pairwise entropy minimizer

Back in the day, Paul Erlich was working on finding scales for
which the sum of the harmonic entropy of their dyads was low.
He found that this was a hard problem. He had a tool which
would relax a scale to a dyadic entropy minimum, but which could
not find a global minimum for a given cardinality. He attempted
to seed this program with random scales, hoping to find global
minima by the Monte Carlo method. Last I heard, he believed
that the global minimum for 5-tone scales was the usual meantone
pentatonic. I can not remember if...

(1) There were ever results for other cardinalities.

(2) If there were significant runners-up for the 5-tone case.

...does anyone have information on this?

Paul, do you prefer if this is posted to the harmonic entropy list?

-Carl

Hmmm ... multi-dimensional optimization isn't a particularly difficult
problem, as long as the function to be optimized is reasonably well behaved.
It does require a decent-speed CPU and some attention to software details,
but there are actually packaged codes out there that do a pretty good job of
it. I assume we're talking about chords here -- worst case being, say, five
simultaneous notes??

BTW, Monte Carlo is *not* particularly efficient -- genetic search and
simulated annealing are much better. Let me do some digging; there used to
be a guy who was giving away a first-rate simulated annealing code. If he's
still around and still giving it away, I'll grab a copy.
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: Carl Lumma [mailto:carl@lumma.org]
> Sent: Sunday, June 24, 2001 10:11 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] pairwise entropy minimizer
>
>
> Back in the day, Paul Erlich was working on finding scales for
> which the sum of the harmonic entropy of their dyads was low.
> He found that this was a hard problem. He had a tool which
> would relax a scale to a dyadic entropy minimum, but which could
> not find a global minimum for a given cardinality. He attempted
> to seed this program with random scales, hoping to find global
> minima by the Monte Carlo method. Last I heard, he believed
> that the global minimum for 5-tone scales was the usual meantone
> pentatonic. I can not remember if...
>
> (1) There were ever results for other cardinalities.
>
> (2) If there were significant runners-up for the 5-tone case.
>
> ...does anyone have information on this?
>
> Paul, do you prefer if this is posted to the harmonic entropy list?
>
> -Carl
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

Well ... it didn't take me long to find the Adaptive Simulated Annealing
code. The URL is http://www.ingber.com/. It's basically free, unless you
want him to do the work for you and hire him as a consultant. I downloaded
it an I'm reading through the manual right now. I don't think I need
anything this fancy for my Partch - Xenakis piece, but I might just use it
anyhow just to get used to working with it. It's in C; should run on a Linux
box and on Windows with any of the compilers (Microsoft, Borland, MinGW32).
Given that my Linux box is a boatload faster than the Windows one, I'll
probably just run it on the Linux box.
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

> -----Original Message-----
> From: M. Edward Borasky [mailto:znmeb@aracnet.com]
> Sent: Sunday, June 24, 2001 10:41 PM
> To: tuning-math@yahoogroups.com; harmonic-entropy@yahoogroups.com
> Subject: RE: [tuning-math] pairwise entropy minimizer
>
>
> Hmmm ... multi-dimensional optimization isn't a particularly difficult
> problem, as long as the function to be optimized is reasonably
> well behaved.
> It does require a decent-speed CPU and some attention to software details,
> but there are actually packaged codes out there that do a pretty
> good job of
> it. I assume we're talking about chords here -- worst case being,
> say, five
> simultaneous notes??
>
> BTW, Monte Carlo is *not* particularly efficient -- genetic search and
> simulated annealing are much better. Let me do some digging; there used to
> be a guy who was giving away a first-rate simulated annealing
> code. If he's
> still around and still giving it away, I'll grab a copy.
> --
> M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
> http://www.borasky-research.net http://www.aracnet.com/~znmeb
> mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com
>
> Q: How do you get an elephant out of a theatre?
> A: You can't. It's in their blood.
>
> > -----Original Message-----
> > From: Carl Lumma [mailto:carl@lumma.org]
> > Sent: Sunday, June 24, 2001 10:11 PM
> > To: tuning-math@yahoogroups.com
> > Subject: [tuning-math] pairwise entropy minimizer
> >
> >
> > Back in the day, Paul Erlich was working on finding scales for
> > which the sum of the harmonic entropy of their dyads was low.
> > He found that this was a hard problem. He had a tool which
> > would relax a scale to a dyadic entropy minimum, but which could
> > not find a global minimum for a given cardinality. He attempted
> > to seed this program with random scales, hoping to find global
> > minima by the Monte Carlo method. Last I heard, he believed
> > that the global minimum for 5-tone scales was the usual meantone
> > pentatonic. I can not remember if...
> >
> > (1) There were ever results for other cardinalities.
> >
> > (2) If there were significant runners-up for the 5-tone case.
> >
> > ...does anyone have information on this?
> >
> > Paul, do you prefer if this is posted to the harmonic entropy list?
> >
> > -Carl
> >
> > To unsubscribe from this group, send an email to:
> > tuning-math-unsubscribe@yahoogroups.com
> >
> >
> >
> > Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
> >
> >
> >
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Back in the day, Paul Erlich was working on finding scales for
> which the sum of the harmonic entropy of their dyads was low.
> He found that this was a hard problem. He had a tool which
> would relax a scale to a dyadic entropy minimum, but which could
> not find a global minimum for a given cardinality. He attempted
> to seed this program with random scales, hoping to find global
> minima by the Monte Carlo method. Last I heard, he believed
> that the global minimum for 5-tone scales was the usual meantone
> pentatonic. I can not remember if...
>
> (1) There were ever results for other cardinalities.

