[MMM] Re: various - any tuning is potentially a good tuning - for someone.

--- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

>> When you get up to scales, even this clarity criterion no longer
>> *necessarily* maximized by JI tuning. IMHO.

Gene wrote,

>Could you expand on this

Well, I've used the 5-limit pentatonic and diatonic examples many
times in the past. It's essentially the whole point of temperament,
so surely you know what I mean (i.e., the *wolves* in those scales
are very "unclear", but spreading the error around makes a larger
number of intervals "clear" than would be possible in JI). In fact,
in the archives of this list you can find my experiments with
my "harmonic entropy minimizer", where many small scales were drawn
into Just Intonation, while larger scales would be drawn into
somewhat irregular temperaments.

>(possibly on tuning, which MMM is rapidly
>becoming.)

There's no reason for that. It's not like the traffic is so heavy
here that we need to rely on the spin-off lists to alleviate it (at
the moment . . .)

on 4/29/04 4:47 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
>>> When you get up to scales, even this clarity criterion no longer
>>> *necessarily* maximized by JI tuning. IMHO.
>
> Gene wrote,
>
>> Could you expand on this
>
> Well, I've used the 5-limit pentatonic and diatonic examples many
> times in the past. It's essentially the whole point of temperament,
> so surely you know what I mean (i.e., the *wolves* in those scales
> are very "unclear", but spreading the error around makes a larger
> number of intervals "clear" than would be possible in JI). In fact,
> in the archives of this list you can find my experiments with
> my "harmonic entropy minimizer", where many small scales were drawn
> into Just Intonation, while larger scales would be drawn into
> somewhat irregular temperaments.

Curious, I was just wondering something related to this, and you gave me the
language to ask the question.

Are there answers to the questions:

What is the 12-tone scale with the most clear intervals?

What is the 12-tone scale that maximizes the clarity of the least clear
interval?

In both cases it is implicit that there is no other restriction or goal for
the resulting tuning. What I was wondering yesterday is what tuning would I
use if I wanted all of the intervals to be as just as possible, and I was
thinking maybe this would translate to having the lowest odd limit.
Basically I was wondering whether it would be possible to "improve" (in this
particular sense) on 5-limit 12-tone tunings by raising the limit of the
lowest-limit intervals in order to lower the limit of the higest-limit
intervals. So in other words I'd expect to lose some pure 5ths and major
3rds compared to, say, Duodene. But I have the suspicion that Duodene or
something similar may already be the optimal in this regard. But I'm still
hoping for a surprise.

As stated, the above questions are asking for a pure just solution. What
could be achieved by allowing tempering with similar goals in mind?

>> (possibly on tuning, which MMM is rapidly
>> becoming.)
>
> There's no reason for that. It's not like the traffic is so heavy
> here that we need to rely on the spin-off lists to alleviate it (at
> the moment . . .)

Thanks for posting this here.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 4/29/04 4:47 PM, wallyesterpaulrus <paul@s...> wrote:
>
> > --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> >
> >>> When you get up to scales, even this clarity criterion no longer
> >>> *necessarily* maximized by JI tuning. IMHO.
> >
> > Gene wrote,
> >
> >> Could you expand on this
> >
> > Well, I've used the 5-limit pentatonic and diatonic examples many
> > times in the past. It's essentially the whole point of
temperament,
> > so surely you know what I mean (i.e., the *wolves* in those scales
> > are very "unclear", but spreading the error around makes a larger
> > number of intervals "clear" than would be possible in JI). In
fact,
> > in the archives of this list you can find my experiments with
> > my "harmonic entropy minimizer", where many small scales were
drawn
> > into Just Intonation, while larger scales would be drawn into
> > somewhat irregular temperaments.
>
> Curious, I was just wondering something related to this, and you
gave me the
> language to ask the question.
>
> Are there answers to the questions:
>
> What is the 12-tone scale with the most clear intervals?
>
> What is the 12-tone scale that maximizes the clarity of the least
clear
> interval?

It depends on exactly what you mean and how you want to define and
constrain things . . . but my harmonic entropy minimizer gives 12-
equal (actually it gave some funny well-temperaments, but it was
simply "getting stuck" in the high-dimensional landscape). This is no
surprise since it treats all intervals equally, not preferring some
over others.

For a 7-note tuning, it gave something like a meantone diatonic
scale, but the fifths were tempered a little more in the middle of
the chain and a little less toward the ends.

Same thing for a 5-note tuning, IIRC.

> In both cases it is implicit that there is no other restriction or
goal for
> the resulting tuning. What I was wondering yesterday is what
tuning would I
> use if I wanted all of the intervals to be as just as possible, and
I was
> thinking maybe this would translate to having the lowest odd limit.

Is the "odd limit" of 3000001:2000001 equal to 3000001? Is that a
reasonable assessment of this interval's "clarity"? How
about "justness"?

> Basically I was wondering whether it would be possible to "improve"
(in this
> particular sense) on 5-limit 12-tone tunings by raising the limit
of the
> lowest-limit intervals in order to lower the limit of the higest-
limit
> intervals.

That doesn't seem likely. If you're talking about JI, the most
compact scales -- the ones where the maximum complexity or "odd
limit" of any interval is kept as low as possible -- are loaded with
low-limit intervals.

> So in other words I'd expect to lose some pure 5ths and major
> 3rds compared to, say, Duodene. But I have the suspicion that
Duodene or
> something similar may already be the optimal in this regard.

The Duodene is certainly nothing special in this regard. There was a
thread on this recently on one of the lists. "Marpurg's Monochord #1"
and other 5-prime-limit JI options look equally appealing from
various angles.

And of course these tuning systems had a historical importance of
approximately nil.

Anyhoo . . .

If you want to stick with a pure JI philosophy, such that
3000001:2000001 gets rated as far more complex than 3:2, then the
best tool for solving these kinds of problems is the lattice.

If you want to be more realistic, then you should use a smooth,
continuous function of interval size that has local minima at the
simple ratios (harmonic entropy works great for this purpose) and
software that can find minima of functions of several variables (I'm
lucky enough to be provided with Matlab by my employer . . .).

on 4/29/04 7:19 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

>> Curious, I was just wondering something related to this, and you
> gave me the
>> language to ask the question.
>>
>> Are there answers to the questions:
>>
>> What is the 12-tone scale with the most clear intervals?
>>
>> What is the 12-tone scale that maximizes the clarity of the least
> clear
>> interval?
>
> It depends on exactly what you mean and how you want to define and
> constrain things . . . but my harmonic entropy minimizer gives 12-
> equal (actually it gave some funny well-temperaments, but it was
> simply "getting stuck" in the high-dimensional landscape). This is no
> surprise since it treats all intervals equally, not preferring some
> over others.

Ah, how dissappointing!

> For a 7-note tuning, it gave something like a meantone diatonic
> scale, but the fifths were tempered a little more in the middle of
> the chain and a little less toward the ends.
>
> Same thing for a 5-note tuning, IIRC.
>
>> In both cases it is implicit that there is no other restriction or
> goal for
>> the resulting tuning. What I was wondering yesterday is what
> tuning would I
>> use if I wanted all of the intervals to be as just as possible, and
> I was
>> thinking maybe this would translate to having the lowest odd limit.
>
> Is the "odd limit" of 3000001:2000001 equal to 3000001? Is that a
> reasonable assessment of this interval's "clarity"? How
> about "justness"?

Well, I suppose "Yes" in the case when a pure just solution is asked for.
"No" when tempering is allowed. That was the destinction I was trying to
make.

>> Basically I was wondering whether it would be possible to "improve"
> (in this
>> particular sense) on 5-limit 12-tone tunings by raising the limit
> of the
>> lowest-limit intervals in order to lower the limit of the higest-
> limit
>> intervals.
>
> That doesn't seem likely. If you're talking about JI, the most
> compact scales -- the ones where the maximum complexity or "odd
> limit" of any interval is kept as low as possible -- are loaded with
> low-limit intervals.

