That tuning method you've been pitching

[Paul Erlich, TD 454.8:]
>The adaptive tuning solution that Vicentino would have chosen using his
>second tuning of 1555, and that I've been trying to pitch to John
>deLaubenfels, involves tuning the roots horizontally in meantone
>temperament while maintaining the same vertical proportions as JI.

I can see that we're still not clear to each other. We tend to come
from different directions to the question of adaptive tuning, and I
think that when you see both, a better picture will emerge.

Your method is to begin with 1/4 comma meantone (and/or 31-tET, which is
virtually identical), then fix the intervals as a last step to be true
JI.

My approach is to begin with the knowledge of the tuning of each chord,
then to place them ("center" them) in position relative to each other.

Since the final destination is presumably the same vertical tunings
(setting aside the very real, separate, question of the particular
tuning chosen for a given set of notes: let's say I agree to your
pick(s) in that regard), the only thing left to be concerned about is
how to place each chord in absolute tuning space.

You have pointed out that your method results in small shifts in notes
continuously sounding (in the classic "comma pump" sequence at least),
and I have several times acknowledged the accuracy of this assertion. I
use a slightly different calculation to judge what is ideal: it is not a
"minimum maximum" formulation, but more of a "weighted minimum sum of
squares". If, for the sake of argument, I adopted your calculation of
ideal, it would be possible to achieve the SAME tight motion without
ever going through meantone.

I'm not trying to diss your idea (though as I've said, I see it breaking
with the 19th century way of modulating). I'm saying that it's only one
path to the tightest possible motion, and that the method I'm already
using is another (given changes in the optimum drift calculation, which
I'm not actually willing to do).

Am I saying this clearly?

JdL

John deLaubenfels wrote,

>If, for the sake of argument, I adopted your calculation of
>ideal, it would be possible to achieve the SAME tight motion without
>ever going through meantone.

Of course.

(1) You may recall how supportive I was of all your theoretical concepts
when you first posted them to the list.
(2) I keep emphasizing meantone because you keep emphasizing 12-tET, while
many composers were nice enough to distinguish G# from Ab, etc., in their
scores.

>I'm not trying to diss your idea (though as I've said, I see it breaking
>with the 19th century way of modulating).

I've said that too, very repeatedly.

>I'm saying that it's only one
>path to the tightest possible motion, and that the method I'm already
>using is another (given changes in the optimum drift calculation, which
>I'm not actually willing to do).

>Am I saying this clearly?

Yup!

[Paul Erlich, TD 455.9:]
>(1) You may recall how supportive I was of all your theoretical
>concepts when you first posted them to the list.

Yes, you answered back immediately to my introductory post, and
contrasted favorably what I said I was doing to others (unnamed). You
agreed with several specific details of my practice (when to use 12-tET,
for example). That post, along with a couple of others, made me feel
welcome on the list from the moment I joined. I have not forgotten, and
by the way, thanks very much!!

[Paul:]
>(2) I keep emphasizing meantone because you keep emphasizing 12-tET,
>while many composers were nice enough to distinguish G# from Ab, etc.,
>in their scores.

OK, but what actual implication does this have for actual tuning? I'm
still struggling with this one.

[JdL:]
>>I'm not trying to diss your idea (though as I've said, I see it
>>breaking with the 19th century way of modulating).

[Paul:]
>I've said that too, very repeatedly.

OK...

So, if this is all clear to both of us, what exactly are you "pitching"
(your word)? Apparently, the G#/Ab distinction, but... I still don't
see what in the final analysis will change.

Unless, perhaps, the calculation of "ideal centering"(?)...

JdL

>>(2) I keep emphasizing meantone because you keep emphasizing 12-tET,
>>while many composers were nice enough to distinguish G# from Ab, etc.,
>>in their scores.

>OK, but what actual implication does this have for actual tuning? I'm
>still struggling with this one.

I guess we'll have to take it on a case-by-case basis.

>So, if this is all clear to both of us, what exactly are you "pitching"
>(your word)?

Well, I was just pointing out that my (Vicentino's) simplistic method of
viewing adaptive JI would seem to produce better results than what was
coming up in your Bach files.

>Unless, perhaps, the calculation of "ideal centering"(?)...

Maybe that's part of it. Please elaborate.

John deLaubenfels wrote,

>You have pointed out that your method results in small shifts in notes
>continuously sounding (in the classic "comma pump" sequence at least),
>and I have several times acknowledged the accuracy of this assertion. I
>use a slightly different calculation to judge what is ideal: it is not a
>"minimum maximum" formulation, but more of a "weighted minimum sum of
>squares". If, for the sake of argument, I adopted your calculation of
>ideal, it would be possible to achieve the SAME tight motion without
>ever going through meantone.

John,

Please demonstrate how any adaptive JI scheme could acheive a lower
"weighted minimum sum of squares" retuning motion in the classic comma pump
sequence than Vicentino's.