Oh yes . . . by the time I got to 12 notes, I was finding that the program was getting "stuck" in
some kind of higher-dimensional "crevices" leading to curious 12-tone well-temperaments which
were not even local minima . . . they could be nudged closer to 12-tET without ever increasing
the total dyadic harmonic entropy at any stage. Monz made a webpage of these
well-temperaments. This was all posted to the tuning list . . . you'll have to dig through the
archives.
>
> (2) If there were significant runners-up for the 5-tone case.

Yes . . . I posted a long list of the number of occurences of, and the rating of, many different
pentatonic scales, obtained by starting the local minimizations from many, many random points.
>
> ...does anyone have information on this?

You'll have to dig through the archives of the tuning list. Search for "relaxed".

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 25, 2001 4:35 AM
> Subject: [tuning-math] Re: pairwise entropy minimizer
>
>
> --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> > Back in the day, Paul Erlich was working on finding scales for
> > which the sum of the harmonic entropy of their dyads was low.
> > ...
> > (1) There were ever results for other cardinalities.
>
> Oh yes . . . by the time I got to 12 notes, I was finding that
> the program was getting "stuck" in some kind of higher-dimensional
> "crevices" leading to curious 12-tone well-temperaments which
> were not even local minima . . . they could be nudged closer
> to 12-tET without ever increasing the total dyadic harmonic
> entropy at any stage. Monz made a webpage of these well-temperaments.
> This was all posted to the tuning list . . . you'll have to dig
> through the archives.

Uh-oh... apparently my webpages must be getting "stuck in some kind
of higher-dimensional crevices"!!! This seems to be another of the
"lost Monzo webpages".

Again, Paul, if you can remember any text from this page I can
probably find it and make a prominent link.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> Hmmm ... multi-dimensional optimization isn't a particularly
> difficult problem, as long as the function to be optimized is
> reasonably well behaved.

IIRC, that's the problem with harmonic entropy.

-Carl

On Mon, 25 Jun 2001 carl@lumma.org wrote:

> --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> > Hmmm ... multi-dimensional optimization isn't a particularly
> > difficult problem, as long as the function to be optimized is
> > reasonably well behaved.
>
> IIRC, that's the problem with harmonic entropy.

Are you saying that harmonic entropy isn't well-behaved? As far as I can tell
from looking at the "dyadic" case, it's very well behaved. It's continuous, and
the minima are very steep -- finding a global minimum is simply a matter of
keeping out the the trap of getting stuck in a local minimum. I haven't looked
at harmonic entropy in more that two notes, mostly because I'm using the
Sethares dissonance function instead of harmonic entropy. At some point in the
not too distant future, I'm going to see what adaptive simulated annealing does
with the Sethares dissonance function for up to 5 simultaneous tones, each
consisting of 11 harmonic partials. This is the test case I've been using.
--
znmeb@aracnet.com (M. Edward Borasky) http://www.aracnet.com/~znmeb

What phrase will you *never* hear Candice Bergen use?
"My daddy didn't raise no dummies!"

