Re: The grooviest linear temperaments for 7-limit music

I haven't read any of the messages about this in tuning-math. I'm purely responding to Paul's summary and subsequent responses by Paul and Gene on the tuning list.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <D.KEENAN@U...> wrote:
> > Thanks for this summary Paul, but ...
>
> You mean you haven't been on tuning-math@y... ? Get thee
> hence :)
>
> > > He proposed a 'badness' measure defined as
> > >
> > > step^3 cent
> > >
> > > where step is a measure of the typical number of notes in a
scale
> > for
> > > this temperament (given any desired degree of harmonic depth),
> >
> > What the heck does that mean?
>
> step is the RMS of the numbers of generators required to get to each
> ratio of the tonality diamond from the 1/1, I think.

This is good. More comprehensive than what Graham and I were using.

> > How does he justify cubing it?
>
--- In tuning@y..., "ideaofgod" <genewardsmith@j...> wrote:
> An order of growth estimate shows there should be an infinite list
> for step^2, but not neccesarily for anything higher, and looking far
> out makes it clear step^3 gives a finite list. What this means, of
> course, is that in some sense step^2 is the right way to measure
> goodness.

Yes! Only squared, not cubed.

> Step^3 weighs the small systems more heavily, and that is
> why we see so many of them to start with.

I believe the way to fix this is not to go to step^3 (I don't think there's any human-perception-or-cognition-based justification for doing that), but instead to correct the raw cents to some kind of dissonance or justness measure (more on this below).

> > > and
> > > cent is a measure of the deviation from JI 'consonances' in
cents.
> >
> > Yes but which measure of deviation? minimum maximum absolute or
> > minimum root mean squared or something else?
>
> RMS

Fine.

> > How does he justify not applying a human sensory correction to
this?
>
> A human sensory correction?

Yes. Once the deviation goes past about 20 cents it's irrelevant how big it is, and a 0.1 cent deviation does not sound 10 times better than a 1.0 cent deviation, it sounds about the same. I suggest this figure-of-demerit.

step^2 * exp((cents/k)^2), where k is somewhere between 5 and 15 cents

I think this will give a ranking of temperaments that corresponds more to how composers or performers would rank them.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com
-- A country which has dangled the sword of nuclear holocaust over the world for half a century and claims that someone else invented terrorism is a country out of touch with reality. --John K. Stoner

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> > An order of growth estimate shows there should be an infinite
list
> > for step^2, but not neccesarily for anything higher, and looking
far
> > out makes it clear step^3 gives a finite list. What this means,
of
> > course, is that in some sense step^2 is the right way to measure
> > goodness.
>
> Yes! Only squared, not cubed.
>
> > Step^3 weighs the small systems more heavily, and that is
> > why we see so many of them to start with.
>
> I believe the way to fix this is not to go to step^3 (I don't think
there's any human-perception-or-cognition-based justification for
doing that),

What human-perception-or-cognition-based justification is there for
using step^2 ???

> Yes. Once the deviation goes past about 20 cents it's irrelevant >
how big it is,

That's not true -- you're ignoring both adaptive tuning and adaptive
timbring.

>and a 0.1 cent deviation does not sound 10 times better than a 1.0
>cent deviation, it sounds about the same.

In my own musical endeavors, this is true, but with all the strict-JI
obsessed people out there, a 0.1 cent deviation may end up being 10
times more interesting than a 1.0 cent deviation.

> I suggest this figure-of->demerit.
>
> step^2 [...]

Again, what on earth does step^2 tell you about how composers and
performers would rate a temperament? OK, step^2 is the number of
possible dyads in the typical scale. Step^3 is the number of possible
triads. Why is the former so much more "human-perception-or-cognition-
based" to you than the latter?

As for the other part, the dissonance measure . . . by doing it
Gene's way, we're going to end up with all the most interesting
temperaments for a wide variety of different ranges, from "you'll
never hear a beat" to "wafso-just" to "quasi-just" to "tempered"
to "needing adaptive tuning/timbring". Thus our top 30 or whatever
will have much of interest to all different schools of microtonal
composers.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > Yes. Once the deviation goes past about 20 cents it's irrelevant >
> how big it is,
>
> That's not true -- you're ignoring both adaptive tuning and adaptive
> timbring.

You can adaptively tune or timbre just about anything, so it seems
like we _should_ ignore it.

> >and a 0.1 cent deviation does not sound 10 times better than a 1.0
> >cent deviation, it sounds about the same.
>
> In my own musical endeavors, this is true, but with all the
strict-JI
> obsessed people out there, a 0.1 cent deviation may end up being 10
> times more interesting than a 1.0 cent deviation.

A strict JI obsessed person will not be the slightest bit interested
in linear temperaments, or at least that has been my experience. If
they are at all interested then think they will be quite happy to have
a 1c error rather than a 0.1c one if it lets them halve (actually
divide by 10^(1/3)) the number of notes in the scale. Given that 1c is
way below the typical accuracy of non-electronic instruments.

> > I suggest this figure-of->demerit.
> >
> > step^2 [...]
>
> Again, what on earth does step^2 tell you about how composers and
> performers would rate a temperament? OK, step^2 is the number of
> possible dyads in the typical scale. Step^3 is the number of
possible
> triads. Why is the former so much more
"human-perception-or-cognition-
> based" to you than the latter?

Ok. Maybe I don't have good argument for that. Try

step^3 * exp((cents/k)^2)

> As for the other part, the dissonance measure . . . by doing it
> Gene's way, we're going to end up with all the most interesting
> temperaments for a wide variety of different ranges, from "you'll
> never hear a beat" to "wafso-just" to "quasi-just" to "tempered"
> to "needing adaptive tuning/timbring". Thus our top 30 or whatever
> will have much of interest to all different schools of microtonal
> composers.

I think it has some extreme cases that are of interest to no one. This
can be fixed.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > > Yes. Once the deviation goes past about 20 cents it's
irrelevant >
> > how big it is,
> >
> > That's not true -- you're ignoring both adaptive tuning and
adaptive
> > timbring.
>
> You can adaptively tune or timbre just about anything,

Not true -- in adaptive tuning, you don't want the horizontal shifts
to be too big, or you lose the melodic coherence of the scale; and in
adaptive timbring, you don't want the partials to deviate too far
from a harmonic series, or you'll lose the sense that each note has a
definite pitch.

> A strict JI obsessed person will not be the slightest bit
interested
> in linear temperaments, or at least that has been my experience. If
> they are at all interested then think they will be quite happy to
have
> a 1c error rather than a 0.1c one if it lets them halve (actually
> divide by 10^(1/3)) the number of notes in the scale.

You don't know that for sure. But look, I myself was trying to get
Gene to adopt some exponential, rather than polynomial, function of
the number of notes in the scale. He resisted . . .

> Given that 1c is
> way below the typical accuracy of non-electronic instruments.

Hey, it won't be the first time a feature of tuning that is highly
removed from most musicians' possible realm of experience has gotten
published!

>
> > > I suggest this figure-of->demerit.
> > >
> > > step^2 [...]
> >
> > Again, what on earth does step^2 tell you about how composers and
> > performers would rate a temperament? OK, step^2 is the number of
> > possible dyads in the typical scale. Step^3 is the number of
> possible
> > triads. Why is the former so much more
> "human-perception-or-cognition-
> > based" to you than the latter?
>
> Ok. Maybe I don't have good argument for that. Try
>
> step^3 * exp((cents/k)^2)

That's the _last_ conclusion I wanted you to reach!

> I think it has some extreme cases that are of interest to no one.
This
> can be fixed.

I tried to argue this point to Gene, but he seems to really like
Ennealimmal. Hey, if we're getting mathematical elegance with this
criterion, and all our favorite systems are showing up (I'm still
waiting for double-diatonic ~26), shouldn't we be willing to pay the
price of letting the guy who's doing all the work get his favorite
system in too?

Personally I'd feel much better if everyone could somehow agree what
was the overall most sensible measure regardless of the results!

In Gene's case, I would hope that it would be some elegant internal
consistency that ties the whole deal together. I'd personally settle
for that even if the results were a tad exotic.

Of course it might help if I understood it all a bit better too! I
feel like I'm getting there though, I just wish Gene were a little bit
more generous with the narrative--either that or someone else besides
him were saying the same things slightly differently... that helps me
sometimes too.

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Wednesday, December 05, 2001 5:33 PM
Subject: [tuning-math] Re: The grooviest linear temperaments for
7-limit music

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > > Yes. Once the deviation goes past about 20 cents it's
> irrelevant >
> > > how big it is,
> > >
> > > That's not true -- you're ignoring both adaptive tuning and
> adaptive
> > > timbring.
> >
> > You can adaptively tune or timbre just about anything,
>
> Not true -- in adaptive tuning, you don't want the horizontal shifts
> to be too big, or you lose the melodic coherence of the scale; and
in
> adaptive timbring, you don't want the partials to deviate too far
> from a harmonic series, or you'll lose the sense that each note has
a
> definite pitch.
>
> > A strict JI obsessed person will not be the slightest bit
> interested
> > in linear temperaments, or at least that has been my experience.
If
> > they are at all interested then think they will be quite happy to
> have
> > a 1c error rather than a 0.1c one if it lets them halve (actually
> > divide by 10^(1/3)) the number of notes in the scale.
>
> You don't know that for sure. But look, I myself was trying to get
> Gene to adopt some exponential, rather than polynomial, function of
> the number of notes in the scale. He resisted . . .
>
> > Given that 1c is
> > way below the typical accuracy of non-electronic instruments.
>
> Hey, it won't be the first time a feature of tuning that is highly
> removed from most musicians' possible realm of experience has gotten
> published!
>
> >
> > > > I suggest this figure-of->demerit.
> > > >
> > > > step^2 [...]
> > >
> > > Again, what on earth does step^2 tell you about how composers
and
> > > performers would rate a temperament? OK, step^2 is the number of
> > > possible dyads in the typical scale. Step^3 is the number of
> > possible
> > > triads. Why is the former so much more
> > "human-perception-or-cognition-
> > > based" to you than the latter?
> >
> > Ok. Maybe I don't have good argument for that. Try
> >
> > step^3 * exp((cents/k)^2)
>
> That's the _last_ conclusion I wanted you to reach!
>
> > I think it has some extreme cases that are of interest to no one.
> This
> > can be fixed.
>
> I tried to argue this point to Gene, but he seems to really like
> Ennealimmal. Hey, if we're getting mathematical elegance with this
> criterion, and all our favorite systems are showing up (I'm still
> waiting for double-diatonic ~26), shouldn't we be willing to pay the
> price of letting the guy who's doing all the work get his favorite
> system in too?
>
>
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--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Personally I'd feel much better if everyone could somehow agree what
> was the overall most sensible measure regardless of the results!

Fat chance :)

> In Gene's case, I would hope that it would be some elegant internal
> consistency that ties the whole deal together. I'd personally settle
> for that even if the results were a tad exotic.

I feel the same way.

> Of course it might help if I understood it all a bit better too! I
> feel like I'm getting there though, I just wish Gene were a little
bit
> more generous with the narrative--either that or someone else
besides
> him were saying the same things slightly differently... that helps
me
> sometimes too.

I think he's the only one who understands abstract algebra around
here, so in a lot of cases, that isn't really possible,
unfortunately . . . of course, I should study up on it, but I should
also make more music, and get more sleep, and . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> You don't know that for sure. But look, I myself was trying to get
> Gene to adopt some exponential, rather than polynomial, function of
> the number of notes in the scale. He resisted . . .

You wanted to have exponential growth for the "step" factor, and Dave
for the "cents" factor, which have opposite tendencies; Dave seems to
want to filter the very things out on the low end that you wanted
included.

If we added an exponential growth to "cents", I would suggest
trying k sinh (cents/k) for various k.

