Adaptive tempering

Suppose you have identified a certain passage as one that used a
particular appproximation. You might then depart from JI to play that
passage in something using that appromimation. For instance, if a
passage treats 81/80 as a unison, you might play it in a 31-et
tuning, and if another passage uses 64/63 as a unison, you might play
it in the 27-et. If it requires both, you could play it in the 12-et.

What would this sound like? I don't know--having just thought of the
idea, I haven't listened to any examples!

If someone is interested in experimenting with this idea, a good
place to start would be testing how quickly one can move from one
tempering to another without it sounding awkward. I would suggest
doing it by moving along the line from one tuning system to another,
in the space defined by the tunings of the primes.

For instance if n3, n5, n7, and n11 are the tuning in cents of 3,
5,7, and 11 in the 22-et, the line to the 31-et in parametric form is
(n3 - 12.3t, n5 + 5.3t, n7 - 14.1t, n11 - 3.5t), where when t=0 we
have the 22-et and when t=1 we have the 31-et. We could also try
stopping over at just intonation on the way; from the 22-et to JI we
would go (n3 - 7.1t, n5 + 4.5t, n7 - 13t, n11 + 5.9t); except for 11
the JI values are more or less on the way from 22 to 31.

We could also move octaves while adapting. If we use the Gram tuning
for the 22-et, octaves are 1198.64689 cents, and if we call the
values we get for the 11-limit primes m2,m3,m5,m7 and m11, the line
from the 22-et Gram tuning to JI is given by
(m2 + 1.4t, m3 - 5t, m5 + 7.6t, m7 - 9.2t, m11 + 10.5t).

--- In tuning@y..., genewardsmith@j... wrote:
> Suppose you have identified a certain passage as one that used a
> particular appproximation. You might then depart from JI to play
that
> passage in something using that appromimation. For instance, if a
> passage treats 81/80 as a unison, you might play it in a 31-et
> tuning, and if another passage uses 64/63 as a unison, you might
play
> it in the 27-et. If it requires both, you could play it in the 12-
et.
>
> What would this sound like? I don't know--having just thought of
the
> idea, I haven't listened to any examples!

John deLaubenfels' program already does this to a great extent.
All "commas" are tempered out, using a combination of melodic and
harmonic tempering.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> John deLaubenfels' program already does this to a great extent.
> All "commas" are tempered out, using a combination of melodic and
> harmonic tempering.

I thought his idea worked on notes. I was proposing to lift a piece
to JI, and then vary the generators slightly. Does his program do
something like this?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > John deLaubenfels' program already does this to a great extent.
> > All "commas" are tempered out, using a combination of melodic and
> > harmonic tempering.
>
> I thought his idea worked on notes.

Entire pieces.

> I was proposing to lift a piece
> to JI,

Even if it drifts? And what about chords like CEGAD?

> and then vary the generators slightly. Does his program do
> something like this?

No, but Herman Miller has done a wonderful job of this on his Warped
Canon page, for a piece that doesn't drift.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > I was proposing to lift a piece
> > to JI,

> Even if it drifts? And what about chords like CEGAD?

Certainly. You lift the whole thing to JI, and then temper out the
drifts. CEGAD could be a polychord, or you could fit it to some
utones or otones; that's the whole trick, isn't it? One twisted
posibility is to try it out in various ets and see what seems best;
after doing something in 12,22, and 31, you can start lifting
automatically in many cases, and if that doesn't work try another
linearly independent et.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > I was proposing to lift a piece
> > > to JI,
>
> > Even if it drifts? And what about chords like CEGAD?
>
> Certainly. You lift the whole thing to JI, and then temper out the
> drifts. CEGAD could be a polychord,

In JI? Please spell out.

> or you could fit it to some
> utones or otones; that's the whole trick, isn't it?

The whole trick of what? I don't like such chords untempered.

> One twisted
> posibility is to try it out in various ets and see what seems best;
> after doing something in 12,22, and 31, you can start lifting
> automatically in many cases, and if that doesn't work try another
> linearly independent et.

Not following you.

[Gene wrote:]
>>>> I was proposing to lift a piece to JI,

[Paul wrote:]
>>> Even if it drifts? And what about chords like CEGAD?

[Gene:]
>>Certainly. You lift the whole thing to JI, and then temper out the
>>drifts. CEGAD could be a polychord,

[Paul:]
>In JI? Please spell out.

