Pure octaves average tuning

Here's another idea a bit like the Tenney-MOS tuning. Suppose we set a
height limit N, and take all p-limit numbers with Tenney height <= N,
call that S. We may take the elements of S to be greater than 1, which
gives the same result for half the work. Now for a rank-two temperament
T take

U(N) = sum_{q in S} generators(q) * monz(q)

Here generators(q) is the number of generator steps for the temperament
T to q, which we may find from the first coefficient of T v q. The
tuning t(N) is the tuning for T with eigenmonzos U(n) and 2. Then, if
t(N)-->t as N-->infinity, t is the pure octaves average tuning. It
appears to exist; that is, it appears there is such a limit. I'll think
about a proof, but first we should check if it looks like anything we
know about already.

For 5-limit meantone, t(10000) = 697.2204 and t(1000000) = 697.2193.
Does that ring any bells?

Gene Ward Smith wrote:
> Here's another idea a bit like the Tenney-MOS tuning. Suppose we set a > height limit N, and take all p-limit numbers with Tenney height <= N, > call that S. We may take the elements of S to be greater than 1, which > gives the same result for half the work. Now for a rank-two temperament > T take

This is a mixture of a prime limit and integer limit, then? I've looked at such things but not got anywhere.

> U(N) = sum_{q in S} generators(q) * monz(q)
> > Here generators(q) is the number of generator steps for the temperament > T to q, which we may find from the first coefficient of T v q. The > tuning t(N) is the tuning for T with eigenmonzos U(n) and 2. Then, if
> t(N)-->t as N-->infinity, t is the pure octaves average tuning. It > appears to exist; that is, it appears there is such a limit. I'll think > about a proof, but first we should check if it looks like anything we > know about already.

Any stable limit you can find for such an average would be useful for the theory. Where do pure octaves come into it?

> For 5-limit meantone, t(10000) = 697.2204 and t(1000000) = 697.2193. > Does that ring any bells?

No.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> This is a mixture of a prime limit and integer limit, then? I've
looked
> at such things but not got anywhere.

Not really. It's a prime-limit tuning which comes to its weighting in a
way which involves looking at all p-limit intervals. It does converge
to a specific tuning, as the eigenmonzo is something you can determine
by integration (that is, the sum converges on an integral, modulo
scaling.)

> Any stable limit you can find for such an average would be useful for
> the theory. Where do pure octaves come into it?

They come in because of the assumption of generator-and-period in the
definition of "generator steps", and because the octave is the other
eigenmonzo (you need two for a rank-two temperament.)