> > Given that 1c is
> > way below the typical accuracy of non-electronic instruments.
>
> Hey, it won't be the first time a feature of tuning that is highly
> removed from most musicians' possible realm of experience has
gotten
> published!

It seems to me it is quite relevant to the strict JI school of
thought. I got roasted for mentioning Partch in such a connection,
but it's hard to see what theoretical objection he could raise to 45
notes of ennealimmal in the 7-limit.

> I tried to argue this point to Gene, but he seems to really like
> Ennealimmal. Hey, if we're getting mathematical elegance with this
> criterion, and all our favorite systems are showing up (I'm still
> waiting for double-diatonic ~26), shouldn't we be willing to pay
the
> price of letting the guy who's doing all the work get his favorite
> system in too?

I think the only way you will get rid of Ennealimmal is to have an
upper-end cut-off, and you said you wanted none. Sorry, you are stuck
with it, and it has nothing to do with my liking it really. I've
never even tried it!

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> In Gene's case, I would hope that it would be some elegant internal
> consistency that ties the whole deal together. I'd personally settle
> for that even if the results were a tad exotic.

Elegant internal consistency suggests to me steps^2 cents as a
measure, but that would need an upper cut-off. We do it for ets,
however, so I don't see that as a bif deal myself.

> Of course it might help if I understood it all a bit better too! I
> feel like I'm getting there though, I just wish Gene were a little
bit
> more generous with the narrative--either that or someone else
besides
> him were saying the same things slightly differently... that helps
me
> sometimes too.

I'm hoping Paul will absorb it all and start coming out with his own
interpretations, but I can't get him to compute a wedge product. :)

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> You wanted to have exponential growth for the "step" factor, and
Dave
> for the "cents" factor,

I think you misunderstood Dave -- he wanted the *goodness* for the
cents factor to be a Gaussian.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
>
> > In Gene's case, I would hope that it would be some elegant
internal
> > consistency that ties the whole deal together. I'd personally
settle
> > for that even if the results were a tad exotic.
>
> Elegant internal consistency suggests to me steps^2 cents as a
> measure, but that would need an upper cut-off. We do it for ets,
> however, so I don't see that as a bif deal myself.

Who's we?
>
> > Of course it might help if I understood it all a bit better too! I
> > feel like I'm getting there though, I just wish Gene were a
little
> bit
> > more generous with the narrative--either that or someone else
> besides
> > him were saying the same things slightly differently... that
helps
> me
> > sometimes too.
>
> I'm hoping Paul will absorb it all and start coming out with his
own
> interpretations, but I can't get him to compute a wedge product. :)

I'll take a look at it again when I get a chance.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> I think you misunderstood Dave -- he wanted the *goodness* for the
> cents factor to be a Gaussian.

I don't think penalizing a system for being good can possibly be
defended, so I'm at a loss here.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Ok. Maybe I don't have good argument for that. Try
>
> step^3 * exp((cents/k)^2)

This looks like hyper-exponential growth penalizing badness, not
goodness.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > I think you misunderstood Dave -- he wanted the *goodness* for the
> > cents factor to be a Gaussian.
>
> I don't think penalizing a system for being good can possibly be
> defended, so I'm at a loss here.

I'm not sure who is confused about what.

gaussian(x) = exp(-(x/k)^2)
goodness = gaussian(cents_error)
badness = 1/goodness
= 1/exp(-(cents_error/k)^2)
= exp((cents_error/k)^2)

sinh might be fine too. I'm not familiar.

The problems, as I see them, are
(a) some temperaments that require ridiculously numbers of notes are
near the top of the list only because they have errors of a fraction of
a cent, but once it's less than about a cent, this should not be enough
to redeeem them. And
(b) some others with ridiculously large errors are near the top of the
list only because they come out needing few notes.

I think that the first can be fixed by applying a function to the cents
error that treats all very small errors as being equal, and the latter
might be fixed by dropping back from steps^3 to steps^2.

-- Dave Keenan

Paul wrote:

> As for the other part, the dissonance measure . . . by doing it
> Gene's way, we're going to end up with all the most interesting
> temperaments for a wide variety of different ranges, from "you'll
> never hear a beat" to "wafso-just" to "quasi-just" to "tempered"
> to "needing adaptive tuning/timbring". Thus our top 30 or whatever
> will have much of interest to all different schools of microtonal
> composers.

Oh, if you think one list can please everybody. I'd rather ask people
what they want, and produce a short list that's likely to have their ideal
temperament on it. That's why I keep up the .key and .micro files. Most
importantly, why I release all the source code for a Free platform so that
anybody can try out their own ideas. Nothing Gene's done so far couldn't
have been done by modifying that code.

Graham

Dave Keenan wrote:

> (b) some others with ridiculously large errors are near the top of the
> list only because they come out needing few notes.
>
> I think that the first can be fixed by applying a function to the cents
> error that treats all very small errors as being equal, and the latter
> might be fixed by dropping back from steps^3 to steps^2.

No, you get ridiculously large errors near the top with steps^2 as well.

Graham

Dan Stearns:
> > Of course it might help if I understood it all a bit better too! I
> > feel like I'm getting there though, I just wish Gene were a little
> bit
> > more generous with the narrative--either that or someone else
> besides
> > him were saying the same things slightly differently... that helps
> me
> > sometimes too.

Paul Erlich:
> I think he's the only one who understands abstract algebra around
> here, so in a lot of cases, that isn't really possible,
> unfortunately . . . of course, I should study up on it, but I should
> also make more music, and get more sleep, and . . .

Most of the results Gene's getting don't require anything I don't
understand. So I said all these things differently a few months ago. If
you want to catch up, try getting the source code from
<http://x31eq.com/temper.html> and an interpreter and try
puzzling it out. I haven't had any feedback at all on readability, so I
don't know easy it'll be for a newbie.

The method shouldn't be difficult for Dan to understand. You generate a
linear temperament from two equal temperaments. That's exactly like
finding an MOS on the scale tree, except you have to do it for all
consonant intervals instead of only the octave.

The wedge products are more difficult, but I don't see them as being at
all important in this context. Working with unison vectors is more
trouble. I've got code for that at
<http://x31eq.com/vectors.html>. Going from temperaments to
unison vectors is an outstanding problem that Gene might have solved, but
I haven't seen any source code yet.

Graham

--- In tuning-math@y..., graham@m... wrote:

> The wedge products are more difficult, but I don't see them as
being at
> all important in this context. Working with unison vectors is more
> trouble.

If working with unison vectors is more trouble, why not wedge
products? The wedgie is good for the following reasons:

(1) It is easy to compute, given a either pair of ets, a pair of
unison vectors, or a generator map.

(2) It uniquely defines the temperament, so that temperaments
obtained by any method can be merged into one list.

(3) It automatically eliminates torsion problems.

(4) Given the wedgie, it is easy to compute assoicated ets, a
generating pair of unison vectors, or a generator map. Hence it is
easy to go from any one of these to any other.

(5) By adding or subtracting wedgies we can produce new temperaments.

Given all of that, I think you are missing a bet by dismissing them;
they could easily be incorporated into your code.

I've got code for that at
> <http://x31eq.com/vectors.html>. Going from
temperaments to
> unison vectors is an outstanding problem that Gene might have
solved, but
> I haven't seen any source code yet.

I don't know what good Maple code will do, but here it is:

findcoms := proc(l)
local p,q,r,p1,q1,r1,s,u,v,w;
s := igcd(l[1], l[2], l[6]);
u := [l[6]/s, -l[2]/s, l[1]/s,0];
v := [p,q,r,1];
w := w7l(u,v);
s := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l[6]-w
[6]});
s := subs(_N1=0,s);
p1 := subs(s,p);
q1 := subs(s,q);
r1 := subs(s,r);
v := 2^p1 * 3^q1 * 5^r1 * 7;
if v < 1 then v := 1/v fi;
w := 2^u[1] * 3^u[2] * 5^u[3];
if w < 1 then w := 1/w fi;
[w, v] end:

coms := proc(l)
local v;
v := findcoms(l);
com7(v[1],v[2]) end:

"w7l" takes two vectors representing intervals, and computes the
wegdge product. "isolve" gives integer solutions to a linear
equation; I get an undeterminded varable "_N1" in this way which I
can set equal to any integer, so I set it to 0. The pair of unisons
returned in this way can be LLL reduced by the "com7" function, which
takes a pair of intervals and LLL reduces them.

--- In tuning-math@y..., graham@m... wrote:
> Dan Stearns:
> > > Of course it might help if I understood it all a bit better
too! I
> > > feel like I'm getting there though, I just wish Gene were a
little
> > bit
> > > more generous with the narrative--either that or someone else
> > besides
> > > him were saying the same things slightly differently... that
helps
> > me
> > > sometimes too.
>
> Paul Erlich:
> > I think he's the only one who understands abstract algebra around
> > here, so in a lot of cases, that isn't really possible,
> > unfortunately . . . of course, I should study up on it, but I
should
> > also make more music, and get more sleep, and . . .
>
> Most of the results Gene's getting don't require anything I don't
> understand. So I said all these things differently a few months
ago. If
> you want to catch up, try getting the source code from
> <http://x31eq.com/temper.html> and an interpreter and
try
> puzzling it out. I haven't had any feedback at all on readability,
so I
> don't know easy it'll be for a newbie.
>
> The method shouldn't be difficult for Dan to understand. You
generate a
> linear temperament from two equal temperaments.

I _really hope_ that that's not what all or even most of Gene's
narrative has been about!!

> That's exactly like
> finding an MOS on the scale tree, except you have to do it for all
> consonant intervals instead of only the octave.

This I don't see at all. Don't you mean "all fractions 1/N of an
octave" rather than "all consonant intervals"?

> The wedge products are more difficult, but I don't see them as
being at
> all important in this context.

Well then, when Dan asks about what is going on here, and you come
back saying you already understood it all a few months ago, you're
actually making a very selective reply to Dan's question, aren't you?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > The wedge products are more difficult, but I don't see them as
> being at
> > all important in this context. Working with unison vectors is
more
> > trouble.
>
> If working with unison vectors is more trouble, why not wedge
> products? The wedgie is good for the following reasons:
>
> (1) It is easy to compute, given a either pair of ets, a pair of
> unison vectors, or a generator map.
>
> (2) It uniquely defines the temperament, so that temperaments
> obtained by any method can be merged into one list.
>
> (3) It automatically eliminates torsion problems.
>
> (4) Given the wedgie, it is easy to compute assoicated ets, a
> generating pair of unison vectors, or a generator map. Hence it is
> easy to go from any one of these to any other.
>
> (5) By adding or subtracting wedgies we can produce new
temperaments.
>
> Given all of that, I think you are missing a bet by dismissing
them;
> they could easily be incorporated into your code.
>
> I've got code for that at
> > <http://x31eq.com/vectors.html>. Going from
> temperaments to
> > unison vectors is an outstanding problem that Gene might have
> solved, but
> > I haven't seen any source code yet.
>
> I don't know what good Maple code will do, but here it is:
>
> findcoms := proc(l)
> local p,q,r,p1,q1,r1,s,u,v,w;
> s := igcd(l[1], l[2], l[6]);
> u := [l[6]/s, -l[2]/s, l[1]/s,0];
> v := [p,q,r,1];
> w := w7l(u,v);
> s := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l[6]-
w
> [6]});
> s := subs(_N1=0,s);
> p1 := subs(s,p);
> q1 := subs(s,q);
> r1 := subs(s,r);
> v := 2^p1 * 3^q1 * 5^r1 * 7;
> if v < 1 then v := 1/v fi;
> w := 2^u[1] * 3^u[2] * 5^u[3];
> if w < 1 then w := 1/w fi;
> [w, v] end:
>
> coms := proc(l)
> local v;
> v := findcoms(l);
> com7(v[1],v[2]) end:
>
> "w7l" takes two vectors representing intervals, and computes the
> wegdge product. "isolve" gives integer solutions to a linear
> equation; I get an undeterminded varable "_N1" in this way which I
> can set equal to any integer, so I set it to 0.

