First melodic spring results

I've got melodic springs wired into the leisure retuning program. Each
time a new note begins sounding, the program looks for other notes a
step away ending. "Step" means, a semitone or whole tone higher or
lower. Thus, if a C begins, the program looks for Bb, B, C#, and D
ending.

For any such notes found, a horizontal non-unison spring is created,
much weaker than horizontal unison springs, but strong enough that if
there are not other compelling forces, melodic steps are guided toward
uniform pitch intervals.

As it is relaxing the full spring matrix, now made up of four spring
types (vertical, grounding, horizontal (== horizontal unison), and
melodic (== horizontal non-unison)), the "ideal" step for semitone and
for whole-tone are calculated. This is done the same way grounding
points are adjusted for dynamic considerations, in a repetitive
relaxation process.

For the Chopin piece on my web site, here are some comparative numbers.
This is a 5-limit, tuning-file-free analysis with weak but non-trivial
tritone springs pulling the 7th degree of a dom 7th down somewhat,
especially when the fifth degree is missing or weak. Vertical springs
are fairly soft.

With melodic springs (100% of nominal):
After relaxing, Total spring pain: 171507.447
nSpring Strength Pain RMS deviation
------- -------- ---- -------------
Vertical 11808 6717.441 118946.200 5.951 cent
Horizontal 8508 304630.599 6503.423 0.207 cent
Melodic 2666 2919.473 23700.113 4.029 cent
Grounding 8520 10665.588 22357.710 2.048 cent
Ideal melodic semitone: 104.3974 cents.
Ideal melodic tone: 197.9118 cents.

Without melodic springs (1% of nominal)
After relaxing, Total spring pain: 134039.224
nSpring Strength Pain RMS deviation
------- -------- ---- -------------
Vertical 11808 6717.441 103479.632 5.551 cent
Horizontal 8508 304630.599 5918.428 0.197 cent
Melodic 2666 29.195 723.912 7.042 cent
Grounding 8520 10665.588 23917.252 2.118 cent
Ideal melodic semitone: 105.2833 cents.
Ideal melodic tone: 197.1829 cents.

Note the new field, RMS deviation. This is the deviation consistent
with the summed spring strength and the summed spring pain, and is a
good measure of the relative strengths of the input coefficients for
the different spring types. With only trivial melody spring stiffness,
there is 7.042 cents RMS deviation, but this is reduced to 4.029 cents
with melody springs at full nominal strength. The improvement is
purchased with only 0.4 cents increase in vertical RMS deviation, .01
cent increase in horizontal RMS deviation, and a slight _reduction_ in
grounding RMS deviation (don't ask ME why!).

If the solution set for 1% melody springs is taken, but the melody
springs themselves are brought up to 100% strength, melodic pain rises
from 723.912 to 72391.2, and total pain is thus 205706, compared to
total pain 171507 when matrix relaxation is done under full strength
melody springs.

More examples will follow.

JdL

My question is: can you hear the difference?

[Paul E wrote:]
>My question is: can you hear the difference?

I've barely begun to listen so far, but I think so. My sensitivity to
uneven steps seems to come and go: in some pieces, like the Gershwin
7-limit "Rhapsody in Blue", I hear it in several places sometimes, but
other times can listen to the whole piece without hearing it much at
all.

I'm toying with allowing the program to adjust the relative strengths
of the coefficients for each spring type, in order to achieve a
user-specified ratio of RMS deviation. The Chopin piece, for example,
has very low grounding deviation, so grounding springs could be loosened
somewhat to reduce vertical (and other) deviations. I don't yet have
a good sense for how much to adjust a given coefficient to yield a
certain change in relative RMS deviation, however. And these dynamic
coefficient adjustments would increase run-time considerably.

Oh, and, at Robert Walker's request, I've allowed negative melodic
spring constants. As predicted, this can cause the matrix to go
unstable beyond some critical value, but smaller values cause the
melodic steps to spread (though why Robert wants this, I'll let _him_
explain!).

JdL

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul E wrote:]
> >My question is: can you hear the difference?
>
> I've barely begun to listen so far, but I think so. My sensitivity
to
> uneven steps seems to come and go: in some pieces, like the Gershwin
> 7-limit "Rhapsody in Blue", I hear it in several places sometimes,
but
> other times can listen to the whole piece without hearing it much at
> all.

