Synched-septimal tuning

The most intriguing of the brats is -1, where everything beats together.
One way to extend that to the 7-limit is what might be called
synch-septimal; if t is an approximate 5/4, f an approximate 3/2, and
s an approximate 7/4, then we can make 2f-3 = 4t-5 = 4s-7 = 5f-6t =
2(7t-5s) = (7f-6s)/2; such a tuning is what I'm defining as
synch-septimal. Given a 7-limit comma, we can find a tuning of the
corresponding temperament which is synch-septimal, which in the good
cases will also be a decent tuning for the temperament.

An example is marvel. If a is the positive real root of

a^4 + 4a^3 + 4a^2 - 32a - 128 = 0

then a gives a 3 of 1899.1723 cents. If b is the positive real root of

b^4 - 4b^3 + 4b^2 - 32b - 64 = 0

then b gives a 5 of 2784.6446 cents; b is simply a+2. Then by marvel
equivalence, we also have a 7 of (ab)^2/32 = (a(a+2))^2/32, of
3367.6339 cents.

This gives a tuning somewhat along the lines of 103 or 175 equal,
which are miracle tunings. In fact, this marvel tuning goes a quite
way towards miracle, as 1029/1024 is shrunk to less than 1/4 of its
original size. 81/80 and 126/125 also shrink, so it moves the tuning
somewhat in the direction of meantone as well.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> The most intriguing of the brats is -1, where everything beats together.
> One way to extend that to the 7-limit is what might be called
> synch-septimal; if t is an approximate 5/4, f an approximate 3/2, and
> s an approximate 7/4, then we can make 2f-3 = 4t-5 = 4s-7 = 5f-6t =
> 2(7t-5s) = (7f-6s)/2; such a tuning is what I'm defining as
> synch-septimal.

The brat=-1 tunings can be defined as those such that if the approximate
3 is r, then the approximate 5 is r+2. The synch-septimal tunings
generalize that to a three-term arithmetic progression; if r is the
approximate 3 of a synch-septimal tuning, then the 5 is r+2, and the 7
is r+4. Hence one might in general define the synch p-limit tuning as
ones where if r is the approximate 3, then the other approximate
primes are arithmetically related; if 3 < q <= p is an odd prime, then
the
aproximate q is r + (q-3).