C-maj7#11, John and Paul

I'd posed the following situation...

> >
> >A pretty chord to end a piece with a feeling
> >of humility, resignation and bittersweet joy.
> >A C-maj7#11 voiced open but straight up
> >
> >(low) C G Eb B F# (high)
> >
> >...and the piece was fairly "C minor tonal"
> >throughout.
> >

...with the following suggested voicings.

> >1) an overtone approach
> >
> > 1/1 3/2 19/16 15/8 11/8
> >
> >2) fifths and thirds (major and minor) in tune
> >
> > 1/1 3/2 6/5 15/8 45/16
> >
> >3) "upper structure triad" in tune warping minor third
> >
> > 1/1 3/2 75/64 15/8 45/16
> >

... (and others which I've deleted) and posed the question

> >Which would singers/strings/winds "naturally" tend to,
> >and which as a composer or director would you coerce
> >them to tennd to?
>

to which John Link replied...

> My conjecture is number 2), to which could be added a D at 9/8 to make it
> even more stable:
>

and Paul Ehrlich answered...

>
> I would use (3) if I wanted to be conventional though (1) would have a nice
> "periodicity buzz" reinforcing the tonic. I think it is helpful (as I was
> just explaining to Jim Savage) to look at all the intervals within the chord
> -- looking at just the ratios relative to the root obscures what is
> happening wih all the other intervals.

Bingo. I plugged them in last night and played with them and the issue
was that ALL the fifths and major thirds were important, and other things
(like "Cminor") less so.

The crux of the problem is that there is an augmented triad. B D# G and
D# is punning Eb. The tuning I ended up preferring was

1/1 3/2 [*] 15/8 45/32

where [*] is 15/16*sqrt(8/5) (which is pretty close to 19/16 and closer
to 32/27).

I believe that singers would somehow 'minimize' the errors across all the
notes, which starts suggesting a temperment. In fact 12tet has a great
solution to both "fifths" and "augmented triads".

53 equal has a minor third available between 6/5 and 7/6, somewhere
in the 13/11, 19/16, 32/27 region. This also perfectly bisects the span
of 3/2 to 15/8. So in 53tet the notes would be

0 31 53+13 53+48 106+26

or in cents

0
702
1200+294
1200+1087
2400+589

For what its worth, the pitch 26 in 53 is "most likely to be heard
as" a 7/5 which is an interesting identity in 53, 15/8*3/2=7/5.

thanks.

Bob Valentine

Robert, I'm glad you independently came to agree with my oft-repeated
assertion that the augmented triad sounds better when tempered (which John
deLaubenfels also agrees with). I wholly agree that the chord in question,
which contains an augmented triad, does not find its "ideal" tuning in just
intonation.

Paul H. Erlich wrote:
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Robert, I'm glad you independently came to agree with my oft-repeated
> assertion that the augmented triad sounds better when tempered (which John
> deLaubenfels also agrees with). I wholly agree that the chord in question,
> which contains an augmented triad, does not find its "ideal" tuning in just
> intonation.

Can you give a short example of tonal music where there is a real
augmented triad? I think that this kind of chord may be mask few
different chords, those chords depend of chord before and chord after
it. I don't work on this chord before today, but I think it's an
interesting subject.
I can imagine this example :
between a I� chord and a IV� chord it may be an altered note (#G in C
major), we suppose that first chord is "4-5-6" (C-E-G), and second chord
is "3-4-5" (C-F-A)
it gives :

18--20

15--16

12--12

then G-A is 18-20

I think it's coherent to say that G is "19"

then the augmented triad is "12-15-19"

12-15 is far of tempered intonation, but 15-19 is just near temperered
third (if I remember well, a little bit bigger, 409 cents?).

What do you think of this example? did you ever try to ear it? (I didn't
try, it's just a hypothesis of this evening...(23h00 near Paris...).

Xavier

> >
> >
> >The crux of the problem is that there is an augmented triad. B D# G and
> >D# is punning Eb. The tuning I ended up preferring was
> >
> >1/1 3/2 [*] 15/8 45/32
> >
> >where [*] is 15/16*sqrt(8/5) (which is pretty close to 19/16 and closer
> >to 32/27).
>
> Robert,
>
> How did you come up with 15/16*sqrt(8/5)? Is that the solution to an
> optimization problem that you posed?

The augmented triad was formed by 3/2 15/8 [*] (3/2).

The 8/5 comes from (3/2)/(15/8). The root seemed like a sensible cut
point (splitting the error on both major thirds), and hte 15/16 steps
it down to the proper point relative to the 15/8.

> How do you think the chord would tune if a D were added at 9/8, as I
> suggested in order to increase stability?
>

The 9/8 certainly is "in the pocket" with all tones EXCEPT for [*]. I
don't think the complexity of the semitone will warrant [*] becoming
a more abrasive major third by slipping in either direction.

Just to show the range of values we're dealing with, the three
choices I thought were "making it", shown with the "minor third"
they are, and the major thirds they create. To my ears, having one
of the major thirds climb up the hill more than offset whatever
plaeasurable effect is given by the other side climbing down the
hill, which is why I ended up preferring the 'root'.

major thirds
"Eb" value from 15/8 and 3/2
19/16 1.1875 1.267 1.2632
32/27 1.1852 1.264 1.2656
[*] 1.1859 1.2649 1.2649

Bob Valentine