The solutions represent?

> The pair of unisons
> returned in this way can be LLL reduced by the "com7" function,
which
> takes a pair of intervals and LLL reduces them.

Why not TM-reduce them?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
>
> > Personally I'd feel much better if everyone could somehow agree
what
> > was the overall most sensible measure regardless of the results!
>
> Fat chance :)
>
> > In Gene's case, I would hope that it would be some elegant
internal
> > consistency that ties the whole deal together. I'd personally
settle
> > for that even if the results were a tad exotic.
>
> I feel the same way.

It's nice to have pretty looking (i.e. simple) fomulae but we can
hardly ignore the fact that we're trying to come up with a list of
linear temperaments that will be of interest to the largest possible
number of human beings. Unfortunately human perception and cognition
is messy to model mathematically, not well established experimentally
and highly variable between individuals. But I'm sure we can come up
with something that is both reasonably elegant mathematically and that
we (in this forum) can all agree isn't too bad. We certainly do it
without trying some out and looking at the results!

We should probably hone the badness metric using 5-limit, where the
most experience exists.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> But I'm sure we can come up
> with something that is both reasonably elegant mathematically and
that
> we (in this forum) can all agree isn't too bad.

I felt that way about steps^3 cents, except where was 12+14?

> We certainly do it
> without trying some out and looking at the results!

You mean a priori? The more arbitrary parameters we put into it, the
more we'll have to rely on particular assumption on how someone is
going to be making music, and this assumtion will be violated for the
next person. The top 25 or 40 according to a very generalized
criterion will best serve to present the _pattern_ of this whole
endeavor, upon which any musician can base their _own_ evaluation,
and if they don't want to, at least pick off one or two temperaments
that interest them.

But I have a nagging suspicion that there are even more "slippery"
ones out there, especially on the ultra-simple end of things . . .

I suspect we can use step^2 cents and cut it off at some point where
there's a long gap in the step-cent plane. For example, the next
point out after Ennealimmal is probably a long way out, so we can
probably put a cutoff there. As for simple temperaments with large
errors, I suspect there are more than Gene and Graham have found so
far that would end up looking good on this criterion, so it may end
up making sense to place another cutoff there . . . but I want to be
sure we've caught all the slippery fish before we decide that.

I would still like to see the "step" thing weighted -- there should
be something very mathematically and acoustically elegant about doing
it that way (if defined correctly) since we are using the Tenney
lattice after all!
>
> We should probably hone the badness metric using 5-limit, where the
> most experience exists.

Yes, I was just going to say we should write the whole paper first in
the 5-limit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > But I'm sure we can come up
> > with something that is both reasonably elegant mathematically and
> that
> > we (in this forum) can all agree isn't too bad.
>
> I felt that way about steps^3 cents, except where was 12+14?
>
> > We certainly do it
> > without trying some out and looking at the results!

Oops! That should have been

We certainly _can't_ do it without trying some out and looking at the
results!

> You mean a priori? The more arbitrary parameters we put into it, the
> more we'll have to rely on particular assumption on how someone is
> going to be making music, and this assumtion will be violated for
the
> next person.

"Not putting in" an arbitrary parameter is usually equivalent to
putting it in but giving it an even more arbitrary value like 0 or 1.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > You mean a priori? The more arbitrary parameters we put into it,
the
> > more we'll have to rely on particular assumption on how someone
is
> > going to be making music, and this assumtion will be violated for
> the
> > next person.
>
> "Not putting in" an arbitrary parameter is usually equivalent to
> putting it in but giving it an even more arbitrary value like 0 or
1.

Well, I think Gene is saying that step^2 cents is clearly the right
measure of "remarkability".

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> The solutions represent?

I take the 5-limit comma defined by the temperament, and then find
another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5-
limit comma is the correct wedgie, that means these two commas define
the temperament.

> > The pair of unisons
> > returned in this way can be LLL reduced by the "com7" function,
> which
> > takes a pair of intervals and LLL reduces them.
>
> Why not TM-reduce them?

I'd always LLL reduce them first.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > The solutions represent?
>
> I take the 5-limit comma defined by the temperament, and then find
> another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5-
> limit comma is the correct wedgie, that means these two commas
define
> the temperament.
>
>
> > > The pair of unisons
> > > returned in this way can be LLL reduced by the "com7" function,
> > which
> > > takes a pair of intervals and LLL reduces them.
> >
> > Why not TM-reduce them?
>
> I'd always LLL reduce them first.

How come?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > I'd always LLL reduce them first.
>
> How come?

Because it makes the TM reduction dead easy.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Well, I think Gene is saying that step^2 cents is clearly the right
> measure of "remarkability".

Huh? "Remarkability" sounds like a kind of goodness. Step^2 * cents is
obviously a form of badness. I think I've already explained why no
product of poynomials of these two things will ever be acceptable to
me, at least not without cutoffs applied to them first. And I
understand Gene to be saying that he wants at least an upper cutoff on
"steps" (which seems like a bad name to me since it suggests scale
steps, I prefer "num_gens" or just "gens").

gens^2 * cents
gives exactly the same ranking as
log(gens^2 * cents) [where the log base is arbitrary]
because log(x) is monotonically increasing. Right?
Now
log(gens^2 * cents)
= log(gens^2) + log(cents)
= 2*log(gens) + log(cents)

So this says that a doubling of the number of generators is twice as
bad as a doubling of the error. And previously someone suggested it
was 3 times as bad. You've arbitrarity decided that only the
logarithms are comparable (when cents is already a logarithmic
quantity) and you arbitrarily decided that the constant of
proportionality between them must be an integer!

So what's wrong with k*steps + cents? The basic idea here is that the
unit of badness is the cent and we decide for a given odd-limit how
many cents the error would need to be reduced for you to tolerate an
extra generator in the width of your tetrads (or whatever), or how
many generators you'd need to reduce the tetrad (or whatever) width by
in order to tolerate another cent of error.

Or maybe you think that a _doubling_ of the number of generators is
worth a fixed number of cents. i.e. badness = k*log(gens) + cents

But always you must decide a value for one parameter k that gives the
proportionality between gens and cents because there is no
relationship between their two units of measurement apart from the one
that comes through human experience. Or at least I can't see any.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Yes, I was just going to say we should write the whole paper first
in
> the 5-limit.

There's not much to the 5-limit--it basically is a mere comma search,
and that can be done expeditiously using a decent 5-limit notation.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Well, I think Gene is saying that step^2 cents is clearly the
right
> > measure of "remarkability".
>
> Huh? "Remarkability" sounds like a kind of goodness. Step^2 * cents
is
> obviously a form of badness.

Right, but it's the _objective_ kind. Not the kind that has anything
to do with any particular musician's desiderata. It's the only
measure that doesn't favor a certain range of acceptable values for
error or for complexity. It only favors the best examples within each
range. The particular users of our findings can then decide what
range suits them best. Within any narrow range, all reasonable
measures will give the same ranking.

This is kind of like using Tenney complexity to determine the seed
set for harmonic entropy -- with different complexity measures the
overall slope of the curve changes, changing the consonance ranking
of intervals of different sizes, but the consonance ranking of nearby
intervals remains the same regardless of how complexity is defined
(as long as the 2-by-2 matrix formed by the numbers in adjacent seed
fractions always has a determinant of 1).

> I think I've already explained why no
> product of poynomials of these two things will ever be acceptable
to
> me, at least not without cutoffs applied to them first.
> And I
> understand Gene to be saying that he wants at least an upper cutoff

Yes -- I discussed the situation a few messages back. We use an
objective measure, and cut things off in a nice wide gap.

> on
> "steps" (which seems like a bad name to me since it suggests scale
> steps, I prefer "num_gens" or just "gens").

Yes.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Yes, I was just going to say we should write the whole paper
first
> in
> > the 5-limit.
>
> There's not much to the 5-limit--it basically is a mere comma
search,
> and that can be done expeditiously using a decent 5-limit notation.

A decent 5-limit notation?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > Well, I think Gene is saying that step^2 cents is clearly the
> right
> > > measure of "remarkability".
> >
> > Huh? "Remarkability" sounds like a kind of goodness. Step^2 *
cents
> is
> > obviously a form of badness.
>
> Right, but it's the _objective_ kind. Not the kind that has anything
> to do with any particular musician's desiderata.

Paul! You seem to have ignored the most of the rest of my message.

What the heck is _objective_ about deciding that a doubling of the
number of generators is twice as bad as a doubling of the error. It's
completely arbitrary.

> It's the only
> measure that doesn't favor a certain range of acceptable values for
> error or for complexity. It only favors the best examples within
each
> range.

What _objective_ reason is there, to choose it over gens^3 * cents or
gens^2.3785 * cents?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Paul! You seem to have ignored the most of the rest of my message.

Not at all.

> > It's the only
> > measure that doesn't favor a certain range of acceptable values
for
> > error or for complexity. It only favors the best examples within
> each
> > range.
>
> What _objective_ reason is there, to choose it over gens^3 * cents
or
> gens^2.3785 * cents?

Because those measures give an overall "slope" to the results, in
analogy to what the Farey series seeding does to harmonic entropy.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > The solutions represent?
>
> I take the 5-limit comma defined by the temperament, and then find
> another comma 2^p 3^q 5^r 7 such that the wedgie of this and the 5-
> limit comma is the correct wedgie, that means these two commas
define
> the temperament.

This should be 2^p 3^q 5^r 7^s where s is gcd(a,b,c), and the 5-limit
comma is 2^a 3^b 5^c.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Yes -- I discussed the situation a few messages back. We use an
> objective measure, and cut things off in a nice wide gap.

You are thinking that gens^2 cents, and Ennealimmal as the shut-off
point, would be a good plan?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > It's the only
> > > measure that doesn't favor a certain range of acceptable values
> for
> > > error or for complexity. It only favors the best examples within
> > each
> > > range.
> >
> > What _objective_ reason is there, to choose it over gens^3 * cents
> or
> > gens^2.3785 * cents?
>
> Because those measures give an overall "slope" to the results, in
> analogy to what the Farey series seeding does to harmonic entropy.

What's objective about that? A certain slope may be _real_. i.e.
humans on average may experience it that way, in which case the "flat"
case will really be favouring one extreme.

I understand what the slope is in the HE case, but what slope are you
talking about re badness of linear temperament? Badness wrt what?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > There's not much to the 5-limit--it basically is a mere comma
> search,
> > and that can be done expeditiously using a decent 5-limit
notation.

> A decent 5-limit notation?

We could search (16/15)^a (25/24)^b (81/80)^c to start out with, and
go to something more extreme if wanted.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Yes -- I discussed the situation a few messages back. We use an
> > objective measure, and cut things off in a nice wide gap.
>
> You are thinking that gens^2 cents, and Ennealimmal as the shut-off
> point, would be a good plan?

Possibly, though since gens and cents are two dimensions, we really
need a shuf-off _curve_, don't we?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I understand what the slope is in the HE case, but what slope are
you
> talking about re badness of linear temperament? Badness wrt what?

What is the problem with a "flat" system and a cutoff? It doesn't
commit to any particular theory about what humans are like and what
they should want, and I think that's a good plan.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > There's not much to the 5-limit--it basically is a mere comma
> > search,
> > > and that can be done expeditiously using a decent 5-limit
> notation.
>
> > A decent 5-limit notation?
>
> We could search (16/15)^a (25/24)^b (81/80)^c to start out with,
and
> go to something more extreme if wanted.

More extreme? I'm not getting this.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Because those measures give an overall "slope" to the results, in
> > analogy to what the Farey series seeding does to harmonic entropy.
>
> What's objective about that? A certain slope may be _real_. i.e.
> humans on average may experience it that way, in which case
the "flat"
> case will really be favouring one extreme.

