Optimum adaptive tuning methods

[JdL, TD 335.2:]
>> Second, the motion from otonal to utonal (as in: G,B,D ->
>> G,B,D,F) drags several pitches across the map in opposite directions
>> simultaneously, which is painful (especially since retuning is still
>> sudden in this program version).

[Paul Erlich, TD 335.10:]
> That doesn't sound very clever.
> Oops, I forgot to bring in the Kahrimanis. I'll try to remember
> tomorrow.

Paul, I'm still waiting; did you change your mind? As far as I can see,
the best tuning strategy focuses first and foremost on the moment: if
the triad C,E,G appears in the sequence (assuming harmonic timbres),
tune it 4:5:6. Any of hundreds of other chords may have come just
before, and the home key may be anywhere, but what does it matter? I
do, as you know, practice a condensation of tunings to reduce the pain
of tuning motion, but this is a highly localized consideration.

I am intrigued but highly skeptical that it might make sense to adjust
the tuning of a piece on some other basis, especially at the expense of
nice integer ratios. Could you provide more substance, please?

JdL

John A. deLaubenfels wrote,

>>> Second, the motion from otonal to utonal (as in: G,B,D ->
>>> G,B,D,F) drags several pitches across the map in opposite directions
>>> simultaneously, which is painful (especially since retuning is still
>>> sudden in this program version).

I wrote,

>> That doesn't sound very clever.
>> Oops, I forgot to bring in the Kahrimanis. I'll try to remember
>> tomorrow.

John deLaubenfels wrote,

>Paul, I'm still waiting; did you change your mind?

I looked closely at Kahrimanis' paper and found the results of his
experiments, and his subsequent analyses of passages from classical works,
totally unconvincing, so I saw no point in trying to go though them on the
list. Again, his principle is that the interval between the Partchian root
of one chord and that of the next chord moves by a consonant 5-limit
interval (errors of 81:80 but not of 64:63 being allowed), and that the
tuning of the chords themselves will be otonal or utonal as best satisfies
this principle. I never agreed with his principle but thought his paper
might present some merely plausible evidence. I haven't been able to see it.

Anyway, if you're simply adding one note to a triad after the triad has
sounded, it doesn't seem to make sense to make radical shifts within the
triad. It seems you agree. Given that we would both tune G,B,D closer to
4:5:6 than to 1/9:1/7:1/6, we'd have to preclude a utonal tuning for
G,B,D,F. But in your usual, otonal scheme, what would happen if, in C minor,
you had F Ab C -> D F Ab C?

>As far as I can see,
>the best tuning strategy focuses first and foremost on the moment: if
>the triad C,E,G appears in the sequence (assuming harmonic timbres),
>tune it 4:5:6. Any of hundreds of other chords may have come just
>before, and the home key may be anywhere, but what does it matter? I
>do, as you know, practice a condensation of tunings to reduce the pain
>of tuning motion, but this is a highly localized consideration.

>I am intrigued but highly skeptical that it might make sense to adjust
>the tuning of a piece on some other basis, especially at the expense of
>nice integer ratios. Could you provide more substance, please?

Perhaps you misunderstood. Kahrimanis worked only with nice integer ratios.
In my view, though, a musically appropriate contrast between consonance and
dissonance combined with a view of horizontal melodic relations is at least
as aesthetically valid, if not more so, than a moment-by-moment maximization
of consonance. Although your example of the major triad would rarely occur
as an example of a contextual dissonance in music after 1450 (although the
raised leading tone may present a counterexample if occuring in the dominant
triad), there are certainly considerations beyond momentary consonance that
play in many people's perceptions of Western music. Have you seen Mark
Nowitzky's dominant seventh chord page?

[JdL, TD 342.4:]
>> I am intrigued but highly skeptical that it might make sense to
>> adjust the tuning of a piece on some other basis, especially at the
>> expense of nice integer ratios. Could you provide more substance,
>> please?

[Paul Erlich, TD 342.10:]
> Perhaps you misunderstood.

Yes; I have only a vague idea of what Kahrimanis, and you, are saying.

