Re: "Symmetrical division" chords.

Paul Erlich clarifies for Joe Monzo:

> Joe, you're remembering incorrectly. I say any meantone is fine for a 'major
> 6/9' chord, while the augmented triad and diminished seventh chord
> definitely cry out for something close to 12-tET. What these chords have in
> common, though, is that their ideal tuning is something different from JI.

Now that I better understand (thanks to you and others) the ongoing context
of this point, I have a question and a comment.

Question: Would not the "crying out for...12-tET" most likely occur when
tonality is not clear--as in a chromatic transitional passage. When the
destination key of a diminished-seventh chord is known (by singers and/or
string players), wouldn't the functional tritone (in an "incomplete dominant
ninth," ala Piston) "refocus" as the leading tone pulls upward toward tonic
while the chord seventh drastically pulls downward (and likely the ninth, as
well)--even when the destination tonic is temporary?

Comment: I suspect that the tonal destination of a well-tuned (JI?)
dominant-function diminished seventh chord is audibly evident even before it
resolves. Also, the functional augmented triad likely offers a clue toward a
destination tonality by means of a "high third" tuning in relation to the
functional root, causing one member (of the three) to emerge as the leading
tone.

(Any grad students out there in need of a topic?)

Jerry

>When the
>destination key of a diminished-seventh chord is known (by singers and/or
>string players), wouldn't the functional tritone (in an "incomplete
dominant
>ninth," ala Piston) "refocus" as the leading tone pulls upward toward tonic
>while the chord seventh drastically pulls downward (and likely the ninth,
as
>well)--even when the destination tonic is temporary?

Maybe, but we were talking simply about maximizing "sensory" consonance
outside of any musical context.

>Also, the functional augmented triad likely offers a clue toward a
>destination tonality by means of a "high third" tuning in relation to the
>functional root, causing one member (of the three) to emerge as the leading
>tone.

So what do you think the ratios are?

I posted:
>
>>When the
>>destination key of a diminished-seventh chord is known (by singers and/or
>>string players), wouldn't the functional tritone (in an "incomplete
> dominant
>>ninth," ala Piston) "refocus" as the leading tone pulls upward toward tonic
>>while the chord seventh drastically pulls downward (and likely the ninth,
> as
>>well)--even when the destination tonic is temporary?

Paul Erlich responded:
>
> Maybe, but we were talking simply about maximizing "sensory" consonance
> outside of any musical context.

Huh?

I continued:
>
>>Also, the functional augmented triad likely offers a clue toward a
>>destination tonality by means of a "high third" tuning in relation to the
>>functional root, causing one member (of the three) to emerge as the leading
>>tone.
>
> So what do you think the ratios are?

I don't know. That's why I'm here. As soon as I get some answers that make
sense I'll leave you brilliant folk alone and go back to writing books. :-)

Jerry

>> Maybe, but we were talking simply about maximizing "sensory" consonance
>> outside of any musical context.

>Huh?

Sensory consonance is a measure of the amount of "roughness" or critical
band interference in a musical sound. It is well-measured and modeled in the
psychoacoustical literature. William Sethares has based most of his musical
publications and theories around this phenomemon. The point is, if you play
an augmented triad or diminished seventh chord outside any specific
voice-leading context, considerations of sensory consonance will lead to a
tuning closer to the 12-tET one than a JI one.