isosceles distance

Here's something from 1999 or abouts. I think it's an exchange
between Manuel and Paul Hahn, as the error in Scala was being
fixed...

>> If so, please verify my results. I have used the base-2 log.
>>
>> 225/224 7.813781
>> 126/125 5.491853
>> 128/125 6.965784
>> 81/80 6.339850
>> 64/63 5.977279
>> 50/49 4.643856
>
>126/125 should be 7.451211.
> 50/49 should be 5.614709.
>
>The others are correct.

Wow. I agree with Paul H. on 126/125 and 50/49 and with
Manuel and Paul H. on 128/125 and 64/63. But I get...

225/224 8.29920801838728
81/80 7.07681559705083

Here's a bit showing what I was just saying about using the
shortest vector in the available odd limit...

>However, the minimum complexity is achieved by assigning the 9
>exponent as large as possible, and the 3 exponent 0 or 1 as
>appropriate.
>
>Okay. The 11-limit prime vectors for 9:5 and 11:5 are (2 -1 0 0)
>and (0 -1 0 1). Converting these to the optimal 11-limit odd-
>factor vectors as described above, we get (0 -1 0 1 0) for 9:5
>and (0 -1 0 0 1) for 11:5.
>
>Now apply the algorithms:
>
>Simple version:
>
> In each interval, the absolute values of the sums of the
> positive and negative exponents are 1. Hence, each of
> these are primary/consonant intervals within the 11-limit.
>
>Weighted version:
>
> (0 -1 0 1 0):
> 1st iteration lg(9) (0 0 0 0 0) done
>
> (0 -1 0 0 1):
> 1st iteration lg(11) (0 0 0 0 0) done
>
>See? It works.

-Carl

Sorry, I'll stop calling this isosceles now.

Meanwhile, the graph that strips the quotes off "distance" is
just the An root lattice. Certainly graphs are more general
than root lattices or Gene's "discrete subgroup of a finite-
dimensional normed real vector spaces", but I don't know what
the equivalent of graph distance is in such a space. Which is
why I said I was mixing metaphors.

-Carl

>Here's something from 1999 or abouts. I think it's an exchange
>between Manuel and Paul Hahn, as the error in Scala was being
>fixed...
>
>>> If so, please verify my results. I have used the base-2 log.
>>>
>>> 225/224 7.813781
>>> 126/125 5.491853
>>> 128/125 6.965784
>>> 81/80 6.339850
>>> 64/63 5.977279
>>> 50/49 4.643856
>>
>>126/125 should be 7.451211.
>> 50/49 should be 5.614709.
>>
>>The others are correct.
>
>Wow. I agree with Paul H. on 126/125 and 50/49 and with
>Manuel and Paul H. on 128/125 and 64/63. But I get...
>
>225/224 8.29920801838728
>81/80 7.07681559705083
>
>Here's a bit showing what I was just saying about using the
>shortest vector in the available odd limit...
>
>>However, the minimum complexity is achieved by assigning the 9
>>exponent as large as possible, and the 3 exponent 0 or 1 as
>>appropriate.
>>
>>Okay. The 11-limit prime vectors for 9:5 and 11:5 are (2 -1 0 0)
>>and (0 -1 0 1). Converting these to the optimal 11-limit odd-
>>factor vectors as described above, we get (0 -1 0 1 0) for 9:5
>>and (0 -1 0 0 1) for 11:5.
>>
>>Now apply the algorithms:
>>
>>Simple version:
>>
>> In each interval, the absolute values of the sums of the
>> positive and negative exponents are 1. Hence, each of
>> these are primary/consonant intervals within the 11-limit.
>>
>>Weighted version:
>>
>> (0 -1 0 1 0):
>> 1st iteration lg(9) (0 0 0 0 0) done
>>
>> (0 -1 0 0 1):
>> 1st iteration lg(11) (0 0 0 0 0) done
>>
>>See? It works.
>
>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> Here's something from 1999 or abouts. I think it's an exchange
> between Manuel and Paul Hahn, as the error in Scala was being
> fixed...
>
> >> If so, please verify my results. I have used the base-2 log.
> >>
> >> 225/224 7.813781
> >> 126/125 5.491853
> >> 128/125 6.965784
> >> 81/80 6.339850
> >> 64/63 5.977279
> >> 50/49 4.643856

What are these the results of?

> >126/125 should be 7.451211.
> > 50/49 should be 5.614709.
> >
> >The others are correct.

What are we calculating?

> Wow. I agree with Paul H. on 126/125 and 50/49 and with
> Manuel and Paul H. on 128/125 and 64/63. But I get...
>
> 225/224 8.29920801838728
> 81/80 7.07681559705083
>
> Here's a bit showing what I was just saying about using the
> shortest vector in the available odd limit...
>
> >However, the minimum complexity is achieved by assigning the 9
> >exponent as large as possible, and the 3 exponent 0 or 1 as
> >appropriate.
> >
> >Okay. The 11-limit prime vectors for 9:5 and 11:5 are (2 -1 0 0)
> >and (0 -1 0 1). Converting these to the optimal 11-limit odd-
> >factor vectors as described above, we get (0 -1 0 1 0) for 9:5
> >and (0 -1 0 0 1) for 11:5.
> >
> >Now apply the algorithms:
> >
> >Simple version:
> >
> > In each interval, the absolute values of the sums of the
> > positive and negative exponents are 1. Hence, each of
> > these are primary/consonant intervals within the 11-limit.
> >
> >Weighted version:
> >
> > (0 -1 0 1 0):
> > 1st iteration lg(9) (0 0 0 0 0) done
> >
> > (0 -1 0 0 1):
> > 1st iteration lg(11) (0 0 0 0 0) done
> >
> >See? It works.
>
> -Carl
>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Sorry, I'll stop calling this isosceles now.
>
> Meanwhile, the graph that strips the quotes off "distance" is
> just the An root lattice.

