11-limit quasi-just

Paul, Dave K (et al.),

Why not try the meantone-like idea of letting the generator make the
most complex consonance in the scales relevant identity just?

Wouldn't the 15th part of an 11/4 generator give something like the
lowest error mean amongst the 4:5:6:7:9:11 anyway?

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul, Dave K (et al.),
>
> Why not try the meantone-like idea of letting the generator make the
> most complex consonance in the scales relevant identity just?
>
> Wouldn't the 15th part of an 11/4 generator give something like the
> lowest error mean amongst the 4:5:6:7:9:11 anyway?

At 116.75c it's not too bad, but there is no necessity about this.

The 11-limit max absolute error optimum occurs with a just 5:9.
The 7-limit max absolute error optimum occurs with a just 5:6.

David Keenan wrote,

<<At 116.75c it's not too bad, but there is no necessity about this.
The 11-limit max absolute error optimum occurs with a just 5:9.>>

No, I don't think that's correct. Not in the 4:5:6:7:9:11 identity, or
primary consonances sense that I posted anyway.

You must mean across the whole 11-limit diamond?

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> No, I don't think that's correct. Not in the 4:5:6:7:9:11 identity,
or
> primary consonances sense that I posted anyway.
>
> You must mean across the whole 11-limit diamond?

He means across the interval matrix of the 4:5:6:7:9:11, which is the
same ratios as the diamond, but interpreted as intervals rather than
pitches.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
>
> > No, I don't think that's correct. Not in the 4:5:6:7:9:11
identity,
> or
> > primary consonances sense that I posted anyway.
> >
> > You must mean across the whole 11-limit diamond?
>
> He means across the interval matrix of the 4:5:6:7:9:11, which is
the
> same ratios as the diamond, but interpreted as intervals rather than
> pitches.

Correct. I mean that the size of Erlich-Keenan generator that gives
the lowest Maximum Absolute error over all 11-limit ratios happens to
result in a just 5:9. A 5:9 is made up of 19 generators (less an
octave), so this MA optimum generator is the 19th root of 18/5 ~=
116.72c. That's the value that I feel is analogous to the 1/4-comma
meantone generator. Since the 1/4-comma fourth (or fifth) is the
meantone generator that gives the lowest MA error at the 5-limit.

I don't know if it makes sense to talk about various fractions of some
11-limit comma in this context. But one difficulty is that the
generator is not itself an approximate consonance in the Erlich-Keenan
(= MIRACLE) case, unlike the meantone, paultone and
chain-of-minor-thirds cases.

Here are various optima alongside some EDOs. MA stands for max
absolute. RMS stands for root-mean-squared.

Cents Type of optimum
------------------------
116.13 31-EDO
.
116.43 7-limit odd-limit-weighted MA
116.55 7-limit odd-limit-weighted RMS
116.57 7-limit RMS
116.59 7-limit MA
116.667 72-EDO
116.674 11-limit odd-limit-weighted RMS
116.68 11-limit RMS
116.71 11-limit odd-limit-weighted MA
116.72 11-limit MA
.
117.07 41-EDO

I don't follow Graham when he says they could range between 10 and
11-EDO. That's like saying that meantone ranges from 5 to 7-EDO.

-- Dave Keenan

Dave Keenan wrote,

<<That's the value that I feel is analogous to the 1/4-comma meantone
generator. Since the 1/4-comma fourth (or fifth) is the meantone
generator that gives the lowest MA error at the 5-limit.>>

But it's completely unanalogous in many other ways, no?

As an example of this I'd say that QCM's just 5/4 is a relevant,
primary consonance -- the primary consonance in fact. In this regard a
meantone like optimizing of the 11/4 is at least as analogous to QCM
as giving the lowest MA error at the 11-limit I think.

At least that's the way I see it at the moment (1:45 AM).

good night,

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> But it's completely unanalogous in many other ways, no?

You're right.

> As an example of this I'd say that QCM's just 5/4 is a relevant,
> primary consonance -- the primary consonance in fact. In this regard
a
> meantone like optimizing of the 11/4 is at least as analogous to QCM
> as giving the lowest MA error at the 11-limit I think.