But I don't feel comfortable deciding that for anyone.

> I understand what the slope is in the HE case, but what slope are
you
> talking about re badness of linear temperament? Badness wrt what?

Both step and cent.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Possibly, though since gens and cents are two dimensions, we really
> need a shuf-off _curve_, don't we?

If we bound one of them and gens^2 cents, we've bound the other;
that's what I'd do.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I understand what the slope is in the HE case, but what slope are
> you
> > talking about re badness of linear temperament? Badness wrt what?
>
> What is the problem with a "flat" system and a cutoff?

Dave is trying to understand why this _is_ a flat system.

> It doesn't
> commit to any particular theory about what humans are like and what
> they should want, and I think that's a good plan.

Thank you.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Possibly, though since gens and cents are two dimensions, we
really
> > need a shuf-off _curve_, don't we?
>
> If we bound one of them and gens^2 cents, we've bound the other;
> that's what I'd do.

Hmm . . . so if we simply put an upper bound on the RMS cents error,
we'll have a closed search? That doesn't seem right . . .

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I understand what the slope is in the HE case, but what slope are
> you
> > talking about re badness of linear temperament? Badness wrt what?
>
> What is the problem with a "flat" system and a cutoff?

I may be able to answer that when someone explains what is flat with
respect to what.

It doesn't
> commit to any particular theory about what humans are like and what
> they should want, and I think that's a good plan.

Don't the cutoffs have to be based on a theory about what humans are
like?

If a "flat" system was miles from anything related what humans are
like, would you still be interested in it?

I don't think you can avoid this choice. You must publish a finite
list. If you include more of certain extremes, you must omit more
of the middle-of-the-road.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > We could search (16/15)^a (25/24)^b (81/80)^c to start out with,
> and
> > go to something more extreme if wanted.
>
> More extreme? I'm not getting this.

(78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives the
5-limit, but is better for finding much smaller commas, to take a
more or less random example.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> > It doesn't
> > commit to any particular theory about what humans are like and
what
> > they should want, and I think that's a good plan.
>
> Don't the cutoffs have to be based on a theory about what humans
are
> like?

I'm suggesting we place the cutoffs where we find big gaps, and
comfortably outside any system that has been used to date.
>
> If a "flat" system was miles from anything related what humans are
> like, would you still be interested in it?

Again, any system that is "best" according to a "human" criterion
will show up as "best in its neighborhood" under a flat criterion.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > We could search (16/15)^a (25/24)^b (81/80)^c to start out
with,
> > and
> > > go to something more extreme if wanted.
> >
> > More extreme? I'm not getting this.
>
> (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives the
> 5-limit, but is better for finding much smaller commas, to take a
> more or less random example.

Once a, b, and c are big enough, the original choice of commas will
do little to induce any tendency of smallness or largeness in the
result, correct?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > Because those measures give an overall "slope" to the results,
in
> > > analogy to what the Farey series seeding does to harmonic
entropy.
> >
> > What's objective about that? A certain slope may be _real_. i.e.
> > humans on average may experience it that way, in which case
> the "flat"
> > case will really be favouring one extreme.
>
> But I don't feel comfortable deciding that for anyone.

But you _are_ deciding it. You can't help but decide it, unless you
intend to publish an infinite list. No matter what you do there will
be someone who thinks there's a lot of fluff in there and you missed
out some others. They aren't going to be impressed by any argument
that "our metric is 'objective' or 'flat'".

> > I understand what the slope is in the HE case, but what slope are
> you
> > talking about re badness of linear temperament? Badness wrt what?
>
> Both step and cent.

Huh? Obviously any badness metric _must_ slope down towards (0,0) on
the (cents,gens) plain. If you make the gens and cents axes
logarithmic then badness = gens^k * cents is simply a tilted plane.
The only way you can decide on whether it should tilt more towards
gens or cents (the exponent k) is through human considerations.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > If a "flat" system was miles from anything related what humans are
> > like, would you still be interested in it?
>
> Again, any system that is "best" according to a "human" criterion
> will show up as "best in its neighborhood" under a flat criterion.

But some neighbourhoods may be so disadvantaged that their best
doesn't even make it into the list.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Hmm . . . so if we simply put an upper bound on the RMS cents
error,
> we'll have a closed search? That doesn't seem right . . .

I was suggesting a *lower* bound on RMS cents as one possibility.

If with all quantities positive we have g^2 c < A and c > B, then
1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably
makes more sense to use g>=1, so that if g^2 c <= A then c <= A.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > > Because those measures give an overall "slope" to the
results,
> in
> > > > analogy to what the Farey series seeding does to harmonic
> entropy.
> > >
> > > What's objective about that? A certain slope may be _real_.
i.e.
> > > humans on average may experience it that way, in which case
> > the "flat"
> > > case will really be favouring one extreme.
> >
> > But I don't feel comfortable deciding that for anyone.
>
> But you _are_ deciding it. You can't help but decide it, unless you
> intend to publish an infinite list. No matter what you do there
will
> be someone who thinks there's a lot of fluff in there and you
missed
> out some others. They aren't going to be impressed by any argument
> that "our metric is 'objective' or 'flat'".

We won't be missing out on anyone's "best" (unless they are really
far out on the plane, beyond the big gap where we will establish the
cutoff). Then they can come up with their own criterion and get their
own ranking. But at least we'll have something for everyone.

> > > I understand what the slope is in the HE case, but what slope
are
> > you
> > > talking about re badness of linear temperament? Badness wrt
what?
> >
> > Both step and cent.
>
> Huh? Obviously any badness metric _must_ slope down towards (0,0)
on
> the (cents,gens) plain.

The badness metric does, but the results don't. The results have a
similar distribution everywhere on the plane, but only when gens^2
cents is the badness metric.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > If a "flat" system was miles from anything related what humans
are
> > > like, would you still be interested in it?
> >
> > Again, any system that is "best" according to a "human" criterion
> > will show up as "best in its neighborhood" under a flat criterion.
>
> But some neighbourhoods may be so disadvantaged that their best
> doesn't even make it into the list.

That won't happen -- that's the point of the "flat" criterion. Only
the neighborhoods outside our cutoff will be disadvantaged, but at
least this will be explicit.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Hmm . . . so if we simply put an upper bound on the RMS cents
> error,
> > we'll have a closed search? That doesn't seem right . . .
>
> I was suggesting a *lower* bound on RMS cents as one possibility.

Oh . . . well I don't think we should frame it _that_ way!

> If with all quantities positive we have g^2 c < A and c > B, then
> 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it probably
> makes more sense to use g>=1, so that if g^2 c <= A then c <= A.

Are you saying that using g>=1 is enough to make this a closed search?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Huh? Obviously any badness metric _must_ slope down towards (0,0)
> on
> > the (cents,gens) plain.
>
> The badness metric does, but the results don't. The results have a
> similar distribution everywhere on the plane, but only when gens^2
> cents is the badness metric.

You're not making any sense. The results are all just discrete points
in the badness surface with respect to gens and cents, so they have
exactly the same slope. The results have a similar distribution of
what? Everywhere on what plane?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I may be able to answer that when someone explains what is flat
with
> respect to what.

Paul did that. An analogy would be to use n^(4/3) cents when seaching
for 7-limit ets; this will give you a list which does not favor
either high or low numbers n, but it has nothing to do with human
perception, and you would use a different exponent in a different
prime limit--n^2 cents in the 3-limit, n^(3/2) cents in the 5-limit,
and so forth.

> It doesn't
> > commit to any particular theory about what humans are like and
what
> > they should want, and I think that's a good plan.
>
> Don't the cutoffs have to be based on a theory about what humans
are
> like?

I don't think you can have much of a theory about what a bunch of
cranky individualists might like, but I hope we could cut it off when
the difference could no longer be percieved. Can anyone hear the
difference between Ennealimmal and just?

> If a "flat" system was miles from anything related what humans are
> like, would you still be interested in it?

I might, most people would not be. I've discovered though that even
the large, "useless" ets have uses.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > Huh? Obviously any badness metric _must_ slope down towards
(0,0)
> > on
> > > the (cents,gens) plain.
> >
> > The badness metric does, but the results don't. The results have
a
> > similar distribution everywhere on the plane, but only when
gens^2
> > cents is the badness metric.
>
> You're not making any sense. The results are all just discrete
points
> in the badness surface with respect to gens and cents, so they have
> exactly the same slope. The results have a similar distribution of
> what? Everywhere on what plane?

I see Gene is, at this very moment, doing a good job explaining these
issues to you; meanwhile, my brain is toast.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >
> > > > We could search (16/15)^a (25/24)^b (81/80)^c to start out
> with,
> > > and
> > > > go to something more extreme if wanted.
> > >
> > > More extreme? I'm not getting this.
> >
> > (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives
the
> > 5-limit, but is better for finding much smaller commas, to take a
> > more or less random example.
>
> Once a, b, and c are big enough, the original choice of commas will
> do little to induce any tendency of smallness or largeness in the
> result, correct?

(78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2

I wouldn't search that far myself.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>

> > > (78732/78125)^a (32805/32768)^b (2109375/2097152)^c also gives
> the
> > > 5-limit, but is better for finding much smaller commas, to take
a
> > > more or less random example.
> >
> > Once a, b, and c are big enough, the original choice of commas
will
> > do little to induce any tendency of smallness or largeness in the
> > result, correct?
>
> (78732/78125)^53 (32805/32768)^(-84) (2109375/2097152)^65 = 2
>
> I wouldn't search that far myself.

How do you know you wouldn't be missing any good ones?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > If with all quantities positive we have g^2 c < A and c > B, then
> > 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it
probably
> > makes more sense to use g>=1, so that if g^2 c <= A then c <= A.

> Are you saying that using g>=1 is enough to make this a closed
search?

All it does is put an upper limit on how far out of tune the worst
cases can be, so we really need to bound c below or g above to get a
finite search.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> How do you know you wouldn't be missing any good ones?

You'd need bounds on what counted for good; I'll think about it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I may be able to answer that when someone explains what is flat
> with
> > respect to what.
>
> Paul did that.

Not in any way that makes any sense to me. I don't think Pauk
really understands it either and may be starting to realise that.

I'm starting to wonder if there's a conspiracy here to make me think
I'm going crazy. :-) Is anyone else getting this "gens^2 * cents is
the only 'flat' metric" thing?

> An analogy would be to use n^(4/3) cents when
seaching
> for 7-limit ets; this will give you a list which does not favor
> either high or low numbers n,

I'm sorry. This makes no sense to me either. _How_ would you use
n^(4/3) cents? Can you prove this to me? Or better still just prove
whatever it is you are trying to say about gens^2 * cents being a
"flat" badness metric for linear temperaments.

> I don't think you can have much of a theory about what a bunch of
> cranky individualists might like, but I hope we could cut it off
when
> the difference could no longer be percieved. Can anyone hear the
> difference between Ennealimmal and just?

Well that is precisely a theory about humans, as opposed to say
grasshoppers or rocks or computers.

If you guys can't explain this to me, I don't think you've got much
chance of getting published in a refereed journal. It doesn't involve
anything beyond high school math.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > An analogy would be to use n^(4/3) cents when
> seaching
> > for 7-limit ets; this will give you a list which does not favor
> > either high or low numbers n,

> I'm sorry. This makes no sense to me either. _How_ would you use
> n^(4/3) cents? Can you prove this to me?

The argument for n^(4/3) is required in order to get the argument for
gens^2 cents, so this is the place to start. The argument comes from
the theory of simultaneous Diophantine approximation, where it is
shown that there is a constant c, depending on d, such that for any d
irrational numbers x1, x2, ... xd there will be an infinite number of
solutions n to

n^(1+1/d) |xi - pi/n| < c

In the case of the 7-limit, we want to simultaneously approximate
log2(3), log2(5) and log2(7), so d=3.