Maybe I should ask what *you* mean by low-limit. But in reality I should
probably have thought more before I posted. It was just the coincidence
that tempted me.

>> So in other words I'd expect to lose some pure 5ths and major
>> 3rds compared to, say, Duodene. But I have the suspicion that
> Duodene or
>> something similar may already be the optimal in this regard.
>
> The Duodene is certainly nothing special in this regard.

Do you mean "nothing special but not particularly bad either"?

> There was a
> thread on this recently on one of the lists. "Marpurg's Monochord #1"
> and other 5-prime-limit JI options look equally appealing from
> various angles.

5-prime-limit is not much of a constraint. Or rather it would seem to me it
constrains the wrong thing. I guess I wanted to allow more 11-odd or 13-odd
intervals (perhaps) if in the process I could get rid of anything higher,
maybe even 25:16 which it strikes me is in some sense "not very just" and
also "not very clear" except perhaps in an otonal chord where in combination
with other things it reinforces the implied fundamental. But my
inexperience is probably showing. And the scale I was *imagining* might in
fact have implied a reduction in the otonal possibilities.

> And of course these tuning systems had a historical importance of
> approximately nil.
>
> Anyhoo . . .
>
> If you want to stick with a pure JI philosophy, such that
> 3000001:2000001 gets rated as far more complex than 3:2, then the
> best tool for solving these kinds of problems is the lattice.

The lattice gets unwieldy beyond 2 or 3 dimensions.

> If you want to be more realistic,

Probably that is the ultimate goal anyway. ;)

> then you should use a smooth,
> continuous function of interval size that has local minima at the
> simple ratios (harmonic entropy works great for this purpose) and
> software that can find minima of functions of several variables (I'm
> lucky enough to be provided with Matlab by my employer . . .).

Well I have a version of Mathematica from a few years back, but I'm not very
proficient with it. :(

Thanks for the reality check. I was hoping someone else had already
discovered a particular "grail" that I could just plop into my software.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 4/29/04 7:19 PM, wallyesterpaulrus <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> >> Curious, I was just wondering something related to this, and you
> > gave me the
> >> language to ask the question.
> >>
> >> Are there answers to the questions:
> >>
> >> What is the 12-tone scale with the most clear intervals?
> >>
> >> What is the 12-tone scale that maximizes the clarity of the least
> > clear
> >> interval?
> >
> > It depends on exactly what you mean and how you want to define and
> > constrain things . . . but my harmonic entropy minimizer gives 12-
> > equal (actually it gave some funny well-temperaments, but it was
> > simply "getting stuck" in the high-dimensional landscape). This
is no
> > surprise since it treats all intervals equally, not preferring
some
> > over others.
>
> Ah, how dissappointing!
>
> > For a 7-note tuning, it gave something like a meantone diatonic
> > scale, but the fifths were tempered a little more in the middle of
> > the chain and a little less toward the ends.
> >
> > Same thing for a 5-note tuning, IIRC.

Are these disappointments, too?

> >> So in other words I'd expect to lose some pure 5ths and major
> >> 3rds compared to, say, Duodene. But I have the suspicion that
> > Duodene or
> >> something similar may already be the optimal in this regard.
> >
> > The Duodene is certainly nothing special in this regard.
>
> Do you mean "nothing special but not particularly bad either"?

Right.

> > There was a
> > thread on this recently on one of the lists. "Marpurg's Monochord
#1"
> > and other 5-prime-limit JI options look equally appealing from
> > various angles.
>
> 5-prime-limit is not much of a constraint. Or rather it would seem
to me it
> constrains the wrong thing. I guess I wanted to allow more 11-odd
or 13-odd
> intervals (perhaps) if in the process I could get rid of anything
higher,
> maybe even 25:16 which it strikes me is in some sense "not very
just" and
> also "not very clear"

The thing is, 25:16 will tend to arise from a stack of two 5:4s, and
if you want to alter 25:16 to something simpler, you'll be forced to
alter one of the 5:4s to something more complex.

But sure, depending on your ears, one can make the scale have
more "consonances" by increasing the prime limit. An old thread that
was revived here fairly recently concerned the number of 7-limit
consonances possible in a 12-note JI scale. Paul Hahn had found 48
examples with 32 of them, hundreds with 31 of them, and hundreds more
with 30 of them. Aaron Johnson took one of the first 48 and made
music with it. Anyway, 32 7-limit consonances is quite a few (8?)
more than you could get if you restricted yourself to a prime limit
of 5 -- though in the latter case, all the 7-limit consonances would
also be 5-limit consonances.

> > And of course these tuning systems had a historical importance of
> > approximately nil.
> >
> > Anyhoo . . .
> >
> > If you want to stick with a pure JI philosophy, such that
> > 3000001:2000001 gets rated as far more complex than 3:2, then the
> > best tool for solving these kinds of problems is the lattice.
>
> The lattice gets unwieldy beyond 2 or 3 dimensions.

Visually, yes. Mathematically, it continues to be useful.

> Thanks for the reality check. I was hoping someone else had already
> discovered a particular "grail" that I could just plop into my
>software.

It's likely. Or perhaps, as you make your desires more specific, it
will be discovered for you.

on 4/29/04 8:08 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 4/29/04 7:19 PM, wallyesterpaulrus <paul@s...> wrote:

>>> For a 7-note tuning, it gave something like a meantone diatonic
>>> scale, but the fifths were tempered a little more in the middle of
>>> the chain and a little less toward the ends.
>>>
>>> Same thing for a 5-note tuning, IIRC.
>
> Are these disappointments, too?

No, actually quite interesting. I just lost track of it in the process of
quoting and replying. But besides dissappointment, I'm initially surprised
that 12-tone went toward total symmetry rather than away from it. It might
be interesting to look at the general result as a function of N. For which
values of N does the result tend toward ET? For which values does a
periodic symmetry occur? Etc. Some interesting patterns might emerge.

I'm definitely not going to be fully satisfied if I can't learn how to do
some of my own numerical research. Has anyone used Mathematica for RMS
optimization kinds of solutions?

>>> There was a
>>> thread on this recently on one of the lists. "Marpurg's Monochord
> #1"
>>> and other 5-prime-limit JI options look equally appealing from
>>> various angles.
>>
>> 5-prime-limit is not much of a constraint. Or rather it would seem
> to me it
>> constrains the wrong thing. I guess I wanted to allow more 11-odd
> or 13-odd
>> intervals (perhaps) if in the process I could get rid of anything
> higher,
>> maybe even 25:16 which it strikes me is in some sense "not very
> just" and
>> also "not very clear"
>
> The thing is, 25:16 will tend to arise from a stack of two 5:4s,

I was stacking 5:4s all day and I *kept* getting 25:16. Remarkable! ;)

> and
> if you want to alter 25:16 to something simpler, you'll be forced to
> alter one of the 5:4s to something more complex.

Yes, that's exactly what I was looking for! Or so I thought. But...

Only now does it hit me that this implies I am looking for a temperament,
not a just solution. Because if I replace one of the 5:4s with an 11:8 for
example stacking yields 55:32 which I want to function as a 5:3. So I would
temper everything to make the best of that. Maybe that's a slightly ugly
example but since I don't know yet what to expect it is not out of the realm
of possibility.

My mistake was to think that there would be a way to improve the 25:16 while
not making one of the 5:4's as *bad* as the 25:16. I think now that a just
solution probably just pushes the complexity around.

> But sure, depending on your ears, one can make the scale have
> more "consonances" by increasing the prime limit. An old thread that
> was revived here fairly recently concerned the number of 7-limit
> consonances possible in a 12-note JI scale.

Yes, searching for "Hahn" I think I've found this stuff including a message
by you on 10/29/03 that refers to:

http://www.sonic-arts.org/td/1599.htm#1599-9

hmm wonder how long *that* link will continue to work.

