Re: 13-limit mappings

In-Reply-To: <9hb6q8+1nsp@eGroups.com>
Dave Keenan wrote:

> 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.

No, it means there was an error in my error report:) 13:6 was wrong.
Actually, I don't think it would have made any difference because however
bad 13:6 is either 13:8 or 13:9 will be worse.

You mean this one:

56/135, 497.807 cent generator

basis:
(1.0, 0.41483937003845572)

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

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

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

highest interval width: 37
complexity measure: 37 (41 for smallest MOS)
highest error: 0.003609 (4.331 cents)
unique

The number 9 temperament has a complexity of 29 and error of 4.1 cents.
So it wins both ways. Number 10 is complexity 26 and a 5.1 cent error,
so you might be able to find an FoD to change the order.

> > 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?

If that's what "conflate" means ;)

Uniqueness is in Monzo's dictionary for ETs. I think it was originally
Carl Lumma's idea.

> > 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.

Then the 13-limit equivalent of Partch's scale would need around 75
notes. After a while, diminishing returns start to apply. The best
15-limit mapping for a diamond comes out as this:

19/69, 165.033 cent generator

basis:
(0.5, 0.13752748238381826)

mapping by period and generator:
([2, 0], ([4, 3, 7, 5, 3], [-3, 6, -5, 7, 16]))

mapping by steps:
[(80, 58), (127, 92), (186, 135), (225, 163), (277, 201), (296, 215)]

unison vectors:
[[11, -4, -2, 0, 0, 0], [5, -1, 3, 0, -3, 0], [2, 2, -7, 0, 0, 3], [1,
10, 0, -6, 0, 0]]

highest interval width: 22
complexity measure: 44 (58 for smallest MOS)
highest error: 0.005692 (6.830 cents)
unique

So the Partch-like 15-limit scale would have around 90 notes.

> > 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?

It'll be 36 for a complete chord, so at least 70 for the diamond I think.
I still don't know if anybody else understands these printouts, so it's
unlikely a mapping's going to be matched to it, unless you've done the
work. I could work it out, but I'd have more incentive if somebody
planned to tune it up.

This one, that's been hiding at the top of the 9-limit list, is much more
inviting:

13/41, 380.391 cent generator

basis:
(1.0, 0.31699250014423125)

mapping by period and generator:
([1, 0], ([0, 2, -1], [5, 1, 12]))

mapping by steps:
[(22, 19), (35, 30), (51, 44), (62, 53)]

unison vectors:
[[-10, -1, 5, 0], [5, -12, 0, 5]]

highest interval width: 12
complexity measure: 12 (13 for smallest MOS)
highest error: 0.004936 (5.923 cents)
unique

Does it have a history? I should be able to work out a 24 note mapping
for the 22 note scale.

Graham

> ----- Original Message -----
> From: <graham@microtonal.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, June 27, 2001 3:02 AM
> Subject: [tuning] Re: 13-limit mappings
>
> ...
> I still don't know if anybody else understands these printouts, so it's
> unlikely a mapping's going to be matched to it, unless you've done the
> work. I could work it out, but I'd have more incentive if somebody
> planned to tune it up.

Graham, I've been struggling to make sense of these printouts and
just don't get it.

I understand the top line and the "basis" line, and I'm pretty
sure the "unison vector" line is simply a prime-factorization
of the unison-vectors, right?

But I'm mystified by the "mapping" lines. Can you explain one
example in detail?, say... this one:

>
>
> This one, that's been hiding at the top of the 9-limit list, is much more
> inviting:
>
> 13/41, 380.391 cent generator
>
> basis:
> (1.0, 0.31699250014423125)
>
> mapping by period and generator:
> ([1, 0], ([0, 2, -1], [5, 1, 12]))
>
> mapping by steps:
> [(22, 19), (35, 30), (51, 44), (62, 53)]
>
> unison vectors:
> [[-10, -1, 5, 0], [5, -12, 0, 5]]

-monz
http://www.monz.org
"All roads lead to n^0"

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In-Reply-To: <001101c0fef4$13860740$4448620c@att.com>
monz wrote:

> Graham, I've been struggling to make sense of these printouts and
> just don't get it.
>
> I understand the top line and the "basis" line, and I'm pretty
> sure the "unison vector" line is simply a prime-factorization
> of the unison-vectors, right?

Yes, but the unison vectors are calculated, rather than being supplied.
Even the program that generates temperaments from unison vectors still
tries to work backwards.

> But I'm mystified by the "mapping" lines. Can you explain one
> example in detail?, say... this one:

Okay, and I'll explain all of it for the benefit for those who haven't
got as far as you. I'll use ratios for tempered intervals to save
writing "approximation of" all the time.

> > 13/41, 380.391 cent generator

13/41 is the ratio of the generator to period as pitches. So it's 13
steps from a 41 note scale. And the optimum generator is as shown. In
some cases the period won't be an octave, but here it is, so it's 41
notes to the octave.

> > basis:
> > (1.0, 0.31699250014423125)

1.0 is the size of the period in octaves. Sometimes this will be a
fraction. 0.5 is common. The other number is the generator again, but
in octaves instead of cents.

> > mapping by period and generator:
> > ([1, 0], ([0, 2, -1], [5, 1, 12]))

The first two-element list shows the mapping of the octave. The second
element is always zero for both my scripts, as the period is always a
fraction of an octave. So the first number tells you how many equal
parts the octave is being divided into. Here it's 1 which is the
simplest case.

[5, 1, 12], the second of the pair of lists, is the more useful one. It
specifies each prime interval in terms of the generator. So 3:1 is 5
generators, 5:1 is 1 generator and 7:1 is 12 generators.

The [0, 2, -1] says how many periods you need to correct by. So 3:1 is
exactly 5 generators. 5:1 is 1 generator plus 2 octaves. 7:1 is 12
generators minus an octave.

In more familiar terms, the generator is a 5:4 major third. 5 major
thirds are a 3:1 perfect twelfth. An octave less two major thirds is a
9:7 supermajor third. (2*(5,0) - (12,-1) = (-2, 1))

I know third-generated MOS scales have been looked at before, so does
anybody have results they'd like to share with us?

> > mapping by steps:
> > [(22, 19), (35, 30), (51, 44), (62, 53)]

Each pair shows the size of a prime interval in terms of scale steps.
Call the steps x and y. An octave is 22x+19y. For the case where x=y,
you have 41-equal. Where x=0, you have 19-equal. Where y=0, you have
22-equal. So 19, 22 and 41-equal are all members of this temperament
family.

3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any
7-prime limit interval in terms of x and y by combining these.

For visualising the scale, it can be simpler to reduce each prime
interval to be within the octave.

> > [(22, 19), (35, 30), (51, 44), (62, 53)]

3:2 is 13x + 11y
5:4 is 7x + 6y
8:7 is 4x + 4y

and use simpler coordinates. Here, q=x+y and p=x

3:2 is 11q + 2p
5:4 is 6q + p
8:7 is 4q

So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal and p is 1 step
in 41-or 22-equal, and no steps in 19-equal.

> > unison vectors:
> > [[-10, -1, 5, 0], [5, -12, 0, 5]]

These probably aren't in their simplest terms. That bit of the program
still needs working on. Dan Stearns says he can do it ...

They're the ratio-space vectors for intervals that are tempered out to
become unisons.

Graham