This follows from the thread "Re: 225:224".

Carl Lumma <clumma@...> wrote:

>Does anybody know the best 5-limit shape for taking advantage of the
>225:224 at the 7-limit? When looking for the best 7-limit shape to take
>advantage of the comma in the 5-limit, I came up with...
>
> /
> /
> 5/3--------5/4......28/15-----
> /|\ / |. / \
> / | \ / | . / \
> / 7/6--------7/4 . /
> / // \\ \ / / / . /
> 4/3-/---\-1/1../...112/75----
> /|\/ \/| /
> / |/\ /\| /
> /28/15------7/5
> / // \\ \ / //
> 16/15/---\-8/5 /
> | / \ | /
> |/ \|/
> 112/75-----28/25
>
>
>...can anyone find a better one that's still reasonable as a scale?

Paul H. Erlich" <PErlich@...> replied:

>You mean among 12-tone scales, right? I see 6 consonant tetrads here
>(you didn't even show one of them). That's pretty impressive! I don't
>think that can be beat. Why not use 72-tET and reduce the maximum error
>from almost 8 cents to under 3 cents?

Yes. There are 6 tetrads plus 4 triads. I'm pretty sure it can't be beat (pun
intended) too. I did some serious searching by considering the 225/128 (~= 7/4)
as a bridge on a 5-limit lattice (easier to think and draw in 2D). The best one
I can find is

75/64----225/128
. / \ . /
. / \. u /
. / . .\ /
. /. . \ /
5/3-------5/4------15/8------45/32
/ \ . / \ . ./ \ /
/ \. / \. u / \ u /
/ o \ / o .\ / .\ /
/ \ /. . \ / . \ /
4/3-------1/1-------3/2-------9/8
/ \ . ./ .
/ \. . / .
/ o .\ / .
/ . \ / .
16/15------8/5

Which, when the septimal kleismas (225/224) disappear, is entirely equivalent to
your (Carl's) 7-limit lattice above. Is this what Terry Riley uses on "The Harp
of New Albion"?

It can be notated as
D# ------ A#
. / \ . /
. / \. u /
. / . .\ /
. /. . \ /
A ------- E ------- B ------- F#
/ \ . / \ . ./ \ /
/ \. / \. u / \ u /
/ o \ / o .\ / .\ /
/ \ /. . \ / . \ /
F ------- C ------- G ------- D
/ \ . ./ .
/ \. . / .
/ o .\ / .
/ . \ / .
Db------- Ab

and we see that it agrees with meantone spelling where the 7:4 is an augmented
sixth, e.g. C-A#. It also contains a diatonic scale.

I agree with Paul that one should distribute the septimal kleisma over the
intervals involved. But forcing the scale to be a mode of such a large ET (72)
seems of minor value (although it is only 1c worse than optimum). Minimum errors
are obtained when the 7:4's and 5:3's are just. In this case all the other
7-limit intervals have only a quarter kleisma (1.93 cent) error. Wow! Do you
agree that this seriously blurrs the distinction between JI and temperaments?

Note that there are 4 wolves in three different sizes. D-A is a comma-narrow
wolf (-21.5c). If this were optimally tempered as well (to give two more
triads), we'd be in 1/4-comma meantone, the max error would go up to 5.4 cents,
and I've just described the 7-limit 12-of-meantone scale with which I joined
this list last October. :-)

Carl's scale (with kleisma distributed) has 3 step sizes (as does my "strange"
9-limit temperament) of approximately 69, 84 and 116 cents.

The next best 12-tone scale I could find, having no comma steps ("reasonable as
a scale"), also has 6 tetrads. But it only has 2 additional triads (not 4). It
has 5 wolves (not 4) and no diatonic scale. Its 5-limit lattice looks like:
Cx
G#D#A#
A E B F#
F C G
Db

Unless Terry Riley can claim priority, here's a proposed Scala archive entry:

! lumma.scl
!
Carl Lumma, 7-limit, 6 tetrads + 4 triads within 2c of Just, TL 19-2-99
12
! 5-limit 7-limit
115.5870 ! 16/15 +3.9c, 15/14 -3.9c
200.0542 ! 9/8 +3.9c, 28/25 -3.9c
268.7988 ! 75/64 -5.8c, 7/6 +1.9c
384.3858 ! 5/4 -1.9c, 56/45 +5.8c
499.9729 ! 4/3 +1.9c 75/56 -5.8c
584.4401 ! 45/32 -5.8c 7/5 +1.9c
700.0271 ! 3/2 -1.9c, 112/75 +5.8c
815.6142 ! 8/5 +1.9c, 45/28 -5.8c
5/3 ! 884.3587c, 224/135+7.7c
7/4 !225/128-7.7c, 968.8259c
1084.4130! 15/8 -3.9c, 28/15 +3.9c
2/1

Talk about JI-meets-12-tET! Notice that 2 of the scale degrees are given as
exact ratios, while 3 others are essentially the same as they are in 12-tET.

Here are offsets from 12-tET:
C 0.0
Db 15.6
D 0.1
D# -31.2
E -15.6
F 0.0
F# -15.6
G 0.0
Ab 15.6
A -15.6
A# -31.2
B -15.6

This scale can of course be considered 9-limit with a max error of 3.9c. 4 of
the 6 tetrads extend to pentads. That's 4-pentads + 2 tetrads + 4 triads.
Beautiful!

This effectively unifies a bunch of 12-tone 5 and 7 limit-scales. Why would
anyone bother with those scales now? Fokker's was the only such scale I could
find in the Scala archive, but there are several others that would be unified,
but for one note (handblue, gamelan-om, just7_12).

People refer to 1/4-comma meantone as quasi-just. What should we call *this*?
wafso-just? (within a fly excrement of)

This is great Carl!

Regards,
-- Dave Keenan
http://uq.net.au/~zzdkeena