--- In tuning-math@y..., carl@l... wrote:
> --- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
> > Hmmm ... multi-dimensional optimization isn't a particularly
> > difficult problem, as long as the function to be optimized is
> > reasonably well behaved.
>
> IIRC, that's the problem with harmonic entropy.
>
Huh?

--- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote:

>At some point in the
> not too distant future, I'm going to see what adaptive simulated
annealing does
> with the Sethares dissonance function for up to 5 simultaneous
tones, each
> consisting of 11 harmonic partials. This is the test case I've been
using.

It would seem that you would want to construct an octave-equivalent
version of the Sethares dissonance function for these sorts of
exercises. Because if you're interested in scales that repeat
themselves every octave, you're really interested in the _interval
classes_ from 0 to 600 cents, with octave inversion and/or extension
not affecting the dissonance value. You can probably construct such a
curve by considering each tone to be a large chord of octave-
equivalent notes, equally "loud" in every octave from the lowest
registers on up to the highest registers. If you then assume 12 or
more harmonic partials for each, you should have no problem obtaining
Partch's 29-per-octave diamond as the set of local minima of dyadic
dissonance (if you tweak the parameters in a suitable way).

On Mon, 25 Jun 2001, Paul Erlich wrote:

> It would seem that you would want to construct an octave-equivalent
> version of the Sethares dissonance function for these sorts of
> exercises. Because if you're interested in scales that repeat
> themselves every octave, you're really interested in the _interval
> classes_ from 0 to 600 cents, with octave inversion and/or extension
> not affecting the dissonance value. You can probably construct such a
> curve by considering each tone to be a large chord of octave-
> equivalent notes, equally "loud" in every octave from the lowest
> registers on up to the highest registers. If you then assume 12 or
> more harmonic partials for each, you should have no problem obtaining
> Partch's 29-per-octave diamond as the set of local minima of dyadic
> dissonance (if you tweak the parameters in a suitable way).

1. I'm using Sethares' formulas as given in the book, specifically the Matlab
code version, since there are errors in the Basic code in the book (but not on
his web site.) Rather than break things up as "intrinsic" dissonance,
"interval" dissonance, "triad" dissonance, etc., I combine the partials in a
list first, and then evaluate the dissonance of the combined "sound".

2. I started with 7 partials with 1/n scaled amplitude and was unable to
reproduce Partch's scale. I replaced this by 11 partials, all with unit
amplitude, and now I have nice minima at many points of the Partch scale. The
curves have been posted to the files area of this list (I think it's this list
:-). When I retain the 11 partials but scale their amplitudes by 1/n, the
whole curve becomes lower -- less dissonant throughout the 1/1 - 2/1 range --
and all those nice minima disappear.

3. I'm interested in the Sethares algorithm essentially as written, with the
built-in adjustments for the spectrum of the tones and the actual physical
frequencies, not in "octave-equivalent" measures or measures based on "small
integer ratios". Since I'm doing algorithmic composition, I have the "luxury"
of fairly complex processing -- I don't need to deliver a note in real time at
the command of some MIDI source. Even so, I think I can code the Sethares
dissonance measure to operate at the *control* rate of a digital synthesizer,
given that the amplitudes and frequencies needed for input are also available
at the control rate. The only thing that *might* be tricky is the exponentials,
but there are some good approximations, because EXP(z) is a very well-behaved
beast. A simple Pade approximation is two multiplies and a divide, for example,
and that might even be overkill.

I'm in the process of re-organizing my Derive code and will post the latest
version sometime this week. At some point I will need to translate all of it
to C, which Derive can do more or less automatically, so I can integrate it
with ASA or whatever optimizer I want to use. I picked ASA because it can
handle difficult problems relatively easily, and of course because it's free.
There are *lots* of optimizers out there, and if ASA can't cut this, I'll
find something that can. But I suspect ASA will be able to handle this, and in
fact is most likely overkill.
--
znmeb@aracnet.com (M. Edward Borasky) http://www.aracnet.com/~znmeb

What phrase will you *never* hear Candice Bergen use?
"My daddy didn't raise no dummies!"

