A higher dimensional continued fraction

There are many analogues to the continued fraction algorithm for
multiple Diophantine approximation, none very good. After asnwering a
question on the main list which involved the fact that being the first
smaller value |q x - p| is equivalent to being the next convergent
p/q to x, it occured to me that we could extend the definition, by
taking the sums of the |q x - p| for fixed q and x log2(3), log2(5) and
log2(5/3). This is not an algorithm, but rather a definition. In other
and more precise words, we take the sums

|q*log2(3) - h(q, 3)| + |q*log2(5) - h(q, 5)| +
|log2(5)*h(q, 3)-log2(3)*(h(q, 5)|

and take the successive minima, which corresponds to making the product
of the {2,3}-comma, the {2,5}-comma and the {3,5}-comma minimal.

When I do this, I get the following list of ets and products of commas:

1 2.210897
2 1.485097
3 .794231
7 .530043
12 .418729
19 .311429
53 .156774
118 .117232
171 .090692
441 .087385
612 .061480
1171 .045611
1783 .038109
2513 .032660
4296 .011628

This can, of course, be extended to higher limits. I'd better ask on
sci.math if anyone's heard of this. Or maybe email Jeff Shallet.

--- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> There are many analogues to the continued fraction algorithm for
> multiple Diophantine approximation, none very good.
...
> This can, of course, be extended to higher limits.
...

This is interesting. I am curious to see the results for higher limits.

--- In tuning-math@yahoogroups.com, d.keenan@b... wrote:
> --- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> > There are many analogues to the continued fraction algorithm for
> > multiple Diophantine approximation, none very good.
> ...
> > This can, of course, be extended to higher limits.
> ...
>
> This is interesting. I am curious to see the results for higher limits.

I've concluded that the whole thing makes more sense if we treat 2 the
same as the odd primes. Let me make some definitions.

If v is a val and a and b are relatively prime positive integers, I'll
define the {a,b}-comma, or a/b-comma, for v as either a^v(b)/b^v(a) or
b^v(a)/a^v(b) depending on which is greater than 1. If either of a or
b is 1, the a/b-comma is also 1, so we can ignore this case and assume
both a and b are greater than 1 if we wish.

For a given postive integer L, the L-even-limit, or L-cap consonances,
are a/b, a, b <= L, a and b relatively prime and where we may
additionally assume a/b > 1. The L-cap badness measure for v is then
defined as the product of all a/b-commas for a/b an L-cap consonance.

Below I list successively better standard vals for cap limits from
5 to 13, and n from 1 to 1000; as usual it makes no real difference to
restrict to standard vals because we are looking at best of breed
anyway. I list the equal division n, and next to it the log base 2 of
the L-cap badness measure (which is an ugly looking rational number I
don't want to mess with.)

5 cap

1 1 3.684828
2 2.537235
3 1.352888
7 1.226512
12 .731555
34 .669239
53 .287177
118 .193352
612 .107366

6 cap

1 5.480687
2 3.852407
3 1.833575
7 1.661817
12 1.130734
19 .809398
34 .786874
53 .440938
118 .285008
441 .213453
612 .165899
730 .164162

7 cap

1 8.936697
2 7.674194
3 7.312043
4 4.175933
12 3.815278
19 3.065070
31 2.008584
72 1.808742
99 1.392254
171 .568097

8 cap

1 11.725529
2 10.408271
3 9.416222
4 6.746879
5 6.049309
10 5.867766
12 5.219740
19 4.777064
22 4.702956
31 2.554792
99 2.261336
171 .976047

9 cap

1 21.373486
2 16.540848
3 16.101527
4 12.672183
5 8.378538
12 7.076935
19 6.744799
31 5.509181
41 4.969997
53 4.356189
99 3.598852
171 1.366268

10 cap

1 28.584111
2 20.207791
3 19.879698
4 14.736610
5 10.703392
12 8.573892
19 8.130354
31 7.088977
41 6.302737
53 5.494689
72 5.153786
99 3.992756
171 1.466939

11 cap

1 42.871697
2 26.752811
4 25.759778
5 20.792831
7 17.489047
12 17.427374
15 17.181518
22 14.358025
31 11.034810
72 7.688143
270 5.353860
342 3.990495

12 cap

1 48.546856
2 31.161133
4 27.891159
5 23.685995
7 20.139916
15 18.691086
22 16.281841
31 12.099355
72 8.314304
270 5.991772
342 4.719771

