13-limit mappings (was Re: Partch's scales on the Miracle keyboard)

In-Reply-To: <9h92qm+81o1@eGroups.com>
Paul wrote:

> > this one does at 53.
>
> With even more overlaps. I'm inclined to dislike this mapping since
> 53-tET is not consistent through the 13-limit (or even the 11-limit).

Okay, I've fixed a bug in my temperament program. I had 11:6 defined as
4:3. This explains why no 13-limit temperaments came out as unique. It
also means my 13- and 15-limit rankings will have been wrong before.

The best unique 13-limit temperaments are listed at
<http://x31eq.com/limit13.unique>

So, where are we on this? I think Wilson's mapping is:

56/135, 497.891 cent generator

basis:
(1.0, 0.41490927070342681)

mapping by period and generator:
([1, 0], ([2, -1, -3, 13, 12], [-1, 8, 14, -23, -20]))

mapping by steps:
[(94, 41), (149, 65), (218, 95), (264, 115), (325, 142), (348, 152)]

unison vectors:
[[5, -3, 0, 0, 1, -1], [-7, -1, 1, 1, 1, 0], [-3, 6, 0, -1, 0, -1], [0,
0, -1, 1, 2, -2]]

highest interval width: 37
complexity measure: 37 (41 for smallest MOS)
highest error: 0.004029 (4.835 cents)
unique

It doesn't matter that 53 isn't consistent, because 94 is. That should
be 74 notes for the diamond as well, isn't that less than Dave said? If
it's correct, it does beat the multiple-29 mapping for
notes-to-the-diamond. But even by that criterion, this looks slightly
better again:

9/52, 103.897 cent generator

basis:
(0.5, 0.086580634742799478)

mapping by period and generator:
([2, 0], ([3, 5, 7, 9, 10], [1, -2, -8, -12, -15]))

mapping by steps:
[(58, 46), (92, 73), (135, 107), (163, 129), (201, 159), (215, 170)]

unison vectors:
[[1, 2, -3, 1, 0, 0], [4, 0, -2, -1, 1, 0], [-1, 3, -2, -1, 0, 1], [2,
-1, 0, 1, -2, 1]]

highest interval width: 17
complexity measure: 34 (46 for smallest MOS)
highest error: 0.004911 (5.893 cents)
unique

Graham

--- In tuning@y..., graham@m... wrote:
> Okay, I've fixed a bug in my temperament program. I had 11:6
defined as
> 4:3. This explains why no 13-limit temperaments came out as unique.
It
> also means my 13- and 15-limit rankings will have been wrong before.

Err, doesn't it mean the 11-limit ones were wrong too? I was surprised
that the 498c generator didn't appear in the 11-limit top 10.
>
> The best unique 13-limit temperaments are listed at
> <http://x31eq.com/limit13.unique>

So "unique" here means they don't conflate any ratios of the diamond?

> So, where are we on this? I think Wilson's mapping is:
... 497.891 cent generator
... [-1, 8, 14, -23, -20]
> highest interval width: 37
> complexity measure: 37 (41 for smallest MOS)
> highest error: 0.004029 (4.835 cents)
> unique

Agreed.

> It doesn't matter that 53 isn't consistent, because 94 is. That
should
> be 74 notes for the diamond as well, isn't that less than Dave said?

Yes. You've got it wrong. That's 74 generators, which means 75 notes,
which is the same as for the 11-limit diamond. This may be ok for the
13-limit diamond, but it is a pretty poor mapping for Partch's 43
because it has so many holes.

> If
> it's correct, it does beat the multiple-29 mapping for
> notes-to-the-diamond. But even by that criterion, this looks
slightly
> better again:
... 103.897 cent generator
... 0.5 octave period
>
> mapping by ... generator:
>... [1, -2, -8, -12, -15]))
> highest error: 0.004911 (5.893 cents)
> unique

That's equivalent to two chains of 703.9 cent fifths, a half octave
apart. Anyone got a picture of a keyboard mapping for that?
So is that 68 notes or 70 notes for the 13-limit diamond?

-- Dave Keenan