Wilson's got class

A while back there was a discussion about 12-note scales with high numbers
of 7-limit dyads. Paul Hahn found the winner, with 32 dyads...

5:3---------5:4
/|\ /|\
/ | \ / | \
/ | \ / | \
/ 7:6---------7:4 \
/.-'/ \'-.\ /.-'/ \'-.\
4:3--/---\--1:1--/---\--3:2
|\ / \ /|\ / \ /|
| / \ | / \ |
|/ \ / \|/ \ / \|
28:15--------7:5--------21:20
'-.\ /.-' '-.\ /.-'
8:5---------6:5

...four tetrads (3 otonal), three 8:10:12:15 "ASS" chords, three
10:12:15:18 ASS's, and 2 hexanies. Unfortunately it is so melodically
un-even that its chords float around scale degrees like small boats.
Recently, I noticed that I had come across this structure (in this same
position) last July, when I was looking for 12-tone scales that contained
bits of the 7-limit diamond.

I had also discovered this scale...

35/32-----105/64
,'/ \`. ,' /
5/4-/---\-15/8/
/|\/ \/| /
/ |/\ /\|/
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
/|\/ \/| /
/ |/\ /\|/
/ 7/5------21/20
/,' `.\ /.'
8/5-------6/5

...Paul Hahn was the first to correctly count the number of consonant dyads
in this structure; 31. He was also able to count all possible versions of
the two-hexany 12-plex, and found that only three lacked an interval
smaller than 25/24 (the smallest interval in my version is 525/512, or 43
cents)...

a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/16, 7/4, 15/8

...of these, he noticed that (a) and (b) are not covered by their tetrads,
but (c) is, although he seemed to prefer (a) because of its lower diameter.
I played a bit with these scales, and found I like (c) the best. Today,
whilst going thru the Scala archive, I found this...

! wilson_class.scl
!
Class Scale, Erv Wilson, 9 july 1967
12
!
25/24
28/25
7/6
5/4
4/3
7/5
35/24
8/5
5/3
7/4
28/15
2/1

...which is identical to (c) when transposed by its 5th degree...

25/16
/|\
/ | \
/35/32\
/,'/ \`.\
5/4-/---\-15/8
/|\/ \/|
/ |/\ /\|
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
|\/ \/| /
|/\ /\|/
7/5------21/20
\`.\ /,'/
\ 6/5 /
\ | /
\|/
42/25

...so there you have it. Nothing new under the sun (at least since 1967).

-C.

I found a small error in my post to TD166 (May 8th). I mixed up the scales
in this list...

>a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
>b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
>c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/16, 7/4, 15/8

...which should read...

a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/64, 7/4, 15/8
c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8

...(b) and (c) were switched. The rest of the post was fine.

a) 35/32
,'/ \`.
5/4-/---\-15/8
/:\/ \/:
/ :/\ /\:
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
/:\/ \/: /
/ :/\ /\:/
/ 7/5------21/20
/,' \`.\ /,'/
8/5-----\-6/5 /
\ : /
\:/
42/25

c) 25/16
/:\
/ : \
/35/32\
/,'/ \`.\
5/4-/---\-15/8
/:\/ \/:
/ :/\ /\:
/ 7/4------21/16
/,'/ \`.\ /,'/
1/1-/---\-3/2 /
:\/ \/: /
:/\ /\:/
7/5------21/20
\`.\ /,'/
\ 6/5 /
\ : /
\:/
42/25

None of these scales are proper, and they all have the same largest and
smallest intervals. They each have six 5-limit ASS's, but (c) is biased
toward the 10:12:15:18.

-C.

Carl!
Where are these from again, I forgot!

Carl Lumma wrote:

> From: Carl Lumma <clumma@nni.com>
>
> I found a small error in my post to TD166 (May 8th). I mixed up the scales
> in this list...
>
> >a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
> >b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
> >c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/16, 7/4, 15/8
>
> ...which should read...
>
> a)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 15/8
> b)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 105/64, 7/4, 15/8
> c)1/1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8
>
> ...(b) and (c) were switched. The rest of the post was fine.
>
> a) 35/32
> ,'/ \`.
> 5/4-/---\-15/8
> /:\/ \/:
> / :/\ /\:
> / 7/4------21/16
> /,'/ \`.\ /,'/
> 1/1-/---\-3/2 /
> /:\/ \/: /
> / :/\ /\:/
> / 7/5------21/20
> /,' \`.\ /,'/
> 8/5-----\-6/5 /
> \ : /
> \:/
> 42/25
>
> c) 25/16
> /:\
> / : \
> /35/32\
> /,'/ \`.\
> 5/4-/---\-15/8
> /:\/ \/:
> / :/\ /\:
> / 7/4------21/16
> /,'/ \`.\ /,'/
> 1/1-/---\-3/2 /
> :\/ \/: /
> :/\ /\:/
> 7/5------21/20
> \`.\ /,'/
> \ 6/5 /
> \ : /
> \:/
> 42/25
>
> None of these scales are proper, and they all have the same largest and
> smallest intervals. They each have six 5-limit ASS's, but (c) is biased
> toward the 10:12:15:18.
>
> -C

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