another "special" comma and a puzzle

There's a comma I've long thought is quite special, that was mentioned
here recently in passing a couple of times: the difference between the
just minor third 6:5 (or 78:65) and the interval that is the sum,
octave-reduced, of a just 7th and just 11th (= 77:64). It's 385:384,
or about 4.5 cents. Why do I think it's so special? Because you can
also describe it using exactly one each of 3,5,7 and 11:

5 * 7 * 11 "=" 3 (modulo several octaves), where "=" means the
approximation is off by 385:384.

In other words its monzo consists of all 1s and -1s up to the 4th odd
prime, and 0 for every other entry.

Puzzle: if we consider all the commas whose monzos have uniquely 1s
and -1s after the factors of 2, up to the nth odd prime, how large
does n have to be before you can find a resulting comma that is better
than 385:384?

(by the way: hi, long-time (since eartha.mills days) on-and-off
reader, don't think I've posted before though.)

Jeremy Targett

--- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...> wrote:

> Puzzle: if we consider all the commas whose monzos have uniquely 1s
> and -1s after the factors of 2, up to the nth odd prime, how large
> does n have to be before you can find a resulting comma that is better
> than 385:384?

715/714 has all prime factors to the +-1 power, *including* 2.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...>
wrote:
>
> > Puzzle: if we consider all the commas whose monzos have uniquely 1s
> > and -1s after the factors of 2, up to the nth odd prime, how large
> > does n have to be before you can find a resulting comma that is better
> > than 385:384?
>
> 715/714 has all prime factors to the +-1 power, *including* 2.

A property these commas have is that they are what might be called
universal xenharmonic bridges. That is, take the primes {3,5,7,11}. If
you remove any one odd prime, 385/384 is a xenharmonic bridge from the
remaining primes, including 2, to the 11-limit. 715/714 works the
same, but now if you like you can add 2, and remove any one prime from
{2,3,5,7,11,17}.

--- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...> wrote:

> Puzzle: if we consider all the commas whose monzos have uniquely 1s
> and -1s after the factors of 2, up to the nth odd prime, how large
> does n have to be before you can find a resulting comma that is better
> than 385:384?

After 715/714, the next smaller one I find is not superparticular:
676039/675840.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...>
wrote:
>
> > Puzzle: if we consider all the commas whose monzos have uniquely 1s
> > and -1s after the factors of 2, up to the nth odd prime, how large
> > does n have to be before you can find a resulting comma that is better
> > than 385:384?
>
> After 715/714, the next smaller one I find is not superparticular:
> 676039/675840.

The only reasonable superparticular universal bridge commas are
385/384 and 715/714. I checked up to 94612/94611 and found no others,
and on the small side the best one can do is 21/20. It is quite
possible that there are no more on the large side, and 3/2, 4/3, 6/5,
15/14, 16/15, 21/20, 385/384, 715/714 is the complete list. If we
filter this list to universal bridges in the strict sense, including
2, we get only
3/2, 6/5, 15/14 and 715/714. It looks like 715/714 is pretty much unique.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...>
> wrote:
> >
> > > Puzzle: if we consider all the commas whose monzos have uniquely 1s
> > > and -1s after the factors of 2, up to the nth odd prime, how large
> > > does n have to be before you can find a resulting comma that is
better
> > > than 385:384?
> >
> > After 715/714, the next smaller one I find is not superparticular:
> > 676039/675840.
>
> The only reasonable superparticular universal bridge commas are
> 385/384 and 715/714. I checked up to 94612/94611 and found no others,
> and on the small side the best one can do is 21/20. It is quite
> possible that there are no more on the large side, and 3/2, 4/3, 6/5,
> 15/14, 16/15, 21/20, 385/384, 715/714 is the complete list. If we
> filter this list to universal bridges in the strict sense, including
> 2, we get only
> 3/2, 6/5, 15/14 and 715/714. It looks like 715/714 is pretty much
unique.

You beat me to 715/714 :-) but I emailed this list to Jeremy Targett,
so as not to spoil it for anyone else who wanted to try (before I saw
that you'd already posted the answer).

Answer:

n = 7, (i.e. 17 limit)
[-1 -1, 1 -1 1, 1 -1> 714:715 2.4 c

Here are all others up to n=11 (i.e. 31-limit), but I don't think they
are of any real interest for tuning.

23-limit
[-12 -1, -1 1 -1, 1 1 1, 1> 0.5 c

29-limit
[-8 1, 1 1 -1, 1 -1 -1, 1 1> 1.7 c
[-2 1, -1 1 1, -1 1 -1, -1 1> 4.0 c

31-limit
[6 1, 1 1 -1, 1 -1 -1, -1 1 -1> 0.1 c
[-10 1, 1 1 -1, 1 1 1, 1 -1 -1> 2.4 c
[-19 1, 1 1 1, 1 1 -1, -1 1 1> 2.7 c
[9 1, -1 1 -1, -1 -1 -1, 1 1 -1> 3.0 c
[-5 1, 1 1 -1, -1 -1 1, -1 1 1> 4.1 c

-- Dave Keenan

Gene wrote, quoting me:

> > Puzzle: if we consider all the commas whose monzos have uniquely 1s
> > and -1s after the factors of 2, up to the nth odd prime, how large
> > does n have to be before you can find a resulting comma that is better
> > than 385:384?
>
> After 715/714, the next smaller one I find is not superparticular:
> 676039/675840.

Yes, this last one is at the 23-limit - pretty close at half a cent! I
also heard off-list from Dave Keenan who didn't want to spoil the
puzzle for anyone else by giving the answer.

Anyway it's amazing to me that the 11-limit (385:384) and 17-limit
(715:714) cases work so well for this. I think the general problem of
partitioning a set of n numbers so that the two resulting sets have
products (or sums, if you take logarithms) that are as close as
possible, is quite a tough one. Though it's easy to brute-force when n
is low - I guess that's how you solved the puzzle? It seems to me that
given random sets of numbers below a similar bound, you'd be unlikely
to find such close fits. Of course the prime numbers aren't "random"
so I have no way of measuring how unlikely these coincidences are.

Here are the other commas of this form that Dave sent me if anyone
else is interested:

> Here are all others up to n=11 (i.e. 31-limit), but I don't think they
> of any real interest for tuning.
>
> 23-limit
> [-12 -1, -1 1 -1, 1 1 1, 1> 0.5 c
>
> 29-limit
> [-8 1, 1 1 -1, 1 -1 -1, 1 1> 1.7 c
> [-2 1, -1 1 1, -1 1 -1, -1 1> 4.0 c
>
> 31-limit
> [6 1, 1 1 -1, 1 -1 -1, -1 1 -1> 0.1 c
> [-10 1, 1 1 -1, 1 1 1, 1 -1 -1> 2.4 c
> [-19 1, 1 1 1, 1 1 -1, -1 1 1> 2.7 c
> [9 1, -1 1 -1, -1 -1 -1, 1 1 -1> 3.0 c
> [-5 1, 1 1 -1, -1 -1 1, -1 1 1> 4.1 c

Regards, Jeremy

--- In tuning@yahoogroups.com, Jeremy Targett <jeremy.targett@g...>
wrote:

> It seems to me that
> given random sets of numbers below a similar bound, you'd be unlikely
> to find such close fits. Of course the prime numbers aren't "random"
> so I have no way of measuring how unlikely these coincidences are.

Number theorists have ways of measuring these things. Jeremy, this is
truly a topic for the tuning-math list, which split off from this list
a few years ago because some people were sick of seeing such
mathematically-oriented posts here.