>(though why Robert wants this, I'll let _him_
> explain!).

Well, why do you think you like the 34-tET Pachelbel canon so much?
Couldn't the fact that you do totally contradict the idea
of "melodic" springs with positive coefficients? (Personally, I like
the idea of "melodic" springs)

[Paul E wrote:]
>>>My question is: can you hear the difference?

[I wrote:]
>>I've barely begun to listen so far, but I think so. My sensitivity to
>>uneven steps seems to come and go: in some pieces, like the Gershwin
>>7-limit "Rhapsody in Blue", I hear it in several places sometimes, but
>>other times can listen to the whole piece without hearing it much at
>>all.

>>(though why Robert wants [negative melodic spring constants], I'll let
>>_him_ explain!).

[Paul:]
>Well, why do you think you like the 34-tET Pachelbel canon so much?
>Couldn't the fact that you do totally contradict the idea
>of "melodic" springs with positive coefficients? (Personally, I like
>the idea of "melodic" springs)

I really don't know! But my guess is that the reason I like 34-tET is
vertical, not melodic. I'm modifying the program so that "tuning file
free" versions can target intervals other than exact JI, and maybe I'll
post a blind A/B (one JI-targetted tuning file free, the other with
slightly widened thirds and fifths (== narrowed fourths and sixths)),
much like 34-tET. Could be interesting!

JdL

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> I really don't know! But my guess is that the reason I like 34-tET
is
> vertical, not melodic. I'm modifying the program so that "tuning
file
> free" versions can target intervals other than exact JI, and maybe
I'll
> post a blind A/B (one JI-targetted tuning file free, the other with
> slightly widened thirds and fifths (== narrowed fourths and
sixths)),
> much like 34-tET. Could be interesting!

Hi John,

Perhaps you prefer wide major thirds and fifths to just ones. Brian
McLaren often cites a preference for stretched intervals in the
psychoacoustic literature. But then you should probably stretch the
octaves quite a bit too, especially since you'll probably prefer just
(or perhaps wide) fourths and minor sixths to narrow ones. Don't you
think?

-Paul

[I wrote:]
>>I really don't know! But my guess is that the reason I like 34-tET is
>>vertical, not melodic. I'm modifying the program so that "tuning file
>>free" versions can target intervals other than exact JI, and maybe
>>I'll post a blind A/B (one JI-targetted tuning file free, the other
>>with slightly widened thirds and fifths (== narrowed fourths and
>>sixths)), much like 34-tET. Could be interesting!

[Paul E:]
>Perhaps you prefer wide major thirds and fifths to just ones. Brian
>McLaren often cites a preference for stretched intervals in the
>psychoacoustic literature. But then you should probably stretch the
>octaves quite a bit too, especially since you'll probably prefer just
>(or perhaps wide) fourths and minor sixths to narrow ones. Don't you
>think?

I think it's a possibility, definitely. A straight stretch of maybe 6
cents per octave would approximate the wideness of 34-tET thirds and
fifths, while also stretching their inversions. Another combination
would stretch thirds and fifths, while leaving their inversions
approximately just. I'll try that too! (I'll have to add a linear
stretch option in addition to the non-linear one I put in recently).

But then how to explain the sensation of 34-tET? It _does_ have narrow
fourths (4 cents) and sixths (2 cents apiece). Must experiment...

JdL

> But then how to explain the sensation of 34-tET? It _does_ have narrow
> fourths (4 cents) and sixths (2 cents apiece).

Maybe the fourths and minor sixths are "less important" than the fifths and major thirds.

[I wrote:]
>>But then how to explain the sensation of 34-tET? It _does_ have
>>narrow fourths (4 cents) and sixths (2 cents apiece).

[Paul E:]
>Maybe the fourths and minor sixths are "less important" than the fifths
>and major thirds.

Maybe, but what possible unifying theory could be behind such a thing?
I've just created an adaptive version that targets 34-tET intervals
(in a tuning-file-free fashion, no templates) for thirds thru sixths
(no stretch). Sounded great on first hearing; listened to the whole
Canon. Then listened to other versions and returned; not so sure the
second time around. The ear often takes a lot of time to make up its
mind, I find...