> If you guys can't explain this to me, I don't think you've got much
> chance of getting published in a refereed journal. It doesn't
involve
> anything beyond high school math.

Explain what? Diophantine approximation, or why to use that
theoretical basis, or what? *What* doesn't involve more than high
school math? The theorem I mentioned isn't hard to prove but it does
use Dirichlet's pidgeonhole principle, which is also not hard but
which you probably did not learn in high school and which I would not
propose to discuss in the pages of a music journal, given that I have
reason to think that there is a limit to how much math they would
find acceptable.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > An analogy would be to use n^(4/3) cents when
> > seaching
> > > for 7-limit ets; this will give you a list which does not favor
> > > either high or low numbers n,
>
> > I'm sorry. This makes no sense to me either. _How_ would you use
> > n^(4/3) cents? Can you prove this to me?
>
> The argument for n^(4/3) is required in order to get the argument
for
> gens^2 cents, so this is the place to start. The argument comes from
> the theory of simultaneous Diophantine approximation,

Oh damn. Ok forget about proving it to me. Just please try to get me
to understand what it is you are saying. I just thought that getting
you to prove it to me my be the easiest way for me to understand what
it was I had asked you to prove. Apparently not.

So ... What is n? What is a 7-limit et? How does one use n^(4/3) to
get a list of them? How would one check to see whether the list
favours high or low n.

> > If you guys can't explain this to me, I don't think you've got
much
> > chance of getting published in a refereed journal. It doesn't
> involve
> > anything beyond high school math.
>
> Explain what? Diophantine approximation, or why to use that
> theoretical basis, or what? *What* doesn't involve more than high
> school math?

Your (and Paul's) statements so far about badness metrics and
flatness.

> The theorem I mentioned isn't hard to prove but it does
> use Dirichlet's pidgeonhole principle, which is also not hard but
> which you probably did not learn in high school and which I would
not
> propose to discuss in the pages of a music journal, given that I
have
> reason to think that there is a limit to how much math they would
> find acceptable.

Agreed.

But surely you can get me to understand what you actually mean by
"flat" here. I may well be prepared to just believe the theorem as
stated, if I can understand what it means.

But no matter what you come up with I can't see how you can get past
the fact that gens and cents are fundamentally incomensurable
quantities, so somewhere there has to be a parameter that says how bad
they are relative to each other.

Currently you are saying that doubling gens is twice as bad as
doubling cents. Why? What if 99% of humans don't experience it like
that.

And why should they both be treated logarithmically? k*log(gens) +
log(cents) gives the same ranking as gens^2 * cents when k=2. Why not
use k*gens + cents. e.g. if badness was simply gens + cents and you
listed everything with badness not more than 30 then you don't need
any additional cutoffs. You automatically eliminate anything with gens
> 30 or cents > 30 (actually cents > 29 because gens can't go below
1).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> So ... What is n? What is a 7-limit et? How does one use n^(4/3) to
> get a list of them? How would one check to see whether the list
> favours high or low n.

"n" is how many steps to the octave, or in other words what 2 is
mapped to. By a "7-limit et" I mean something which maps 7-limit
intervals to numbers of steps in a consistent way. Since we are
looking for the best, we can safely restrict these to what we get by
rounding n*log2(3), n*log2(5) and n*log2(7) to the nearest integer,
and defining the n-et as the map one gets from this.

Let's call this map "h"; for the 12-et, h(2)=12, h(3)=19, h(5)=28 and
h(7)=34; this entails that h(5/3) = h(5)-h(3) = 9, h(7/3)=15 and
h(7/5)=6. I can now measure the relative badness of "h" by taking the
sum, or maximum, or rms, of the differences of |h(3)-n*log2(3)|,
|h(5)-n*log2(5)|, |h(7)-n*log2(7)|, |h(5/3)-n*log2(5/3)|,
|h(7/3)-n*log2(7/3)| and |h(7/5)-n*log2(7/5)|.

This measure of badness is flat in the sense that the density is the
same everywhere, so that we would be selecting about the same number
of ets in a range around 12 as we would in a range around 1200. I
don't really want this sort of "flatness", so I use the theory of
Diophantine approximation to tell we that if I multiply this badness
by the cube root of n, so that the density falls off at a rate of
n^(-1/3), I will still get an infinite list of ets, but if I make it
fall off faster I probably won't. I can use either the maximum of the
above numbers, or the sum, or the rms, and the same conclusion holds;
in fact, I can look at the 9-limit instead of the 7-limit and the
same conclusion holds. If I look at the maximum, and multiply by 1200
so we are looking at units of n^(4/3) cents, I get the following list
of ets which come out as less than 1000, for n going from 1 to 2000:

1 884.3587134
2 839.4327178
4 647.3739047
5 876.4669184
9 920.6653451
10 955.6795096
12 910.1603254
15 994.0402775
31 580.7780905
41 892.0787789
72 892.7193923
99 716.7738001
171 384.2612749
270 615.9368489
342 968.2768986
441 685.5766666
1578 989.4999106

This list just keeps on going, so I cut it off at 2000. I might look
at it, and decide that it doesn't have some important ets on it, such
as 19,22 and 27; I decide to put those on, not really caring about
any other range, by raising the ante to 1200; I then get the
following additions:

3 1154.683345
6 1068.957518
19 1087.886603
22 1078.033523
27 1108.589256
68 1090.046322
130 1182.191130
140 1091.565279
202 1143.628876
612 1061.222492
1547 1190.434242

My decision to add 19,22, and 27 leads me to add 3 and 6 at the low
end, and 68 and so forth at the high end. It tells me that if I'm
interested in 27 in the range around 31, I should also be interested
in 68 in the range around 72, in 140 and 202 around 171, 612 around
441, and 1547 near 1578. That's the sort of "flatness" Paul was
talking about; it doesn't favor one range over another.

> But no matter what you come up with I can't see how you can get
past
> the fact that gens and cents are fundamentally incomensurable
> quantities, so somewhere there has to be a parameter that says how
bad
> they are relative to each other.

"n" and cents are incommeasurable also, and n^(4/3) is only right for
the 7 and 9 limits, and wrong for everything else, so I don't think
this is the issue if we adopt this point of view.

Why not
> use k*gens + cents. e.g. if badness was simply gens + cents and you
> listed everything with badness not more than 30 then you don't need
> any additional cutoffs. You automatically eliminate anything with
gens
> > 30 or cents > 30 (actually cents > 29 because gens can't go below
> 1).

Gens^3 cents also automatically cuts things off, but I rather like
the idea of keeping it "flat" in the above sense and doing the
cutting off deliberately, it seems more objective.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > If with all quantities positive we have g^2 c < A and c > B,
then
> > > 1/c < 1/B, and so g^2 < A/B and g < sqrt(A/B). However, it
> probably
> > > makes more sense to use g>=1, so that if g^2 c <= A then c <= A.
>
> > Are you saying that using g>=1 is enough to make this a closed
> search?
>
> All it does is put an upper limit on how far out of tune the worst
> cases can be, so we really need to bound c below or g above to get
a
> finite search.

So do you still stand by this statement:

"If we bound one of them and gens^2 cents, we've bound the other;
that's what I'd do."

(which you wrote after I said that a single cufoff point wouldn't be
enough, that we would need a cutoff curve)?

Thanks Gene, for taking the time to explain this in a way that a
mere computer scientist can understand. :-)

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > So ... What is n? What is a 7-limit et? How does one use n^(4/3)
to
> > get a list of them? How would one check to see whether the list
> > favours high or low n.
>
> "n" is how many steps to the octave, or in other words what 2 is
> mapped to. By a "7-limit et" I mean something which maps 7-limit
> intervals to numbers of steps in a consistent way. Since we are
> looking for the best, we can safely restrict these to what we get by
> rounding n*log2(3), n*log2(5) and n*log2(7) to the nearest integer,
> and defining the n-et as the map one gets from this.

OK so far.

> Let's call this map "h";
> for the 12-et, h(2)=12, h(3)=19, h(5)=28
and
> h(7)=34; this entails that h(5/3) = h(5)-h(3) = 9, h(7/3)=15 and
> h(7/5)=6.

Fine.

> I can now measure the relative badness of "h" by taking the
> sum, or maximum, or rms, of the differences of |h(3)-n*log2(3)|,
> |h(5)-n*log2(5)|, |h(7)-n*log2(7)|, |h(5/3)-n*log2(5/3)|,
> |h(7/3)-n*log2(7/3)| and |h(7/5)-n*log2(7/5)|.

I'd say this is just one component of badness. Its the error expressed
as a proportion of the step size. The number of steps in the octave n
has an effect on badness independent of the relative error.

> This measure of badness is flat in the sense that the density is the
> same everywhere, so that we would be selecting about the same number
> of ets in a range around 12 as we would in a range around 1200.

Yes. I believe this. See the two charts near the end of
http://dkeenan.com/Music/EqualTemperedMusicalScales.htm
although it uses a weighting error that only includes the primes
(only the "rooted" intervals) that I now find dubious.

> I don't really want this sort of "flatness",

Hardly anyone would. Not without some additional penalty for large n,
even if it's just a crude sudden cutoff. But _why_ don't you want this
sort of flatness? Did you reject it on "objective" grounds? Is there
some other sort of flatness that you _do_ want? If so, what is it? How
many sorts of flatness are there and how did you choose between them?

> so I use the theory of
> Diophantine approximation to tell we that if I multiply this badness
> by the cube root of n, so that the density falls off at a rate of
> n^(-1/3), I will still get an infinite list of ets, but if I make it
> fall off faster I probably won't.

Here's where the real leap-of-faith occurs.

First of all, I take it that when you say you will (or wont) "get an
infinite list of ets", you mean "when the list is limited to ETs whose
badness does not exceed a given badness limit, greater than zero".

There are an infinite number of ways of defining badness to achieve a
finite list with a cutoff only on badness itself. Most of these will
produce a finite list that is of of absolutely no interest to 99.99%
of the population (of people who are interested in the topic at all).

Why do you immediately leap to the theory of Diophantine approximation
as giving the best way to achieve a finite list?

I think a good way to achieve it is simply to add an amount k*n to the
error in cents (absolute, not relative to step size). I suggest
initially trying a k of about 0.5 cents per step.

The only way to tell if this is better than something based on the
theory of Diophantine equations is to suck it and see. Some of us have
been on the tuning lists long enough to know what a lot of other
people find useful or interesting, even though we don't necessarily
find them so ourselves.

> I can use either the maximum of
the
> above numbers, or the sum, or the rms, and the same conclusion
holds;
> in fact, I can look at the 9-limit instead of the 7-limit and the
> same conclusion holds. If I look at the maximum, and multiply by
1200
> so we are looking at units of n^(4/3) cents, I get the following
list
> of ets which come out as less than 1000, for n going from 1 to 2000:
>
> 1 884.3587134
> 2 839.4327178
> 4 647.3739047
> 5 876.4669184
> 9 920.6653451
> 10 955.6795096
> 12 910.1603254
> 15 994.0402775
> 31 580.7780905
> 41 892.0787789
> 72 892.7193923
> 99 716.7738001
> 171 384.2612749
> 270 615.9368489
> 342 968.2768986
> 441 685.5766666
> 1578 989.4999106
>
> This list just keeps on going, so I cut it off at 2000. I might look
> at it, and decide that it doesn't have some important ets on it,
such
> as 19,22 and 27; I decide to put those on, not really caring about
> any other range, by raising the ante to 1200; I then get the
> following additions:
>
> 3 1154.683345
> 6 1068.957518
> 19 1087.886603
> 22 1078.033523
> 27 1108.589256
> 68 1090.046322
> 130 1182.191130
> 140 1091.565279
> 202 1143.628876
> 612 1061.222492
> 1547 1190.434242
>
> My decision to add 19,22, and 27 leads me to add 3 and 6 at the low
> end, and 68 and so forth at the high end. It tells me that if I'm
> interested in 27 in the range around 31, I should also be interested
> in 68 in the range around 72, in 140 and 202 around 171, 612 around
> 441, and 1547 near 1578. That's the sort of "flatness" Paul was
> talking about; it doesn't favor one range over another.