In some of these messages the issue of "consistency" came up, and I don't
understand it yet but I'm wondering whether it is related to my desire to
flatten out the odd-limit variations across all the intervals of a scale
which I have been talking about here.

> Paul Hahn had found 48
> examples with 32 of them, hundreds with 31 of them, and hundreds more
> with 30 of them. Aaron Johnson took one of the first 48 and made
> music with it. Anyway, 32 7-limit consonances is quite a few (8?)
> more than you could get if you restricted yourself to a prime limit
> of 5 -- though in the latter case, all the 7-limit consonances would
> also be 5-limit consonances.

Yes, I see no reason not to leave the prime limit completely open. I'm not
sure why you even bring up prime limit. I see you have "consonances" in
quotes, and I certainly didn't want to "stretch" the meaning of consonance
to a concept that is merely mathematical and not audible, although perhaps
the word consonance is a bit that way, as opposed to (Monz's?) word
"accordance".

Except it remains an unknown to me what it takes to create what I might call
"otonal clarify" meaning (tentatively) the degree to which a single or a
very small number (like 2) of implied fundamentals are clearly present. But
my guess is that this still relates more to the odd limit than to the prime
limit, which in its unboundedness seems to have no predictive value at all
in relation to consonance, clarity, justness, whatever.

It would be good to have a better handle on something that relates to this
"otonal clarify" in order to have a practical sense of how audible
consonance is generated in a chord. You didn't bite last time, so I'm
asking again more carefully. I know I'm grasping at something and I don't
know for sure whether there would end up being any practicality to what I am
grasping for. I'm guessing that harmonic entropy might deal with it but I'm
not sure whether harmonic entropy predicts anything about clarity of implied
fundamental in a chord. I don't recall this from when I perused H.E.
material in the past, but that was a while ago. I'm not at all sure that
clarify of implied fundamental is any kind of grail either, but I have a
strong inclination to believe that there is *something* beyond individual
intervals that needs to be understood to understand the "consonance" of a
chord, and the presence of implied fundamental is currently the only thing I
have experienced that appears to relate to my perception of "justness".

As far as appropriateness of the above material for this list, I'm not sure
whether it is on the one hand "too newbie/nieve" to interest most people or
"too theoretical" and belonging on that basis more on tuning math. I *can*
say that I'm very confortable in this terrain which is both very close to my
experience, very tied to a very intuitively active realm for me, and also
leading at least *me* to further clarification of the language I use, which
I hope will have the effect that it will enhance my capacity to communicate
with others about things that can be experienced. But if nobody else chimes
in maybe I should assume this belongs in private communication with Paul?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> No, actually quite interesting. I just lost track of it in the
process of
> quoting and replying. But besides dissappointment, I'm initially
surprised
> that 12-tone went toward total symmetry rather than away from it.
It might
> be interesting to look at the general result as a function of N.
For which
> values of N does the result tend toward ET? For which values does a
> periodic symmetry occur? Etc. Some interesting patterns might
emerge.

It's going to depend on what consonance fuction you use, of course.
I'm surprised that a 9-odd-limit solution would not converge to an
irregular meantone, getting in the subminor and supermajor thirds
resulting from that.
> > The thing is, 25:16 will tend to arise from a stack of two 5:4s,
>
> I was stacking 5:4s all day and I *kept* getting 25:16.
Remarkable! ;)
>
> > and
> > if you want to alter 25:16 to something simpler, you'll be forced
to
> > alter one of the 5:4s to something more complex.
> Only now does it hit me that this implies I am looking for a
temperament,
> not a just solution. Because if I replace one of the 5:4s with an
11:8 for
> example stacking yields 55:32 which I want to function as a 5:3.
So I would
> temper everything to make the best of that. Maybe that's a slightly
ugly
> example but since I don't know yet what to expect it is not out of
the realm
> of possibility.
>
> My mistake was to think that there would be a way to improve the
25:16 while
> not making one of the 5:4's as *bad* as the 25:16. I think now
that a just
> solution probably just pushes the complexity around.

The most obvious solution is to equate 25/16 with 14/9, and temper
out 225/224. The major thirds resulting from this are not at all bad.

> It would be good to have a better handle on something that relates
to this
> "otonal clarify" in order to have a practical sense of how audible
> consonance is generated in a chord.

The augmented triad we get from marvel (225/224-planar) would be a
good starting point if someone wanted to try to figure out what the
heck otonal clarity was. Then there's utonals to worry about.

But if nobody else chimes
> in maybe I should assume this belongs in private communication with
Paul?

Please do not assume that.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 4/29/04 8:08 PM, wallyesterpaulrus <paul@s...> wrote:
> >
> > But sure, depending on your ears, one
> > can make the scale have more "consonances"
> > by increasing the prime limit. An old thread
> > that was revived here fairly recently
> > concerned the number of 7-limit consonances
> > possible in a 12-note JI scale.
>
> Yes, searching for "Hahn" I think I've found
> this stuff including a message by you on
> 10/29/03 that refers to:
>
> http://www.sonic-arts.org/td/1599.htm#1599-9
>
> hmm wonder how long *that* link will continue to work.

at least as long as Sonic Arts keeps paying for the website
... and longer, if that tiny Tuning List archive i created
migrates to Tonalsoft.

what's unfortunate is that i only archived a few weeks
of tuning list messages ... because it was later that
month (December 1998) that the list migrated from the
old Mills server, and everything since then has been
archived here on Yahoo. there are some of the old Mills
messages archived somewhere, but unfortunately not all
of them.

> Except it remains an unknown to me what it
> takes to create what I might call "otonal clarify"
> meaning (tentatively) the degree to which a single
> or a very small number (like 2) of implied
> fundamentals are clearly present. But my guess
> is that this still relates more to the odd limit
> than to the prime limit, which in its unboundedness
> seems to have no predictive value at all in
> relation to consonance, clarity, justness, whatever.
>
> It would be good to have a better handle on
> something that relates to this "otonal clarify"
> in order to have a practical sense of how audible
> consonance is generated in a chord. You didn't
> bite last time, so I'm asking again more carefully.
> I know I'm grasping at something and I don't
> know for sure whether there would end up being
> any practicality to what I am grasping for.
> I'm guessing that harmonic entropy might deal
> with it but I'm not sure whether harmonic entropy
> predicts anything about clarity of implied fundamental
> in a chord. I don't recall this from when I perused
> H.E. material in the past, but that was a while ago.
> I'm not at all sure that clarify of implied fundamental
> is any kind of grail either, but I have a strong
> inclination to believe that there is *something*
> beyond individual intervals that needs to be understood
> to understand the "consonance" of a chord, and the
> presence of implied fundamental is currently the
> only thing I have experienced that appears to relate
> to my perception of "justness".

i would say that you are definitely on the right track.
if i'm not mistaken, the leading edge on this theory
is by Terhardt. Paul can give references and/or corrections.

harmonic entropy theoretically can work on sonic
aggregates larger than intervals (which by definition
always have 2 notes), but AFAIK Paul wanted to define
H.E. for triads but never got around to it. Paul?

-monz

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 4/29/04 8:08 PM, wallyesterpaulrus <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >> on 4/29/04 7:19 PM, wallyesterpaulrus <paul@s...> wrote:
>
> >>> For a 7-note tuning, it gave something like a meantone diatonic
> >>> scale, but the fifths were tempered a little more in the middle
of
> >>> the chain and a little less toward the ends.
> >>>
> >>> Same thing for a 5-note tuning, IIRC.
> >
> > Are these disappointments, too?
>
> No, actually quite interesting. I just lost track of it in the
process of
> quoting and replying. But besides dissappointment, I'm initially
surprised
> that 12-tone went toward total symmetry rather than away from it.
It might
> be interesting to look at the general result as a function of N.
For which
> values of N does the result tend toward ET? For which values does a
> periodic symmetry occur? Etc. Some interesting patterns might
emerge.

