Commas and the ABC conjecture

I had thought of examining the Farey sequence for useful intervals,
but it occurred to me that the same thing may be accomplised more
easily simply by looking at ratios belonging to a p-limit. If b is a
fixed small integer, it follows from a very deep and recent
conjecture in elementary number theory called the ABC conjecture that
in the p-limit, there will be only a finite number of ratios (a+b)/a.
From this we get the following list:

3-limit

2, 3/2, 4/3, 9/8

5-limit

5/4, 6/5, 10/9, 16/15, 25/24, 81/80

7-limit

7/6, 8/7, 15/14, 21/20, 28/27, 36/35, 49/48, 50/49, 64/63, 126/125,
225/224, 2401/2400, 4375/4374

11-limit

11/10, 12/11, 22/21, 33/32, 45/44, 55/54, 56/55, 99/98, 100/99,
121/120, 176/175, 243/242, 385/384, 441/440, 540/539, 3025/3024,
9801/9800

There is no guarantee that these lists are complete, or even finite,
but I went out much past the final value in all cases without finding
another one. If the ABC conjecture is true, these lists are finite,
and an effective form of it would tell us when they are complete.

If we look only at numerators which are squares, which cover a lot of
these cases, it is easy to carry this out farther; these lists will
at least contain all the jumping jacks:

3-limit

4/3, 9/8

5-limit

16/15, 25/24, 81/80

7-limit

36/35, 49/48, 64/63, 225/224, 2401/2400

11-limit

100/99, 121/120, 441/440, 3025/3024, 9801/9800

13-limit

144/143, 169/168, 196/195, 625/624, 676/675, 729/728, 4096/4095,
4225/4224, 123201/123200

17-limit

256/255, 289/288, 1089/1088, 1156/1155, 1225/1224, 2500/2499,
2601/2600, 14400/14399, 28561/28560, 194481/194480

19-limit

324/323, 361/360, 400/399, 1521/1520, 3136/3135, 5776/5775,
5929/5928, 23409/23408, 28900/28899, 43681/43680, 104976/104975,
5909761/5909760

23-limit

484/483, 529/528, 576/575, 2025/2024, 4761/4760, 8281/8280,
25921/25920, 43264/43263, 104329/104328, 152881/152880,
4096576/4096575

I find it fascinating that deep mathematical conjectures of number
theory such as ABC and the Riemann hypothesis can have some sort of
connection with music theory.

The most interesting intervals on these lists are the larger ones,
and these are a fruitful place to look to find notations, scales and
the like. For instance, the four smallest 7-limit intervals on the
list are 126/125, 225/224, 2401/2400, and 4375/4374, and this is the
basis for the notation [h72, h27, -h19, h31]; from which, for
instance, we could contruct a 72-note PB, or tempered PBs such as the
miracle system of h72 and h31 and the meantone system of h31 and h19.

I posted this over on tuning by mistake; it case my cancel doesn't
work it probably should be responded to here.

Gene wrote:

> I had thought of examining the Farey sequence for useful intervals,
> but it occurred to me that the same thing may be accomplised more
> easily simply by looking at ratios belonging to a p-limit. If b is a
> fixed small integer, it follows from a very deep and recent
> conjecture in elementary number theory called the ABC conjecture that
> in the p-limit, there will be only a finite number of ratios (a+b)/a.

Can you define the p-limit?

> If we look only at numerators which are squares, which cover a lot of
> these cases, it is easy to carry this out farther; these lists will
> at least contain all the jumping jacks:
>
> 3-limit
>
> 4/3, 9/8

So these define a Pythagorean system

> 5-limit
>
> 16/15, 25/24, 81/80

And these are the usual intervals for defining a 5-limit diatonic scale!

> 7-limit
>
> 36/35, 49/48, 64/63, 225/224, 2401/2400

These are linearly dependent. I think 49/48*63/64=225/224*2401/2400. So
the first four can be inverted to give

[10 12 9 5]
[16 19 14 8]
[23 28 21 12]
[28 34 25 14]

which I think means we can define either Paultone or meantone from these
unison vectors.

> 11-limit
>
> 100/99, 121/120, 441/440, 3025/3024, 9801/9800

These give

[ 46 26 8 -14 15]
[ 73 41 13 -22 24]
[107 60 19 -32 35]
[129 73 23 -39 42]
[159 90 28 -48 52]

So all bar the largest define 46-equal.

I haven't looked at the higher limits.

Graham

--- In tuning-math@y..., graham@m... wrote:

> > If b is a
> > fixed small integer, it follows from a very deep and recent
> > conjecture in elementary number theory called the ABC conjecture
that
> > in the p-limit, there will be only a finite number of ratios
(a+b)/a.

> Can you define the p-limit?