Paul's point is that chains of fifths with four or more notes have
inherent conflicts between fifths and thirds. With such chords, there
can never be a "JI" tuning that satisfies all the intervals we normally
want to designate as consonant (thirds, fifths, and inversions). The
best compromises in such cases (according to what I would consider to be
a reasonable measure) look a lot like meantone: the fifths are squeezed
a bit narrow in order to keep the thirds from being Pythagorean.

JdL

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > drifts. CEGAD could be a polychord,
>
> In JI? Please spell out.

It could be 1-5/4-3/2-5/3-20/9, for instance. Or it could even be
1-5/4-20/13-5/3-10/9.

> > or you could fit it to some
> > utones or otones; that's the whole trick, isn't it?

> The whole trick of what? I don't like such chords untempered.

Untempered as what? That's the whole trick--finding the what.

> > One twisted
> > posibility is to try it out in various ets and see what seems
best;
> > after doing something in 12,22, and 31, you can start lifting
> > automatically in many cases, and if that doesn't work try another
> > linearly independent et.

> Not following you.

It relates to my definition of notations on tuning-math. The 12,22,
and 31 ets taken together are a notation for the 5-limit, so if you
know how you want a piece to sound in all three scales and what the
result in the 5-limit you are done. If we write them as a matrix, we
have

[12 22 31]
[19 35 49]
[28 51 72]

if we invert this, we get

[-21 3 7]
[ -4 4 -1]
[ 11 -4 -2]

So if u = 2^(-21) * 3^3 * 5^7, v = 2^(-4) * 3^4 * 5^(-1), and
w = 2^11 * 3^(-4) * 5^(-2), then if we decide a certain note should
be a steps in 12-et, b steps in 22 et, and c steps in 31-et, in the
5-limit the note is u^a * v^b * w^c; we have lifted it automatically.
If we need to continue on to the 7-limit, we could then decide what
the piece should be in the 55-et; if we invert

[12 22 31 55]
[19 35 49 87]
[28 51 72 128]
[34 62 87 154]

we get

[-11 -1 3 2]
[-4 4 -1 0]
[16 -6 -4 1]
[-5 2 2 -1]

Calculating the commas corresponding to the rows as before, we get
that if u,v,w,x are the number of steps for a note in the 12,22,31
and 55 et respectively, then the 7-limit JI value is

(6125/6144)^u * (81/80)^v * (458752/455625)^w * (225/224)^x

It would be interesting to test this and see how practical it was as
a way of lifting 12-et music to JI, the neat feature is that a
musician would need no mathematics to accomplish such a lifting. The
12, 31 and 55 ets are all mean-tone, but the 22-et sets 81/80 to a
quarter-tone, and will force some kind of decision about that
notorious comma pump.

What's good music to play when feeling depressed, I wonder?

--- In tuning@y..., genewardsmith@j... wrote:

> Calculating the commas corresponding to the rows as before, we get
> that if u,v,w,x are the number of steps for a note in the 12,22,31
> and 55 et respectively, then the 7-limit JI value is
>
> (6125/6144)^u * (81/80)^v * (458752/455625)^w * (225/224)^x

This also could help make adaptive tempering work. If in three of the
systems things seem to be working well, but in one we have a problem,
then we could temper that system out of the mix. For instance if 22
is giving us fits with a surly comma drift, we may decide to send
81/80 to 1; now we want

A^u * B^w * C^x

to approximate 2, 3, 5, and 7; one way to do this would be to solve
for a weighted least-squares solution in the logs of 2,3,5,7 and A,
B, C, with weights perhaps being 1/ln(2)^2, 1/ln(3)^2, 1/ln(5)^2 and
1/ln(7)^2. We could then traverse a line from the generators above to
A, 1, B, C and we will have adaptively tempered away the problem.
Again, it would be interesting to see how well this works in practice.

--- In tuning@y..., genewardsmith@j... wrote:
<snip>
> What's good music to play when feeling depressed, I wonder?

Bach, Beethoven or Scarlatti work for me.

John Starrett

--- In tuning@y..., "John Starrett" <jstarret@c...> wrote:

> > What's good music to play when feeling depressed, I wonder?
>
> Bach, Beethoven or Scarlatti work for me.

Sounds too cheery. I formerly went for things like the Vaughan
Williams 4th when in a certain mood, but today I'm finding the 15et
is the thing.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > drifts. CEGAD could be a polychord,
> >
> > In JI? Please spell out.
>
> It could be 1-5/4-3/2-5/3-20/9, for instance.

