A 31-ET puzzle

I recently learned the extraordinary fact that one can obtain every
possible interval-size that's available in 31-EDO, from an octave-
repeating "scale" having only 6 notes per octave!

The puzzle is for you to find that scale, which is unique up to a mode
(cyclic permutation) or inversion (reversal), and list its sequence of
step-sizes, as measured in steps of 31-EDO.

Hint: The scare-quotes around "scale" above, are because it isn't a
melodically useful scale by any stretch of the imagination. It's very
uneven and no two steps are the same size.

It turns out that any n-EDO where n = q*(q+1)+1 and q is either prime
or a whole-number power of a prime, has complete interval coverage
from a scale with only q+1 notes. However, 31-EDO is the only such EDO
of any great musical interest.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> It turns out that any n-EDO where n = q*(q+1)+1 and q is either prime
> or a whole-number power of a prime, has complete interval coverage
> from a scale with only q+1 notes. However, 31-EDO is the only such EDO
> of any great musical interest.

That's what happens when you bail out of tuning-math--this was
discussed a while back there. Actually, my Ostinato on a Difference
Set is related to this business, in connection with 13 (the number of
elements in the 7-limit tonality diamond) rather than 31.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > It turns out that any n-EDO where n = q*(q+1)+1 and q is either
prime
> > or a whole-number power of a prime, has complete interval coverage
> > from a scale with only q+1 notes. However, 31-EDO is the only such
EDO
> > of any great musical interest.
>
> That's what happens when you bail out of tuning-math--this was
> discussed a while back there. Actually, my Ostinato on a Difference
> Set is related to this business, in connection with 13 (the number of
> elements in the 7-limit tonality diamond) rather than 31.

My humble apologies. Yes this would have been more appropriate for
tuning-math. I see now that this very question was posed by Bob
Valentine and answered by Kees van Prooijen, in April 2002.

I fully expected that _you_ would know the answer, Gene. But I'm a
little surprised that you didn't pick me up on my mistaken assertion
that such a scale was unique (ignoring rotations and reversals). Kees
found 5. One of them is a little more melodically-even than the one I
was looking at.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> I fully expected that _you_ would know the answer, Gene. But I'm a
> little surprised that you didn't pick me up on my mistaken assertion
> that such a scale was unique (ignoring rotations and reversals). Kees
> found 5. One of them is a little more melodically-even than the one I
> was looking at.

Sorry, I was reading with my usual inattention. In fact, these kinds
of difference sets are sometimes called planar, as they lead to finite
projective planes, and projective automorphisms will give you various
scales. I had the impression you were saying that the scales are to be
regarded as unique up to rotations and reversals. Anyway, the 31 case
is the projective plane of order 5, which has 31 points but also 31
lines; it's the lines we want.

If I'm supposed to be correcting things, everyone believes that they
all must be of the form n^2+n+1 with n a prime power, but that has not
yet been proven. It is true for anything of conceivable musical use.