Yes. I guess so. It's just that in any arbitrary 11-limit temperament,
making the 4:11 just might well throw everything else so severely out
of whack that it just wouldn't be worth it. It just so happens that in
this case it gives quite good results. So by all means make one with a
just 8:11 (preferably slowly beating, not phase-locked).

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> As an example of this I'd say that QCM's just 5/4 is a relevant,
> primary consonance -- the primary consonance in fact.

I don't see that at all. Thus the relevance of tunings like 1/3-comma
meantone and 1/6-comma meantone. In the 5-limit, 1/4-comma only
stands out from these because of its lower MA error.

Dave Keenan wrote:

> I don't know if it makes sense to talk about various fractions of some
> 11-limit comma in this context. But one difficulty is that the
> generator is not itself an approximate consonance in the Erlich-Keenan
> (= MIRACLE) case, unlike the meantone, paultone and
> chain-of-minor-thirds cases.

You could cover the mistuning of the fifth from 3:2 or the neutral third
from 11:9. Probably a continuous scale from one to the other would work,
so 0 is just fifth, 1 is just 11:9.

Incidentally, what do you get by optimizing for the 7-limit plus 11:9?

> I don't follow Graham when he says they could range between 10 and
> 11-EDO. That's like saying that meantone ranges from 5 to 7-EDO.

Easley Blackwood covered everything from 5 to 7-EDO as "recognizable
diatonic tunings" so we seem to agree. He used another term (which I
couldn't find by flicking through) to (roughly) cover 19 to 12. I think
both categories are useful, and the latter would here be 31 to 41.

At <http://x31eq.com/decimal_notation.html#strange> I use this
concept to notate 11-equal, and adapt it for 22-equal. I even go so far
as to notate 12-equal with the same idea, although it's even outside the
11-10 range.

The scale tree for recognizable decimal tunings would then be:

11 10

21
32 31
43 53 52 41
54 75 85 74 73 83 72 51

It's interesting to see 43 and 53 on there. I haven't looked to see if
they're meaningful from this perspective. It's also interesting to see 32
on there, relating to this paragraph from Patrick Ozzard-Low's paper
(possibly still at <http://www.lgu.ac.uk/mit/21corch.html>):

"At this point caution should be suggested. In my own compositional work,
using electronic simulations of acoustic instruments, one of the ET
systems to which I have been particularly drawn is 32-ET. A quick glance
at APPENDIX II Table 32 shows that 32-ET does not satisfy particularly
well the criteria which are being suggested as advantageous. I have found
its exotic and expressive palette of tones, including the wide fifth and
narrow third, rather interesting, and that particular timbres (such as a
very bright piano) give especially attractive consonances, and bright
sharp dissonances."

So even this tuning lies in the broader range of scales which look to me
ideal for the project in question.

Graham

--- In tuning@y..., graham@m... wrote:
> Incidentally, what do you get by optimizing for the 7-limit plus
11:9?

You didn't say what kind of optimum.

RMS is at 116.55 cents.

Errors are:
2:3 4:5 5:6 4:7 5:7 6:7 9:11
-2.6 -2.2 -0.5 -1.9 0.3 0.7 2.3 cents

But why would you care if the neutral third approximates 9:11 in this
context? Wouldn't you simply want it to be half the fifth (which is
guaranteed for any reasonable size of this generator)? So you'd just
use the 7-limit optimum.

-- Dave Keenan

On 5/13/01 7:00 AM, "graham@microtonal.co.uk" <graham@microtonal.co.uk>
wrote:

> "At this point caution should be suggested. In my own compositional work,
> using electronic simulations of acoustic instruments, one of the ET
> systems to which I have been particularly drawn is 32-ET. A quick glance
> at APPENDIX II Table 32 shows that 32-ET does not satisfy particularly
> well the criteria which are being suggested as advantageous. I have found
> its exotic and expressive palette of tones, including the wide fifth and
> narrow third, rather interesting, and that particular timbres (such as a
> very bright piano) give especially attractive consonances, and bright
> sharp dissonances."

Ah... 32...
so low a number
and so daring a bouquet.

I cast a vacant glare
at this pile of fretboards sometimes...
the little numbers
at the bottom of them
looking up at me...
it's so nice to hear
people in other bodies
use these numbers in sentences.

Thank you.

Marc