I dislike that sound. The 40:27 fifth hurts.

> Or it could even be
> 1-5/4-20/13-5/3-10/9.

Pretty out there!
>
> > > or you could fit it to some
> > > utones or otones; that's the whole trick, isn't it?
>
> > The whole trick of what? I don't like such chords untempered.
>
> Untempered as what? That's the whole trick--finding the what.

I think that in chords like CEGAD, the fifths want to be 3:2, the
fourths want to be 4:3, the major sixth 5:3, the major third 5:4, and
the minor third 6:5. The best solution is some compromise so that all
these goals can be met as closely as possible.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> I think that in chords like CEGAD, the fifths want to be 3:2, the
> fourths want to be 4:3, the major sixth 5:3, the major third 5:4,
and
> the minor third 6:5. The best solution is some compromise so that
all
> these goals can be met as closely as possible.

Your requirements entail that two tones give a major third, so
your "some compromise" is called a meantone system. Here's a question
for you--if you had to pick the best way of playing this chord in a
transcription to the 22-et, what would you pick?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I think that in chords like CEGAD, the fifths want to be 3:2, the
> > fourths want to be 4:3, the major sixth 5:3, the major third 5:4,
> and
> > the minor third 6:5. The best solution is some compromise so that
> all
> > these goals can be met as closely as possible.
>
> Your requirements entail that two tones give a major third, so
> your "some compromise" is called a meantone system.

Exactly.

> Here's a question
> for you--if you had to pick the best way of playing this chord in a
> transcription to the 22-et, what would you pick?

I wouldn't. 22-tET is totally incompatible with common-pratice
(diatonic 5-limit) music. Only meantone-like temperaments work for
conventional musical notation and thinking (as you well know).

In a message dated 2004:01:04 11:39:35 PM, gwsmith@svpal.org writes:

>Graham Breed <graham@m...> wrote:
>> I've played with this in Kyma, and it's good for
>> experimenting, so you can find your preferred generator size by ear.
>
>Hey, someone who has actually used adaptive tempering!

Hey, I am _limited_ to "adaptive" tunings.
My 2 metre long futon monochord is more accurate than my damn antique Mac
(glorified PDA, desktop-sized PowerMac OS 9.0, ex-Berkeley System's LAN
terminal) and freeware/shareware from as "recent" and "fair&good" as 1990 it runs!
... so go figger how phuqed that be...)
And my temper oh does flare upon witnessin' still yet another systems
crash... my coolie amused BBC (Sino-British) temperament is usually less prone
to such vuglar displays of self-central _sturm-und-drang angst_
snap-crack-pop-mindedness.
Not that I am complaining (much)...
bein' the new music mutant - that I am -
coat of arms xenotonal Militant &
tone&noise-colour&cluster Rampant,
I actually like being able to face such
_p(r)etty_ materialistic SOCIO-ECONO-TECHNO
obstacles head-on
and just freakin' _adapt_ & _mutate_
By Any Means Necessary, Possible and Imaginable

+<imperative : adapt/mutate: continuo * I:we>
OR
- <state : non-function: total * I:we>

^--- translated "binary, Borg-like" paraphrase of Albert Camus & Le
Legion aVantGarde Existentialiste

(BTW I use/abuse a Schoenhut "Baby Bleu Grand" Toy Piano as a "sketchpad" for
somethings diatonic ... or, *googolplexgigglabyte!*, diabolique... a la
Pram's _sono-cinema noir_
Seig Heil Verfremdungseffektkamikazeblitzkriegmelodie! ;)

---|-----|--------|-------------|---------------------|
Hanuman Zhang, musical mad scientist/linguafracasmangalanger

The German word for "noise" _Geräusch_ is derived from _rauschen_ "the
sound of the wind," related to _Rausch_ "ecstasy, intoxication" hinting at some
of the possible aesthetic, bodily effects of noise in music. In Japanese
Romaji: _uchu_ = "universe"... _uchoten_ = "ecstasty," "rapture"..._uchujin_ =
[space] alien!

"For twenty-five centuries, Western knowledge has tried to look upon the
world. It has failed to understand that the world is not for the beholding. It
is for the hearing. It is not legible, but audible. ... Music is a herald,
for change is inscribed in noise faster than it transforms society. ...
Listening to music is listening to all noise, realizing that its appropriation and
control is a reflection of power, that is essentially political." - Jacques
Attali, _Noise: The Political Economy of Music_