> Kahrimanis worked only with nice integer ratios.

OK, good...

> I looked closely at Kahrimanis' paper and found the results of his
> experiments, and his subsequent analyses of passages from classical
> works, totally unconvincing, so I saw no point in trying to go though
> them on the list. Again, his principle is that the interval between
> the Partchian root of one chord and that of the next chord moves by a
> consonant 5-limit interval (errors of 81:80 but not of 64:63 being
> allowed), and that the tuning of the chords themselves will be otonal
> or utonal as best satisfies this principle. I never agreed with his
> principle but thought his paper might present some merely plausible
> evidence. I haven't been able to see it.

Ok... I'm still vague on exactly what K is saying, but since you feel
it's not really all that useful, it's a moot point.

> Anyway, if you're simply adding one note to a triad after the triad
> has sounded, it doesn't seem to make sense to make radical shifts
> within the triad. It seems you agree.

Yes.

> In my view, though, a musically appropriate contrast between
> consonance and dissonance combined with a view of horizontal melodic
> relations is at least as aesthetically valid, if not more so, than a
> moment-by-moment maximization of consonance.

I'm still open on that question, but it'd be great to have a concrete
example...

> ...there are certainly considerations beyond momentary consonance that
> play in many people's perceptions of Western music. Have you seen Mark
> Nowitzky's dominant seventh chord page?

Yes; I've ref'd it on the list several times, but I don't remember it
containing any considerations other than the moment.

JdL

>Paul, I'm still waiting; did you change your mind? As far as I can see,
>the best tuning strategy focuses first and foremost on the moment: if
>the triad C,E,G appears in the sequence (assuming harmonic timbres),
>tune it 4:5:6. Any of hundreds of other chords may have come just
>before, and the home key may be anywhere, but what does it matter? I
>do, as you know, practice a condensation of tunings to reduce the pain
>of tuning motion, but this is a highly localized consideration.

From what Paul has told me of K's theory, I think it must be hogwash. Root
motion by irrational and extra-scalar intervals ought to be very important
in adaptive tuning; I believe Barbershop quartets do it all the time.

-Carl

I missed responding to the following:

[Paul Erlich, TD 342.10:]
> But in your usual, otonal scheme, what would happen if, in C minor,
> you had F Ab C -> D F Ab C?

You have identified a point of tuning conflict. F Ab C would be tuned
10:12:15, but D F Ab C would push the F Ab C into a subminor. As my
software now stands, the two chords would at first be tuned differently;
then, during the condensation phase, each would be invited to consider
taking the tuning of its neighbor. The pain of transition would be
compared to the pain of less-than-ideal tuning, and a decision would be
made either to merge or not to merge (and, if merged, which tuning to
take). This is as it should be, I think, but is still an extremely
local decision. I THINK you're arguing for a broader consideration than
this, are you not?

JdL

I wrote,

>> But in your usual, otonal scheme, what would happen if, in C minor,
>> you had F Ab C -> D F Ab C?

John A. deLaubenfels wrote,

>You have identified a point of tuning conflict. F Ab C would be tuned
>10:12:15, but D F Ab C would push the F Ab C into a subminor. As my
>software now stands, the two chords would at first be tuned differently;
>then, during the condensation phase, each would be invited to consider
>taking the tuning of its neighbor. The pain of transition would be
>compared to the pain of less-than-ideal tuning, and a decision would be
>made either to merge or not to merge (and, if merged, which tuning to
>take). This is as it should be, I think, but is still an extremely
>local decision. I THINK you're arguing for a broader consideration than
>this, are you not?

I'm really just trying to get you to consider the utonal tetrad. What about
the opening of Tristan? The clear missing fundamental of the 5:6:7:9 just
doesn't seem appropriate there.

Actually, some timbres pull a minor chord with added major sixth toward a
10:12:15:17 tuning, to my ears.

I wrote,

>> ...there are certainly considerations beyond momentary consonance that
>> play in many people's perceptions of Western music. Have you seen Mark
>> Nowitzky's dominant seventh chord page?