By the time I got to 12, the program was getting stuck, so I didn't
push it much higher. But that was the only ET, IIRC. It's all here in
the archives . . .

> I'm definitely not going to be fully satisfied if I can't learn how
to do
> some of my own numerical research. Has anyone used Mathematica for
RMS
> optimization kinds of solutions?

I'm sure someone has.

> Only now does it hit me that this implies I am looking for a
temperament,
> not a just solution.

Phew! It's a lot easier to operate when you're not constrained by
unrealistic assumptions.

> Because if I replace one of the 5:4s with an 11:8 for
> example stacking yields 55:32 which I want to function as a 5:3.
>So I would
> temper everything to make the best of that.

Cool! This would require quite a large amount of tempering, but could
be done.

> Maybe that's a slightly ugly
> example but since I don't know yet what to expect it is not out of
the realm
> of possibility.

Before I reply further on this, I'll see what others have written to
you . . .

> In some of these messages the issue of "consistency" came up, and I
don't
> understand it yet but I'm wondering whether it is related to my
desire to
> flatten out the odd-limit variations across all the intervals of a
scale
> which I have been talking about here.

You'll have to show me the context, but most likely it was about ETs.

> > Paul Hahn had found 48
> > examples with 32 of them, hundreds with 31 of them, and hundreds
more
> > with 30 of them. Aaron Johnson took one of the first 48 and made
> > music with it. Anyway, 32 7-limit consonances is quite a few (8?)
> > more than you could get if you restricted yourself to a prime
limit
> > of 5 -- though in the latter case, all the 7-limit consonances
would
> > also be 5-limit consonances.
>
> Yes, I see no reason not to leave the prime limit completely open.
I'm not
> sure why you even bring up prime limit.

You brought up the Duodene (stricly 5-prime-limit) in this
discussion, so it seemed pertinent . . . Also, the "harmonic entropy
minimizer" only models dyadic consonance, so it might miss out on
some of the higher-odd-limit (and thus, often, higher-prime-limit)
intervals that become important in large otonal chords.

> I see you have "consonances" in
> quotes, and I certainly didn't want to "stretch" the meaning of
consonance
> to a concept that is merely mathematical and not audible, although
perhaps
> the word consonance is a bit that way, as opposed to (Monz's?) word
> "accordance".

Concordance.

> Except it remains an unknown to me what it takes to create what I
might call
> "otonal clarify" meaning (tentatively) the degree to which a single
or a
> very small number (like 2) of implied fundamentals are clearly
present.

Out of all the note-combinations possible in the scale? Then maybe
it's a very different type of scale you're looking for . . . though
the Duodene is certainly capable of at least 3 clear implied
fundamentals . . .

> But
> my guess is that this still relates more to the odd limit than to
the prime
> limit, which in its unboundedness seems to have no predictive value
at all
> in relation to consonance, clarity, justness, whatever.

True . . . however, it's often easier to solve these problems if
you're *given* a prime limit first . . .

> It would be good to have a better handle on something that relates
to this
> "otonal clarify" in order to have a practical sense of how audible
> consonance is generated in a chord. You didn't bite last time, so
I'm
> asking again more carefully.

So you're asking about chords? When comparing simple otonal JI ones,
I find the size of the numbers works great. But you rapidly reach the
limit of usefulness with that formula, particularly where tempered
chords are concerned.

Triadic and tetradic harmonic entropy (which would address this
directly) are things I am working on on the harmonic entropy list,
and the former is well-studied in part . . . I guess my work there is
halted for now since no one else wanted to get involved in the
calculations . . .

> I know I'm grasping at something and I don't
> know for sure whether there would end up being any practicality to
what I am
> grasping for. I'm guessing that harmonic entropy might deal with
it but I'm
> not sure whether harmonic entropy predicts anything about clarity
of implied
> fundamental in a chord.

It does for dyads. Triadic version to come.

> I don't recall this from when I perused H.E.
> material in the past, but that was a while ago. I'm not at all
sure that
> clarify

You keep saying that. You mean "clarity"?

? of implied fundamental is any kind of grail either, but I have a
> strong inclination to believe that there is *something* beyond
individual
> intervals that needs to be understood to understand
the "consonance" of a
> chord, and the presence of implied fundamental is currently the
only thing I
> have experienced that appears to relate to my perception
of "justness".

Yes, I think it's that *as well as* the individual intervals, as well
as the individual triads, etc. (if its a chord with more than 3
notes).

> As far as appropriateness of the above material for this list, I'm
not sure
> whether it is on the one hand "too newbie/nieve" to interest most
people or
> "too theoretical" and belonging on that basis more on tuning math.

It's right in line with the harmonic entropy list too.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> > No, actually quite interesting. I just lost track of it in the
> process of
> > quoting and replying. But besides dissappointment, I'm initially
> surprised
> > that 12-tone went toward total symmetry rather than away from
it.

>
> It's going to depend on what consonance fuction you use, of course.
> I'm surprised that a 9-odd-limit solution would not converge to an
> irregular meantone, getting in the subminor and supermajor thirds
> resulting from that.

I don't know what you mean by "9-odd-limit solution would not
converge to" . . . A "solution" would seem to refer to the tuning you
converge to, not to something that converges to a tuning.

For the consonance function, I used a harmonic entropy curve that did
have a tiny local minimum for 9:7, and none for more complex ratios.
You have to use a smooth function if you expect to converge to the
important solutions -- otherwise you get stuck in
insignificant "cracks".

I think the problem with the 12-tone irregular meantone would be that
wolf fifth. Eliminating the wolf probably means riding down the
gradient in most, if not all, situations, regardless of what's
happening with the other intervals.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i would say that you are definitely on the right track.
> if i'm not mistaken, the leading edge on this theory
> is by Terhardt. Paul can give references and/or corrections.

Something a little closer to the leading edge today:

http://homepage.mac.com/cariani/CarianiWebsite/TramoCarianiNYAS2001.pd
f

http://homepage.mac.com/cariani/CarianiWebsite/TramoHarmony.pdf

http://homepage.mac.com/cariani/CarianiWebsite/JNMR2001.pdf

> harmonic entropy theoretically can work on sonic
> aggregates larger than intervals (which by definition
> always have 2 notes), but AFAIK Paul wanted to define
> H.E. for triads but never got around to it. Paul?

The next edition of Sethares's _Tuning, Timbre, Spectrum, Scale_ will
detail the 2-note version, and the edition after that will detail the
3-note version. If people take a deep enough interest on the harmonic
entropy list, I'll do it there.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Only now does it hit me that this implies I am looking for a
temperament,
> not a just solution. Because if I replace one of the 5:4s with an
11:8 for
> example stacking yields 55:32 which I want to function as a 5:3.

So the 11:8 would be "identical" to a 4:3 . . . 33:32 is tempered out.

> So I would
> temper everything to make the best of that. Maybe that's a slightly
ugly
> example but since I don't know yet what to expect it is not out of
the realm
> of possibility.

It is a "slightly ugly example", but let's try it (the TOP way). If
you temper prime 2 to 1205.3 cents, prime 3 to 1893.55, and prime 11
to 4132.97 cents, then it works out. 4:3 = 2*1205.3 - 1893.55 =
517.06 cents, and 11:8 = 4132.97 - 3*1205.3 = 517.06 cents. Check!
Now, it what context can you hear 11:8 *as* 11:8? In a big otonal
chord, right? Try a big otonal chord with this tuning (the other
primes, but of course not their octave-reductions, are just) and see
if it sounds acceptable to you.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> I think the problem with the 12-tone irregular meantone would be that
> wolf fifth. Eliminating the wolf probably means riding down the
> gradient in most, if not all, situations, regardless of what's
> happening with the other intervals.

