RE: [tuning] two open questions

Carl Lumma wrote,

>1. Paul Erlich's excellent result that the "domain" of a dyad in a N-order
>Farey series is inversely proportional to its denominator (digest 650.23)
--
>does anybody think, or not, that this could be generalized to chords by
using
>the smallest number in their harmonic series representation? Like 4:5:6 =
4,
>10:12:15 = 10, etc.?

I would really like to know! More interesting, though, and perhaps easier to
extend to triads (?), is my very recent (Friday?) posting to the effect that
the domain of a dyad in the set of dyads within a certain
numerator-times-denominator limit is inversely proportional to the square
root of the numerator times the denominator.

>2. Woolhouse and Erlich found the optimal meantone for 5-limit harmony...
>what's the optimal meantone for 7-limit harmony?

That depends on your mapping from 7-limit to meantone. For example, using
the Huygens mapping

(3:2)^10 ~ 7:4
(3:2)^9 ~ 7:6
(3:2)^6 ~ 7:5

we can create a chart like the one for 5-limit at

http://www.ixpres.com/interval/dict/meantone.htm

For the Huygens mapping, we get:

| Max. error |Sum-squared error|Sum-absolute error|
---------+------------+-----------------+------------------+
Inverse | 697.3465 | 696.6717 | 696.5784 |
Limit | | | |
Weighted | | | |
---------+------------+-----------------+------------------+
Equal | 696.5784 | 696.6480 | 696.8826 |
Weighted | | | |
| | | |
---------+------------+-----------------+------------------+
Limit | 696.5126 | 696.6672 | 696.8826 |
Weighted | | | |
| | | |
---------+------------+-----------------+------------------+
7/4 & | 696.6255 | 696.7059 | 696.8826 |
7/5 & | | | |
7/6 | | | |
only | | | |
---------+------------+-----------------+------------------+

As far as I'm concerned, these are all "close enough" to the 31-tET fifth,
696.7742¢