>>> Hmmm ... multi-dimensional optimization isn't a particularly
>>> difficult problem, as long as the function to be optimized is
>>> reasonably well behaved.
>>
>> IIRC, that's the problem with harmonic entropy.
>>
> Huh?

You wrote...

>John's spring model works because the objective
>function (the function being minimized) is
>quadratic in all the input parameters. The full
>harmonic entropy curve is clearly not quadratic;
>hence this wouldn't work. Based on my education
>as a physicist and my profession as a statistician/
>financial engineer, I can tell you that global
>optimization of functions with many local optima is
>a very icult problem to attack rigorously and is
>typically approached with Monte Carlo methods.

-Carl

--- In tuning-math@y..., "M. Edward (Ed) Borasky" <znmeb@a...> wrote:
> On Mon, 25 Jun 2001, Paul Erlich wrote:
>
> > It would seem that you would want to construct an octave-
equivalent
> > version of the Sethares dissonance function for these sorts of
> > exercises. Because if you're interested in scales that repeat
> > themselves every octave, you're really interested in the _interval
> > classes_ from 0 to 600 cents, with octave inversion and/or
extension
> > not affecting the dissonance value. You can probably construct
such a
> > curve by considering each tone to be a large chord of octave-
> > equivalent notes, equally "loud" in every octave from the lowest
> > registers on up to the highest registers. If you then assume 12 or
> > more harmonic partials for each, you should have no problem
obtaining
> > Partch's 29-per-octave diamond as the set of local minima of
dyadic
> > dissonance (if you tweak the parameters in a suitable way).
>
> 1. I'm using Sethares' formulas as given in the book, specifically
the Matlab
> code version, since there are errors in the Basic code in the book
(but not on
> his web site.) Rather than break things up as "intrinsic"
dissonance,
> "interval" dissonance, "triad" dissonance, etc., I combine the
partials in a
> list first, and then evaluate the dissonance of the
combined "sound".

Naturally.

>
> 2. I started with 7 partials with 1/n scaled amplitude and was
unable to
> reproduce Partch's scale. I replaced this by 11 partials, all with
unit
> amplitude, and now I have nice minima at many points of the Partch
scale. The
> curves have been posted to the files area of this list (I think
it's this list
> :-). When I retain the 11 partials but scale their amplitudes by
1/n, the
> whole curve becomes lower -- less dissonant throughout the 1/1 -
2/1 range --
> and all those nice minima disappear.

Exactly the kind of behavior I found with the Sethares stuff.
>
> 3. I'm interested in the Sethares algorithm essentially as written,
with the
> built-in adjustments for the spectrum of the tones and the actual
physical
> frequencies, not in "octave-equivalent" measures

Well, Partch's scale and most other scales are conceived in octave-
equivalent form . . . I'm just pointing out that it would be more
theoretically consistent to use tones doubled and tripled at multiple
octaves for the Sethares calculation when evaluating
Partch's "consonances".

> or measures based on "small
> integer ratios".

Huh? What are you referring to here?

> Since I'm doing algorithmic composition, I have the "luxury"
> of fairly complex processing -- I don't need to deliver a note in
real time at
> the command of some MIDI source. Even so, I think I can code the
Sethares
> dissonance measure to operate at the *control* rate of a digital
synthesizer,
> given that the amplitudes and frequencies needed for input are also
available
> at the control rate.

Why would you need to do that?

Also, keep in mind that Sethares/Plomp dissonance (due to critical
band roughness) is only one component of the sensory dissonance we
perceive (as amply demonstrated by our listening experiments on the
Tuning Lab). This will become very noticeable when you compare otonal
hexads with utonal hexads.

--- In tuning-math@y..., carl@l... wrote:
> >>> Hmmm ... multi-dimensional optimization isn't a particularly
> >>> difficult problem, as long as the function to be optimized is
> >>> reasonably well behaved.
> >>
> >> IIRC, that's the problem with harmonic entropy.
> >>
> > Huh?
>
> You wrote...
>
> >John's spring model works because the objective
> >function (the function being minimized) is
> >quadratic in all the input parameters. The full
> >harmonic entropy curve is clearly not quadratic;
> >hence this wouldn't work. Based on my education
> >as a physicist and my profession as a statistician/
> >financial engineer, I can tell you that global
> >optimization of functions with many local optima is
> >a very icult problem to attack rigorously and is
> >typically approached with Monte Carlo methods.
>
> -Carl

Oh, come on, Carl. Name a dissonance function that does _not_ have
many local optima.

