Two tunings I've recently noticed

I'm sure someone must have commented on this before, but I'm curious.
It has come to my attention during my mathematical investigations on
my HP programmable scientific that 31-EDO with its 38.71-cent steps
divided by seven comes out within 0.09 cents of being 1/4 of the
average between the Paythagorean and syntonic commas. Note that 53-
EDO steps are quite close to the average of the two commas. Paul E.'s
last comments in a post about adaptive JI and Vicento's approach to
it made me look at this.

7 X 31 = 217

217-EDO is a large numbered scale, but it is an extremely accurate
finite or "atomic" model of up to 19-limit JI. Has Gene or anyone
looked at this. It also contains 31-EDO within it, so the options for
comma drift avoidance are always there. What are its drawback other
than its enormous size?

Using the same approach with 19-EDO, you get:

9 X 19 = 171

171-EDO has somewhat less accuracy, but it's still very accurate.
Same questions apply.

Cheers,

Bob

In-Reply-To: <a9vuvu+t2re@eGroups.com>
robert_wendell wrote:

> 7 X 31 = 217
>
> 217-EDO is a large numbered scale, but it is an extremely accurate
> finite or "atomic" model of up to 19-limit JI. Has Gene or anyone
> looked at this. It also contains 31-EDO within it, so the options for
> comma drift avoidance are always there. What are its drawback other
> than its enormous size?

It's been discussed on tuning-math a bit. It's consistent up to the
31-limit, and so the simplest 31-limit consistent ET.

> Using the same approach with 19-EDO, you get:
>
> 9 X 19 = 171
>
> 171-EDO has somewhat less accuracy, but it's still very accurate.
> Same questions apply.

Something like this has come up as well.

Graham

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

/tuning/topicId_36532.html#36532

>
> Using the same approach with 19-EDO, you get:
>
> 9 X 19 = 171
>
> 171-EDO has somewhat less accuracy, but it's still very accurate.
> Same questions apply.
>

***Hello Bob!

I'm almost certain 171 has been discussed. I believe Joe Monzo was
talking about it. Paul Erlich will probably have all the details...

J. Pehrson

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> I'm sure someone must have commented on this before, but I'm
curious.
> It has come to my attention during my mathematical investigations
on
> my HP programmable scientific that 31-EDO with its 38.71-cent steps
> divided by seven comes out within 0.09 cents of being 1/4 of the
> average between the Paythagorean and syntonic commas. Note that 53-
> EDO steps are quite close to the average of the two commas. Paul
E.'s
> last comments in a post about adaptive JI and Vicento's approach to
> it made me look at this.

quite a few months ago, joe monzo realized that 217-equal is
excellent for implementing Vicentino's adaptive JI approach, and was
planning to use it to render some Mahler MIDI files he was working
on. i think he mentioned that on tuning-math.

> 7 X 31 = 217
>
> 217-EDO is a large numbered scale, but it is an extremely accurate
> finite or "atomic" model of up to 19-limit JI.

i agree that it stands out in this regard. i'm pretty disappointed at
myself for not noticing that it excels in both these aspects.

> Has Gene or anyone
> looked at this.

217-equal has been getting a huge amount of 'airplay' on tuning-math
very recently. george secor and dave keenan seem to be, in different
ways, using 217-equal as some kind of 'benchmark' for their JI
notation scheme, with 1600-equal an additional reference ET.

> It also contains 31-EDO within it, so the options for
> comma drift avoidance are always there. What are its drawback other
> than its enormous size?
>
> Using the same approach with 19-EDO, you get:
>
> 9 X 19 = 171
>
> 171-EDO has somewhat less accuracy,

or much more, if one only looks up through the 9-limit.

> but it's still very accurate.
> Same questions apply.

171-equal has been discussed quite a lot historically, and on this
list. search this list for "171-tET", "171-tone", etc.

both of these, as you've presented them, are great examples of
the 'bicycle concept' that comes up from time to time. i brought both
of them up in a brief survey of 'bikes' that i posted here some time
ago, in response to joseph pehrson. see if you can find that
post . . .

personally, as i've mentioned before, if i were to choose a large ET
for composing, it would be 152-equal (8*19) . . . in addition to
supporting a 1/3-comma version of adaptive (5-limit) JI, it's really
good through 11-limit strict JI. combine this with the fact that 76-
equal (152/2) comes up as 'universally' important in my paper (near
the end), and that 152 affords the opportunity to 'adaptively'
realize the resources of 76-equal but with far better vertical
harmony, and you get an idea of a few of the reasons i like 152.

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <a9vuvu+t2re@e...>
> robert_wendell wrote:
>
> > 7 X 31 = 217
> >
> > 217-EDO is a large numbered scale, but it is an extremely
accurate
> > finite or "atomic" model of up to 19-limit JI. Has Gene or anyone
> > looked at this. It also contains 31-EDO within it, so the options
for
> > comma drift avoidance are always there. What are its drawback
other
> > than its enormous size?
>
> It's been discussed on tuning-math a bit. It's consistent up to
the
> 31-limit, and so the simplest 31-limit consistent ET.

unfortunately, graham, this is not the case. if it were, you'd see
217-equal on this table:

http://library.wustl.edu/~manynote/consist2.txt

even 94-equal is consistent through a higher odd limit than 217-equal
is.

In-Reply-To: <aa2979+lctg@eGroups.com>
Me:
> > It's been discussed on tuning-math a bit. It's consistent up to
> the
> > 31-limit, and so the simplest 31-limit consistent ET.

Paul:
> unfortunately, graham, this is not the case. if it were, you'd see
> 217-equal on this table:
>
> http://library.wustl.edu/~manynote/consist2.txt
>
> even 94-equal is consistent through a higher odd limit than 217-equal
> is.

Yes, I thought we had a 31-limit ET that was a multiple of 31-equal. Oh
well. 217= is only 21-limit consistent, and the simplest 31-limit
consistent ETs are 311 and 388, neither of them multiples of 31. 217= has
a worst error of 0.58 scale steps in the 31-limit, and is the simplest ET
to nail the 31-limit to within 0.6 scale steps. Perhaps that's important.

Graham