Wilson 19-tone scale

I found the complete Fokker block collection for 81/80 and 3125/3072,
which seemed like a good choice for marvel tempering in some 7 and 11
limit harmony. It turns out that there are 25 of these, with one scale
and its inverse beating the rest in terms of quantity of 11-limit
dyads (100) and triads (209.) This turns out to be a known scale; the
Scala archive calls it "wilson1", and describes it as "Wilson 19-tone,
1976."
Nothing else is said about it, and I don't know how Erv arrived at it.

It has (assuming marvel) one otonal and one utonal 11-limit hexad,
five otonal and five utonal 9-limit pentads, five otonal and five
utonal tetrads, and 11 otonal and 12 utonal 5-limit triads. The
inverted form, obviously, will be the same except for having one more
major, and one less minor, triad. Since it (slightly) favors otonality
it might be preferred.

{81/80, 3125/3072} is the TM basis for <19 30 44| and it could be
5-limit considerations were all Erv Wilson had in mind, but I'd like
to know if anyone knows or can find out.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I found the complete Fokker block collection for 81/80
> and 3125/3072, which seemed like a good choice for marvel
> tempering in some 7 and 11 limit harmony. It turns out
> that there are 25 of these, with one scale and its inverse
> beating the rest in terms of quantity of 11-limit
> dyads (100) and triads (209.) This turns out to be
> a known scale; the Scala archive calls it "wilson1", and
> describes it as "Wilson 19-tone, 1976."
> Nothing else is said about it, and I don't know how Erv
> arrived at it.
>
> It has (assuming marvel) one otonal and one utonal
> 11-limit hexad, five otonal and five utonal 9-limit
> pentads, five otonal and five utonal tetrads, and 11
> otonal and 12 utonal 5-limit triads. The inverted form,
> obviously, will be the same except for having one more
> major, and one less minor, triad. Since it (slightly)
> favors otonality it might be preferred.
>
> {81/80, 3125/3072} is the TM basis for <19 30 44| and
> it could be 5-limit considerations were all Erv Wilson
> had in mind, but I'd like to know if anyone knows or
> can find out.

is it "A Scale for Scott", which you can find in here?

http://www.anaphoria.com/xen456.PDF

there are two other 19-tone scales diagrammed right after
that one.

Erv doesn't say anything about them here (but he may
have elsewhere) ... but knowing him, i doubt if he only
had 5-limit in mind. more likely 11-limit.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> is it "A Scale for Scott", which you can find in here?
>
> http://www.anaphoria.com/xen456.PDF

Thanks, Joe, that seems to be right. If you ran it through your
rectangular lattice diagramming, you would presumably find what Wilson
shows on this page, which is that the scale is a 4x5 rectangle with
one corner left off.

> there are two other 19-tone scales diagrammed right after
> that one.

One of these is just meantone; the other is an 11-limit scale I give
below. It is said to have one "harmonic hexad" and one "subharmonic
hexad" in the 11-limit; my count is one otonal and no utonal hexads,
so someone may want to check my work before believing I've gotten it
right. It is a rather irregular scale with a 5-limit marvel reduction
nothing like the Scott scale. On the cover is the cps for
{1,3,5,7,9,11}, which has 20 notes, laid out in a mandala-like graph.

! wilclav.scl
Erv Wilson's clavochord scale from Xenharmonikon 4
19
!
45/44
12/11
9/8
40/33
5/4
14/11
21/16
15/11
11/8
45/32
3/2
18/11
27/16
12/7
7/4
20/11
15/8
21/11
2

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

On the cover is the cps for
> {1,3,5,7,9,11}, which has 20 notes, laid out in a mandala-like graph.

That's products of these six taken three at a time, sorry.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > is it "A Scale for Scott", which you can find in here?
> >
> > http://www.anaphoria.com/xen456.PDF
>
> Thanks, Joe, that seems to be right. If you ran it through your
> rectangular lattice diagramming,

i did! ;-)

> you would presumably find what Wilson shows on this page,
> which is that the scale is a 4x5 rectangle with
> one corner left off.

basically, yes.

the version of Musica i'm running right now
(which will be replaced by the latest rewrite in a
couple of weeks) centers periodicity-blocks on the origin
(1/1 ratio ... i think the next version will let the
user put the periodicity-block boundaries anywhere).

so the one i got was shifted slightly to the left
in lattice-space than Wilson's, and cuts of two
diagonal corners instead of only one.

25/18 ---- 25/24 ---- 25/16 ---- 75/64
. | ........ | ........ | ........ |
10/9 ------ 5/3 ------ 5/4 ----- 15/8
............ | ........ | ........ |
----------- 4/3 ------ 1/1 ------ 3/2
............ | ........ | ........ |
---------- 16/15 ----- 8/5 ------ 6/5 ------ 9/5
............ | ........ | ........ |
--------- 128/75 ---- 32/25 ---- 48/25 ---- 36/25

> On the cover is the cps for {1,3,5,7,9,11}, which
> has 20 notes, laid out in a mandala-like graph.

hence, Erv called it the "eikosany".

-monz