open question

What's the biggest ET that tempers out all four of
these commas: 2401/2400, 225/224, 126/125, and 81/80?

-Carl

>What's the biggest ET that tempers out all four of
>these commas: 2401/2400, 225/224, 126/125, and 81/80?

...with its patent val?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >What's the biggest ET that tempers out all four of
> >these commas: 2401/2400, 225/224, 126/125, and 81/80?
>
> ...with its patent val?

Scala gives you the answer (93):
divide/consistent "2401/2400 225/224 126/125 81/80" 2/1 0

Manuel

>> >What's the biggest ET that tempers out all four of
>> >these commas: 2401/2400, 225/224, 126/125, and 81/80?
>>
>> ...with its patent val?
>
>Scala gives you the answer (93):
>divide/consistent "2401/2400 225/224 126/125 81/80" 2/1 0
>
>Manuel

Wow! Can it give the smallest? Curious abuot the syntax.

I did something similar in Scheme last night. According to
it, the complete list of ETs < 1200 tempering these commas
out with their patent vals is (93 62 31).

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >What's the biggest ET that tempers out all four of
> >> >these commas: 2401/2400, 225/224, 126/125, and 81/80?
> >>
> >> ...with its patent val?
> >
> >Scala gives you the answer (93):
> >divide/consistent "2401/2400 225/224 126/125 81/80" 2/1 0
> >
> >Manuel
>
> Wow! Can it give the smallest? Curious abuot the syntax.

> I did something similar in Scheme last night. According to
> it, the complete list of ETs < 1200 tempering these commas
> out with their patent vals is (93 62 31).

Yes, same result. The last three commas are not independent by the way
so you could have omitted one of them.

I have to look up the syntax too every time, but there's also a
dialog: Approximate:Pitch with equal division, then select the last
option.

Manuel

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> What's the biggest ET that tempers out all four of
> these commas: 2401/2400, 225/224, 126/125, and 81/80?

I took the kernel of these and got what I expected to get, namely
<31 49 72 87|.

At 01:52 PM 12/30/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> What's the biggest ET that tempers out all four of
>> these commas: 2401/2400, 225/224, 126/125, and 81/80?
>
>I took the kernel of these and got what I expected to get, namely
><31 49 72 87|.

Funny, that doesn't agree with Manuel or my lists. At least
I think Manuel was agreeing with my list of 31 62 93. Perhaps
you missed the bit about it having to be the patent val?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 01:52 PM 12/30/2006, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >>
> >> What's the biggest ET that tempers out all four of
> >> these commas: 2401/2400, 225/224, 126/125, and 81/80?
> >
> >I took the kernel of these and got what I expected to get, namely
> ><31 49 72 87|.
>
> Funny, that doesn't agree with Manuel or my lists.

Sure it does. It proves any such val must be of the form
n*<31 49 72 87|.

At least
> I think Manuel was agreeing with my list of 31 62 93. Perhaps
> you missed the bit about it having to be the patent val?

No, I was just adding a comment which I thought was relevant, namely,
that you are looking at multiples of 31 and those only.

>> >> What's the biggest ET that tempers out all four of
>> >> these commas: 2401/2400, 225/224, 126/125, and 81/80?
>> >
>> >I took the kernel of these and got what I expected to get, namely
>> ><31 49 72 87|.
>>
>> Funny, that doesn't agree with Manuel or my lists.
>
>Sure it does. It proves any such val must be of the form
>n*<31 49 72 87|.
>
> At least
>> I think Manuel was agreeing with my list of 31 62 93. Perhaps
>> you missed the bit about it having to be the patent val?
>
>No, I was just adding a comment which I thought was relevant, namely,
>that you are looking at multiples of 31 and those only.

Oh! Right!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> asked:
>
> What's the biggest ET that tempers out all four of
> these commas: 2401/2400, 225/224, 126/125, and 81/80?
>
For the smallest ratio among that 4 values we do obtain:
ln(2)/ln(2401/2400) =~ 1663.89978...
Round that to the next integer for:

1664=13*2^7 ET

Hence
(2401/2400)^1664 =~ 2.0000835...
fits well a little bit sharper above an 8th.

Also using 2^(1/1664) ET-steps suit well
in order to approximatate the other above ratios by:

1.
2401/2400 ~one single step per definition

2.
225/224 =~ 1.00446429...
(1664/ln(2))*ln(225/224) =~ 10.6933119...
11 steps
2^(11/1664) =~ 1.00459262... >~ 225/224

3.
126/125 = 1.008 exactlty
(1664/ln(2))*ln(126/125) =~ 19.128743...
19 steps
2^(19/1664) =~ 1.00794594... <~ 126/125

4. 81/80 = 1.0125 the SC
(1664/ln(2))*ln(81/80) =~ 29.8220549...
by 30 steps or 15 double-steps
2^(30/1664) = 2^(15/832) =~ 1.01257505... >~ 81/80

A.S.

If I have an n-ET, the max error for any set of targets
will be 600/n cents. What will the mean error be over
all possible targets? My guess is 300/n.

-Carl

At 02:52 PM 3/18/2007, you wrote:
>If I have an n-ET, the max error for any set of targets
>will be 600/n cents. What will the mean error be over
>all possible targets? My guess is 300/n.

I tested this, and my guess seems to be right.

-Carl

At 02:52 PM 3/18/2007, you wrote:
>If I have an n-ET, the max error for any set of targets
>will be 600/n cents. What will the mean error be over
>all possible targets? My guess is 300/n.

I tested this, and my guess seems to be right.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> At 02:52 PM 3/18/2007, you wrote:
> >If I have an n-ET, the max error for any set of targets
> >will be 600/n cents. What will the mean error be over
> >all possible targets? My guess is 300/n.
>
> I tested this, and my guess seems to be right.

How could it not be?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@> wrote:
> >
> > At 02:52 PM 3/18/2007, you wrote:
> > >If I have an n-ET, the max error for any set of targets
> > >will be 600/n cents. What will the mean error be over
> > >all possible targets? My guess is 300/n.
> >
> > I tested this, and my guess seems to be right.
>
> How could it not be?
>

Sorry, I shouldn't be oracular. Check this out:

http://en.wikipedia.org/wiki/Equidistribution_theorem

http://mathworld.wolfram.com/WeylsCriterion.html

>>>If I have an n-ET, the max error for any set of targets
>>>will be 600/n cents. What will the mean error be over
>>>all possible targets? My guess is 300/n.
>>
>> I tested this, and my guess seems to be right.
>
>How could it not be?

It seemed like it should. I didn't feel like working it out,
so I tested it instead.

-Carl