Neutral second scales

You know how it is -- you wait years for a new 11-limit temperament and
then 6 all turn up at once. Well, here are 4 more for your collection.

I played with unison vectors that include (-1 -1 0 2)h which defines the
neutral seconds 12:11 and 11:10 as being the same. That also means two
11:8 make a 15:8. The 11-limit ontonality can be shown like this on the
lattice:

5
/ \
/ \
/ 11
/ 7 \
/ \
1-----------3-----------9

If you add the 243:242 unison vector which defines a 3:2 as two 11:9
steps, you get the meantone-like neutral third scales 31, 38 and 24. And
for 11-limit consistency, you end up with 31.

So, now for some other scales with neutral seconds. In each case, I added
the syntonic comma, schisma and Pythagorean comma to get some ETs. First
off, these unison vectors:

|-2 2 1 0|
|-1 1 1 1|
|-1 -1 0 2|

covers 22-, 31- and 53-equal. I'm not sure how much use 53 is in the
11-limit, but it might be useful to have it there. You get 7 generators
to a 3:1, or 4 to a 15:8. This is the simplest temperament in Dave
Keenan's table, with only 17 notes required for an 11-limit hexad. The
conversion matrix is:

( 9 4)
(14 7)(s)
H' = (21 9)(r)
(25 12)
(31 14)

I use 9 notes to the octave. The nominals are:

0 1 2 3 4 5 6 7 8 0
t s t s t s t s s

Where t=s+r. Using ^ and v for shifts by r, an 11-limit hexad is

3v
/ \
/ \
/ 4
/ 7 \
/ \
0-----------5-----------1^

Next up, these unison vectors:

|-2 -2 1 0|
|-2 2 0 1|
|-1 -1 0 2|

It covers 7-, 22- and 29-equal. There are 3 generators to a 4:3. The
conversion matrix is:

( 7 1)
(11 2)(t)
H' = (16 3)(q)
(19 5)
(24 4)

Only 16 steps needed for an 11-limit hexad. But it isn't on Dave's chart,
so is probably as bad an approximation as you'd expect with it combining
22 and 29. Still, it can be written as a 7 note scale, and therefore with
a familiar notation:

C D E F G A B C
t+q t t t t t t

and that 11-limit chord

E
/ \
/ \
/ F
/ A^^ \
/ \
C-----------G-----------D

where ^ is a shift by q.

| 0 -2 -1 1|
| 2 -3 1 0|
|-1 -1 0 2|

Covers 31- and 46-equal. It has 9 steps to a 3:2 and needs 21 steps for
an 11-limit hexad. A definition matrix is

(14 1)
(23 1)(r)
H' = (33 2))s)
(39 3)
(49 3)

I get this 15 note scale:

r r r r r r r r r r s r r r r
C D E F G A B H C

Using # to raise by r, and v to lower by s-r the hexad is

E#
/ \
/ \
/ F#
/ Hb \
/ \
C-----------G#----------Dv

Lastly,

| 4 0 -1 1|
| 2 -3 1 0|
|-1 -1 0 2|

took a lot of work to make sense of, which is why I took so long to post
this. It covers 7, 46 and 53. I eventually got this definition out, with
8 steps to a 6:5:

( 4 10)
( 7 15)(s)
H' = (11 21)(r)
(15 23)
(15 33)

That might be 4 generators to a 6:5, but I forgot to count how many of
them are needed for a hexad. As it isn't on Dave's chart, probably quite
a few. After some more tinkering, I found this scale:

C D E F G A B C
T t T t T t T

That's interesting, as it's the complement to the neutral third MOS.
Let's try another conversion matrix

( 4 3)
( 7 4)(T)
H' = (10 7)(t)
(15 4)
(15 9)

You may be able to derive that from the previous one. It isn't obvious to
me how. Using ^ to raise by T-t and # for ^^, the 11-limit hexad is:

E#
/ \
/ \
/ F\
/ A## \
/ \
C-----------G^----------D^

The A## can be replaced with Bb in 46=. It looks like a simplification,
but isn't really. I don't know what the new unison vectors are.

E#
/ \
/ \
/ F\
/ Bb \
/ \
C-----------G^----------D^

This may be a useful way of notating 46=. You might be able to do
something using the nominals as a diatonic.

So there you go. You could try some more by guessing more unison vectors.
But this seems to cover everything likely from Dave's table.

Graham

--- In tuning@y..., graham@m... wrote:
> You know how it is -- you wait years for a new 11-limit temperament
and
> then 6 all turn up at once. Well, here are 4 more for your
collection.
>
> I played with unison vectors that include (-1 -1 0 2)h which defines
the
> neutral seconds 12:11 and 11:10 as being the same.

So obviously you can't get the min MA error below 7.2 c (half of
120:121), so they aint gonna ge quasi-just. My search was only
interested if they were better than 1/4 comma meantone's 11 cents.

...

> Next up, these unison vectors:
>
> |-2 -2 1 0|
> |-2 2 0 1|
> |-1 -1 0 2|
>
> It covers 7-, 22- and 29-equal. There are 3 generators to a 4:3.
The
> conversion matrix is:
>
> ( 7 1)
> (11 2)(t)
> H' = (16 3)(q)
> (19 5)
> (24 4)
>
> Only 16 steps needed for an 11-limit hexad. But it isn't on Dave's
chart,
> so is probably as bad an approximation as you'd expect with it
combining
> 22 and 29.

Yes. Min MA is 17.2 cents with a 165.22 cent generator. It's not
really 11-limit anything.

> | 0 -2 -1 1|
> | 2 -3 1 0|
> |-1 -1 0 2|
>
> Covers 31- and 46-equal. It has 9 steps to a 3:2 and needs 21 steps
for
> an 11-limit hexad. A definition matrix is
>
> (14 1)
> (23 1)(r)
> H' = (33 2))s)
> (39 3)
> (49 3)
>
>
> I get this 15 note scale:
>
> r r r r r r r r r r s r r r r
> C D E F G A B H C
>
> Using # to raise by r, and v to lower by s-r the hexad is
>
>
> E#
> / \
> / \
> / F#
> / Hb \
> / \
> C-----------G#----------Dv

So that's my 78.25c generator with MA err of 8.5c.

> Lastly,
>
> | 4 0 -1 1|
> | 2 -3 1 0|
> |-1 -1 0 2|
>
> took a lot of work to make sense of, which is why I took so long to
post
> this. It covers 7, 46 and 53. I eventually got this definition
out, with
> 8 steps to a 6:5:
>
>
> ( 4 10)
> ( 7 15)(s)
> H' = (11 21)(r)
> (15 23)
> (15 33)
>
> That might be 4 generators to a 6:5, but I forgot to count how many
of
> them are needed for a hexad. As it isn't on Dave's chart, probably
quite
> a few. After some more tinkering, I found this scale:
>
> C D E F G A B C
> T t T t T t T
>
> That's interesting, as it's the complement to the neutral third MOS.

It's either 339.44 cents with MA err 8.0 cents and 30 gens to the
hexad. Or just possibly 338.37 with MA err 14.9 cents and 23 gens to
the hexad. I doubt it's the latter because 46 is not a denominator of
even a semiconvergents, whereas both 46 and 53 are denoinators of
convergents of the former.

-- Dave Keenan