What I mean by p-limit is that all of the intervals on the list can
be factored by primes less than or equal to p. I was looking at
ratios (a+b)/a in the particularly interesting case where b=1. The
first list of 7-limit epimoric ratios are all such numbers which
require a 7 to factor, the ones not requiring a 7 having already
appeared on a previous list. In the second list I had only those
ratios with a square numerator, but searched much farther.

> > 7-limit
> >
> > 36/35, 49/48, 64/63, 225/224, 2401/2400
>
> These are linearly dependent. I think
49/48*63/64=225/224*2401/2400.

Well, that's certainly not true. :) But it's equally true there must
be some dependency, since we have five numbers and only four primes.

So
> the first four can be inverted to give
>
> [10 12 9 5]
> [16 19 14 8]
> [23 28 21 12]
> [28 34 25 14]
>
> which I think means we can define either Paultone or meantone from
these
> unison vectors.

Paultone? Are those 15, 22, 27?

> [ 46 26 8 -14 15]
> [ 73 41 13 -22 24]
> [107 60 19 -32 35]
> [129 73 23 -39 42]
> [159 90 28 -48 52]
>
> So all bar the largest define 46-equal.

Anyone want a 46 PB? This would be a good way to get one.

--- In tuning-math@y...,
genewardsmith@j... wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > > If b is a
> > > fixed small integer, it follows from a very deep and recent
> > > conjecture in elementary number theory called the ABC conjecture
> that
> > > in the p-limit, there will be only a finite number of ratios
> (a+b)/a.

John Chalmers published a list of
all the superparticular ratios,
with numbers up to ten billion or
something, in every limit up to
the 23-limit, in Xenharmonikon 17
(the same issue my TTTTTT paper
is in). You've reproduced the first
few entries. By the way, Rami
Vitale responded to you on the
main list . . . you should check it
out.
>
> > Can you define the p-limit?
>
> What I mean by p-limit is that all of the intervals on the list can
> be factored by primes less than or equal to p. I was looking at
> ratios (a+b)/a in the particularly interesting case where b=1. The
> first list of 7-limit epimoric ratios are all such numbers which
> require a 7 to factor, the ones not requiring a 7 having already
> appeared on a previous list. In the second list I had only those
> ratios with a square numerator, but searched much farther.
>
> > > 7-limit
> > >
> > > 36/35, 49/48, 64/63, 225/224, 2401/2400
> >
> > These are linearly dependent. I think
> 49/48*63/64=225/224*2401/2400.
>
> Well, that's certainly not true. :) But it's equally true there must
> be some dependency, since we have five numbers and only four primes.
>
> So
> > the first four can be inverted to give
> >
> > [10 12 9 5]
> > [16 19 14 8]
> > [23 28 21 12]
> > [28 34 25 14]
> >
> > which I think means we can define either Paultone or meantone from
> these
> > unison vectors.
>
> Paultone? Are those 15, 22, 27?

Hmm . . . Paultone means
commatic unison vectors of 50:49,
64:63, and 225:224; and chromatic
unison vectors of 25:24, 28:27, and
49:48. So what set of ETs does it
correspond to? 10, 22, and . . . ?

>
> > [ 46 26 8 -14 15]
> > [ 73 41 13 -22 24]
> > [107 60 19 -32 35]
> > [129 73 23 -39 42]
> > [159 90 28 -48 52]
> >
> > So all bar the largest define 46-equal.
>
> Anyone want a 46 PB? This would be a good way to get one.

Awesome! I'm thinking that 22, 31,
46, and 72 would be a good choice
of four cardinalities to base my
future instruments on. My
present plan for the four
fingerboards that come with the
Rankin system is 22-tET, 31-tET,
22-out-of-46-tET, and 31-out-of-
72-tET.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> John Chalmers published a list of
> all the superparticular ratios,
> with numbers up to ten billion or
> something, in every limit up to
> the 23-limit, in Xenharmonikon 17
> (the same issue my TTTTTT paper
> is in). You've reproduced the first
> few entries.

Wonderful news! That means I can publish in mathematical forums, and
cite this as proof that the question is, in fact, musically
interesting.

By the way, Rami
> Vitale responded to you on the
> main list . . . you should check it
> out.

I cancelled that about a minute after posting it, but I'm not
surprised.

> > Paultone? Are those 15, 22, 27?

> Hmm . . . Paultone means
> commatic unison vectors of 50:49,
> 64:63, and 225:224; and chromatic
> unison vectors of 25:24, 28:27, and
> 49:48. So what set of ETs does it
> correspond to? 10, 22, and . . . ?

The dual group to the one generated by 50/49, 64/63 and 225/224 is
generated by h2 and h10, and contains h12 and h22; that's about the
size of it--we get two elements since we have four primes and a
linear dependency 50/49 = 225/224 64/63.