But this is nonsense. It simply isn't true that 3, 6, 612 and 1547 are
of approximately equal interest to 19, 22 and 27. Sure you'll always
be able to find one person who'll say they are. But ask anyone who has
actually used 19-tET or 22-tET when they plan to try 3-tET or
1547-tET. It's just a joke. I suspect you've been seduced by the
beauty of the math and forgotten your actual purpose. This metric
clearly favours both very small and very large n over middle ones.

> > But no matter what you come up with I can't see how you can get
> past
> > the fact that gens and cents are fundamentally incomensurable
> > quantities, so somewhere there has to be a parameter that says how
> bad
> > they are relative to each other.
>
> "n" and cents are incommeasurable also,

Yes.

> and n^(4/3) is only right for
> the 7 and 9 limits, and wrong for everything else, so I don't think
> this is the issue if we adopt this point of view.
>
> Why not
> > use k*gens + cents. e.g. if badness was simply gens + cents and
you
> > listed everything with badness not more than 30 then you don't
need
> > any additional cutoffs. You automatically eliminate anything with
> gens
> > > 30 or cents > 30 (actually cents > 29 because gens can't go
below
> > 1).
>
> Gens^3 cents also automatically cuts things off, but I rather like
> the idea of keeping it "flat" in the above sense and doing the
> cutting off deliberately, it seems more objective.

_Seems_ more objective? You mean that subjectively, to you, it seems
more objective?

Well I'm afraid that it seems to me that this quest for an "objective"
badness metric (with ad hoc cutoffs) is the silliest thing I've heard
in quite a while.

If you're combining two or more incomensurable quantities into a
single badness metric, the choice of the constant of proportionality
between them (and the choice of whether this constant should relate
the plain quantities or their logarithms or whatever) should be
decided so that as many people as possible agree that it actually
gives something like what they perceive as badness, even if its only
roughly so.

An isobad that passes near 3, 6, 19, 22, 612 and 1547, isn't one. The
fact that its based on the theory of Diophantine equations is utterly
irrelevant.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> But this is nonsense. It simply isn't true that 3, 6, 612 and 1547
are
> of approximately equal interest to 19, 22 and 27. Sure you'll
always
> be able to find one person who'll say they are. But ask anyone who
has
> actually used 19-tET or 22-tET when they plan to try 3-tET or
> 1547-tET. It's just a joke.

For the third or fourth time Dave, this isn't intended to appeal to
any one person, but rather to the widest possible audience. Since
this is a "flat" measure, it will rank the systems in the _vicinity_
of *your* #1 system, the same way you would, whoever *you* happen to
be. But it makes absolutely no preference for one end of the spectrum
over another, or the middle. That's what makes it flat
and "objective". Look at Gene's list for 7-limit ETs again. Can it be
denied that 31-tET is by far the best _in its vicinity_, and 171-tET
is by far the best _in its vicinity_?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > So do you still stand by this statement:
> >
> > "If we bound one of them and gens^2 cents, we've bound the other;
> > that's what I'd do."
> >
> > (which you wrote after I said that a single cufoff point wouldn't
> be
> > enough, that we would need a cutoff curve)?
>
> Sure. I think bounding g makes the most sense, since we can
calculate
> it more easily. I've been thinking about how one might calculate
> cents without going through the map stage, but for gens we can get
it
> immediately from the wedgie with no trouble.

I don't immediately know what "the map stage" means, but I've been
thinking that, in regarding to "standardizing the wedge product", we
might want to use something that has the Tenney lattice built in.

> We could then toss
> anything with too high a gens figure before even calculating
anything
> else, which should help.

So I'm not getting where g>=1 comes into all this.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> So I'm not getting where g>=1 comes into all this.

What I wrote was confused, but you've already replied, I see. Bounding
g from below is easy, since it bounds itself. Bounding it from above
could mean just setting a bound, or bounding g^2 c; I think just
setting an upper bound to it makes a lot of sense.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > But this is nonsense. It simply isn't true that 3, 6, 612 and 1547
> are
> > of approximately equal interest to 19, 22 and 27. Sure you'll
> always
> > be able to find one person who'll say they are. But ask anyone who
> has
> > actually used 19-tET or 22-tET when they plan to try 3-tET or
> > 1547-tET. It's just a joke.
>
> For the third or fourth time Dave, this isn't intended to appeal to
> any one person, but rather to the widest possible audience.

But that's exactly my intention too. I'm trying to help you find a
metric that will appeal, not to me, but to all those people whose
divergent views I've read on the tuning list over the years. I'm
simply claiming that your metric is seriously flawed in acheiving your
intended goal. Practically _nobody_ thinks 3,6,612,1547 are equally as
good or bad or interesting as 19 or 22. If you include fluff like that
then there will be less room for ETs of interest to actual humans.

> Since
> this is a "flat" measure, it will rank the systems in the _vicinity_
> of *your* #1 system, the same way you would, whoever *you* happen to
> be. But it makes absolutely no preference for one end of the
spectrum
> over another, or the middle. That's what makes it flat
> and "objective".

You seem to be arguing in circles.

> Look at Gene's list for 7-limit ETs again. Can it
be
> denied that 31-tET is by far the best _in its vicinity_, and 171-tET
> is by far the best _in its vicinity_?

Of course I don't deny that. I claim that it is irrelevant. _Any_ old
half-baked way of monotonically combining steps and cents into a
badness metric will be the same as any other, _locally_. You said the
same yourself in regard to your HE curves. Maybe you need more sleep.
:-)

Since when does merely local behaviour determine if something is
_flat_ or not?

In any case, I don't think you understand Gene's particular kind of
flatness, you certainly weren't able to explain it to me, as Gene has
now done. This particular kind of "flatness" is just one of many.
There's nothing objective about a decision to favour it, and then to
ad hoc introduce additional cutoffs besides the one for badness.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > So I'm not getting where g>=1 comes into all this.
>
> What I wrote was confused, but you've already replied, I see.
Bounding
> g from below is easy, since it bounds itself. Bounding it from
above
> could mean just setting a bound, or bounding g^2 c; I think just
> setting an upper bound to it makes a lot of sense.

Yes -- g could play the role than N plays in your ET lists. One would
order the results by g, give the g^2 c score for each (or not), and
give about a page of nice musician-friendly information on each.

Gene, there are a lot of outstanding questions and comments . . . I
wanted to know if there would have been a lot more "slippery" ones
had you included simpler unison vectors in your source list . . . I
want to use a Tenney-distance weighted "gens" measure . . . but for
now, a master list would be great. Can someone produce such a list,
with columns for "cents" and "gens" at least as currently defined?
I'd like to try to find omissions.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I'd say this is just one component of badness. Its the error
expressed
> as a proportion of the step size. The number of steps in the
octave n
> has an effect on badness independent of the relative error.

Then you should be happier with an extra cube root of n adjustment.

> Hardly anyone would. Not without some additional penalty for large
n,
> even if it's just a crude sudden cutoff. But _why_ don't you want
this
> sort of flatness?

Because my interest isn't independent of size--you need more at
higher levels to make me care.

Did you reject it on "objective" grounds? Is there
> some other sort of flatness that you _do_ want? If so, what is it?
How
> many sorts of flatness are there and how did you choose between
them?

You could use the Riemann Zeta function and the omega estimates based
on the assumption of the Riemann hypothesis and do it that way, if
you liked. Or there are no doubt other ways; this one seems the
simplest and it gets the job done, and the alternatives would have a
certain family resemblence.

> Why do you immediately leap to the theory of Diophantine
approximation
> as giving the best way to achieve a finite list?

It gives me a measure which is connected to the nature of the
problem, which is a Diophantine approximation problem, which seems to
make a lot of sense both in practice and theory to me, if not to you.

> I think a good way to achieve it is simply to add an amount k*n to
the
> error in cents (absolute, not relative to step size). I suggest
> initially trying a k of about 0.5 cents per step.

Should I muck around in the dark until I make this measure behave in
a way something like the measure I already have behaves, which would
be both pointless and inelegant, or is there something about it to
recommend it?

> The only way to tell if this is better than something based on the
> theory of Diophantine equations is to suck it and see.

Better how? The measure I already have does exactly what I'd want a
measure to do.

Some of us have
> been on the tuning lists long enough to know what a lot of other
> people find useful or interesting, even though we don't necessarily
> find them so ourselves.

One of the advantages of the measure I'm using is that it accomodates
this well.

> But this is nonsense. It simply isn't true that 3, 6, 612 and 1547
are
> of approximately equal interest to 19, 22 and 27.

I'm not trying to measure your interest, I'm only saying if you want
to look at a certain range, look at these.

Sure you'll always
> be able to find one person who'll say they are. But ask anyone who
has
> actually used 19-tET or 22-tET when they plan to try 3-tET or
> 1547-tET. It's just a joke.

The 4-et is actually interesting in connection with the 7-limit, as
the 3-et is with the 5-limit, and the large ets have uses other than
tuning up a set of marimbas as well.

I suspect you've been seduced by the
> beauty of the math and forgotten your actual purpose. This metric
> clearly favours both very small and very large n over middle ones.

In other words, the range *you* happen to care about is the only
interesting range; it's that which I was regarding as not objective.

> An isobad that passes near 3, 6, 19, 22, 612 and 1547, isn't one.

An isobad which passes near 3, 6, 19, 22, 612 and 1547 makes a lot of
sense to me, so I think I would probably *not* like your alternative
as well.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > But this is nonsense. It simply isn't true that 3, 6, 612 and
1547
> > are
> > > of approximately equal interest to 19, 22 and 27. Sure you'll
> > always
> > > be able to find one person who'll say they are. But ask anyone
who
> > has
> > > actually used 19-tET or 22-tET when they plan to try 3-tET or
> > > 1547-tET. It's just a joke.
> >
> > For the third or fourth time Dave, this isn't intended to appeal
to
> > any one person, but rather to the widest possible audience.
>
> But that's exactly my intention too. I'm trying to help you find a
> metric that will appeal, not to me, but to all those people whose
> divergent views I've read on the tuning list over the years. I'm
> simply claiming that your metric is seriously flawed in acheiving
your
> intended goal. Practically _nobody_ thinks 3,6,612,1547 are equally
as
> good or bad or interesting as 19 or 22. If you include fluff like
that
> then there will be less room for ETs of interest to actual humans.

Dave, if you don't have a cutoff, you'd have an infinite number of
ETs better than 1547. Of course there has to be a cutoff.

>
> > Look at Gene's list for 7-limit ETs again. Can it
> be
> > denied that 31-tET is by far the best _in its vicinity_, and 171-
tET
> > is by far the best _in its vicinity_?
>
> Of course I don't deny that. I claim that it is irrelevant. _Any_
old
> half-baked way of monotonically combining steps and cents into a
> badness metric will be the same as any other, _locally_. You said
the
> same yourself in regard to your HE curves. Maybe you need more
sleep.
> :-)

I mean that only Gene's measure tells you exactly _how much_ better a
system is than the systems in their vicinity, _in units of_ the
average differences between different systems in their vicinity.

> Since when does merely local behaviour determine if something is
> _flat_ or not?

It doesn't.

> In any case, I don't think you understand Gene's particular kind of
> flatness, you certainly weren't able to explain it to me, as Gene
has
> now done. This particular kind of "flatness" is just one of many.

I'd like to see a list of ETs, as far as you'd like to take it, above
some cutoff different from Gene's, that shows this kind of behavior
(not just the flatness of the measure itself, but also the flatness
of the size of the wiggles).