John A. DeLaubenfels wrote,

>Yes; I've ref'd it on the list several times, but I don't remember it
>containing any considerations other than the moment.

Actually, I recalled that Mark preferred the tuning where the seventh of the
dominant seventh was exactly a perfect fourth above the tonic to which the
chord resolved. This would be a cross-chordal consideration, no?

[Carl Lumma:]
>Root motion by irrational and extra-scalar intervals ought to be very
important in adaptive tuning; I believe Barbershop quartets do it all
the time.

This really makes a lot of sense to me both intuitively, and as it
relates to my own experiences. I'm also quite convinced that non-fixed
pitch ensembles would tend to execute identical sorts of harmonic and
melodic spellings quite differently depending on the context and
makeup of their musical character (i.e., tempo, rhythm, dynamic
markings, etc., etc., etc., etc.), and endless "other accidents of
life..." But as John has to have a program that executes a relatively
simple set of fixed algorithms, what would one that addresses "root
motion by irrational and extra-scalar intervals" look like? (I would
imagine that it would be something along the lines of horizontal or
melodic planes moving in some near Pythagorean manner?)

Dan

[Paul Erlich, TD 342.10:]
>>> ...there are certainly considerations beyond momentary consonance
>>> that play in many people's perceptions of Western music. Have you
>>> seen Mark Nowitzky's dominant seventh chord page?

[JdL, TD 343.8:]
>> Yes; I've ref'd it on the list several times, but I don't remember it
>> containing any considerations other than the moment.

[Paul Erlich, TD 343.18:]
> Actually, I recalled that Mark preferred the tuning where the seventh
> of the dominant seventh was exactly a perfect fourth above the tonic
> to which the chord resolved. This would be a cross-chordal
> consideration, no?

Yes, you are right: Mark does make that point, and yes, that would be
a cross-chordal consideration.

On the other hand, my own ears do not care for a dom7 with 9/16: it
sounds horribly out of tune to my ears, which have tasted well of 7/4
(and 7/5 and 7/6 and...). Also, the alleged connection from the seventh
of the dominant to the fourth of tonic doesn't click with me. The
fourth degree is a "tonic killer" to my ear more than any other note, I
think because the harmonic overtones of the fourth include tonic but
not visa versa, so that the fourth co-opts tonic in a way that other
scale degrees don't.

But I digress. I'm still open on the subject; please pass along other
examples as they occur to you.

JdL

>But as John has to have a program that executes a relatively
>simple set of fixed algorithms, what would one that addresses "root
>motion by irrational and extra-scalar intervals" look like? (I would
>imagine that it would be something along the lines of horizontal or
>melodic planes moving in some near Pythagorean manner?)

I do believe the adaptive tuning has one central, cut-and-dried issue:
common tones vs. pitch drift; pythagorean manner, blah.

Other than that, I don't know much about adaptive tuning; it isn't one of
my main interests. But I'll indulge myself. Keep in mind I'm pulling this
out of the air.

First, we need a lookup table that will relate 12-tone chords and just
ones. I can imagine a table based on scalar intervals, but it would likely
be quite a trick to get it to work well, and it wouldn't work at all unless
the composer was paying attention to his spellings (99% of midi files out
the window). So the table would be based on absolute intervals. [For most
music post 1850, a well-working scalar table would be equivalent to an
absolute table for 12 anyway.]

Start by taking all dyads, triads, tetrads, and pentads of the set [1 3 5 7
9 15 17 19]. That's less than 210 things. Now, tune them each in 12tET.
12 may be consistent with respect to this set. If not, just use MAD (mean
absolute deviation) to pick the closest tuning. Now, sort by the 12tET
side, and you should have multiple just versions for each 12tET chord. For
each of these, rank just versions inversely to their (MAD from 12) times
(largest number in them). Keep only the top four ranking chords in each
class.

Now, assuming that your midi file has only one timbre per channel, as all
good midi files do, come up with a decay time for notes on each channel, so
you can measure from note-on how long a note will be heard. Now step thru
the midi file tick by tick. At each tick...

a) if < 2 or > 5 octave-equivalent notes are sounding, do nothing,
b) otherwise, look up the chord in your table and tune it accordingly,
c) if there's no match, ignore the softest note and goto step 1.

