Steinhaus conjecture

While perusing the Wilson archives just now, I ran across this:
http://anaphoria.com/steinhaus.PDF

It apparently refers to this:
http://anziamj.austms.org.au/JAMSA/V45/Part3/Ravenstein.html

If this abstract is to be believed, there's nothing worse
than Trihill for scales of linear construction. I didn't
realize that.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> While perusing the Wilson archives just now, I ran across this:
> http://anaphoria.com/steinhaus.PDF
>
> It apparently refers to this:
> http://anziamj.austms.org.au/JAMSA/V45/Part3/Ravenstein.html
>
> If this abstract is to be believed, there's nothing worse
> than Trihill for scales of linear construction. I didn't
> realize that.

I don't know what Trihill is and googling fails to turn it up, but the really relevant theorem is the three distance theorem, and I thought we all knew that. Don't we?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I don't know what Trihill is and googling fails to turn it up, but the really relevant theorem is the three distance theorem, and I thought we all knew that. Don't we?
>

I don't know what the heck Erv is up too, but for an illustration of the three gap (NOT three distance) theorem, consider the list of ets with relative error, size*error, for the fifth between -24 and 0. So 12, with a fifth less than 2 cents flat, just makes the list. If you look at the gaps, there are gaps of size 12, size 41, and size 53. Three gaps!

At 09:40 PM 6/27/2010, you wrote:

>> While perusing the Wilson archives just now, I ran across this:
>> http://anaphoria.com/steinhaus.PDF
>>
>> It apparently refers to this:
>> http://anziamj.austms.org.au/JAMSA/V45/Part3/Ravenstein.html
>>
>> If this abstract is to be believed, there's nothing worse
>> than Trihill for scales of linear construction. I didn't
>> realize that.
>
>I don't know what Trihill is and googling fails to turn it up,

Googling should turn up tuning list posts. I think Dan Stearns
made it up. I don't know the exact definition or if there is one,
but it's some kind of generalization of Myhill.

>but the
>really relevant theorem is the three distance theorem, and I thought
>we all knew that. Don't we?

I didn't even know a T[n] scale can't have more than 3 sizes per
generic interval, if indeed that's true.

Wikipedia and mathworld don't have entries for the "three distance
theorem". Google produces this
http://demonstrations.wolfram.com/ThreeDistanceTheorem/
which sounds like the same thing as the Steinhaus conjecture.

-Carl

I wrote:

>I didn't even know a T[n] scale can't have more than 3 sizes per
>generic interval, if indeed that's true.

Crud, I didn't mean that, but rather, three sizes of 2nd (step).

-Carl

At 10:22 PM 6/27/2010, you wrote:

>I don't know what the heck Erv is up too, but for an illustration of
>the three gap (NOT three distance) theorem, consider the list of ets
>with relative error, size*error, for the fifth between -24 and 0. So
>12, with a fifth less than 2 cents flat, just makes the list. If you
>look at the gaps, there are gaps of size 12, size 41, and size 53.
>Three gaps!

Hm, I think that actually does have something to do with what he's
doing

http://anaphoria.com/steinhaus.PDF

which is drawing a horogram and showing the contained scales in a
table below, with their 2nds exposed. The MOS have two sizes of
2nds and all the other scales have only three. The rays on the
horogram are numbered by position on the generator chain, and the
scale degrees are numbered the same way on the table.

-Carl

At 10:28 PM 6/27/2010, you wrote:

>Wikipedia and mathworld don't have entries for the "three distance
>theorem". Google produces this
>http://demonstrations.wolfram.com/ThreeDistanceTheorem/
>which sounds like the same thing as the Steinhaus conjecture.

This says they're the same:
http://myweb1.lsbu.ac.uk/~whittyr/MathSci/TheoremOfTheDay/NumberTheory/ThreeDistance/TotDThreeDistance.pdf

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 10:28 PM 6/27/2010, you wrote:
>
> >Wikipedia and mathworld don't have entries for the "three distance
> >theorem". Google produces this
> >http://demonstrations.wolfram.com/ThreeDistanceTheorem/
> >which sounds like the same thing as the Steinhaus conjecture.
>
> This says they're the same:
> http://myweb1.lsbu.ac.uk/~whittyr/MathSci/TheoremOfTheDay/NumberTheory/ThreeDistance/TotDThreeDistance.pdf

It is the same as the Steinhaus conjecture. But it's not the same as another theorem called the "Three Gaps Theorem", and to add to the confusion, apparently people sometimes call the three distances theorem the three gaps theorem--at least Ravenstein does. Both theorems have musical applications, but not the same ones.