Distinct p-limit intervals and ets

Here's an interesting computation--the first ets which distinguish all the p-limit intervals for p = 5,7,11,13 and 17. Anyone who cares to may join in the fun by filling in the gaps, or extending the list of p-limits.

5: 9,11,12,14,15,16,18,19 ...

7: 27,31,35,36,37,40,41,42 ...

11: 58,65,72,73,80,84,87,89 ...

13: 87,94,95,103,111,113,118,120 ...

17: 149,151,159,161,163,169,170,171 ...

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here's an interesting computation--the first ets which distinguish
all the p-limit intervals for p = 5,7,11,13 and 17.

There are an infinite number of 5-prime limit intervals. You must
mean odd limit, in which case you should include 9 and 15.

>Anyone who cares to may join in the fun by filling in the gaps, or
>extending the list of p-limits.
>
> 5: 9,11,12,14,15,16,18,19 ...
>
> 7: 27,31,35,36,37,40,41,42 ...
>
> 11: 58,65,72,73,80,84,87,89 ...
>
> 13: 87,94,95,103,111,113,118,120 ...
>
> 17: 149,151,159,161,163,169,170,171 ...

This is what we call "uniqueness", and Manuel has compiled vast
tables of both this and "consistency" for both odd and integer
limits, and fractional ETs. Please see

http://www.ixpres.com/interval/dict/unique.htm

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > Here's an interesting computation--the first ets which distinguish
> all the p-limit intervals for p = 5,7,11,13 and 17.
>
> There are an infinite number of 5-prime limit intervals. You must
> mean odd limit, in which case you should include 9 and 15.

Thus my mention of gaps.

> http://www.ixpres.com/interval/dict/unique.htm

Monz, the links to the tables are outdated. Manuel, could you provide
the updated links?

>Monz, the links to the tables are outdated. Manuel, could you provide
>the updated links?

http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
> http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html

These don't contain the same information as I was looking at; I only considered the standard et val which rounds to the nearest integer for each prime, and then looked at the first eight unique examples.
I already went farther than Manual's tables, and was pondering such questions as whether 311 would turn out unique in the 41-limit.

We can define a funtion unq(n) from odd numbers>1, which tells us the first unique standard et for odd limit n. So, unq(3)=3, unq(5)=9,
unq(7)=27, unq(9)=?, unq(11)= 58 ... calculating unq to some point (49?) might be an interesting project sometime.

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> >Monz, the links to the tables are outdated. Manuel, could you
provide
> >the updated links?
>
> http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
> http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html

Monz, would you update your links in the "unique" definition, please?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> > http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
> > http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html
>
> These don't contain the same information as I was looking at; I
>only considered the standard et val which rounds to the nearest
>integer for each prime,

I don't like considering this "the standard et val" . . . perhaps you
can call this the genewardsmith val or something.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> I don't like considering this "the standard et val" . . .

Why not? I can think of other standards, but this one is easy and is the first thing anyone would think of, I should imagine.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > I don't like considering this "the standard et val" . . .
>
> Why not? I can think of other standards, but this one is easy and >
is the first thing anyone would think of, I should imagine.

I would hope not. For example, in 64-tET for the 5-limit, it's
probably only the third-best mapping. You should look at Stoney's
article, for example, before assuming that this should be seen as
some kind of "standard".

Hi Paul,

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 10, 2002 11:45 AM
> Subject: [tuning-math] Re: Distinct p-limit intervals and ets
>
>
> --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> >
> > > Monz, the links to the tables are outdated. Manuel, could you
> > > provide the updated links?
> >
> > http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
> > http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html
>
> Monz, would you update your links in the "unique" definition, please?

I started to do this, but I see that the original links both
point to the first URL listed here. But I've never fully understood
these tables, so I'm not sure how to link to them. Please clarify.
Feel free to expand your "unique" definition if needed.
Diagrams *always* help me. :)

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> Hi Paul,
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Thursday, January 10, 2002 11:45 AM
> > Subject: [tuning-math] Re: Distinct p-limit intervals and ets
> >
> >
> > --- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> > >
> > > > Monz, the links to the tables are outdated. Manuel, could you
> > > > provide the updated links?
> > >
> > > http://www.xs4all.nl/~huygensf/doc/consist_limits.html and
> > > http://www.xs4all.nl/~huygensf/doc/cons_limit_bounds.html
> >
> > Monz, would you update your links in the "unique" definition,
please?
>
>
> I started to do this, but I see that the original links both
> point to the first URL listed here.

Huh? I'm looking at http://www.ixpres.com/interval/dict/unique.htm ,
and I see that the links are to
ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt . . . You
should change that to
http://www.xs4all.nl/~huygensf/doc/consist_limits.html . . .

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 10, 2002 6:34 PM
> Subject: [tuning-math] Re: Distinct p-limit intervals and ets
>
>
> Huh? I'm looking at http://www.ixpres.com/interval/dict/unique.htm ,
> and I see that the links are to
> ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt . . . You
> should change that to
> http://www.xs4all.nl/~huygensf/doc/consist_limits.html . . .

OK, that's done, and it's been uploaded. I just wasn't sure
if the other link should have been in the definition.

Anyway, my request still stands: can you please explain these
tables in more detail? I don't quite understand what's in them.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> Anyway, my request still stands: can you please explain these
> tables in more detail? I don't quite understand what's in them.

I'll leave that to Manuel for now . . . good night.

>OK, that's done, and it's been uploaded. I just wasn't sure
>if the other link should have been in the definition.

You still need to change the second link, it's the same as the
first one.

>Anyway, my request still stands: can you please explain these
>tables in more detail? I don't quite understand what's in them.

Let's take 31-tET as example. If you do EQUAL/DATA 31 in Scala,
then you see
Highest harmonic represented consistently: 12
This is the first and fourth column in the table. If you find
31.0 in the third column, you see it's in the range between
30.85557 and 31.07329 with consistency 12.
Next in Scala you get
Highest harmonic represented uniquely: 9
This is the fifth column. In 31-tET it's 9, because it
rounds to 5 steps. One ratio with the 10th harmonic, 10/9,
rounds to the same 5 steps so it doesn't have a unique representation.
Next you get
Highest harm. represented uniquely inv. equiv.: 8
which means with inversional equivalence. In 31-tET it's 8,
because a ratio with the 9th harmonic, 9/8, rounds to 5 steps and
therefore its inverse to 31-5=26 steps. However 9/5 also rounds
to 26 steps and so 9 is not unique.

The other file contains the same information in a different
arrangement.

Manuel