adaptive JI question

What would be an ideal type of temperament to use with adaptive JI?

I suppose one would want to minimize the retuning motion. It seems
that would mean minimizing the absolute errors of intervals.

So maybe TOP isn't that good for adaptive JI or what do you think?

Kalle

>What would be an ideal type of temperament to use with adaptive JI?
>
>I suppose one would want to minimize the retuning motion. It seems
>that would mean minimizing the absolute errors of intervals.
>
>So maybe TOP isn't that good for adaptive JI or what do you think?

The TOP method would be fine, but the particular temperament depends
on the music and ultimately on the listener's taste. AFAIK, the
criteria for adaptive tuning are the same as for fixed tuning as
far as temperament choice is concerned.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >What would be an ideal type of temperament to use with adaptive JI?
> >
> >I suppose one would want to minimize the retuning motion. It seems
> >that would mean minimizing the absolute errors of intervals.
> >
> >So maybe TOP isn't that good for adaptive JI or what do you think?
>
> The TOP method would be fine, but the particular temperament depends
> on the music and ultimately on the listener's taste.

Yes, I should have formulated my question more correctly. I didn't
mean to talk about types of temperament such as equal, linear or
planar but types of tempering such as RMS, minimax or TOP.

> AFAIK, the
> criteria for adaptive tuning are the same as for fixed tuning as
> far as temperament choice is concerned.

Why? Remember that in general TOP has worse absolute errors (and thus
also more retuning motion) in more complex intervals than other types
of tempering.

Kalle

>> >What would be an ideal type of temperament to use with adaptive JI?
>> >
>> >I suppose one would want to minimize the retuning motion. It seems
>> >that would mean minimizing the absolute errors of intervals.
>> >
>> >So maybe TOP isn't that good for adaptive JI or what do you think?
>>
>> The TOP method would be fine, but the particular temperament depends
>> on the music and ultimately on the listener's taste.
>
>Yes, I should have formulated my question more correctly. I didn't
>mean to talk about types of temperament such as equal, linear or
>planar but types of tempering such as RMS, minimax or TOP.

It depends on which error function you prefer and on whether you
want temperd octaves.

>> AFAIK, the
>> criteria for adaptive tuning are the same as for fixed tuning as
>> far as temperament choice is concerned.
>
>Why? Remember that in general TOP has worse absolute errors (and thus
>also more retuning motion) in more complex intervals than other types
>of tempering.

Why would weighted error be less relevant for adaptive tuning than
fixed tuning?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Why? Remember that in general TOP has worse absolute errors (and
thus
> >also more retuning motion) in more complex intervals than other
types
> >of tempering.

> Why would weighted error be less relevant for adaptive tuning than
> fixed tuning?

Read my last sentence again. Do you think that tolerance for retuning
motion is exactly the same as tolerance for mistuning of consonances?

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> What would be an ideal type of temperament to use with adaptive JI?

One where the deviations from Just are all either zero or a fixed
size. Example: 1/4-comma meantone, which is why the "Vicentino's
second of 1555" variety of adaptive JI works so well.

> I suppose one would want to minimize the retuning motion. It seems
> that would mean minimizing the absolute errors of intervals.

Well, there are several independent absolute errors, since there are
several consonant intervals being approximated in a temperament, in
general. So there's no precise sense in which you can minimize them
all, but you can minimize some function of them, such as their
maximum or their sum. 1/4-comma meantone actually satisfies both
criteria in the 5-odd-limit.

> So maybe TOP isn't that good for adaptive JI

You're probably right.

>> >Why? Remember that in general TOP has worse absolute errors (and
>> >thus also more retuning motion) in more complex intervals than
>> >other types of tempering.
>
>> Why would weighted error be less relevant for adaptive tuning than
>> fixed tuning?
>
>Read my last sentence again. Do you think that tolerance for retuning
>motion is exactly the same as tolerance for mistuning of consonances?

Read my sentence again -- I'm not aware of any principle, and I'm
asking you what you think it is.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Why? Remember that in general TOP has worse absolute errors
(and
> >> >thus also more retuning motion) in more complex intervals than
> >> >other types of tempering.
> >
> >> Why would weighted error be less relevant for adaptive tuning
than
> >> fixed tuning?
> >
> >Read my last sentence again. Do you think that tolerance for
retuning
> >motion is exactly the same as tolerance for mistuning of
consonances?
>
> Read my sentence again -- I'm not aware of any principle, and I'm
> asking you what you think it is.

minimizing the amount of retuning motion

>> >> >Why? Remember that in general TOP has worse absolute errors
>> >> >(and thus also more retuning motion) in more complex intervals
>> >> >than other types of tempering.
>> >
>> >> Why would weighted error be less relevant for adaptive tuning
>> >> than fixed tuning?
>> >
>> >Read my last sentence again. Do you think that tolerance for
>> >retuning motion is exactly the same as tolerance for mistuning
>> >of consonances?
>>
>> Read my sentence again -- I'm not aware of any principle, and I'm
>> asking you what you think it is.
>
>minimizing the amount of retuning motion

This doesn't answer our question. You have to say why it should
be different than minimizing errors of vertical consonances.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >minimizing the amount of retuning motion
>
> This doesn't answer our question. You have to say why it should
> be different than minimizing errors of vertical consonances.

