 / / `. /
/ / `. ,' `. / / 20
/ / 6--------24 / / `.
10 / / \`. ,'/ `. / / 13
`. / / \ 30--/-----17 / /
3 / \/| / `. / /
/ `. / /\|/ 10 /
/ 27--------14 / `. /
/ / `. /,' / 3
/ / 20 / `.

Clearly this is one of those scales (like the diatonic) that goes beyond any
one rational interpretation. (One interesting feature is a chain of 7:6s
that includes ten of the notes -- only 14 is left out.)

Here's the simplest JI interpretion -- since 6 and 24 are the most connected
to other ratios, take 6 as the 1/1 and 24 as the 3/2 and using each note's
closest occurrence in the lattice to 6 and 24:

6 1/1
10 35/32 or 54/49
13 7/6
14 6/5
17 9/7
20 48/35 or 49/36
24 3/2
27 8/5
30 12/7
0 7/4
3 15/8 or 28/15

but this of course leaves out many of the possibilities in the lattice
above.

Since 7-limit harmony is much better than 9- or 11-limit in 31-tET, this
lattice is probably the best view of the situation.

Jon, here's my stab at an 11-limit JI interpretation:

0 3 6 10 13 14 17 20 24 27 30 31

1/1 15/14 8/7 5/4 4/3 11/8 16/11 11/7 12/7 11/6 88/45 2/1

David Canright http://www.mbay.net/~anne/david/

> -----Original Message-----
> Message: 15
> Date: Thu, 9 Sep 1999 19:43:44 -0400 (EDT)
> From: Jon Wild <wild@fas.harvard.edu>
> Subject: 11-out-of-31
>
> can anyone come up with a rational "interpretation" of the following
> 11-note subset of 31-equal--or even any rotation of it?
>
> 0, 3, 6, 10, 13, 14, 17, 20, 24, 27, 30, (31)
>
> 3 3 4 3 1 3 3 4 3 3 1
>
> I'm looking for a "nice" (though it'll have to be non-standard) set of
> ratios that are best approximated by this scale or one of its modes.
>

Paul Erhlich wrote:
>
> >0, 3, 6, 10, 13, 14, 17, 20, 24, 27, 30, (31)
>
> (One interesting feature is a chain of 7:6s
> that includes ten of the notes -- only 14 is left out.)

Paul, thanks for the response and diagram. I guess another way to see it
is as a chain of *twelve* 7:6s with just one note - 7 - left out.

Thanks also to David Canright for his 11-limit interpretation.

[Paul Erlich:]
>(One interesting feature is a chain of 7:6s that includes ten of the
notes -- only 14 is left out.)

Though it's easy to understand (and wonderfully clear to see in the
lattice) what is meant by this, couldn't calling this a chain of 7/6s,
instead of a chain of 7/31s, possibly lead to some confusion...
inasmuch as the chain of 7/31s approximations begin to overshoot the
chain of 7/6s at five, or as the 7/31 chain would no longer be
congruent with an actual 7/6 chain at 7^5/6^5 & 7*5(mod 31), where
(log7^5-log6^5)*(31/log2) would be 3, and five 7/31s would be 4
(etc.)?

Dan

>Jon, here's my stab at an 11-limit JI interpretation:

> 0 3 6 10 13 14 17 20 24 27 30 31

>1/1 15/14 8/7 5/4 4/3 11/8 16/11 11/7 12/7 11/6 88/45 2/1

>David Canright http://www.mbay.net/~anne/david/

Rather than 16/11, which harmonizes only weakly with 1/1 and 4/3, I'd use
22/15, which harmonizes very strongly with 11/6 and 88/45, and is also
closer to the original 31-equal value (2^(17/31)).