two open questions

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.?

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

-Carl

<1.>
>>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.

You included something about that in your original post on the derrivation
of denominator complexity from the Farey series, if memory serves. You
said it was better behaved than denominator limit. I've been using n*d
complexity since '96. I'm glad to know that it is self-consistent in this
way (predicts domain in an n*d series). [The fuzz factor of 2 for the Farey
stuff -- is that related an original odd-limit assumption (is there one?)
which is lacking from the n*d stuff? Hmm! Looks like I need to read your
derrivation again.]

How n*d is easier to apply to triads is beyond me. Since mediants are
not defined for triads, the strategy used in your denominator proof is
seems un-usable. If I remember, you left the n*d derrivation to the
reader. Looks like I should take you up on that...

<2.>
>That depends on your mapping from 7-limit to meantone.

Naturally.

> | 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 | | | |
>---------+------------+-----------------+------------------+

Solid!

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

Agree. I suspected it wasn't too far off, by Dave Keenan's charts.

-Carl