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

--- 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Monz, the links to the tables are outdated.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Monz, the links to the tables are outdated.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