Gene wrote:

> Sure. I think bounding g makes the most sense, since we can calculate
> it more easily. I've been thinking about how one might calculate
> cents without going through the map stage, but for gens we can get it
> immediately from the wedgie with no trouble. We could then toss
> anything with too high a gens figure before even calculating anything
> else, which should help.

My program throws out bad temperaments before doing the optimization, if
that's what you're suggesting. It's on of the changes I made this, er,
yesterday morning. It does make a difference, but not much now my
optimization's faster. Big chunks of time are being spent generating the
ETs and formatting the results currently.

Graham

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > I'd say this is just one component of badness. Its the error
> expressed
> > as a proportion of the step size. The number of steps in the
> octave n
> > has an effect on badness independent of the relative error.
>
> Then you should be happier with an extra cube root of n adjustment.

Yes I am. But still way from as happy as I think most people would be
with something not based on k*log(gens) + log(cents) but instead on
k*gens + cents (or maybe something else).

> > But _why_ don't you want this
> > sort of flatness?
>
> Because my interest isn't independent of size--you need more at
> higher levels to make me care.

Indeed.

> Did you reject it on "objective" grounds? Is there
> > some other sort of flatness that you _do_ want? If so, what is it?
> How
> > many sorts of flatness are there and how did you choose between
> them?
>
> You could use the Riemann Zeta function and the omega estimates
based
> on the assumption of the Riemann hypothesis and do it that way, if
> you liked. Or there are no doubt other ways; this one seems the
> simplest and it gets the job done, and the alternatives would have a
> certain family resemblence.

But there's nothing "objective" about these decisions. You're just
finding stuff so it matches what you think everyone likes. Right?

> > Why do you immediately leap to the theory of Diophantine
> approximation
> > as giving the best way to achieve a finite list?
>
> It gives me a measure which is connected to the nature of the
> problem, which is a Diophantine approximation problem, which seems
to
> make a lot of sense both in practice and theory to me, if not to
you.

There are probably many such things "connected to the nature of the
problem" which give entirely different results.

> > I think a good way to achieve it is simply to add an amount k*n to
> the
> > error in cents (absolute, not relative to step size). I suggest
> > initially trying a k of about 0.5 cents per step.
>
> Should I muck around in the dark until I make this measure behave in
> a way something like the measure I already have behaves, which would
> be both pointless and inelegant, or is there something about it to
> recommend it?

Yes. The fact that I've been reading the tuning list and thinking
about and discussing these things with others for many years. So it's
hardly groping in the dark. I'm not saying this particular one I
pulled out of the air is the one most representative of all views, but
I do know that we can do a lot better than your current proposal.

> > The only way to tell if this is better than something based on the
> > theory of Diophantine equations is to suck it and see.
>
> Better how? The measure I already have does exactly what I'd want a
> measure to do.

Answered below.

> Some of us have
> > been on the tuning lists long enough to know what a lot of other
> > people find useful or interesting, even though we don't
necessarily
> > find them so ourselves.
>
> One of the advantages of the measure I'm using is that it
accomodates
> this well.

How do you know that?

> > But this is nonsense. It simply isn't true that 3, 6, 612 and 1547
> are
> > of approximately equal interest to 19, 22 and 27.
>
> I'm not trying to measure your interest,

I keep saying that I'm trying to consider as wide a set of interests
as possible. You and Paul keep accusing me of only trying to serve my
own interests. I accept that you're trying to consider as wide a set
of interests as possible, I just claim that you're failing.

> I'm only saying if you want
> to look at a certain range, look at these.

Yes, but some _ranges_ are more interesting than others and so if you
include an equal number in every range then you won't be including
enough in the most interesting ranges. It isn't just _my_ prejudice
that there are more ETs of interest in the vicinity of 26-tET than
there are in the vicinity of 3-tET or 1550-tET. It's practically
everyone's.

> Sure you'll always
> > be able to find one person who'll say they are. But ask anyone who
> has
> > actually used 19-tET or 22-tET when they plan to try 3-tET or
> > 1547-tET. It's just a joke.
>
> The 4-et is actually interesting in connection with the 7-limit, as
> the 3-et is with the 5-limit, and the large ets have uses other than
> tuning up a set of marimbas as well.

Those are good points, which maybe says that my metric is too harsh on
the extremes, but I still say yours is way too soft. There's got to be
something pretty damn exceptional about an ET greater than 100 for it
to be of interest. But note that our badness metric is only based on
steps and cents (or gens and cents for temperaments) so we can't claim
that our metric should include some exceptional high ET if it's
exceptional property has nothing to do with the magnitude of the
number of steps or the cents error.

> I suspect you've been seduced by the
> > beauty of the math and forgotten your actual purpose. This metric
> > clearly favours both very small and very large n over middle ones.
>
> In other words, the range *you* happen to care about is the only
> interesting range; it's that which I was regarding as not objective.

There you go again. Accusing me of only trying to serve my own
interests.

> > An isobad that passes near 3, 6, 19, 22, 612 and 1547, isn't one.
>
> An isobad which passes near 3, 6, 19, 22, 612 and 1547 makes a lot
of
> sense to me, so I think I would probably *not* like your alternative
> as well.

Whether you or I would like it, isn't the point. The only way this
could be settled is by some kind of experiment or survey, say on the
tuning list.

We could put together two lists of ETs of roughly equally "badness".
One using your metric, one using mine. They should contain the same
number of ETs (you've already given a suitable list of 11). They
should have as many ETs as possible in common. We would tell people
the 7-limit rms error of each and the number of steps per octave in
each, but nothing more. Then we'd ask them to choose which list was a
better example of a list of ETs of approximately equal 7-limit
goodness, badness, usefulness, interestingness or whatever you want to
call it, based only on considerations of the number of steps and the
error.

We could even ask them to rate each list on a scale of 1 to 10
according to how well they think each list manages to capture equal
7-limit interestingness or whatever, based only on considerations of
the number of steps and the error.

Here they are:

ET List 1

Steps 7-limit
per RMS
octave error (cents)
---------------------
3 176.9
6 66.9
19 12.7
22 8.6
27 7.9
68 2.4
130 1.1
140 1.0
202 0.61
612 0.15
1547 0.040

ET list 2

Steps 7-limit
per RMS
octave error (cents)
---------------------
15 18.5
19 12.7
22 8.6
24 15.1
26 10.4
27 7.9
31 4.0
35 9.9
36 8.6
37 7.6
41 4.2

Do we really need to do the experiment? Paul?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Dave, if you don't have a cutoff, you'd have an infinite number of
> ETs better than 1547. Of course there has to be a cutoff.

Yes. This just shows that this isn't a very good badness metric.
A decent badness metric would not need a cutoff in anything but
badness in order to arrive at a finite list.

> I mean that only Gene's measure tells you exactly _how much_ better
a
> system is than the systems in their vicinity,

How do you know it does that? "Exactly"?

> _in units of_ the
> average differences between different systems in their vicinity.

I don't understand that bit. Can you explain.

> I'd like to see a list of ETs, as far as you'd like to take it,
above
> some cutoff different from Gene's, that shows this kind of behavior
> (not just the flatness of the measure itself, but also the flatness
> of the size of the wiggles).

But why ever do you think the size of the wiggles should be flat? I
think it is quite expected that the size of the wiggles in badness
around 1-tET to 9-tET are _much_ bigger than the wiggles around 60-tET
to 69-tET. Apparently you agree that the wiggles around 100000-tET are
completely irrelevant, since you're happy to have a cutoff in
steps, somewhere below that.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> ET list 2
>
> Steps 7-limit
> per RMS
> octave error (cents)
> ---------------------
> 15 18.5
> 19 12.7
> 22 8.6
> 24 15.1
> 26 10.4
> 27 7.9
> 31 4.0
> 35 9.9
> 36 8.6
> 37 7.6
> 41 4.2

If you're going to do this, let's at least do it right and use the
right list:

1 884.3587134
2 839.4327178
4 647.3739047
5 876.4669184
9 920.6653451
10 955.6795096
12 910.1603254
15 994.0402775
31 580.7780905
41 892.0787789
72 892.7193923
99 716.7738001
171 384.2612749
270 615.9368489
342 968.2768986
441 685.5766666
1578 989.4999106

The first point to note is that the two lists are clearly not
intended to do the same thing. The second is that while you object to
this characterization, your list seems to want to do our thinking for
us more than mine; you've decided the important place to look is
around 27. The third thing to notice is that if you want to look at a
limited range, you always can. Suppose I look from 10 to 50 and see
what the top 11 are, using my measure:

10 .796
12 .758
15 .828
16 1.113
19 .906
22 .898
26 1.122
27 .924
31 .484
41 .743
46 1.181

I'm afraid I like this list better than yours, but your milage may
vary.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > ET list 2
> >
> > Steps 7-limit
> > per RMS
> > octave error (cents)
> > ---------------------
> > 15 18.5
> > 19 12.7
> > 22 8.6
> > 24 15.1
> > 26 10.4
> > 27 7.9
> > 31 4.0
> > 35 9.9
> > 36 8.6
> > 37 7.6
> > 41 4.2
>
> If you're going to do this, let's at least do it right and use the
> right list:
>
> 1 884.3587134
> 2 839.4327178
> 4 647.3739047
> 5 876.4669184
> 9 920.6653451
> 10 955.6795096
> 12 910.1603254
> 15 994.0402775
> 31 580.7780905
> 41 892.0787789
> 72 892.7193923
> 99 716.7738001
> 171 384.2612749
> 270 615.9368489
> 342 968.2768986
> 441 685.5766666
> 1578 989.4999106

But this doesn't look like an approximate isobad. It looks like a list
of ETs less than a certain badness. i.e. it's a top 17. Right?

We can do it that way if you like. So I'll have to give my top 17. I
wasn't proposing that we give the badness measure (since it was meant
to be an isobad). But I guess we could if it's a top 17. However I
don't want people distracted by 9 significant digits of badness.
Couldn't we normalise to a 10 point scale and only give whole numbers.
And you need to supply the RMS error.

> The first point to note is that the two lists are clearly not
> intended to do the same thing.

Mine is intended to pack the maximum number of ETs likely to be of
interest to musicians, composers, music theorists etc. who are
interested in 7-limit music, into a list of a given size. Maybe you
need to explain what yours is intended to do.

> The second is that while you object to
> this characterization, your list seems to want to do our thinking
for
> us more than mine; you've decided the important place to look is
> around 27.

Not at all. It just comes out that way. I simply decided that an extra
note per octave was worth about the same badness as an increase of 0.5
cent in the RMS error. This comes thru experience and tuning list
discussions.

> The third thing to notice is that if you want to look at
a
> limited range, you always can. Suppose I look from 10 to 50 and see
> what the top 11 are, using my measure:
>
> 10 .796
> 12 .758
> 15 .828
> 16 1.113
> 19 .906
> 22 .898
> 26 1.122
> 27 .924
> 31 .484
> 41 .743
> 46 1.181

Sure. I can do that too.

> I'm afraid I like this list better than yours, but your milage may
> vary.

I might like it better than mine too. Mine's still got problems. But
you had to arbitrarily limit it to 10<n<50 to get this list. This is
clearly doing our thinking for us.

I thought we we're talking about a single published list, not a piece
of software that lets you enter your favourite limits.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> But this doesn't look like an approximate isobad. It looks like a
list
> of ETs less than a certain badness. i.e. it's a top 17. Right?

Right, but your list looked like a top 11 in a certain range also.

>
> We can do it that way if you like. So I'll have to give my top 17.
I
> wasn't proposing that we give the badness measure (since it was
meant
> to be an isobad).

The things on your list didn't make sense to me as an isobad, and I
didn't know that was what it was supposed to be. Trying a top n and
comparing makes more sense to me, but I need to pick a range.

> Mine is intended to pack the maximum number of ETs likely to be of
> interest to musicians, composers, music theorists etc. who are
> interested in 7-limit music, into a list of a given size.