If your consonance function went up to the 13 limit and had a downtick
for 20/13 you might snag a version of ratwolf.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > I think the problem with the 12-tone irregular meantone would be
that
> > wolf fifth. Eliminating the wolf probably means riding down the
> > gradient in most, if not all, situations, regardless of what's
> > happening with the other intervals.
>
> If your consonance function went up to the 13 limit and had a
downtick
> for 20/13 you might snag a version of ratwolf.

True. It still wouldn't be the global minimum that 12-equal would
still probably be.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > If your consonance function went up to the 13 limit and had a
> downtick
> > for 20/13 you might snag a version of ratwolf.
>
> True. It still wouldn't be the global minimum that 12-equal would
> still probably be.

Why not? It seems to be it would likely depend on the consonance function.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > If your consonance function went up to the 13 limit and had a
> > downtick
> > > for 20/13 you might snag a version of ratwolf.
> >
> > True. It still wouldn't be the global minimum that 12-equal would
> > still probably be.
>
> Why not? It seems to be it would likely depend on the consonance
>function.

Let's try one which isn't smooth, and *does* have a local minimum for
20/13. Start with
/tuning/files/Erlich/george.txt
. The first 601 numbers are the dissonances for 0 cents - 600 cents.
There's a local minimum at 13/10. Now, for each interval in the
scale, use the smallest member of its octave-equivalence class, which
will necessarily be 600 cents or less. So 20/13 will be treated as
13/10.

Ignoring the unisons, a 12-note scale will have 66 intervals to check.

I get a sum of 441.51 for these 66 intervals in 12-equal.

What is the sum in ratwolf? Or give me the intervals and I'll do it.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> Ignoring the unisons, a 12-note scale will have 66 intervals to check.
>
> I get a sum of 441.51 for these 66 intervals in 12-equal.
>
> What is the sum in ratwolf? Or give me the intervals and I'll do it.

Ratwolf is a meantone with a fifth of (416/5)^(1/11), or 695.8376
cents, little different from 2/7-comma meantone. You of course get 11
ratwolf fifths, plus a wolf of 20/13. Do you want more than that, and
if so in what format?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > Ignoring the unisons, a 12-note scale will have 66 intervals to
check.
> >
> > I get a sum of 441.51 for these 66 intervals in 12-equal.
> >
> > What is the sum in ratwolf? Or give me the intervals and I'll do
it.
>
> Ratwolf is a meantone with a fifth of (416/5)^(1/11), or 695.8376
> cents, little different from 2/7-comma meantone. You of course get
11
> ratwolf fifths, plus a wolf of 20/13. Do you want more than that,
and
> if so in what format?

I get a sum, over the 66 intervals, of 441.52 -- an incredibly close
call.

The true global minimum is actually all 12 notes in unison, which
scores a 364.03.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > Ignoring the unisons, a 12-note scale will have 66 intervals to
> check.
> > >
> > > I get a sum of 441.51 for these 66 intervals in 12-equal.
> > >
> > > What is the sum in ratwolf? Or give me the intervals and I'll
do
> it.
> >
> > Ratwolf is a meantone with a fifth of (416/5)^(1/11), or 695.8376
> > cents, little different from 2/7-comma meantone. You of course
get
> 11
> > ratwolf fifths, plus a wolf of 20/13. Do you want more than that,
> and
> > if so in what format?
>
> I get a sum, over the 66 intervals, of 441.52 -- an incredibly
close
> call.

1/4-comma meantone gives 441.16.

This is probably because this consonance function, as you can see if
you plot it, has abs-exponential, not quadratic, minima. So tempering
out a comma won't improve things here, it'll just worsen them. So a
JI scale would look even better than 1/4-comma meantone, I'm sure.

I think smooth functions with quadratic minima, which is what I used
originally, make more sense. But then having 20/13 as a local
minimum, I'm afraid, doesn't.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> I get a sum, over the 66 intervals, of 441.52 -- an incredibly close
> call.

Yow. That suggests that if you ran a cubic spline through your points
and sought the local minima, it could go either way--and it wouldn't
matter, since what we have is in effect a Florida vote situation. An
interesting result! Can you tell us how your consonance function was
defined?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > I get a sum, over the 66 intervals, of 441.52 -- an incredibly
close
> > call.
>
> Yow. That suggests that if you ran a cubic spline through your
points

I don't get this part . . .

> and sought the local minima, it could go either way--and it wouldn't
> matter, since what we have is in effect a Florida vote situation. An
> interesting result! Can you tell us how your consonance function was
> defined?

Did you try plotting it? It's one of those harmonic entropy thingies,
specifically, the one that satisfied each and every one of George
Secor's requirements for a consonance function, which he had been
drawing by hand before he encountered harmonic entropy. See the
harmonic entropy list for more details.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > Yow. That suggests that if you ran a cubic spline through your
> points
>
> I don't get this part . . .

It's a way of smoothly interpolating; however I was assuming the
function in question was supposed to be smooth.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > Yow. That suggests that if you ran a cubic spline through your
> > points
> >
> > I don't get this part . . .
>
> It's a way of smoothly interpolating; however I was assuming the
> function in question was supposed to be smooth.

The one I was using for today's examples was continuous but not
differentiable. The older stuff used a function that might be more
amenable to "splining", since it is differentiable everywhere (like
that depicted here:
/harmonic_entropy/

on 4/29/04 10:51 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

>>> and
>>> if you want to alter 25:16 to something simpler, you'll be forced
> to
>>> alter one of the 5:4s to something more complex.
>> Only now does it hit me that this implies I am looking for a
> temperament,
>> not a just solution. Because if I replace one of the 5:4s with an
> 11:8 for
>> example stacking yields 55:32 which I want to function as a 5:3.
> So I would
>> temper everything to make the best of that. Maybe that's a slightly
> ugly
>> example but since I don't know yet what to expect it is not out of
> the realm
>> of possibility.
>>
>> My mistake was to think that there would be a way to improve the
> 25:16 while
>> not making one of the 5:4's as *bad* as the 25:16. I think now
> that a just
>> solution probably just pushes the complexity around.
>
> The most obvious solution is to equate 25/16 with 14/9, and temper
> out 225/224. The major thirds resulting from this are not at all bad.

I'll have to try this. It is not something I am quite trained in yet to
implement such a specification (convert it to a scale spec), though I think
I can figure it out given an hour or so. Maybe Carl can come over here and
show me how to do this in scala.

>> It would be good to have a better handle on something that relates
> to this
>> "otonal clarify" in order to have a practical sense of how audible
>> consonance is generated in a chord.
>
> The augmented triad we get from marvel (225/224-planar) would be a
> good starting point if someone wanted to try to figure out what the
> heck otonal clarity was.

In the past my tendency was to dislike augmented triads, so this sounds
interesting. I might be able to try this pretty quickly.

> Then there's utonals to worry about.

I think I'll have more to say about this in another message.

> But if nobody else chimes
>> in maybe I should assume this belongs in private communication with
> Paul?
>
> Please do not assume that.

Thanks. This is fun.

-Kurt

on 4/30/04 12:23 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 4/29/04 8:08 PM, wallyesterpaulrus <paul@s...> wrote:
>>
>>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>>> on 4/29/04 7:19 PM, wallyesterpaulrus <paul@s...> wrote:
>>
>>>>> For a 7-note tuning, it gave something like a meantone diatonic
>>>>> scale, but the fifths were tempered a little more in the middle
> of
>>>>> the chain and a little less toward the ends.
>>>>>
>>>>> Same thing for a 5-note tuning, IIRC.
>>>
>>> Are these disappointments, too?
>>
>> No, actually quite interesting. I just lost track of it in the
> process of
>> quoting and replying. But besides dissappointment, I'm initially
> surprised
>> that 12-tone went toward total symmetry rather than away from it.
> It might
>> be interesting to look at the general result as a function of N.
> For which
>> values of N does the result tend toward ET? For which values does a
>> periodic symmetry occur? Etc. Some interesting patterns might
> emerge.
>
> By the time I got to 12, the program was getting stuck, so I didn't
> push it much higher. But that was the only ET, IIRC. It's all here in
> the archives . . .