I thought you were talking about weighted.

For unweighted, this is sort of the case, though you have to specify
the configuration of the "rungs" too (just saying An doesn't do
this), and there are separate axes for each odd, which introduces
redundancy and other problems. I can go with it to an extent but it
doesn't yet tell me enough to know how to show balls of some of the
other measures on it, which would be required for your table. This is
why I'm asking you all these questions that we sort of went through
years ago.

> Certainly graphs are more general
> than root lattices or Gene's "discrete subgroup of a finite-
> dimensional normed real vector spaces", but I don't know what
> the equivalent of graph distance is in such a space.

In some cases, there is a norm that gives exactly the same result.
This is what I've been talking about in connection with deleting all
the rungs in the Tenney and Kees lattices (and using square (,
cubic, . . .) and regular-hexagonal (, cuboctahedral, . . .) norms
respectively. Stop thinking about them as graphs, and a lot of
different bases open up for you.

>> Here's something from 1999 or abouts. I think it's an exchange
>> between Manuel and Paul Hahn, as the error in Scala was being
>> fixed...
>>
>> >> If so, please verify my results. I have used the base-2 log.
>> >>
>> >> 225/224 7.813781
>> >> 126/125 5.491853
>> >> 128/125 6.965784
>> >> 81/80 6.339850
>> >> 64/63 5.977279
>> >> 50/49 4.643856
>
>What are these the results of?
>
>> >126/125 should be 7.451211.
>> > 50/49 should be 5.614709.
>> >
>> >The others are correct.
>
>What are we calculating?

I assume these are "7-limit" distances.

-Carl

>In some cases, there is a norm that gives exactly the same result.
>This is what I've been talking about in connection with deleting all
>the rungs in the Tenney and Kees lattices (and using square (,
>cubic, . . .) and regular-hexagonal (, cuboctahedral, . . .) norms
>respectively. Stop thinking about them as graphs, and a lot of
>different bases open up for you.

Sure, I'm all ears.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> Here's something from 1999 or abouts. I think it's an exchange
> >> between Manuel and Paul Hahn, as the error in Scala was being
> >> fixed...
> >>
> >> >> If so, please verify my results. I have used the base-2 log.
> >> >>
> >> >> 225/224 7.813781
> >> >> 126/125 5.491853
> >> >> 128/125 6.965784
> >> >> 81/80 6.339850
> >> >> 64/63 5.977279
> >> >> 50/49 4.643856
> >
> >What are these the results of?
> >
> >> >126/125 should be 7.451211.
> >> > 50/49 should be 5.614709.
> >> >
> >> >The others are correct.
> >
> >What are we calculating?
>
> I assume these are "7-limit" distances.

So you mean the weighted Hahn complexities of these ratios in an odd
limit of 7?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >In some cases, there is a norm that gives exactly the same result.
> >This is what I've been talking about in connection with deleting all
> >the rungs in the Tenney and Kees lattices (and using square (,
> >cubic, . . .) and regular-hexagonal (, cuboctahedral, . . .) norms
> >respectively. Stop thinking about them as graphs, and a lot of
> >different bases open up for you.
>
> Sure, I'm all ears.

I don't know what else to say about this. Perhaps some of my posts that
confused you might make more sense with this in mind . . . (?)

At 01:21 PM 12/6/2005, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>
>> >> Here's something from 1999 or abouts. I think it's an exchange
>> >> between Manuel and Paul Hahn, as the error in Scala was being
>> >> fixed...
>> >>
>> >> >> If so, please verify my results. I have used the base-2 log.
>> >> >>
>> >> >> 225/224 7.813781
>> >> >> 126/125 5.491853
>> >> >> 128/125 6.965784
>> >> >> 81/80 6.339850
>> >> >> 64/63 5.977279
>> >> >> 50/49 4.643856
>> >
>> >What are these the results of?
>> >
>> >> >126/125 should be 7.451211.
>> >> > 50/49 should be 5.614709.
>> >> >
>> >> >The others are correct.
>> >
>> >What are we calculating?
>>
>> I assume these are "7-limit" distances.
>
>So you mean the weighted Hahn complexities of these ratios in an odd
>limit of 7?

Yes. -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> At 01:21 PM 12/6/2005, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>
> >> >> Here's something from 1999 or abouts. I think it's an
exchange
> >> >> between Manuel and Paul Hahn, as the error in Scala was being
> >> >> fixed...
> >> >>
> >> >> >> If so, please verify my results. I have used the base-2
log.
> >> >> >>
> >> >> >> 225/224 7.813781
> >> >> >> 126/125 5.491853
> >> >> >> 128/125 6.965784
> >> >> >> 81/80 6.339850
> >> >> >> 64/63 5.977279
> >> >> >> 50/49 4.643856
> >> >
> >> >What are these the results of?
> >> >
> >> >> >126/125 should be 7.451211.
> >> >> > 50/49 should be 5.614709.
> >> >> >
> >> >> >The others are correct.
> >> >
> >> >What are we calculating?
> >>
> >> I assume these are "7-limit" distances.
> >
> >So you mean the weighted Hahn complexities of these ratios in an
odd
> >limit of 7?
>
> Yes. -Carl

OK, I'll try to check this when I get a chance. Now, I'm off. Good
night!