>>>>> Hmmm ... multi-dimensional optimization isn't a particularly
>>>>> difficult problem, as long as the function to be optimized is
>>>>> reasonably well behaved.
>>>>
>>>> IIRC, that's the problem with harmonic entropy.
>>>>
>>> Huh?
>>
>> You wrote...
>>
>>>John's spring model works because the objective
>>>function (the function being minimized) is
>>>quadratic in all the input parameters. The full
>>>harmonic entropy curve is clearly not quadratic;
>>>hence this wouldn't work. Based on my education
>>>as a physicist and my profession as a statistician/
>>>financial engineer, I can tell you that global
>>>optimization of functions with many local optima is
>>>a very icult problem to attack rigorously and is
>>>typically approached with Monte Carlo methods.
>>
>> -Carl
>
> Oh, come on, Carl. Name a dissonance function that does _not_ have
> many local optima.

Sounds like you're reading me to say there was a problem
with harmonic entropy. Maybe I should have said, that's
a property of harmonic entropy which makes the optimization
problem difficult.

Either way, I wouldn't know. I was simply recalling
something you had said in response to my asking why the
Monte Carlo approach was necessary, why something like
John's approach wouldn't work.

Hopefully, this won't obscure the two questions I asked
originally, which were...

(1) Are there results for scales with numbers of tones
other than five?

(2) Are there runners-up for the 5-tone case? Generally,
are there many significantly-different scales close to
the global minima at each cardinality?

I think the fact that meantone pentatonics won is fairly
interesting, but IIRC they didn't win by very much.

In the long run, I'd be interested in finding scales where
the total entropy is low and the entropy of the modes are
nearly the same.

-Carl

--- In tuning-math@y..., carl@l... wrote:
>
> Sounds like you're reading me to say there was a problem
> with harmonic entropy. Maybe I should have said, that's
> a property of harmonic entropy which makes the optimization
> problem difficult.

Again, can you name a dissonance function which would make the
optimization problem easier?
>
> (1) Are there results for scales with numbers of tones
> other than five?

Yes . . . see the archives.
>
> (2) Are there runners-up for the 5-tone case? Generally,
> are there many significantly-different scales close to
> the global minima at each cardinality?

Yes . . . see the archives.
>
> I think the fact that meantone pentatonics won is fairly
> interesting, but IIRC they didn't win by very much.
>
> In the long run, I'd be interested in finding scales where
> the total entropy is low

Meaning total dyadic entropy, or entropy of larger chords?

> and the entropy of the modes are
> nearly the same.

How could they not be the same?

> -----Original Message-----
> From: Paul Erlich [mailto:paul@stretch-music.com]
> Sent: Monday, June 25, 2001 5:32 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: pairwise entropy minimizer

> > Since I'm doing algorithmic composition, I have the "luxury"
> > of fairly complex processing -- I don't need to deliver a note in
> real time at
> > the command of some MIDI source. Even so, I think I can code the
> Sethares
> > dissonance measure to operate at the *control* rate of a digital
> synthesizer,
> > given that the amplitudes and frequencies needed for input are also
> available
> > at the control rate.
>
> Why would you need to do that?
>
> Also, keep in mind that Sethares/Plomp dissonance (due to critical
> band roughness) is only one component of the sensory dissonance we
> perceive (as amply demonstrated by our listening experiments on the
> Tuning Lab). This will become very noticeable when you compare otonal
> hexads with utonal hexads.

The general plan is to do more or less free stochastic music as Xenakis
defines it, except that all the tones will be constrained to the Partch
11-limit JI. What I want to do is define a musical space in terms of the
Partch concepts of Monophony -- 11-limit JI, the 28 tonalities, the
one-footed bride (this is where Sethares comes in -- I'm using the Sethares
algorithm because I understand it), etc. Then the algorithm will explore
that space. If the algorithm produces utonal hexads that sound worse than
otonal hexads, I'll tweak it.
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:

> The general plan is to do more or less free stochastic music as
Xenakis
> defines it, except that all the tones will be constrained to the
Partch
> 11-limit JI. What I want to do is define a musical space in terms
of the
> Partch concepts of Monophony -- 11-limit JI, the 28 tonalities, the
> one-footed bride (this is where Sethares comes in -- I'm using the
Sethares
> algorithm because I understand it), etc. Then the algorithm will
explore
> that space. If the algorithm produces utonal hexads that sound
worse than
> otonal hexads, I'll tweak it.

