Regarding consistency and 24et

Consistency is only important if the "JI math" you might do
in an ET is important to you. Regarding 24, it is not
consistent in the 7 limit, meaning representing all arithmatic
of {3,5,7} (2 is assumed) will not work.

24 IS consistant in {3,5} (same as twelve) and in {3,7}
with better tuning than 12. It is only if you use 5-limit and
7-limit intervals at the same time that you MIGHT find an
unexpected choice to make.

So as I have mentioned before, the septimal "natural
minor scale"

1/1 9/8 7/6 4/3 3/2 14/9 7/4

is logical (consistant) and kinda nice sounding in 24

4154145 (0 200 250 500 700 750 950)

with a somewhat exagerated septimal "buzz".

Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

/tuning/topicId_34086.html#34086

>
> Consistency is only important if the "JI math" you might do
> in an ET is important to you. Regarding 24, it is not
> consistent in the 7 limit, meaning representing all arithmatic
> of {3,5,7} (2 is assumed) will not work.
>
> 24 IS consistant in {3,5} (same as twelve) and in {3,7}
> with better tuning than 12. It is only if you use 5-limit and
> 7-limit intervals at the same time that you MIGHT find an
> unexpected choice to make.
>
> So as I have mentioned before, the septimal "natural
> minor scale"
>
> 1/1 9/8 7/6 4/3 3/2 14/9 7/4
>
> is logical (consistant) and kinda nice sounding in 24
>
> 4154145 (0 200 250 500 700 750 950)
>
> with a somewhat exagerated septimal "buzz".
>
> Bob Valentine

***Hello Bob!

This really "sums up" my questions concerning 24-tET. Frankly, it's
looking better and better!

J. Pehrson

Heres a little more for you Joseph.

I have a little program to try to look at consistency
and I believe that, excluding 5, 24 is 11-limit
consistent. Those who write JI music or some other
(for instance Miracle derived) music that matches that
description might try this RI mapping and see if it
has anything to do with their musical intentions. There
are players for it.

(In the following table, square
brackets mean its the best approximation compared to
fractions with simpler numbers. Step 8 at 14/11 is
correct in the system, however 5/4 is actually a
better approximation, but not consistant in the
system).

Summary of 24 et at the {3,7,9,11}-limit

1 :
2 :
3 : [12/11]
4 : [9/8]
5 : [7/6] 8/7
6 :
7 : [11/9]
8 : 14/11
9 : [9/7]
10 : [4/3]
11 : [11/8]
12 :
13 : [16/11]
14 : [3/2]
15 : [14/9]
16 : 11/7
17 : [18/11]
18 :
19 : [7/4] [12/7]
20 : [16/9]
21 : [11/6]
22 :
23 :

Bob Valentine

Hi there,

Just a thought that occurred -

if 12-tet is a consistent subset of 24-tet,
are there any other twelve note consistent
subsets?

What about maximally consistent subsets -
lets define a subset as maximally consistent
if adding one note to it turns it into an
inconsistent subset.

So, a maximally consistent subset needn't
necessarily have the maximum possible
nmber of notes for a consistent subset
- 24-tet might well have maximally consistent
subsets with much less than 24 notes.

It would be very amenable to a computer search
to find all the maximally consistent subsets
of an n-tet.

I wonder if there are any theoretical approaches
to get them too?

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi there,
>
> Just a thought that occurred -
>
> if 12-tet is a consistent subset of 24-tet,
> are there any other twelve note consistent
> subsets?

??? consistency (in this sense) is only well-defined for equal
temperaments. i can't see how you could possibly define it for an
unequal temperament.

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

/tuning/topicId_34086.html#34149

>
> Heres a little more for you Joseph.
>
> I have a little program to try to look at consistency
> and I believe that, excluding 5, 24 is 11-limit
> consistent. Those who write JI music or some other
> (for instance Miracle derived) music that matches that
> description might try this RI mapping and see if it
> has anything to do with their musical intentions. There
> are players for it.
>

***This is really interesting, Bob, and thanks so much! Now I can
get an idea what degrees of 24-tET the "spectral" school of composers
are looking at to get their JI effects!

Joseph