Yahoo Tuning Groups Ultimate Backup tuning Carl Lumma's 7-limit, 6 tetrads + 4 triads within 2c of Just

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

>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@Acadian-Asset.com> 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://dkeenan.com

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

On Thu, 25 Feb 1999, Dave Keenan wrote:
> 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. :-)

I thought the lattice diagram looked rather familiar. As I've mentioned
before, that scale is not all that different from some of the scales I
came up with during Carl's 7-limit max-consonance JI scale challenge.

As I've discovered when messing in the 7-limit with my 225:224-based
9-tone scales, using too many 3:2s bumps you up agains the 64:63 awfully
quickly. If you want scale steps closer in size to a semitone, you have
to orient yourself more along the 5-axis. Hence the three-tetrachord
technique that I described some time back, which delimits the
tetrachords with 5:4s instead of 4:3s.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "

NOTE: dehyphenate node to remove spamblock. <*>

🔗Dave Keenan <d.keenan@uq.net.au>

Paul Hahn <Paul-Hahn@library.wustl.edu> wrote:

>I thought the lattice diagram looked rather familiar. As I've mentioned
>before, that scale is not all that different from some of the scales I
>came up with during Carl's 7-limit max-consonance JI scale challenge.

I guess the question is: Was Carl the first to describe this scale as being that comma-free 12-of-7-limit-just scale that has the greatest harmonic resources when 225/224 errors (septimal kleismas) are ignored? Thus paving the way for the more-or-less obvious (at least to Carl, Paul Erlich and myself) tempering to distribute the errors.

Of course no one has *proved* that it is the best such scale. But it's highly suggestive that I came up with an equivalent scale when prompted by Carl (to the list) to come at it from the 5-limit. But then you (Paul Hahn) surprised us last time with that 31 interval scale.

>As I've discovered when messing in the 7-limit with my 225:224-based
>9-tone scales ...

Where can I read about these?

"Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote:

>Here's a note I wrote to myself on Lumma's tuning before Keenan's post
>appeared: ...

"Great minds ..." and all that. :-)

>Well, it appears I was right that Dave Keenan found this scale in
>October.

No. I see this one as distinct from the meantone one I described. But certainly they are a progression. This one has an extra wolf and two less triads as the price of its very low max error. I'd never heard of a temperament with less than 2c error before. Have 7-limit JI-ists previously realised that the 225:224 could be so thinly distributed?

Personally, I'd prefer the meantone version but I expect there are many (like Mr Lumma) who would not.

I (Dave Keenan) wrote:
>>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).

Paul Erlich asked:

>What do you mean here?

Look at the comments in the Scala file regarding which ratios are approximated by each degree. Choose almost any ratio for any degree and you'll have a Just scale which, when its kleismas are distributed, will produce the same scale as Carl's. The only such scale I could find in the archive is Fokker's 12 of 7-limit. I think that Handblue, gamelan-om, and just7_12 do not generate Carl's scale by this method, however I think that if they are "meantoned" they all (incl. Fokker's) generate the same 12-of-meantone scale (the one I described last October).

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

I (Dave Keenan) wrote,

>>Look at the comments in the Scala file regarding which ratios are
>>approximated by each degree. Choose almost any ratio for any degree and
>>you'll have a Just scale which, when its kleismas are distributed, will
>>produce the same scale as Carl's. The only such scale I could find in
>>the archive is Fokker's 12 of 7-limit. I think that Handblue,
>>gamelan-om, and just7_12 do not generate Carl's scale by this method,
>>however I think that if they are "meantoned" they all (incl. Fokker's)
>>generate the same 12-of-meantone scale (the one I described last
>>October).

"Paul H. Erlich" <PErlich@acadian-asset.com> replied:

>I'm still not following, but for a start, can you give these scales
>explicitly?

I'm sorry. I assumed you had access to Manuel's archive. ftp://ella.mills.edu/ccm/tuning/software/scales/scales.zip (645K)

Here are three scales (the first two are not in the archive):

Carl's 7-limit 12-tone just scale that started this thread

16/15
28/25
7/6
5/4
4/3
7/5
112/75
8/5
5/3
7/4
28/15
2/1

The 5-limit 12-tone just scale I found at Carl's prompting

16/15
9/8
75/64
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2/1

! fokker_12.scl
!
Fokker's 7-limit 12-tone just scale
12
!
15/14
9/8
7/6
5/4
4/3
45/32
3/2
45/28
5/3
7/4
15/8
2/1

When the 225/224 kleismas are optimally distributed, these three scales and many others, all become:

! 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

Everything after ! on a line is a comment.

It is in that sense that this scale unifies all those others. And with only 2c errors why would anyone bother with those Just scales? Actually, I was expecting a bite from some strict JI-ist when I first said that. :-)

Now consider the following three 7-limit just scales (all from the archive):

! gamelan_om.scl
!
Other Music gamelan (7 limit black keys)
12
!
15/14
9/8
7/6
5/4
4/3
7/5
3/2
14/9 !*
5/3
7/4
15/8
2/1

! handblue.scl
!
"Handy Blues" of Pitch Palette, 7-limit
12
!
16/15
9/8
7/6
5/4
4/3
7/5
3/2
14/9 !*
5/3
7/4
15/8
2/1

! just7_12.scl
!
7-limit 12 tone scale
12
!
16/15
9/8
7/6
5/4
4/3
7/5
3/2
8/5
12/7 !*
7/4
15/8
2/1

These do not map to Carl's scale when the kleismas are distributed. In each case a single note (shown with !*) disqualifies it.

However when the syntonic commas are distributed as well i.e. by mapping to 1/4-comma meantone (or 31-tET), all 7 of the above scales, and many others, map to

! keenan.scl
!
Dave Keenan 31-tet mode with three 4:5:6:7 tetrads plus three inverted
12
!
116.12903
193.54839
270.96774
387.09677
503.22581
580.64516
696.77419
812.90323
890.32258
967.74194
1083.87097
2/1

This scale has the same number of tetrads (6) but 2 more triads than Carl's scale. Now with 5.4c (or 6c) errors I can understand that a JI-ist might prefer one of the previous 3 scales to this one, despite having fewer consonances.

But I'd find it very hard to understand why anyone would prefer any of the first three just scales to Carl's and my temperament with only 2c errors.

Regards,

-- Dave Keenan
http://dkeenan.com

Carl Lumma's 7-limit, 6 tetrads + 4 triads within 2c of Just

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Dave Keenan <d.keenan@uq.net.au>

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Dave Keenan <d.keenan@xx.xxx.xxx>

🔗Joseph L Monzo <monz@xxxx.xxxx>