13 cap

1 61.688418
2 43.938450
4 42.273300
5 35.573172
7 28.108502
15 27.801661
19 26.452925
31 23.684039
41 22.121134
53 19.854742
72 16.495624
224 14.411084
270 10.995631
494 8.532451

--- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> Below I list successively better standard vals for cap limits from
> 5 to 13, and n from 1 to 1000; as usual it makes no real difference to
> restrict to standard vals because we are looking at best of breed
> anyway. I list the equal division n, and next to it the log base 2 of
> the L-cap badness measure (which is an ugly looking rational number I
> don't want to mess with.)
...

I'd appreciate it if you could give these sequences for integer-limits
to 31 and ETs to 5000.

The reason I'm interested is because it may help us decide the best
places to site the various precision-levels of Sagittal JI notation.
Only one such level is effectively set in stone now, and that's the
medium-precision "athenian" set which is similar (epimorphic?) to 224-ET.

I notice 224-ET first appears in the 13-integer-limit list.

Of course, integer limit is not really what would be most relevant to
this application, but I'm interested nonetheless.

On the subject of what sort of "limit" would be most relevant to this:

If anyone can come up with a simple real-valued function of the
following 2,3-free ratios (or rather their exponent vectors) that
would give roughly monotonically increasing values for the following
ratio sequence I'd be very interested.

This might then be extrapolated back to include powers of 2 and 3 in
some reasonable way. These are listed in popularity order from the
Scala archive.

1
5
7
25
5:7
11
35
125
49
13
5:11
7:11
17
7:25
5:49
5:13
175
19
245
7:13
625
23
25:49
55
77
5:17
5:19
11:35
11:13
31
343
29
7:125
7:55
11:17
5:77
7:19
385
49:55
7:17
1225
37
121
5:23
13:19
13:17
7:23
11:25
11:31
65
47
13:25
3125
11:49
11:19
5:343
91
43
5:29
11:37
11:23
13:23
25:77
49:125
15625
143
5:31
13:35
25:343
5:37
17:19
41
95
53
17:23

--- In tuning-math@yahoogroups.com, d.keenan@b... wrote:
> --- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> > Below I list successively better standard vals for cap limits from
> > 5 to 13, and n from 1 to 1000; as usual it makes no real
difference to
> > restrict to standard vals because we are looking at best of breed
> > anyway. I list the equal division n, and next to it the log base
2 of
> > the L-cap badness measure (which is an ugly looking rational
number I
> > don't want to mess with.)
> ...
>
> I'd appreciate it if you could give these sequences for integer-
limits
> to 31 and ETs to 5000.
>
> The reason I'm interested is because it may help us decide the best
> places to site the various precision-levels of Sagittal JI notation.

But why is this particular ET badness measure of so much interest to
you, Dave? I mean, I'm *very* interested in learning more about it,
and I may end up advocating it myself (who knows?), but there are so
many others we've used . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, d.keenan@b... wrote:
> > --- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> > > Below I list successively better standard vals for cap limits from
> > > 5 to 13, and n from 1 to 1000; as usual it makes no real
> difference to
> > > restrict to standard vals because we are looking at best of breed
> > > anyway. I list the equal division n, and next to it the log base
> 2 of
> > > the L-cap badness measure (which is an ugly looking rational
> number I
> > > don't want to mess with.)
> > ...
> >
> > I'd appreciate it if you could give these sequences for integer-
> limits
> > to 31 and ETs to 5000.
> >
> > The reason I'm interested is because it may help us decide the best
> > places to site the various precision-levels of Sagittal JI notation.
>
> But why is this particular ET badness measure of so much interest to
> you, Dave? I mean, I'm *very* interested in learning more about it,
> and I may end up advocating it myself (who knows?), but there are so
> many others we've used . . .

The badness measure wasn't of interest at all. I don't even have any
idea what it is. I probably misunderstood, but I imagined that the
actual sequence of ETs wasn't dependent on the badness measure, and
was something like the sequence of convergents for a ratio. Since Gene
claimed he was using a higher-D generalisation of continued fraction
approximation (which I also don't understand).

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> The badness measure wasn't of interest at all. I don't even have any
> idea what it is. I probably misunderstood, but I imagined that the
> actual sequence of ETs wasn't dependent on the badness measure, and
> was something like the sequence of convergents for a ratio. Since
Gene
> claimed he was using a higher-D generalisation of continued fraction
> approximation (which I also don't understand).

