Compromised thirds

In response to my post:

>>Aside, I know very well that the 6:7:9 minor triad locks vividly. Creating
> a
>>sound illustration for my book CD, I was moving the third of a C triad
>>downward gradually from a "locked" major third position to a "locked" minor
>>third position while holding the root and fifth constant. When I moved it
>>"too far" down to what was apparently a 6:7 ratio, I got a very audible low
>>F in my recording. It took considerable time to find a "compromise"
> position
>>that would both illustrate the point and avoid the low F.

Paul Erlich asked:

> Why don't you think the tuning of the major triad with high third involves
> similar compromises? If you insist it doesn't, what did you think of my post
> where I suggested 1/24:1/19:1/16?
>
At this point, I don't insist on anything except to report my observation
that the "high third" appears to lock into an "optimum consonance" tuning
when the root and fifth are sounding. Beyond that I'm wide open to ideas,
theories, and rationalizations. The only problem I have with viewing the
"high third" as a compromise is that it locks (tunes) so vividly. It feels
as "right" as a well-tuned perfect fifth, which makes me suspect that there
is more to the matter than simply cultured preference.

Regarding the "compromise" in my minor triad illustration, I felt the
frustration was a result of the inability of my M-1 to produce a "locked"
minor triad as one would experience it in a vocal ensemble. While it clearly
locked in the 14:18:21 triad (since it produced the audible combination
tone), it seemed unable to lock in the more basic(?) 5:6, 4:5, 2:3
components of the minor triad as I had hoped.

In order to offer a response to your 1/24:1/19:1/16 suggestion, I will need
a clarification of this expression in what appears to be fractions. I'm
rather a novice at mathematical acoustics and this form is new to me. If
it's not too much trouble, I'd appreciate it.

Also, what did you think of my suggestion that perhaps the consonance of the
high third is a result of the combined partial structures of both the root
and the dominant pitches? I haven't done any numbers on this, but you can
clearly do so much faster and intuitively than I. The reason it occurred to
me (as you may remember) is that the high third only seems to be preferred
when both root and fifth are sounding. It simply seems logical to look here
for a clue.

Jerry