primes and ETs

Most good ETs I can think of are either prime, or adjacent
to a prime. 12 and 72 are sandwiched between adjacent primes.
99, 171, and perhaps 34 are the only exceptions that come
to mind.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma wrote:
>
> Most good ETs I can think of are either prime, or adjacent
> to a prime. 12 and 72 are sandwiched between adjacent primes.
> 99, 171, and perhaps 34 are the only exceptions that come
> to mind.

Numerology! ;-)

What of you make of the fact that the sequences
11, 12, 13
and
31, 34, 37
both follow the AP pattern 12k-|k|, 12k, 12k+|k|,
where the function |k| = [k+0.5] is the nearest integer to k
and the two extreme terms of the AP are both prime
when k takes the values 1 and 17/6 ?

Perhaps the pattern generalises to some other 'good' compound ET?
But suppose it did: what is a 'good' ET anyway, and would such a
pattern have any musical utility?

Yahya

At 07:21 AM 8/22/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma wrote:
>>
>> Most good ETs I can think of are either prime, or adjacent
>> to a prime. 12 and 72 are sandwiched between adjacent primes.
>> 99, 171, and perhaps 34 are the only exceptions that come
>> to mind.
>
>
>Numerology! ;-)
>
>What of you make of the fact that the sequences
> 11, 12, 13
>and
> 31, 34, 37
>both follow the AP pattern 12k-|k|, 12k, 12k+|k|,
> where the function |k| = [k+0.5] is the nearest integer to k
> and the two extreme terms of the AP are both prime
> when k takes the values 1 and 17/6 ?

I'm drawing a blank. - C .

Carl Lumma wrote:
> Most good ETs I can think of are either prime, or adjacent
> to a prime. 12 and 72 are sandwiched between adjacent primes.
> 99, 171, and perhaps 34 are the only exceptions that come
> to mind.
> > -Carl

What, 15 and 26 aren't good ETs? :-)

Could be just that there aren't many small integers that aren't adjacent to primes. 9, 15, 21, 25, 26, 27, 33, 34, 35, 39, 45, 49, 50, 51, 55, 56, 57.... come to think of it, 50-ET and 55-ET are useful as meantones, and 99-ET gets mentioned once in a while as well.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Carl Lumma wrote:
> > Most good ETs I can think of are either prime, or adjacent
> > to a prime. 12 and 72 are sandwiched between adjacent primes.
> > 99, 171, and perhaps 34 are the only exceptions that come
> > to mind.
> >
> > -Carl
>
> What, 15 and 26 aren't good ETs? :-)

Well, for 15 we have 2*15-1 = 29 and 2*15+1 = 31, which is pretty
good. We get the same for 99: 2*99-1 and 2*99+1 are both prime.

> Could be just that there aren't many small integers that aren't
adjacent
> to primes.

You've hit on a key fact: for small n, a lot of n are prime. Still,
270 is getting up there, and 269 and 271 are a prime pair. 441 is
larger yet, and 2*441-1 and 2*441+1 is also a prime pair.

What about Sophie Germain pairs such that 2*(p, 2*p+1) are both good
scale divisions? (10, 22), (22, 46), (46, 94), (58, 118)

Numerology rules, I guess.

Carl,

I guess that was my point! Ww can often discern patterns; or
shoehorn facts into patterns; but that fact does not itself make the
pattern *significant* in any way.

Regards,
Yahya

--- In tuning-math@yahoogroups.com, you wrote:
>
> At 07:21 AM 8/22/2006, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma wrote:
> >>
> >> Most good ETs I can think of are either prime, or adjacent
> >> to a prime. 12 and 72 are sandwiched between adjacent primes.
> >> 99, 171, and perhaps 34 are the only exceptions that come
> >> to mind.
> >
> >
> >Numerology! ;-)
> >
> >What of you make of the fact that the sequences
> > 11, 12, 13
> >and
> > 31, 34, 37
> >both follow the AP pattern 12k-|k|, 12k, 12k+|k|,
> > where the function |k| = [k+0.5] is the nearest integer to k
> > and the two extreme terms of the AP are both prime
> > when k takes the values 1 and 17/6 ?
>
> I'm drawing a blank. - C .
>

At 05:24 AM 8/24/2006, you wrote:
>Carl,
>
>I guess that was my point! Ww can often discern patterns; or
>shoehorn facts into patterns; but that fact does not itself make
>the pattern *significant* in any way.
>
>Regards,
>Yahya

My observation had two parts:
. That good ETs tend to be prime
. or adjacent to primes.

For the first part, consider the following brainstorms:

. If there's an ET that's good, larger ETs in which it is a factor
are less likely to be good because of torsion (since torsion is bad).
Going up from zero, we get a sieve effect. For example, the first
multiple of 12 that's good is 72.

. In rational intonation, all 'ratios of x' are relatively prime
to 'ratios of y'. In the limit of accuracy ETs will obey this, and
especially at higher limits this means they're more likely to be
prime, since there's less chance one of the consonances will divide
the octave.

We can test it by comparing the average badness of a bunch of
prime ETs with a bunch of random ones. Let's compare 30 ETs smaller
than 10,000 at a time, using untruncated 13-limit Hahn consistency
as a badness measure, ten runs each.

(mean
(map (lambda (x)
(consist x '(1 3 5 7 11 13)))
(random-ls 31 10000)))

0.8091539225278565
0.8279930157083801
0.8823700958192148
0.913896737618406
0.8621411277942301
0.7687775352579252
0.7523147522249154
0.7714483171507516
0.7476024053364617
0.8196603232022545

Now primes

(mean
(map (lambda (x)
(consist x '(1 3 5 7 11 13)))
(rand-pick 30 (primes 10000))))

0.8921026848772283
0.8346476151331039
0.8851908528576399
0.8843665373852679
0.8230317519483468
0.7171379883971037
0.9444732776219547
0.9044881700090625
0.8233044474918735
0.8160732186927563

These scores are slightly higher... mean of 0.85 compared
to 0.81... don't know if that's significant.

What's that large ET that Paul found that just kills everything for
miles around? And didn't Gene say it was important to something to
do with primes?

I don't know about the second part of my observation.

-Carl

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

> What's that large ET that Paul found that just kills everything for
> miles around?

You probably mean 103169:

http://www.research.att.com/~njas/sequences/A117555

And didn't Gene say it was important to something to
> do with primes?

It's not a prime: 103169 = 11*83*113. I don't recall saying anything
like the above, but it was mentioned in connection with the 2401/2400
periodicity phenomenon.

At 01:51 AM 8/25/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
>> What's that large ET that Paul found that just kills everything for
>> miles around?
>
>You probably mean 103169:
>
>http://www.research.att.com/~njas/sequences/A117555
>
> And didn't Gene say it was important to something to
>> do with primes?
>
>It's not a prime: 103169 = 11*83*113. I don't recall saying anything
>like the above, but it was mentioned in connection with the 2401/2400
>periodicity phenomenon.

Yes, thanks. -C.