"Tune it accordingly". Well, one way would be...

1) is there an option that preserves all the common tones in the score?
2) if so, use it, if not, pick the one that preserves the most,
3) in case of a tie, use the just chord with the highest rank.
4) be sure to pick frequencies that give the common tones you wound up with!

Then...

5) was there a chord change in the score at this tick?
6) if not, do nothing,
7) if so, did the tuning I picked keep all the common tones?
8) if so, do nothing, if not,
9) adjust the frequency of the root of the chosen chord to minimize the MAD
between the scored and potential common tones.

...Now this version pretty much ignores pitch-drift. Which I maintain
isn't that important in music that modulates a lot. For music that
doesn't, well, I'm still working on a solution. Any ideas?

I might sneak a few 9-limit utonal chords, and probably the 12:14:18:21
tetrad, into the lookup table since they are not generated by the method I
described.

-C.

"John A. deLaubenfels" wrote:

> On the other hand, my own ears do not care for a dom7 with 9/16: it

Is that a 9/8 or 16/9? A 16/9 would have a nasty clang but
a 9/8 is just fine to me. Use it all the time with a dom7 (1/1, 5/4, 3/2,
7/4) chord.

> sounds horribly out of tune to my ears, which have tasted well of 7/4
> (and 7/5 and 7/6 and...). Also, the alleged connection from the seventh
> of the dominant to the fourth of tonic doesn't click with me. The
> fourth degree is a "tonic killer" to my ear more than any other note, I
> think because the harmonic overtones of the fourth include tonic but
> not visa versa, so that the fourth co-opts tonic in a way that other
> scale degrees don't.

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* J u x t a p o s i t i o n N e t R a d i o
*
*
* http://www.virtulink.com/immp/lookhere.htm

I wrote...

>"Tune it accordingly". Well, one way would be...
>
>1) is there an option that preserves all the common tones in the score?

That is, common notes from the last tick. Sorry if that wasn't clear.

-C.

I wrote...

>Other than that, I don't know much about adaptive tuning; it isn't one of
my >main interests. But I'll indulge myself. Keep in mind I'm pulling
this out >of the air.

I've gone over my froth, and improved things quite a bit...

a) If < 2 or > 5 octave-equivalent notes are sounding, do nothing.
b) Otherwise, look up the chord in your table and tune it accordingly.
c) If there is no match, ignore the softest note and goto step 1.

"Tune it accordingly". Well, one way would be...

1) If the the "previous-chord record" is empty, tune the current chord with
the highest-ranking just version and record the frequencies in the record.

2) Otherwise, if there are common tones in the score between this and the
previous tick,
a) If there's an option for the current chord that preserves all
scored common tones with the chord in the record, use it, rooting
the new chord appropriately and over-writing the record with the
resulting frequencies.

b) Otherwise, pick the option that preserves the most, or take
the highest-ranking one in case of a tie, root the new chord
to minimize the MAD (of log-frequency) of the scored common tones
(between the record and the current choice), and over-write the
record with the resulting frequencies.

3) Otherwise, make a table of the log-frequency changes between the 12tET
values in the score, "tare" these into fake common tones, and run it thru
the procedure above! Clever, eh?

-> Keep in mind that everything here assumes octave equivalence.
-> Trial might suggest improvements to the ranking method.
-> I would probably be tempted to sneak a few 9-limit utonal chords, almost
surely the 12:14:18:21 tetrad, into the lookup table since they are not
generated by the method I described. Still working on a way to measure the
dissonance of the utonal chords...

-Carl

To: tuning@onelist.com
From: John deLaubenfels
Date: 10-08-99
Subject: Re: Optimum adaptive tuning methods

[JdL, TD 344.8:]
>> On the other hand, my own ears do not care for a dom7 with 9/16.