If you want to approximate log2(3), and take as your badness measure
the size of the Pythagorean comma, you get exactly the convergents of
the continued fraction. I generalized that, but whether anyone finds
it interesting is another question. I asked Jeff Shallit, who didn't
know and suggested I ask Jeff Lagarias, who hasn't answered my email
yet. Chances are it's a new idea.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > The badness measure wasn't of interest at all. I don't even have any
> > idea what it is. I probably misunderstood, but I imagined that the
> > actual sequence of ETs wasn't dependent on the badness measure, and
> > was something like the sequence of convergents for a ratio. Since
> Gene
> > claimed he was using a higher-D generalisation of continued fraction
> > approximation (which I also don't understand).
>
> If you want to approximate log2(3), and take as your badness measure
> the size of the Pythagorean comma, you get exactly the convergents of
> the continued fraction. I generalized that, but whether anyone finds
> it interesting is another question. I asked Jeff Shallit, who didn't
> know and suggested I ask Jeff Lagarias, who hasn't answered my email
> yet. Chances are it's a new idea.

OK. Well it still sounds pretty interesting to me.

Any chance of going up to the 32-integer-limit and 5000-ET?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> Any chance of going up to the 32-integer-limit and 5000-ET?

Maple is slow, and I use the computer overnight for other things.
Others on this list could compute it much faster.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > The badness measure wasn't of interest at all. I don't even have
any
> > idea what it is. I probably misunderstood, but I imagined that the
> > actual sequence of ETs wasn't dependent on the badness measure,
and
> > was something like the sequence of convergents for a ratio. Since
> Gene
> > claimed he was using a higher-D generalisation of continued
fraction
> > approximation (which I also don't understand).
>
> If you want to approximate log2(3), and take as your badness
measure
> the size of the Pythagorean comma, you get exactly the convergents
of
> the continued fraction. I generalized that, but whether anyone
finds
> it interesting is another question. I asked Jeff Shallit, who
didn't
> know and suggested I ask Jeff Lagarias, who hasn't answered my
email
> yet. Chances are it's a new idea.

J. Murray Barbour used multi-term continued fractions too, as did
many other people. There's actually been a huge amount of discussion
surrounding this topic here and on the tuning list, as it's connected
to so many issues. The problem is that there's no clear choice for
the 'best' way of defining them, and there have been some proofs of
this statement (made more precise, of course), but then there's the
ferguson-forcade algorithm . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> There's actually been a huge amount of discussion
> surrounding this topic here

or at least the harmonic entropy list.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> J. Murray Barbour used multi-term continued fractions too, as did
> many other people.

I know. That was an algorithm, and not a very good one. This is a
definition.

I might be mistaken, but it looks at least related to what I was
doing here, half way down the page.

http://www.kees.cc/tuning/perbl.html

--- In tuning-math@yahoogroups.com, gwsmith@s... wrote:
> There are many analogues to the continued fraction algorithm for
> multiple Diophantine approximation, none very good. After asnwering
a
> question on the main list which involved the fact that being the
first
> smaller value |q x - p| is equivalent to being the next convergent
> p/q to x, it occured to me that we could extend the definition, by
> taking the sums of the |q x - p| for fixed q and x log2(3), log2(5)
and
> log2(5/3). This is not an algorithm, but rather a definition. In
other
> and more precise words, we take the sums
>
> |q*log2(3) - h(q, 3)| + |q*log2(5) - h(q, 5)| +
> |log2(5)*h(q, 3)-log2(3)*(h(q, 5)|
>
> and take the successive minima, which corresponds to making the
product
> of the {2,3}-comma, the {2,5}-comma and the {3,5}-comma minimal.
>
> When I do this, I get the following list of ets and products of
commas:
>
> 1 2.210897
> 2 1.485097
> 3 .794231
> 7 .530043
> 12 .418729
> 19 .311429
> 53 .156774
> 118 .117232
> 171 .090692
> 441 .087385
> 612 .061480
> 1171 .045611
> 1783 .038109
> 2513 .032660
> 4296 .011628
>
> This can, of course, be extended to higher limits. I'd better ask on
> sci.math if anyone's heard of this. Or maybe email Jeff Shallet.