Interesting. I hope we can work on that consonance function to make it more
well-behaved. Though in fact I suspect a really good consonance function
may *not* be well-behaved. More on that another time.

>> I'm definitely not going to be fully satisfied if I can't learn how
> to do
>> some of my own numerical research. Has anyone used Mathematica for
> RMS
>> optimization kinds of solutions?
>
> I'm sure someone has.
>
>> Only now does it hit me that this implies I am looking for a
> temperament,
>> not a just solution.
>
> Phew! It's a lot easier to operate when you're not constrained by
> unrealistic assumptions.

Well I wasn't constrained. I was offering it as one possible question,
along with the other tempered variant. The question was reasonable (to the
inexperienced) and it just turns out that the solution set is probably nil.
I had to think more about prime factorization to figure that out.

>> In some of these messages the issue of "consistency" came up, and I
> don't
>> understand it yet but I'm wondering whether it is related to my
> desire to
>> flatten out the odd-limit variations across all the intervals of a
> scale
>> which I have been talking about here.
>
> You'll have to show me the context, but most likely it was about ETs.

Ok, I found it you're right. I found as good a context as any in your reply
to me on 12/2/03:

/tuning/topicId_48659.html#48931

in which you referred to this page:

http://library.wustl.edu/~manynote/consist3.txt

which is indeed about ETs. (The SETI people doing googling will have an
interesting time with our list archives. ;)

[That message 48931 of yours is still flagged in my local email folder as
something that has information for me to assimilate, a piece of rich
material for my ongoing training. I never replied to it but it is not
lost.]

>>> Paul Hahn had found 48
>>> examples with 32 of them, hundreds with 31 of them, and hundreds
> more
>>> with 30 of them. Aaron Johnson took one of the first 48 and made
>>> music with it. Anyway, 32 7-limit consonances is quite a few (8?)
>>> more than you could get if you restricted yourself to a prime
> limit
>>> of 5 -- though in the latter case, all the 7-limit consonances
> would
>>> also be 5-limit consonances.
>>
>> Yes, I see no reason not to leave the prime limit completely open.
> I'm not
>> sure why you even bring up prime limit.
>
> You brought up the Duodene (stricly 5-prime-limit) in this
> discussion, so it seemed pertinent . . .

Ah, yes, but I brought it up as something to improve upon if possible.

> Also, the "harmonic entropy
> minimizer" only models dyadic consonance, so it might miss out on
> some of the higher-odd-limit (and thus, often, higher-prime-limit)
> intervals that become important in large otonal chords.

Yes, we definitely have to work on that.

>> Except it remains an unknown to me what it takes to create what I
> might call
>> "otonal clarify" meaning (tentatively) the degree to which a single
> or a
>> very small number (like 2) of implied fundamentals are clearly
> present.
>
> Out of all the note-combinations possible in the scale?

In my sentence above when refer to 2 implied fundamentals I was talking
about a single chord. Applying this to my original goal would mean I would
like to see that achieved in a lot of different keys, with hopefully a
choice of several chords achieving that in most keys.

> Then maybe
> it's a very different type of scale you're looking for . . . though
> the Duodene is certainly capable of at least 3 clear implied
> fundamentals . . .

It has 6 pure major triads. Seems to me a pure major triad is sufficient to
have a clear implied fundamental though perhaps it gets clearer when a good
4:7 approximation is available, and Duodene has a 4:5:6:7 approximation in
two keys, with the 7 being around 7 cents off, but it is still adequate to
reinforce the implied fundamental. This is a good example chord with which
to look at a generalized consonance function (adequate for 4 notes in this
case). Compare what you get with the 4:5:6 pure and the 7 approximated to
what you get when you temper Duodene in some tentative way to optimize all
2:3, 4:5, 5:6, 4:7, 5:7, and 6:7 occurrences within the scale. Or pick a
different list if you have a better idea. Does the result do better with
the 4:5:6 exact and the 7 off or does it do better with an RMS optimization
across a list of dyads or does it do better with an equal-beating or
exact-multiple-beating solution? By "do better" I clearly mean in listening
tests here.

>> But
>> my guess is that this still relates more to the odd limit than to
> the prime
>> limit, which in its unboundedness seems to have no predictive value
> at all
>> in relation to consonance, clarity, justness, whatever.
>
> True . . . however, it's often easier to solve these problems if
> you're *given* a prime limit first . . .

Ah, I see. And if no prime limit is relevant then you would just run at the
highest prime-limit that your computer and algorithm can handle, right?

>> It would be good to have a better handle on something that relates
> to this
>> "otonal clarify" in order to have a practical sense of how audible
>> consonance is generated in a chord. You didn't bite last time, so
> I'm
>> asking again more carefully.
>
> So you're asking about chords?

Yes, I'm not inclined to simply project dyad results into a chord scenario.
I believe something else is going on. Thoughts about neural responses
reinforce that intuition.

> When comparing simple otonal JI ones,
> I find the size of the numbers works great.

I can't follow what you mean here.

> But you rapidly reach the
> limit of usefulness with that formula, particularly where tempered
> chords are concerned.
>
> Triadic and tetradic harmonic entropy (which would address this
> directly) are things I am working on on the harmonic entropy list,
> and the former is well-studied in part . . . I guess my work there is
> halted for now since no one else wanted to get involved in the
> calculations . . .

Yes, I just refreshed myself briefly on H.E. and realize this is probably
very in line with my own intuition, even moreso based on what I perused this
time than what I had seen before.

My first inclination might be to use some specific timbres and do some
nonlinear things with them in order to hopefully discover something
predictive, something perhaps involving how beating signal movements cross
each other. My impression is you are starting with a simpler distillation
to see what it predicts well. But I still haven't read enough. Is the best
resource the H.E. list archives? Or the tuning dictionary excerpts?

>> I don't recall this from when I perused H.E.
>> material in the past, but that was a while ago. I'm not at all
>> sure that
>> clarify
>
> You keep saying that. You mean "clarity"?

Sorry, yes, its stuck in my fingers apparently and my eyes don't notice the
difference.

> ? of implied fundamental is any kind of grail either, but I have a
>> strong inclination to believe that there is *something* beyond
> individual
>> intervals that needs to be understood to understand
> the "consonance" of a
>> chord, and the presence of implied fundamental is currently the
> only thing I
>> have experienced that appears to relate to my perception
> of "justness".
>
> Yes, I think it's that *as well as* the individual intervals, as well
> as the individual triads, etc. (if its a chord with more than 3
> notes).

Yes, exactly.

>> As far as appropriateness of the above material for this list, I'm
> not sure
>> whether it is on the one hand "too newbie/nieve" to interest most
> people or
>> "too theoretical" and belonging on that basis more on tuning math.
>
> It's right in line with the harmonic entropy list too.

How is the volume on that list?

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> /tuning/topicId_48659.html#48931
>
> in which you referred to this page:
>
> http://library.wustl.edu/~manynote/consist3.txt

I mentioned over on MMM that if you can get within around 7 cents
things seem to sound pretty good; assuming that holds for higher
limits, and defining "around 7 cents" as less than 7.5 cents off, we
get the following from the above table, where the first number is an
et and the second is how far you can go, keeping the error under 7.5
cents. None of these are hamster and duct tape tunings, in other
words, and I am not sure if anyone, ever, has tried 58, 80 or 94, so
it's not as if we are running out of possibilities to explore new
tunings. 94 makes a nice septimal schismic, by the way.