JdL

Hi John,

> Oh, and, at Robert Walker's request, I've allowed negative melodic
> spring constants. As predicted, this can cause the matrix to go
> unstable beyond some critical value, but smaller values cause the
> melodic steps to spread (though why Robert wants this, I'll let _him_
> explain!).

Thanks!

I'm not that directly aware of equalness of step sizes when listening
to music, but when quite a fair amount is played, I think do get
some impression of equalness to it.

As it happens, I find a kind of raggedness of step size rather
attractive. Or at least, I think I do, hard to really say
if one isn't that directly aware of it as such.

So, as I like your original retuning of Chopin with the irregular
step sizes, would be intereseting to see what it sounds like
with the steps even more irregular.

That's my explanation, as far as it goes, and I'll be interested
to hear it if you do retune the Chopin like that.

While on the subject of adaptive tuning:

I've seen some posts suggesting targetting a particular
temperament as well as adaptively tuning.

I wonder if one could do that by taking some particular
12 tone temperament and making semitone and tone springs
that depend on the position in the scale, e.g. maybe
stronger for C - C# than C# to D, or vice versa, etc,
and adjust the strength of all those springs so that
when on their own, with rest set to 0, causes the
tuning to settle to the desired temperament.

No idea if this is feasible / desirable etc., just
suggesting it as an idea that occurred in case
it suggests anything useful or interesting.

Robert

[I wrote:]
>>Oh, and, at Robert Walker's request, I've allowed negative melodic
>>spring constants. As predicted, this can cause the matrix to go
>>unstable beyond some critical value, but smaller values cause the
>>melodic steps to spread (though why Robert wants this, I'll let _him_
>>explain!).

>Thanks!

You're welcome! My methods would be a small shadow of their current
capabilities but for requests and comments such as yours. To be sure,
they're still just a small shadow of what the future will hold in
refinement, in somebody's hands...

>I'm not that directly aware of equalness of step sizes when listening
>to music, but when quite a fair amount is played, I think do get
>some impression of equalness to it.

>As it happens, I find a kind of raggedness of step size rather
>attractive. Or at least, I think I do, hard to really say
>if one isn't that directly aware of it as such.

>So, as I like your original retuning of Chopin with the irregular
>step sizes, would be interesting to see what it sounds like
>with the steps even more irregular.

>That's my explanation, as far as it goes, and I'll be interested
>to hear it if you do retune the Chopin like that.

I'll send you a spread version off-list. Please comment on-list what
you think!

>While on the subject of adaptive tuning:

>I've seen some posts suggesting targetting a particular
>temperament as well as adaptively tuning.

>I wonder if one could do that by taking some particular
>12 tone temperament and making semitone and tone springs
>that depend on the position in the scale, e.g. maybe
>stronger for C - C# than C# to D, or vice versa, etc,
>and adjust the strength of all those springs so that
>when on their own, with rest set to 0, causes the
>tuning to settle to the desired temperament.

>No idea if this is feasible / desirable etc., just
>suggesting it as an idea that occurred in case
>it suggests anything useful or interesting.

Lemme see if I'm understanding you - it sounds like you'd be referring
to grounding springs, their strength and target tuning, yes? No, that
doesn't fit what you wrote. There won't be springs between C and C#
except where that interval is sounding vertically and/or the new melodic
springs attempt to impose a uniform step.

I've targeted 34-tET by adding a feature to the "tuning-file free"
methods in which 3 semitones doesn't necessarily target 315.64 cents,
etc. The new targets are applied to vertical intervals only in this
procedure (i.e., simultaneously sounding notes).

So I'm _not_ understanding what you're suggesting - would you try again?
Thanks for all your comments!

JdL

--- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [I wrote:]
> >>But then how to explain the sensation of 34-tET? It _does_ have
> >>narrow fourths (4 cents) and sixths (2 cents apiece).
>
> [Paul E:]
> >Maybe the fourths and minor sixths are "less important" than the
fifths
> >and major thirds.
>
> Maybe, but what possible unifying theory could be behind such a
thing?

The 4:3 less important than the 3:2 . . . the 8:5 less important than
the 5:4 . . . higher-number ratios less important than lower-number
ratios.

John, what did you think of Herman's "1/7-comma meantone with 1/7-
comma-stretched octaves" version, as compared with, say, 55-tET?