-Paul

[I wrote:]
>>You have pointed out that your method results in small shifts in notes
>>continuously sounding (in the classic "comma pump" sequence at least),
>>and I have several times acknowledged the accuracy of this assertion.
>>I use a slightly different calculation to judge what is ideal: it is
>>not a "minimum maximum" formulation, but more of a "weighted minimum
>>sum of squares". If, for the sake of argument, I adopted your
>>calculation of ideal, it would be possible to achieve the SAME tight
>>motion without ever going through meantone.

[Paul Erlich, TD 477.14:]
>Please demonstrate how any adaptive JI scheme could acheive a lower
>"weighted minimum sum of squares" retuning motion in the classic comma
>pump sequence than Vicentino's.

Simple: hypothesize, as in real sequences, notes of unequal volume. The
louder ones, in my model, should move less than the softer ones. Or,
take an actual performance in which some notes are more connected than
others; again, I want to consider this in the fine shading of retuning.
Vicentino's scheme is rigid in these matters.

And what about a softening of strict JI, which I'm working on
implementing right now? Vicentino's scheme requires exactly just major
thirds under all circumstances. Now, I *love* exactly just major
thirds, but I think you and I agree that a slight mistuning, around
1 or 2 cents, say, is almost free of "pain", but, depending upon the
context, can make a big difference in reducing painful retuning motion.

If you want numbers, I need a specific sequence to throw at my program:
it considers enough that hand-calculating is difficult. When I get the
new model going, it'll be all but impossible (of course, we'll still
be able to tinker to better please our ears).

JdL

>Simple: hypothesize, as in real sequences, notes of unequal volume. The
>louder ones, in my model, should move less than the softer ones. Or,
>take an actual performance in which some notes are more connected than
>others; again, I want to consider this in the fine shading of retuning.
>Vicentino's scheme is rigid in these matters.

Excellent. However, I would "tune" these considerations carefully, perhaps
by making tests of the smallest discernable motion under these various
conditions.

>And what about a softening of strict JI, which I'm working on
>implementing right now? Vicentino's scheme requires exactly just major
>thirds under all circumstances. Now, I *love* exactly just major
>thirds, but I think you and I agree that a slight mistuning, around
>1 or 2 cents, say, is almost free of "pain", but, depending upon the
>context, can make a big difference in reducing painful retuning motion.

Absolutely! I wouldn't mind even larger mistunings in faster passages.

>If you want numbers, I need a specific sequence to throw at my program:
>it considers enough that hand-calculating is difficult. When I get the
>new model going, it'll be all but impossible (of course, we'll still
>be able to tinker to better please our ears).

I'll still offer the classic comma-pump sequence as a "test case".

[I wrote:]
>>You have pointed out that your method results in small shifts in notes
>>continuously sounding (in the classic "comma pump" sequence at least),
>>and I have several times acknowledged the accuracy of this assertion.
>>I use a slightly different calculation to judge what is ideal: it is
>>not a "minimum maximum" formulation, but more of a "weighted minimum
>>sum of squares". If, for the sake of argument, I adopted your
>>calculation of ideal, it would be possible to achieve the SAME tight
>>motion without ever going through meantone.

[Paul Erlich, TD 477.14:]
>Please demonstrate how any adaptive JI scheme could acheive a lower
>"weighted minimum sum of squares" retuning motion in the classic comma
>pump sequence than Vicentino's.

My first response appeared in TD 479.9. Now I've got my new model semi
up and running, and thought I'd share the first "comma pump" results.
All possible notes are tied, all notes are the same volume, and all
chord durations are the same (256 msec). The results:

C major:
C +5.41
E -7.17
G +6.25 (change: +3.51 cents)

A minor:
C +7.17 (change: +1.76 cents)
E -5.41 (change: +1.76 cents)
A -6.25

D minor:
D -3.14
F +10.33
A -2.74 (change: +3.51 cents)

G major:
B -10.33
D +3.14 (change: +6.28 cents)
G +2.74

Pretty kyool, eh? Notice that this model allows some flexibility of
the intervals from strict JI, and it makes good use of it! Notice, too,
that a symmetry pops out which you might have noticed but I haven't
before.

Paul, did you once post the bends you'd use for this sequence using your
"modified Vicentino" method? I have to believe this has got that beat.
I'll have some redone versions of larger sequences on my web site in a
day or two.

JdL

John A. deLaubenfels wrote,

>Paul, did you once post the bends you'd use for this sequence using your
>"modified Vicentino" method? I have to believe this has got that beat.

I'm pretty sure it does, but that's because you're allowing the
simultaneities to deviate from JI (which I'm in favor of) while Vicentino's
second tuning method (nothing need be modified about it) keeps all
simultaneities in JI. What do you get if you don't allow deviations from JI
vertically?

The "relaxed" comma pump that I generated is:

C major:
C +5.41
E -7.17
G +6.25 (change: +3.51 cents)

A minor:
C +7.17 (change: +1.76 cents)
E -5.41 (change: +1.76 cents)
A -6.25

D minor:
D -3.14
F +10.33
A -2.74 (change: +3.51 cents)

G major:
B -10.33
D +3.14 (change: +6.28 cents)
G +2.74

[Paul E, TD 493.18:]
>What do you get if you don't allow deviations from JI vertically?