12: 3
19: 5
31: 7
41: 9
58: 13
72: 17
80: 19
94: 23

Gene Ward Smith wrote:

> I mentioned over on MMM that if you can get within around 7 cents > things seem to sound pretty good; assuming that holds for higher > limits, and defining "around 7 cents" as less than 7.5 cents off, we > get the following from the above table, where the first number is an > et and the second is how far you can go, keeping the error under 7.5 > cents. None of these are hamster and duct tape tunings, in other > words, and I am not sure if anyone, ever, has tried 58, 80 or 94, so > it's not as if we are running out of possibilities to explore new > tunings. 94 makes a nice septimal schismic, by the way.

i've got a 58 note ztar tuning. not usually et.

graham

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 4/30/04 12:23 PM, wallyesterpaulrus <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >> on 4/29/04 8:08 PM, wallyesterpaulrus <paul@s...> wrote:
> >>
> >>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >>>> on 4/29/04 7:19 PM, wallyesterpaulrus <paul@s...> wrote:
> >>
> >>>>> For a 7-note tuning, it gave something like a meantone
diatonic
> >>>>> scale, but the fifths were tempered a little more in the
middle
> > of
> >>>>> the chain and a little less toward the ends.
> >>>>>
> >>>>> Same thing for a 5-note tuning, IIRC.
> >>>
> >>> Are these disappointments, too?
> >>
> >> No, actually quite interesting. I just lost track of it in the
> > process of
> >> quoting and replying. But besides dissappointment, I'm initially
> > surprised
> >> that 12-tone went toward total symmetry rather than away from it.
> > It might
> >> be interesting to look at the general result as a function of N.
> > For which
> >> values of N does the result tend toward ET? For which values
does a
> >> periodic symmetry occur? Etc. Some interesting patterns might
> > emerge.
> >
> > By the time I got to 12, the program was getting stuck, so I
didn't
> > push it much higher. But that was the only ET, IIRC. It's all
here in
> > the archives . . .
>
> Interesting. I hope we can work on that consonance function to
make it more
> well-behaved.

I don't think the problem necessarily was the consonance function
(although a cubic spline, as Gene suggested, might have helped), I
think it was the optimization routine. The 11-dimensional landscape,
derived from 66 instances of this consonance function, might have
simply been too gnarly for the optimizer to navigate. But it was
clear by "inspection" or "plug&chug" that the optimizer was just
getting stuck at some funny kinks near 12-equal, and that 12-equal
was in reality the optimum.

> Though in fact I suspect a really good consonance function
> may *not* be well-behaved. More on that another time.

Are you speaking of the "pointy" ones I was just talking about here
with Gene?

> >> I'm definitely not going to be fully satisfied if I can't learn
how
> > to do
> >> some of my own numerical research. Has anyone used Mathematica
for
> > RMS
> >> optimization kinds of solutions?
> >
> > I'm sure someone has.
> >
> >> Only now does it hit me that this implies I am looking for a
> > temperament,
> >> not a just solution.
> >
> > Phew! It's a lot easier to operate when you're not constrained by
> > unrealistic assumptions.
>
> Well I wasn't constrained. I was offering it as one possible
question,
> along with the other tempered variant. The question was reasonable
(to the
> inexperienced) and it just turns out that the solution set is
probably nil.

Hmm . . . the idea of JI optima is certainly tenable, and
the "solution set" in that regard is certainly not "nil" (not sure if
that's what you meant). Did you see this message:

/tuning/topicId_53249.html#53280

?

Did you understand it?

This regards the use of "pointy" rather than "smooth" consonance
functions. You can see some pictures here:

smooth: /harmonic_entropy/

pointy:
/tuning/files/dyadic/sec
orts2.gif

The first sort will favor temperament, the second sort won't. This is
because distrubuting an error among several intervals will look
better than leaving it concentrated in one interval when the minima
are quadratic (as in the "smooth" case), but worse when the minima
are abs-exponential (as in the "pointy" case).

> I had to think more about prime factorization to figure that out.

I'm afraid I've lost you. To figure what out?

> >>> Paul Hahn had found 48
> >>> examples with 32 of them, hundreds with 31 of them, and hundreds
> > more
> >>> with 30 of them. Aaron Johnson took one of the first 48 and made
> >>> music with it. Anyway, 32 7-limit consonances is quite a few
(8?)
> >>> more than you could get if you restricted yourself to a prime
> > limit
> >>> of 5 -- though in the latter case, all the 7-limit consonances
> > would
> >>> also be 5-limit consonances.
> >>
> >> Yes, I see no reason not to leave the prime limit completely
open.
> > I'm not
> >> sure why you even bring up prime limit.
> >
> > You brought up the Duodene (stricly 5-prime-limit) in this
> > discussion, so it seemed pertinent . . .
>
> Ah, yes, but I brought it up as something to improve upon if
>possible.

Maybe the pointy consonance function gives us one way to immediately
address this, without forcing us to leave the strict-JI realm.

> >> Except it remains an unknown to me what it takes to create what I
> > might call
> >> "otonal clarify" meaning (tentatively) the degree to which a
single
> > or a
> >> very small number (like 2) of implied fundamentals are clearly
> > present.
> >
> > Out of all the note-combinations possible in the scale?
>
> In my sentence above when refer to 2 implied fundamentals I was
talking
> about a single chord. Applying this to my original goal would mean
I would
> like to see that achieved in a lot of different keys, with
hopefully a
> choice of several chords achieving that in most keys.

Phew! So there are three levels: chord, key, and scale? Sounds
complicated. How do you define key? Presumably using some sort of sub-
scale . . . ??

> > Then maybe
> > it's a very different type of scale you're looking for . . .
though
> > the Duodene is certainly capable of at least 3 clear implied
> > fundamentals . . .
>
> It has 6 pure major triads.

Whoops, yes I meant at least 6 . . .

> Seems to me a pure major triad is sufficient to
> have a clear implied fundamental though perhaps it gets clearer
when a good
> 4:7 approximation is available, and Duodene has a 4:5:6:7
approximation in
> two keys,

In two keys? Perhaps you really just meant "positions"
or "transpositions", not "keys"? But if so, what would your remark
above -- "with hopefully a choice of several chords achieving that in
most keys" -- mean? Did you really just mean "on most possible
roots/tonics"?

> >> But
> >> my guess is that this still relates more to the odd limit than to
> > the prime
> >> limit, which in its unboundedness seems to have no predictive
value
> > at all
> >> in relation to consonance, clarity, justness, whatever.
> >
> > True . . . however, it's often easier to solve these problems if
> > you're *given* a prime limit first . . .
>
> Ah, I see. And if no prime limit is relevant then you would just
run at the
> highest prime-limit that your computer and algorithm can handle,
right?

Hopefully there's be no limitation there -- instead I was thinking
that if you wanted to construct a scale out of pure JI consonances,
and had a list of them, then the highest prime found in the list
would be the best choice for a prime limit in which to address the
problem. Clearly you won't be generating any higher primes by
combining intervals in your list, so you might as well cap the
dimensionality of the problem using the number of primes present (and
thus, the number that could possibly arise).

> > So you're asking about chords?
>
> Yes, I'm not inclined to simply project dyad results into a chord
scenario.
> I believe something else is going on. Thoughts about neural
responses
> reinforce that intuition.

Sure, but one can do a lot with the false assumption that the dyad
results are everything. All of Sethares's work is founded on this
assumption, and would never see the difference between a 4:5:6:7 and
1/(7:6:5:4) chord. Clearly, these assumptions are flawed, but they
led to some nice music and a theoretical treatise that seems to have
a large number of people convinced (whatever that's worth).

> > When comparing simple otonal JI ones,
> > I find the size of the numbers works great.
>
> I can't follow what you mean here.

Aside from dyad and triad considerations, which usually appear less
important anyway, a:b:c:d will generally be more 'concordant' than
e:f:g:h if a*b*c*d < e*f*g*h, and all eight of these are quite small
integers.

