Kleismic[8] and Hemifourths[9]

Enough 49/48 for now--on to what I wanted to do, 15/14 as a chroma
from which I expect great things.

Kleismic[8]
rms error 12.274

[6, 5, 3, -6, -12, -7]
[1, 21/20, 7/6, 6/5, 7/5, 10/7, 5/3, 7/4]
[7/6, 7/5, 5/3, 1, 6/5, 10/7, 7/4, 21/20]

6/5 generator tetrad circles
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 5/4, 3/2,
9/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 5/3] [1, 5/4, 3/2,
5/3]
[1, 5/4, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 5/4, 7/5,
5/3]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 5/3] [1, 7/6, 7/5,
5/3]
[1, 7/6, 7/5, 8/5] [1, 10/9, 7/5, 8/5] [1, 10/9, 7/5, 5/3] [1, 7/6,
7/5, 5/3]

Hemifourths[9]
rms error 12.690

[2, 8, 1, 8, -4, -20]
[1, 9/8, 7/6, 9/7, 4/3, 3/2, 14/9, 7/4, 9/5]
[9/8, 9/7, 3/2, 7/4, 1, 7/6, 4/3, 14/9, 9/5]

7/6 generator tetrad circles
[1, 7/6, 7/5, 8/5] [1, 6/5, 7/5, 8/5] [1, 6/5, 7/5, 9/5] [1, 7/6, 7/5,
14/9]
[1, 7/6, 7/5, 7/4] [1, 6/5, 7/5, 7/4] [1, 6/5, 7/5, 9/5] [1, 7/6, 7/5,
14/9]
[1, 7/6, 3/2, 7/4] [1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2,
14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 14/9]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 5/3]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 7/6,
3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 9/7, 3/2, 7/4] [1, 9/7, 3/2, 27/14] [1, 5/4,
3/2, 5/3]

>Kleismic[8]
>rms error 12.274

Yeah, it's a good 'un. But how'd you pick 8? Once again: where
are you getting the n's????

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Kleismic[8]
> >rms error 12.274
>
> Yeah, it's a good 'un. But how'd you pick 8? Once again: where
> are you getting the n's????

Now that I've sent you my maple programs, you could find them yourself.

First, read "temper". Now get the wedgie for kleismic. If you know it
comes from the standard vals h19 and h34, you can do

a7v(19, 34) = [6, 5, 3, -6, -12, -7]

If you happen to know it has 49/48 and 126/125 as a TM basis, you can
get it as

a7i(49/48, 126/125) = [6, 5, 3, -6, -12, -7]

Or maybe you just know the kleisma is a comma, and think 49/48 is
also. You can try

a7i(49/48, 15625/15552) = [-12, -10, -6, 12, 24, 14]. You can feed
this into the reduction to standard wedgie function, and get

kleisma:=wedgie7(a7i(49/48, 15625/15552));

[6, 5, 3, -6, -12, -7]

>

Now you can check and see what kleisma does with various commas

a7d(kleisma, 25/24);

[4, 6, 9, 11]

Four notes--not much of a scale

a7d(kleisma, 36/35);

[4, 6, 9, 11]

Same four notes. Oh well.

a7d(kleisma, 49/48);

[0, 0, 0, 0]

You didn't forget 49/48 was a comma, did you? :)

a7d(kleisma, 21/20);

[4, 6, 9, 11]

Darn! This is getting annoying!

a7d(kleisma, 15/14);

[8, 12, 18, 22]

Success at last! This is the 8-note scale you asked about. The
torsional aspect is benign.

a7d(kleisma, 28/27);

[-15, -24, -35, -42]

Nice number of notes, but the equivalences from 28/27 are a little goofy.

a7d(kleisma, 16/15);

[-11, -18, -26, -31]

Same comment, but probably worth exploring.

>> Yeah, it's a good 'un. But how'd you pick 8? Once again: where
>> are you getting the n's????
>
>Now that I've sent you my maple programs, you could find them yourself.

Thanks for the tutorial. I glanced at your stuff, but probably won't
get a chance to look at it until next week.

>Same comment, but probably worth exploring.

So, you're doing it by hand, was all I was looking for. I wonder if
there's any way to predict the good stopping points. I suppose they
occur when the chromatic uv is simple? . . .

>Pelogic[7]
>
>135/128
>[1, 9/8, 5/4, 4/3, 3/2, 8/5, 15/8]
>[5/4, 15/8, 4/3, 1, 3/2, 9/8, 8/5]
//
>Dominant sevenths[7]
>rms error 20.163 cents
//
>Hemifourths[9]
>rms error 12.690
//
>Tertiathirds[9]
>rms error 12.189 cents
//
>Hexadecimal[9]
>rms error 18.585

Ahh -- it's all coming too fast!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> So, you're doing it by hand, was all I was looking for. I wonder if
> there's any way to predict the good stopping points. I suppose they
> occur when the chromatic uv is simple? . . .

It's not just when it is simple, it's when it moves chord elements
around in a way which is useful. This is why I favor 15/14, 21/20,
25/24 and 36/35.

> Ahh -- it's all coming too fast!

I could look at ennealimmal[27] and ennealimmal[36] instead--that
would put you back to sleep!