[David Beardsley, TD 344.13:]
> Is that a 9/8 or 16/9? A 16/9 would have a nasty clang but a 9/8 is
> just fine to me. Use it all the time with a dom7 (1/1, 5/4, 3/2, 7/4)
> chord.

Sorry, I had a brain fart (that hasn't happened since.... I forget). I
always think 8/9 of the root above, which of course is 16/9, NOT 9/16,
of the root below. This is Mark Nowitzky's choice for dom 7. Like you,
I love the sound of a chord that goes right on up: 1/1, 5/4, 3/2, 7/4,
2/1, 9/4, 5/2, ... You just gotta have seven, to my ears!

JdL

Carl Lumma wrote,

>Start by taking all dyads, triads, tetrads, and pentads of the set [1 3 5 7
>9 15 17 19]. That's less than 210 things. Now, tune them each in 12tET.
>12 may be consistent with respect to this set

It is!

>I might sneak a few 9-limit utonal chords, and probably the 12:14:18:21
>tetrad, into the lookup table since they are not generated by the method I
>described.

Are there examples in classical music where the dominant ninth would rather
be the 9-limit utonality than the 9-limit otonality? The 12:14:18:21 makes
sense for a minor seventh chord if it is connected to a 4:5:6:7 dominant
seventh on the same root.

>...Now this version pretty much ignores pitch-drift. Which I maintain
>isn't that important in music that modulates a lot. For music that
>doesn't, well, I'm still working on a solution. Any ideas?

How about introducing a slow, constant drift in the opposite direction? I
think that was John deLaubenfels' idea.

[Carl Lumma, TD 344.10:]
> Other than that, I don't know much about adaptive tuning; it isn't one
> of my main interests.

And yet, have you not rhapsodized about the sound of these retuned 19th
century pieces I've gotten ahold of?

> But I'll indulge myself. Keep in mind I'm pulling this out of the
> air.

Where else is there to go? This is new stuff! (One finds no shortage
of writers who say that it would be "easy", but most have not yet gotten
around to it).

Carl, do you program? I would encourage you to jump into this and put
your ideas to work!

In fact, let me challenge everyone on this list who is interested in
adaptive tuning to jump right on in!! By the middle of the next
century, I'm guessing that it'll be the norm; it's just too lovely for
the world to resist for long, IMHO, and the technology and the
programming will be well advanced by that time. But today it is
absolutely WIDE OPEN for innovation. I consider myself a reasonably
clever guy, but I'm just one single guy, and there is room for dozens,
hundreds of people applying themselves to this challenge.

JdL

Carl,

Thanks for coming up with and sharing all those adaptive tuning
strategies. It seems like John deLaubenfels also found them very
interesting, and hopefully someone who's doing adaptive tuning will
eventually be able to try them out.

When you earlier wrote "I believe Barbershop quartets do it all the
time," how do you think that this ("root motion by irrational and
extra-scalar intervals") is done there?

Dan

>> Other than that, I don't know much about adaptive tuning; it isn't one
>> of my main interests.
>
>And yet, have you not rhapsodized about the sound of these retuned 19th
>century pieces I've gotten ahold of?

I love listening. But my interest is composing and playing, and when it
comes to spending time designing and getting an instrument to play music
on, vs. working out how to re-tune existing music, I choose the former.
That isn't to say that playing old music in new tunings won't be very
instructive as I'm learning the new instrument!

>Carl, do you program? I would encourage you to jump into this and put
>your ideas to work!

Can you tell by that first procedure I posted? I learned how to solve
polynomials with LISP 4 years ago at IU. Other than that...

Programming is actually something I plan on learning, as it seems to be a
rewarding way of making a living, and a great tool for, well, just about
everything!

>In fact, let me challenge everyone on this list who is interested in
>adaptive tuning to jump right on in!! By the middle of the next
>century, I'm guessing that it'll be the norm; it's just too lovely for
>the world to resist for long, IMHO, and the technology and the
>programming will be well advanced by that time. But today it is
>absolutely WIDE OPEN for innovation. I consider myself a reasonably
>clever guy, but I'm just one single guy, and there is room for dozens,
>hundreds of people applying themselves to this challenge.