It needs work.

Maybe you
> need to explain what yours is intended to do.

Mine is intended to show what the relatively best 7-limit ets are, in
a measurement which has the logarithmic flatness I describe in
another posting.

> I might like it better than mine too. Mine's still got problems.
But
> you had to arbitrarily limit it to 10<n<50 to get this list. This
is
> clearly doing our thinking for us.

And I can reduce that problem to essentially nil, by putting in a
high cut-off and leaving it at that. You are stuck with it as an
intrinsic feature.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > But this doesn't look like an approximate isobad. It looks like a
> list
> > of ETs less than a certain badness. i.e. it's a top 17. Right?
>
> Right, but your list looked like a top 11 in a certain range also.

It happens to also be the top 11 by the 0.5*steps + cents metric, but
not limited to any range.

> > We can do it that way if you like. So I'll have to give my top 17.
> I
> > wasn't proposing that we give the badness measure (since it was
> meant
> > to be an isobad).
>
> The things on your list didn't make sense to me as an isobad,

Obviously they wouldn't, given what your isobad looked like.

> and I
> didn't know that was what it was supposed to be.

I thought I made that pretty clear.

> Trying a top n and
> comparing makes more sense to me,

Fine.

> but I need to pick a range.

Objectively of course. Ha ha. If you have to pick a range then your
so-called badness metric obviously isn't really a badness metric at
all!

> > Mine is intended to pack the maximum number of ETs likely to be of
> > interest to musicians, composers, music theorists etc. who are
> > interested in 7-limit music, into a list of a given size.
>
> It needs work.

I think I said that.

> Mine is intended to show what the relatively best 7-limit ets are,
in
> a measurement which has the logarithmic flatness I describe in
> another posting.

Even if you and Paul are the only folks on the planet who find that
interesting? In that case I think its very misleading to call it a
badness metric when it only gives relative badness _locally_.

> > I might like it better than mine too. Mine's still got problems.
> But
> > you had to arbitrarily limit it to 10<n<50 to get this list. This
> is
> > clearly doing our thinking for us.
>
> And I can reduce that problem to essentially nil, by putting in a
> high cut-off and leaving it at that.

How high? How will this fix the problem that folks will assume you're
saying that 3-tET and 1547-tET are about as useful as 22-tET for
7-limit.

> You are stuck with it as an
> intrinsic feature.

And a damn fine feature it is too. :-) Seriously, mine was proposed
without any great amount of research or deliberation to show that it
is easy to find alternatives that do _much_ better than yours
_globally_ and about the same _locally_.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > But this doesn't look like an approximate isobad. It looks like
a
> > list
> > > of ETs less than a certain badness. i.e. it's a top 17. Right?
> >
> > Right, but your list looked like a top 11 in a certain range also.
>
> It happens to also be the top 11 by the 0.5*steps + cents metric,
but
> not limited to any range.

You could describe my top 11 in the range from 10 to 50 as the top 11
using a measure which multipies by a function equal to 1 from 10 to
50, and 10^n otherwise, which we multiply by our badness measure and
so end up with a top 11 "not limited by range". The difference is
that you have blurry outlines to your chosen region, which seems to
me to be a bad thing, not a good one. It allows you to imagine you
have not chosen a range, which hardly clarifies matters, since in
effect you have.

> Objectively of course. Ha ha. If you have to pick a range then your
> so-called badness metric obviously isn't really a badness metric at
> all!

See above; I can screw it up in an _ad hoc_ way and make it a screwed-
up, _ad hoc_ measure also, but why should I want to?

> Even if you and Paul are the only folks on the planet who find that
> interesting? In that case I think its very misleading to call it a
> badness metric when it only gives relative badness _locally_.

Global relative badness means what, exactly? This makes no sense to
me.

> How high? How will this fix the problem that folks will assume
you're
> saying that 3-tET and 1547-tET are about as useful as 22-tET for
> 7-limit.

I think you would be one of the very few who looked at it that way.
After all, this is hardly the first time such a thing has been done.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Even if you and Paul are the only folks on the planet who find
that
> > interesting? In that case I think its very misleading to call it a
> > badness metric when it only gives relative badness _locally_.
>
> Global relative badness means what, exactly? This makes no sense to
> me.

It means if two ETs have around the same badness number then are are
about as bad as each other, no matter how far apart they are on the
spectrum.

> > How high? How will this fix the problem that folks will assume
> you're
> > saying that 3-tET and 1547-tET are about as useful as 22-tET for
> > 7-limit.
>
> I think you would be one of the very few who looked at it that way.
> After all, this is hardly the first time such a thing has been done.

Ok. So I'm the only person who will assume that two ETs with about the
same badness number are roughly as bad as each other. In that case, I
shant bother you any more. We are apparently speakimg different
languages.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Global relative badness means what, exactly? This makes no sense
to
> > me.
>
> It means if two ETs have around the same badness number then are
are
> about as bad as each other, no matter how far apart they are on the
> spectrum.

This strikes me as subjective to the point of being meaningless.

Gene wrote:
> I don't know what good Maple code will do, but here it is:
>
> findcoms := proc(l)
> local p,q,r,p1,q1,r1,s,u,v,w;

More descriptive variable names might help. Is l the wedge invariant?

> s := igcd(l[1], l[2], l[6]);
> u := [l[6]/s, -l[2]/s, l[1]/s,0];

Presumably this is simplifying the octave-equivalent part?

> v := [p,q,r,1];

What values do p, q and r have? Is it important?

> w := w7l(u,v);

> "w7l" takes two vectors representing intervals, and computes the
> wegdge product.

So w is the wedge product of u and v, whatever they are.

> s := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l[6]-w
> [6]});

> "isolve" gives integer solutions to a linear
> equation;

Oh, that sounds useful.

> s := subs(_N1=0,s);

> I get an undeterminded varable "_N1" in this way which I
> can set equal to any integer, so I set it to 0.

Okay.

> p1 := subs(s,p);
> q1 := subs(s,q);
> r1 := subs(s,r);

What about this?

> v := 2^p1 * 3^q1 * 5^r1 * 7;

And here ^ is exponentiation instead of a wedge product.

> if v < 1 then v := 1/v fi;

So v must be a ratio, and you want it to be ascending.

> w := 2^u[1] * 3^u[2] * 5^u[3];
> if w < 1 then w := 1/w fi;

Same for w.

> [w, v] end:

And that's the result, is it? Two unison vectors?

> coms := proc(l)
> local v;
> v := findcoms(l);
> com7(v[1],v[2]) end:

> The pair of unisons
> returned in this way can be LLL reduced by the "com7" function, which
> takes a pair of intervals and LLL reduces them.

That makes sense. Return the reduced results of the other function.

> "w7l" takes two vectors representing intervals, and computes the
> wegdge product. "isolve" gives integer solutions to a linear
> equation; I get an undeterminded varable "_N1" in this way which I
> can set equal to any integer, so I set it to 0. The pair of unisons
> returned in this way can be LLL reduced by the "com7" function, which
> takes a pair of intervals and LLL reduces them.

Looks like the magic is being done by "isolve" which I presume is built-in
to Maple.

Graham

--- In tuning-math@y..., graham@m... wrote:
> Gene wrote:
> > I don't know what good Maple code will do, but here it is:
> >
> > findcoms := proc(l)
> > local p,q,r,p1,q1,r1,s,u,v,w;
>
> More descriptive variable names might help. Is l the wedge
invariant?

Yes.

> > s := igcd(l[1], l[2], l[6]);
> > u := [l[6]/s, -l[2]/s, l[1]/s,0];
>
> Presumably this is simplifying the octave-equivalent part?

"s" is the gcd of the first, second and sixth coordinates of the
wedgie, these are the ones used to construct the 5-limit comma. I
divide out by s, and get u, which is a vector representing this comma.

> > v := [p,q,r,1];
>
> What values do p, q and r have? Is it important?

p, q, and r are indeterminates, and the "1" above should be "s", the
gcd I obtained before.

Here is a more recent version, which should be used instead of the
old one as a reference:

findcoms := proc(l)
local p,q,r,p1,q1,r1,s,t,u,v,w;
s := igcd(l[1], l[2], l[6]);
u := [l[6]/s, -l[2]/s, l[1]/s,0];
v := [p,q,r,s];
w := w7l(u,v);
t := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l[6]-w
[6]});
t := subs(_N1=0,t);
p1 := subs(t,p);
q1 := subs(t,q);
r1 := subs(t,r);
v := 2^p1 * 3^q1 * 5^r1 * 7^s;
if v < 1 then v := 1/v fi;
w := 2^u[1] * 3^u[2] * 5^u[3];
if w < 1 then w := 1/w fi;
[w, v] end:

> So w is the wedge product of u and v, whatever they are.

Right, and "u" is the 5-limit comma, while "v" is undetermined aside
from the fact that the power of 7 is "s".

> > s := isolve({l[1]-w[1],l[2]-w[2],l[3]-w[3],l[4]-w[4],l[5]-w[5],l
[6]-w
> > [6]});
>
> > "isolve" gives integer solutions to a linear
> > equation;
>
> Oh, that sounds useful.

It is; a linear Diophantine equation routine would be a good thing to
acquire.

> > p1 := subs(s,p);
> > q1 := subs(s,q);
> > r1 := subs(s,r);
>
> What about this?

I've now re-named "s" (bad programming style if I was going to
publish the code, but I didn't write it with that in mind) to be the
set of solutions of the linear Diophantine equation. In my newer
version, that is "t"; t is a particular solution, and I substitute
this solution into the indeterminates, getting a specific value. It's
Maple-specific idiocy, and you would no doubt do something different
using Python.

> > v := 2^p1 * 3^q1 * 5^r1 * 7;
>
> And here ^ is exponentiation instead of a wedge product.

Right, and 7 should be "7^s".

> > if v < 1 then v := 1/v fi;
>
> So v must be a ratio, and you want it to be ascending.

I just like to standardize things.

> > w := 2^u[1] * 3^u[2] * 5^u[3];
> > if w < 1 then w := 1/w fi;
>
> Same for w.
>
> > [w, v] end:
>
> And that's the result, is it? Two unison vectors?

Correct; two unison vectors free of torsion problems which define the
linear temperament.

> Looks like the magic is being done by "isolve" which I presume is
built-in
> to Maple.

It's a built-in Maple function; however much of the magic can still
be had by solving the system over the rationals, because part of the
magic was to start out in such a way that torsion problems would be
exterminated. One way to solve a linear Diophantine system is to
solve over the rationals, and then solve the congruence conditions
required to give an integer solution, in fact. You might look in
Niven and Zuckerman if you have a copy for linear Diophantine
equations.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Dave, if you don't have a cutoff, you'd have an infinite number
of
> > ETs better than 1547. Of course there has to be a cutoff.
>
> Yes. This just shows that this isn't a very good badness metric.
> A decent badness metric would not need a cutoff in anything but
> badness in order to arrive at a finite list.
>
> > I mean that only Gene's measure tells you exactly _how much_
better
> a
> > system is than the systems in their vicinity,
>
> How do you know it does that? "Exactly"?

Sure, in a limit-probability sense. How many digits did Gene report?
Anyhow, I'll have to refer you to Gene on the details of how it does
that.

I'd just like this paper to have some very simple systems with large
errors, where a combined adaptive-tuning & adaptive-timbre approach
would be needed, as well as systems to satisfy people like Rami
Vitale, for whom even the _melodic_ distinctions of 225:224 cannot be
tempered out.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> But why ever do you think the size of the wiggles should be flat? I
> think it is quite expected that the size of the wiggles in badness
> around 1-tET to 9-tET are _much_ bigger than the wiggles around 60-
tET
> to 69-tET.

The two ranges would gave to be the same size logarithmically, for
example 1-tET to 9-tET and 10-tET to 90-tET.