A simple trick

I've had General MIDI sequences employing adaptive 7-limit and 5-limit
tuning on my web site for several months. I've always tuned the
intervals to true JI to the extent desired by the tuning file used at
any particular moment in the piece.

But, just for fun, I've constructed tuning files that call for only
HALF the correction from 12-tET that I've used so far. Thus, a major
third is narrowed by 6.84 cents, to 393.16 cents, instead of the full
13.69 cents, to 386.31 cents.

Here's what's wild: retuning motion and deviation from 12-tET are cut
in half, but according to the model I'm using, "pain" is proportional
to motion and deviation SQUARED, so should be only 1/4 of the pain
of full retuning. The error of a half-tuned interval is half of the
12-tET error, but again the model I'm using correlates pain with tuning
error squared, so the pain of ill-tuning should be only 1/4 of the pain
of 12-tET.

Confusing? The upshot is that I'm saying the tuning should have only
1/4 the retuning pain of the full JI tunings, yet should have 3/4 of
the gain in well-tuned intervals.

To illustrate, I've done the first movement of Schubert's D894 sonata
in G several ways and posted them on my web site:

http://www.idcomm.com/personal/jadl/

I would invite especially those who have sampled the previous tunings
and found them too "bouncy" to give these half-tuned files a try!

JdL

John deLaubenfels wrote,

>I've always tuned the
>intervals to true JI to the extent desired by the tuning file used at
>any particular moment in the piece.

That means you weren't really trying to minimize the total "pain"
contributed to by vertical deviatons from JI and horizontal retuning shifts;
you were usually just setting the former to zero and using some 12-tET-based
heuristics to get reasonably small results for the latter.

>The upshot is that I'm saying the tuning should have only
>1/4 the retuning pain of the full JI tunings, yet should have 3/4 of
>the gain in well-tuned intervals.

That clearly goes a long way toward acheiving your original stated goal: to
minimize the total "pain".

[I wrote:]
>>I've always tuned the intervals to true JI to the extent desired by
>>the tuning file used at any particular moment in the piece.

[Paul Erlich, TD 463.18:]
>That means you weren't really trying to minimize the total "pain"
>contributed to by vertical deviatons from JI and horizontal retuning
>shifts; you were usually just setting the former to zero and using some
>12-tET-based heuristics to get reasonably small results for the latter.

Paul, you are obsessed with the fact that I'm using 12-tET to establish
the ideal center of tuning, yet every time I try to pin you down as to
what actual practical difference you would introduce, you don't have
a precise answer! IMHO, that issue is relatively unimportant.

But you ARE correct in saying that by setting the vertical deviations
to zero, which I HAVE been doing, I am failing to minimize the total
"pain". I already have a new model in mind that will both allow
relaxation of vertical tuning from strict JI and will allow horizontal
tuning to be considered. This will also address your concerns regarding
strict JI being undesirable because of phase-lock; there will
essentially "never" be strict JI coming out of the new program (as yet
unwritten!).

[JdL:]
>>The upshot is that I'm saying the tuning should have only 1/4 the
>>retuning pain of the full JI tunings, yet should have 3/4 of the gain
>>in well-tuned intervals.

[Paul:]
>That clearly goes a long way toward acheiving your original stated
>goal: to minimize the total "pain".

Well, this particular trick is very crude; it imposes a constant factor
of out-of-tuneness to the entire piece, so I don't consider it going a
"long way" toward my goal, but it gives a hint of the fruits of
considering a larger set of tradeoffs.

(Please note that achieving 1/4 the retuning pain with 3/4 of the gain
in well-tuned intervals does not automatically indicate an improvement!)

JdL