Well, with the spring method, absolute rigidity becomes difficult, not
to model, but to relax with good convergence (without special tricks).
Here's one that comes very close to forcing true JI:

C major:
C +5.68
E -8.01
G +7.49 (change: +5.14 cents)

A minor:
C +7.98 (change: +2.30 cents)
E -5.71 (change: +2.30 cents)
A -7.51

D minor:
D -4.13
F +11.39
A -2.37 (change: +5.14 cents)

G major:
B -11.42
D +4.10 (change: +8.23 cents)
G +2.35

The symmetry is still present, but is slightly obscured here.

JdL

>>What do you get if you don't allow deviations from JI vertically?

>Well, with the spring method, absolute rigidity becomes difficult, not
>to model, but to relax with good convergence (without special tricks).
>Here's one that comes very close to forcing true JI:

> C major:
> C +5.68
> E -8.01
> G +7.49 (change: +5.14 cents)

> A minor:
> C +7.98 (change: +2.30 cents)
> E -5.71 (change: +2.30 cents)
> A -7.51

> D minor:
> D -4.13
> F +11.39
> A -2.37 (change: +5.14 cents)

> G major:
> B -11.42
> D +4.10 (change: +8.23 cents)
> G +2.35

So your total squared shift is

2*(5.14)^2+2*(2.30)^2+(8.23)^2 = 131.15�^2

I can do better with chords tuned vertically just like those. I'll
arbitrarily use the same starting pitches:

C major:
C +5.68
E -8.01
G +7.49 (change: +5.94 cents)

A minor:
C +8.67 (change: +2.99 cents)
E -5.02 (change: +2.99 cents)
A -6.84

D minor:
D -2.66
F +12.86
A -0.90 (change: +5.94 cents)

G major:
B -12.24
D +3.28 (change: +5.94 cents)
G +1.53

Here the total squared shift is 3*(5.94)^2+2*(2.99)^2 = 123.7310�^2

Why did your method not find the optimal solution? Probably because:

>>>Each note's tuning is sprung to "ground", the center expectation for
>>>tuning of that note.

To which I replied:

>>Which is?

>You KNOW what I'm going to answer, just as I know (unless you've changed
>your mind) that you'll object. I wire each note to the center, 0.00
>cents deviation, from 12-tET. As the original sequences I'm working
>with are IN 12-tET, that seems reasonable.

Is there any way to get rid of the springs to "ground"? Isn't it true that
all you really need is to "ground" one note, say C, and let the rest fall as
they may? At least in a piece of music where every note in the scale is
harmonically connected (directly or indirectly) to the overall tonic, which
I would think would take care of all pieces of interest.

>As before, I wouldn't rule out some other center of shift, but have yet
>to see a compelling reason to use anything else. Your proposed meantone
>centering would, as we've discussed at length, have trouble with 19th
>century sequences with their diesis pumps.

True, but I think I've given compelling reasons to use meantone centering
for pre-19th century music. But this should all be moot if, as I suspect,
only one note needs to be "centered."

By the way, Vicentino's method is better only if you weight two simultaneous
shifts as a single shift instead of a double shift, or if you use a
non-linear energy function.

On 12/23/99, John deLaubenfels wrote,

>Your method is to begin with 1/4 comma meantone (and/or 31-tET, which is
>virtually identical), then fix the intervals as a last step to be true
>JI.

>My approach is to begin with the knowledge of the tuning of each chord,
>then to place them ("center" them) in position relative to each other.

The latter sounds like my recent suggestion of removing all "grounding"
springs except one as needed to fix overall pitch level. "Placing them in
position relative to each other" meant, to me, finding pairs of nearby
occurences of the same pitch and connecting each of them with a spring.

So when you wrote,

>I'm not trying to diss your idea (though as I've said, I see it breaking
>with the 19th century way of modulating). I'm saying that it's only one
>path to the tightest possible motion, and that the method I'm already
>using is another

I agreed.

But you have just revealed that you are in fact using a 12-tET grid to
achieve this positioning. Perhaps this is where you're falling short of
optimality. See if you can try abandoning this grid (I know, easier said
than done, but a small step in this direction might make for a big
improvement in tuning). Clearly, a piece that consisted only of two
non-overlapping chords, C E G and D F A, would not be tunable by this
approach. But I think you'll find that in just about any _real_ classical
work, there would always be some other chord nearby, say A C E, which would
allow you to position the chords relative to one another being guided
strictly by the harmonic relations that occur, without imposing any overall
grid of centered tunings for each note.

What I would think is that, in a pre-19th century piece of music, occurences
of G# and Ab, though named identically in the MIDI file, would occur far
enough apart in the music so that they wouldn't be attached with a spring,
or at least not a very strong one. As a result, the two pitches would come
out quite different, as they would in a meantone tuning. In fact, I would
expect most pre-19th century music to _naturally_ approach something close
to meantone tuning (if 5-limit is the overall paradigm expressed in your
tuning files) if this method is used.