Couldn't you decide whether utonal hexads sound worse that otonal
hexads _before_ you implement or even decide upon any algorithm?

But actually this may be a moot point for your artistic goals, since
Partch himself put otonal and utonal on an equal footing, so if
you're trying to emulate Partch, you may wish to retain that status.

> -----Original Message-----
> From: Paul Erlich [mailto:paul@stretch-music.com]
> Sent: Monday, June 25, 2001 7:41 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: pairwise entropy minimizer

> Couldn't you decide whether utonal hexads sound worse that otonal
> hexads _before_ you implement or even decide upon any algorithm?
>
> But actually this may be a moot point for your artistic goals, since
> Partch himself put otonal and utonal on an equal footing, so if
> you're trying to emulate Partch, you may wish to retain that status.

Well ... I'm trying to emulate both Partch and Xenakis ... I was in fact
planning to give otonal and utonal equal status. Now that I think about it
though, aren't utonal and otonal hexads identical? There's only one hexad --
1:3:5:7:9:11 -- right???
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:
>
> > -----Original Message-----
> > From: Paul Erlich [mailto:paul@s...]
> > Sent: Monday, June 25, 2001 7:41 PM
> > To: tuning-math@y...
> > Subject: [tuning-math] Re: pairwise entropy minimizer
>
> > Couldn't you decide whether utonal hexads sound worse that otonal
> > hexads _before_ you implement or even decide upon any algorithm?
> >
> > But actually this may be a moot point for your artistic goals, since
> > Partch himself put otonal and utonal on an equal footing, so if
> > you're trying to emulate Partch, you may wish to retain that status.
>
> Well ... I'm trying to emulate both Partch and Xenakis ... I was in fact
> planning to give otonal and utonal equal status.

Well then dyadic consonance measures will be fine.

> Now that I think about it
> though, aren't utonal and otonal hexads identical? There's only one hexad --
> 1:3:5:7:9:11 -- right???

No -- that, along with its many octave-equivalents (such as 7:8:9:10:11:12) is an otonal hexad.
The utonal hexad is 1/11:1/9:1/7:1/5:1/3:1/1, along with its many octave equivalents (such as
1/12:1/11:1/10:1/9:1/8:1/7). Why don't you listen to various voicings of these and tell us what
you think.

> -----Original Message-----
> From: Paul Erlich [mailto:paul@stretch-music.com]
> Sent: Monday, June 25, 2001 9:07 PM
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Re: pairwise entropy minimizer
> > Now that I think about it
> > though, aren't utonal and otonal hexads identical? There's only
> one hexad --
> > 1:3:5:7:9:11 -- right???
>
> No -- that, along with its many octave-equivalents (such as
> 7:8:9:10:11:12) is an otonal hexad.
> The utonal hexad is 1/11:1/9:1/7:1/5:1/3:1/1, along with its many
> octave equivalents (such as
> 1/12:1/11:1/10:1/9:1/8:1/7).
Yeah -- I remembered that while I was downloading my daily financial data
... ties up the line :(

> Why don't you listen to various
> voicings of these and tell us what
> you think.
So, if these have already been "voiced", where might I go listen to them
(when my download finishes, of course :-)??

I'm wondering if I shouldn't give Sethares equal billing with Xenakis and
Partch as "influences" on the piece ... the effect I'm going for is "weird
but pleasant", which is the way most of the microtonal and xentonal music
I've heard strikes me, especially the compositions of Sethares. What *won't*
be in it is any notion of "musique concrete" or "found music", the two
concepts that dominated my previous compositions, wherever they might
reside.

Or should that be "pleasant but weird"?? :-)
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.

>> Sounds like you're reading me to say there was a problem
>> with harmonic entropy. Maybe I should have said, that's
>> a property of harmonic entropy which makes the optimization
>> problem difficult.
>
> Again, can you name a dissonance function which would make the
> optimization problem easier?

