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:

> 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.