Two versions of 12&34 in the 11-limit

From 12 and 34, we can obtain a unique generator/period for the
46=12+34 et, which I denote by "34+12", by the following procedure:

(1) Find the penultimate convergent to 12/34, obtaining 1/3

(2) Take the mediant of 1/12 and 3/34, obtaining 4/46 = 2/23

(3) Find the mapping to primes of 46, obtaining [46, 73, 107, 129,
159]. Note that this does *not* require us to even look at mappings
for 12 or 34, much less worry about validity!

(4) Taking our period of 23 steps and our generator of 4/46, we
calculate generator steps:

73/4 = 1 mod 23
107/4 = -2 mod 23
129/4 = -8 mod 23
159/4 = 11 mod 23

(5) Our other generator is 1/2, and we find the corresponding number
of steps for it:

73/46 - 2/23 = 3/2
107/46 + 2(2/23) = 5/2
129/46 + 8(2/23) = 7/2
159/46 - 11(2/23) = 9/2

(6) Since the mapping to number of generator steps from an interval
is a val, and we represent vals by column vectors, we can put our
results together in a 5x2 matrix:

[ 0 2]
[ 1 3]
[-2 5]
[-8 7]
[11 5]

(7) We now have all we need so far as the 46-et goes; however we may
also detemper using the above map and linear programming or least
squares to find an optimal tuning. Using least squares in the 11-
limit gives a generator a1 = .08700594368 = 4.002273409 / 46; this is
not much different from the 46-et and gives similar tuning errors.
However, the map to primes was only unique mod 23, and we might have
used instead -12 = 11 mod 23 for the number of steps we mapped 11 to.
We obtain instead the map:

[ 0 2]
[ 1 3]
[ -2 5]
[ -8 7]
[-12 9]

Since the other maps to primes have a negative tendency, this seems
like it is probably the best plan. If we adopt it, we get instead
a2 = .08648628 = 3.97836888 / 46 as our generator, which is quite a
bit farther from 46-et than the other system. A comparison of tunings
shows:

3: 2.39 2.45 1.83
5: 4.99 4.87 6.12
7: -3.61 -4.08 0.91
11: -3.49 -2.84 3.28

Here the first column is the 46-et, the second our first detempering,
and the third the alternative--which looks pretty good!

In-Reply-To: <9ter9b+f1fe@eGroups.com>
Aha, we have an algorithm!

This is the bit that concerns me:

> (7) We now have all we need so far as the 46-et goes; however we may
> also detemper using the above map and linear programming or least
> squares to find an optimal tuning. Using least squares in the 11-
> limit gives a generator a1 = .08700594368 = 4.002273409 / 46; this is
> not much different from the 46-et and gives similar tuning errors.
> However, the map to primes was only unique mod 23, and we might have
> used instead -12 = 11 mod 23 for the number of steps we mapped 11 to.
> We obtain instead the map:

If you can choose two different mappings, that means the mapping isn't
defined by the notation 12&34. Then we get to:

> Since the other maps to primes have a negative tendency, this seems
> like it is probably the best plan.

If it's only "probably" the best, then you obviously don't have a
deterministic algorithm.

Graham

--- In tuning-math@y..., graham@m... wrote:

> If you can choose two different mappings, that means the mapping
isn't
> defined by the notation 12&34.

Correct--you would need to decide what was an optimal version of the
mod 23 reduced mapping.

Then we get to:
>
> > Since the other maps to primes have a negative tendency, this
seems
> > like it is probably the best plan.
>
> If it's only "probably" the best, then you obviously don't have a
> deterministic algorithm.

If you want a deterministic algorithm, you need to decide what to
optimize. In fact, however, both systems work.

In-Reply-To: <9tg210+ge72@eGroups.com>
Gene wrote:

> If you want a deterministic algorithm, you need to decide what to
> optimize. In fact, however, both systems work.

That's okay, so long as we agree that you can't uniquely define all linear
temperaments with a notation like 12&34.

Have you tried doing an exhaustive search like mine, finding the best
generator mappings for a list of equal temperaments? This method should
catch some that mine doesn't.

Graham