No, and I have no desire to; I want harmonic entropy! Maybe
I should have said, "that's a problem of dissonance that makes
the optimization problem difficult." You've completely
misunderstood me here Paul, and it's unfortunate. I am merely
trying to summarize what _you_ once told _me_ for Ed, who was
not a part of the original thread. Now you're playing the part
I was in the original thread? Bizarre.

> > (1) Are there results for scales with numbers of tones
> > other than five?
>
> Yes . . . see the archives.

Obviously, I've already been there. I was unable to find
them.

>> I think the fact that meantone pentatonics won is fairly
>> interesting, but IIRC they didn't win by very much.
>>
>> In the long run, I'd be interested in finding scales where
>> the total entropy is low
>
> Meaning total dyadic entropy, or entropy of larger chords?

Dyadic.

>> and the entropy of the modes are nearly the same.
>
> How could they not be the same?

By modes here, I mean "the set of dyads measured from a given
scale member". In the original thread, I argued that harmonic
series segments would get high marks with total pairwise entropy
(I can't remember if this was a guess, or if this scales really
did show up). But one mode would be lower than the others, in
most cases.

-Carl

--- In tuning-math@y..., carl@l... wrote:
> >> Sounds like you're reading me to say there was a problem
> >> with harmonic entropy. Maybe I should have said, that's
> >> a property of harmonic entropy which makes the optimization
> >> problem difficult.
> >
> > Again, can you name a dissonance function which would make the
> > optimization problem easier?
>
> No, and I have no desire to; I want harmonic entropy! Maybe
> I should have said, "that's a problem of dissonance that makes
> the optimization problem difficult." You've completely
> misunderstood me here Paul, and it's unfortunate. I am merely
> trying to summarize what _you_ once told _me_ for Ed, who was
> not a part of the original thread. Now you're playing the part
> I was in the original thread? Bizarre.

Just wanted to make things clearer for _everyone_.
>
> > > (1) Are there results for scales with numbers of tones
> > > other than five?
> >
> > Yes . . . see the archives.
>
> Obviously, I've already been there. I was unable to find
> them.

Sorry. You'll have to keep looking.

> >> and the entropy of the modes are nearly the same.
> >
> > How could they not be the same?
>
> By modes here, I mean "the set of dyads measured from a given
> scale member".

I didn't do that -- I added all the dyads -- but Robert Valentine's
approach is to do what you've described.

--- In tuning-math@y..., "M. Edward Borasky" <znmeb@a...> wrote:

/tuning-math/message/403

> Or should that be "pleasant but weird"?? :-)

Of course Ed, as I'm sure you're aware, these are ENTIRELY value
judgements. When I hear the works of some so-called "contemporary"
composers which use hackneyed "traditional" 12-tET progressions, the
effect is anything but pleasant.

What you are doing would probably strike me, therefore, as
just "pleasant." :)

______ ________ _______
Joseph Pehrson

> -----Original Message-----
> From: jpehrson@rcn.com [mailto:jpehrson@rcn.com]

> Of course Ed, as I'm sure you're aware, these are ENTIRELY value
> judgements. When I hear the works of some so-called "contemporary"
> composers which use hackneyed "traditional" 12-tET progressions, the
> effect is anything but pleasant.
>
> What you are doing would probably strike me, therefore, as
> just "pleasant." :)

Interestingly enough, over the weekend, I saw a local piano duo called
d.u.o. :-) The female half of the duo is of Greek descent, and spent two
years in Greece collecting Greek classical music of the 19th and 20th
centuries. Some of the music is, as she put it, "Byzantine", though *not*
microtonal -- they had tempered instruments by then. Lots of fifths and
fourths and few major and minor thirds. So ... does anyone on this list know
anything about Greek music from that era? It seems there are plenty of
experts on the older stuff -- I have Xenakis' writings on the subject -- but
would they have really been forced into the equal tempered paradigm, or
would they have stuck with their Byzantine roots somehow? I could have sworn
there were some places where a quarter tone or two would have been
appropriate in the one Greek piece she played.
--
M. Edward (Ed) Borasky, Chief Scientist, Borasky Research
http://www.borasky-research.net http://www.aracnet.com/~znmeb
mailto:znmeb@borasky-research.com mailto:znmeb@aracnet.com

Q: How do you get an elephant out of a theatre?
A: You can't. It's in their blood.