Re: Digest Number 396 Message 20

>Message: 20
> Date: Mon, 15 Nov 1999 10:27:28 +0200 (IST)
> From: Robert C Valentine <bval@iil.intel.com>
>Subject: 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".

Robert,

Good work. As a former econometrician, I think that your idea of minimizing
errors is on the right track. I suspect that such an approach could yield
good solutions (i.e., solutions that sound in tune) to a number of tuning
puzzles (e.g., C6,9 as tonic). It does suggest that good tuning is
sometimes tempered, and that the tempering depends upon harmonic context.

How did you come up with 15/16*sqrt(8/5)? Is that the solution to an
optimization problem that you posed?

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

John Link
ALMOST ACAPPELLA

John Link wrote,

>Good work. As a former econometrician, I think that your idea of minimizing
>errors is on the right track. I suspect that such an approach could yield
>good solutions (i.e., solutions that sound in tune) to a number of tuning
>puzzles (e.g., C6,9 as tonic). It does suggest that good tuning is
>sometimes tempered, and that the tempering depends upon harmonic context.

>How did you come up with 15/16*sqrt(8/5)? Is that the solution to an
>optimization problem that you posed?

Dave Keenan and I had many discussions on this list about using optimization
for solving tuning problems. I think this approach is highly promising.
William Sethares has been kind enough to provide Matlab code to calculate
his roughness function, and I use Matlab for harmonic entropy. I also have
the Matlab optimization toolbox. Someday, I'd like to put all this to work
for some real musical examples. Of course, my ear will always be the final
arbiter and shaper of the procedures I use (as it has been so far).