I'm really glad you're out there doing it!

-C.

>>Start by taking all dyads, triads, tetrads, and pentads of the set [1 3 5 7
>>9 15 17 19]. That's less than 210 things. Now, tune them each in 12tET.
>>12 may be consistent with respect to this set
>
>It is!

Cool! And, thanks!

>Are there examples in classical music where the dominant ninth would rather
>be the 9-limit utonality than the 9-limit otonality?

Yes. The next time I run across one, I'll post it.

>The 12:14:18:21 makes sense for a minor seventh chord if it is connected
to >a 4:5:6:7 dominant seventh on the same root.

Indeed.

>>...Now this version pretty much ignores pitch-drift. Which I maintain
>>isn't that important in music that modulates a lot. For music that
>>doesn't, well, I'm still working on a solution. Any ideas?
>
>How about introducing a slow, constant drift in the opposite direction? I
>think that was John deLaubenfels' idea.

This is something that should be tried, but I'd certainly have a stab at
making the identifying and correction of the drift more precise. I would
have to see how my current proposal does first, before designing the
anti-drift part. I do have some ideas... it may be better to apply the
correction a comma at a time, in certain spots, than to spread it out all
over the place. Then again, it may not. I really'd have to try out what
I've got so far before speculating further.

-C.

>When you earlier wrote "I believe Barbershop quartets do it all the
>time," how do you think that this ("root motion by irrational and
>extra-scalar intervals") is done there?

More or less as I described, with less precision and some sort of fudging
for pitch drift.

-C.

[Paul Erlich, TD 345.13:]
> How about introducing a slow, constant drift in the opposite
> direction? I think that was John deLaubenfels' idea.

That'll likely be bad in the middle, since drift is not uniform. I
exponentially decay the drift out as the piece progresses, using about a
1 second half-life. I find that this tends to limit maximum drift to
about +/- 30 cents, a non-trivial amount, but bearable in context.

More sophisticated methods could be employed, certainly.

[Bill Sethares, TD 345.16:]
> One possibility would be to adjust the tuning based
> on the partials (spectrum) of the notes being played.
> This would be the same as adjusting to simple integer
> ratios if the sounds had harmonic overtones, but would
> be different otherwise.

Quite right, Bill! I am failing to state that I am assuming harmonic
timbres when I speak of integer ratios. As your book, "Tuning, Timbre,
Spectrum, Scale", with its accompanying CD, make so vividly clear,
other timbres require other ratios!

> Another idea would be to adjust the tuning so as
> to achieve a certain rate of beating among the
> partials (i.e, adjusting to beatlessnes is not the
> only sensible goal). This is sometimes stated as a goal of
> gamelan tuners in poetic terms - to make the tones "shimmer".

This makes me smile, because I think of the first 40 years of my life
as being locked into the "shimmering" of 12-tET. For the past ten
years I have tasted JI, and I believe I could stand another 30 with
nothing but JI, or as close as I can come, before wanting anything
else! But of course you're speaking of controlled out-of-tuneness,
which might make perfect sense.

[Dan Stearns, TD 346.7:]
> Carl [Lumma],
> Thanks for coming up with and sharing all those adaptive tuning
> strategies. It seems like John deLaubenfels also found them very
> interesting, and hopefully someone who's doing adaptive tuning will
> eventually be able to try them out.

While it's always possible that Carl will find someone to program his
ideas, it's been my experience that if I want something like this done,
I'd better do it myself (I'm assuming for the sake of discussion that
Carl doesn't have the resources to snap his fingers and have a
programmer or two under his personal hire...).

[Carl Lumma, TD 346.9:]
> I love listening. But my interest is composing and playing...

I feel the same. But the work I've done with retuning can only help me
when it comes to composing and playing.

-------------------------------------------
[Dante Rosati, TD 346.12:]
> I, Dante Rosati, do hereby claim copyright of the interval known as
> 5/4.

Dante, I get up early and read the digest while my wife sleeps. It was
REALLY TOUGH not to break into guffaws when I read this piece! Luckily
she goes back to sleep easily...

JdL