Actually it isn't except for TOP and other temperings that use error
weighting. TOP doesn't minimize the amount of retuning motion. That's
why.

>> >minimizing the amount of retuning motion
>>
>> This doesn't answer our question. You have to say why it should
>> be different than minimizing errors of vertical consonances.
>
>Actually it isn't except for TOP and other temperings that use error
>weighting. TOP doesn't minimize the amount of retuning motion. That's
>why.

But it minimizes the amount of weighted retuning motion. Why isn't
that important? One way it might be important is if the different
consonances in music do not occur with equal frequency. Modulation
by fifths is supposedly more common than modulation by thirds, for
example. In a rich orchestration octaves are the most common
interval, it would seem. Does this make sense?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >minimizing the amount of retuning motion
> >>
> >> This doesn't answer our question. You have to say why it should
> >> be different than minimizing errors of vertical consonances.
> >
> >Actually it isn't except for TOP and other temperings that use
error
> >weighting. TOP doesn't minimize the amount of retuning motion.
That's
> >why.
>
> But it minimizes the amount of weighted retuning motion.

I don't know what that means . . . or even on what basis you say it --
it's far from clear what an adaptive solution based on TOP meantone
would be. The Vicentino solution based on 1/4-comma meantone is only
somewhat unambiguous because 1/4-comma meantone's errors are all
either zero or 1/4-comma.

>> >> This doesn't answer our question. You have to say why it should
>> >> be different than minimizing errors of vertical consonances.
>> >
>> >Actually it isn't except for TOP and other temperings that use
>> >error weighting. TOP doesn't minimize the amount of retuning
>> >motion. That's why.
>>
>> But it minimizes the amount of weighted retuning motion.
>
>I don't know what that means . . . or even on what basis you say
>it --

It means the weighted "shifts" over a piece of music will tally
smaller than if you'd used non-TOP, provided you "tally" in the
appropriate way (which I don't have time to figure out at the
moment -- the present example is made rather annoying by the fact
that TOP meantone is so close to 1/4-comma).

>it's far from clear what an adaptive solution based on TOP
>meantone would be.

You root JI chords to roots in TOP meantone. Am I missing
something?

>The Vicentino solution based on 1/4-comma meantone is only
>somewhat unambiguous because 1/4-comma meantone's errors are all
>either zero or 1/4-comma.

I saw this in your earlier reply, but have no idea what it means,
or on what basis you say it. Can you explain?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >it's far from clear what an adaptive solution based on TOP
> >meantone would be.
>
> You root JI chords to roots in TOP meantone. Am I missing
> something?

It's far from clear to me that this is what it means. The only reason
this roots-to-roots rule works well in the Vicentino case is that you
have those pure major thirds.

> >The Vicentino solution based on 1/4-comma meantone is only
> >somewhat unambiguous because 1/4-comma meantone's errors are all
> >either zero or 1/4-comma.
>
> I saw this in your earlier reply, but have no idea what it means,
> or on what basis you say it. Can you explain?

If it weren't for the pure major thirds of 1/4-comma meantone, the
approach would fall apart. Construct each chain using some other
variety of meantone, and you introduce all kinds of new shifts.
Which, in turn, can be somewhat ameliorated by relaxing the roots-to-
roots rule.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >it's far from clear what an adaptive solution based on TOP
> > >meantone would be.
> >
> > You root JI chords to roots in TOP meantone. Am I missing
> > something?
>
> It's far from clear to me that this is what it means. The only
reason
> this roots-to-roots rule works well in the Vicentino case is that
you
> have those pure major thirds.

*And* pure octaves!!

>If it weren't for the pure major thirds of 1/4-comma meantone, the
>approach would fall apart. Construct each chain using some other
>variety of meantone, and you introduce all kinds of new shifts.
>Which, in turn, can be somewhat ameliorated by relaxing the roots-to-
>roots rule.

The shifts should obey the error function that was used
to choose the temperament.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >If it weren't for the pure major thirds of 1/4-comma meantone, the
> >approach would fall apart. Construct each chain using some other
> >variety of meantone, and you introduce all kinds of new shifts.
> >Which, in turn, can be somewhat ameliorated by relaxing the roots-
to-
> >roots rule.
>
> The shifts should obey the error function that was used
> to choose the temperament.

That doesn't make any sense to me -- and what if you didn't use any
error function but are using golden meantone or metameantone or
lucytuning?

1/4-comma meantone can be derived from minimizing either max-abs or
sum-abs error of 5-odd-limit. In between, you get other varieties of
meantone, such as 7/26-comma if you use rms. Yet most of the shifts
are zero only in 1/4-comma adaptive, assuming you use the roots-to-
roots rule. So in what sense do the shifts obey the error criterion
used to choose the temperament?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > It's far from clear to me that this is what it means. The only
> reason
> > this roots-to-roots rule works well in the Vicentino case is that
> you
> > have those pure major thirds.
>
> *And* pure octaves!!

Presumably 1/3 comma would work also?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > It's far from clear to me that this is what it means. The only
> > reason
> > > this roots-to-roots rule works well in the Vicentino case is
that
> > you
> > > have those pure major thirds.
> >
> > *And* pure octaves!!
>
> Presumably 1/3 comma would work also?

You're right (as I mentioned in regard to 152-equal once)! So only
1/3-comma and 1/4-comma work; if you used Pythagorean, enforcing
roots-to-roots would produce two comma shifts anytime you had a
diatonic chord change up or down a third -- ick.

>> >If it weren't for the pure major thirds of 1/4-comma meantone, the
>> >approach would fall apart. Construct each chain using some other
>> >variety of meantone, and you introduce all kinds of new shifts.
>> >Which, in turn, can be somewhat ameliorated by relaxing the roots-
>> >to-roots rule.

What is the roots-to-roots rule?

>> The shifts should obey the error function that was used
>> to choose the temperament.
>
>That doesn't make any sense to me -- and what if you didn't use any
>error function but are using golden meantone or metameantone or
>lucytuning?

It may be possible to find error functions that give those tunings,
I don't know.

>1/4-comma meantone can be derived from minimizing either max-abs or
>sum-abs error of 5-odd-limit. In between, you get other varieties of
>meantone, such as 7/26-comma if you use rms. Yet most of the shifts
>are zero only in 1/4-comma adaptive, assuming you use the roots-to-
>roots rule. So in what sense do the shifts obey the error criterion
>used to choose the temperament?

How are the shifts different than vertical error? For one thing,
they can occur against any pitch in the tuning, not just the
consonances. . .

Here's a syntonic pump in strict JI...

G ..... A ----- A ..... B ..... C
E ----- E ..... F ..... G ----- G
C ----- C ..... D ----- D ..... E
-----------------------------------
1 .... 5/3 ... 10/9 . 40/27 .. 80/81

...there are no shifts, but a drift of 81/80.

Here's the pump in 19-tET adaptive JI...

702 .. 884 -3c 891 .... 1081 .. 0
386 -- 386 ... 505 .... 695 +7c 702
0 ---- 0 ..... 189 +8c. 197 ... 386
-----------------------------------
0 .... 884 ... 189 .... 695 ... 0

...there is no drift, but 3 shifts totalling 18 cents, the
largest of which is 8 cents.

Here's the pump in TOP meantone (5-limit) adaptive JI...

702 .. 891 +4c 895 .... 1084 .. 1200
386 +7 393 ... 509 .... 698 +4c 702
0 ---- 0 ..... 193 +7c. 200 ... 386
-------------------------------------
0 .... 891 ... 193 .... 698 ... 0

Here there are 4 shifts totalling 22 cents, the largest of
which is 7 cents. Hmm... And I realize my earlier speculation
on weighted shifts was ill-conceived...

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >If it weren't for the pure major thirds of 1/4-comma meantone,
the
> >> >approach would fall apart. Construct each chain using some
other
> >> >variety of meantone, and you introduce all kinds of new shifts.
> >> >Which, in turn, can be somewhat ameliorated by relaxing the
roots-
> >> >to-roots rule.
>
> What is the roots-to-roots rule?

You were the one to bring it up, in message #52863, Carl.

"You root JI chords to roots in TOP meantone" -- or presumably in
whatever variety of meantone.

> > So in what sense do the shifts obey the error criterion
> >used to choose the temperament?
>
> How are the shifts different than vertical error? For one thing,
> they can occur against any pitch in the tuning, not just the
> consonances. . .

Precisely.

> Hmm... And I realize my earlier speculation
> on weighted shifts was ill-conceived...

So it seems.

>> What is the roots-to-roots rule?
>
>You were the one to bring it up, in message #52863, Carl.
>
>"You root JI chords to roots in TOP meantone" -- or presumably in
>whatever variety of meantone.

Since I didn't use that terminology, I didn't know what you
meant.

>> > So in what sense do the shifts obey the error criterion
>> >used to choose the temperament?
>>
>> How are the shifts different than vertical error? For one thing,
>> they can occur against any pitch in the tuning, not just the
>> consonances. . .
>
>Precisely.

The tuning is still generated from the consonances, though...

-Carl