> > But you rapidly reach the
> > limit of usefulness with that formula, particularly where tempered
> > chords are concerned.
> >
> > Triadic and tetradic harmonic entropy (which would address this
> > directly) are things I am working on on the harmonic entropy list,
> > and the former is well-studied in part . . . I guess my work
there is
> > halted for now since no one else wanted to get involved in the
> > calculations . . .
>
> Yes, I just refreshed myself briefly on H.E. and realize this is
probably
> very in line with my own intuition, even moreso based on what I
perused this
> time than what I had seen before.
>
> My first inclination might be to use some specific timbres and do
some
> nonlinear things with them in order to hopefully discover something
> predictive, something perhaps involving how beating signal
movements cross
> each other. My impression is you are starting with a simpler
distillation
> to see what it predicts well. But I still haven't read enough. Is
the best
> resource the H.E. list archives? Or the tuning dictionary excerpts?

The former.

> >> As far as appropriateness of the above material for this list,
I'm
> > not sure
> >> whether it is on the one hand "too newbie/nieve" to interest most
> > people or
> >> "too theoretical" and belonging on that basis more on tuning
math.
> >
> > It's right in line with the harmonic entropy list too.
>
> How is the volume on that list?

Very slow. But I feel moving this discussion there or to the tuning-
math list would be most appropriate now, though any non-technical sub-
discussions could remain here, of course.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> pointy:
>
>/tuning/files/dyadic/se
corts2.gif

/tuning/files/dyadic/vos.gif
has the simple ratios labeled, so may be more meaningful at a glance.

on 5/3/04 2:29 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 4/30/04 12:23 PM, wallyesterpaulrus <paul@s...> wrote:
>>
>>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>>> on 4/29/04 8:08 PM, wallyesterpaulrus <paul@s...> wrote:

>>> By the time I got to 12, the program was getting stuck, so I
> didn't
>>> push it much higher. But that was the only ET, IIRC. It's all
> here in
>>> the archives . . .
>>
>> Interesting. I hope we can work on that consonance function to
>> make it more well-behaved.
>
> I don't think the problem necessarily was the consonance function
> (although a cubic spline, as Gene suggested, might have helped), I
> think it was the optimization routine. The 11-dimensional landscape,
> derived from 66 instances of this consonance function, might have
> simply been too gnarly for the optimizer to navigate. But it was
> clear by "inspection" or "plug&chug" that the optimizer was just
> getting stuck at some funny kinks near 12-equal, and that 12-equal
> was in reality the optimum.

Ok. Well that's interesting. Still Gene also had some other expectations
IIRC and it'd be interesting to follow up on that.

>> Though in fact I suspect a really good consonance function
>> may *not* be well-behaved. More on that another time.
>
> Are you speaking of the "pointy" ones I was just talking about here
> with Gene?

I was thinking possibly worse than that. I was just thinking with my
"neural intuition" and figuring that any expectation of continuity might not
be based in neural reality.

>>>> I'm definitely not going to be fully satisfied if I can't learn
> how
>>> to do
>>>> some of my own numerical research. Has anyone used Mathematica
> for
>>> RMS
>>>> optimization kinds of solutions?
>>>
>>> I'm sure someone has.
>>>
>>>> Only now does it hit me that this implies I am looking for a
>>> temperament,
>>>> not a just solution.
>>>
>>> Phew! It's a lot easier to operate when you're not constrained by
>>> unrealistic assumptions.
>>
>> Well I wasn't constrained. I was offering it as one possible
> question,
>> along with the other tempered variant. The question was reasonable
> (to the
>> inexperienced) and it just turns out that the solution set is
> probably nil.
>
> Hmm . . . the idea of JI optima is certainly tenable, and
> the "solution set" in that regard is certainly not "nil" (not sure if
> that's what you meant).

Oh, I see what you mean. I see that JI optima may well exist in some sense,
but I'm not sure that such optima would get at what I was trying to get at
which I will restate this way:

Goal: to make the *least* consonant intervals in a scale more consonant by
sacrificing the consonance of some of the more consonant ones. In short to
reduce the variation in consonance among all the intervals in the scale such
that the worse ones are less bad and the best ones less good.

I'm not sure whether I should say some other word besides "consonant" here,
but remember one of my original questions was:
>What is the 12-tone scale that maximizes the clarity of the least clear
interval?

Well after this little exchange:

Kurt wrote:
>> Only now does it hit me that this implies I am looking for a
>> temperament,not a just solution.
on 4/30/04 12:23 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:
> Phew! It's a lot easier to operate when you're not constrained by
> unrealistic assumptions.

I thought about it some more, thought about how prime factors work, with the
example of stacking two 5:4s as representative of the typical problem which
is 2 stacked intervals inevitably producing a higher-odd-limit interval,
nothing you can do about that basically, unless you are tempering something
away from just.

That's what I *was* thinking and your comment about "unrealistic" sort of
inspired that thinking and tentatively confirmed it.

But now I'm thinking that if you chose carfully what intervals are available
to stack, maybe you can improve things. But on the other hand you had
apparently thought about it and considered Duodene to be typical, neither
particularly good nor particularly bad, and I took that to apply to my
criterion of optimizing the least clear interval in a scale.

> Did you see this message:
>
> /tuning/topicId_53249.html#53280
>
> ?
>
> Did you understand it?

I understand abs exponential and pointy. So you are saying the points ones
are so pointy that inevitably any change will drive down the value of the
consonance function. But seems to me that depends on how you add them up
over all the intervals being optimized, and doing an ad-hoc smoothing step
might not really be to the point. Can you partition the HE function into
into more gross and more detailed contributions, or is the whole thing kind
of inevitably fractal in nature?

> This regards the use of "pointy" rather than "smooth" consonance
> functions. You can see some pictures here:
>
> smooth: /harmonic_entropy/

Hmm. You just pointed me to the H.E. page. No picture there.

> pointy:
> /tuning/files/dyadic/sec
> orts2.gif
>
> The first sort will favor temperament, the second sort won't. This is
> because distrubuting an error among several intervals will look
> better than leaving it concentrated in one interval when the minima
> are quadratic (as in the "smooth" case), but worse when the minima
> are abs-exponential (as in the "pointy" case).

So I still need to look at a smooth one.

>> I had to think more about prime factorization to figure that out.
>
> I'm afraid I've lost you. To figure what out?

I sort of explained this above.

>>> You brought up the Duodene (stricly 5-prime-limit) in this
>>> discussion, so it seemed pertinent . . .
>>
>> Ah, yes, but I brought it up as something to improve upon if
>> possible.
>
> Maybe the pointy consonance function gives us one way to immediately
> address this, without forcing us to leave the strict-JI realm.

Aha.

>>>> Except it remains an unknown to me what it takes to create what I
>>> might call
>>>> "otonal clarify" meaning (tentatively) the degree to which a
> single
>>> or a
>>>> very small number (like 2) of implied fundamentals are clearly
>>> present.
>>>
>>> Out of all the note-combinations possible in the scale?
>>
>> In my sentence above when refer to 2 implied fundamentals I was
> talking
>> about a single chord. Applying this to my original goal would mean
> I would
>> like to see that achieved in a lot of different keys, with
> hopefully a
>> choice of several chords achieving that in most keys.
>
> Phew! So there are three levels: chord, key, and scale? Sounds
> complicated. How do you define key? Presumably using some sort of sub-
> scale . . . ??

Its probably some sort of minimax thing. I didn't work it all out and if it
isn't intuitive then I will need to do that.

>> Seems to me a pure major triad is sufficient to
>> have a clear implied fundamental though perhaps it gets clearer
> when a good
>> 4:7 approximation is available, and Duodene has a 4:5:6:7
> approximation in
>> two keys,
>
> In two keys? Perhaps you really just meant "positions"
> or "transpositions", not "keys"?

I meant the 128:225 appears twice in any fixed tuning of the scale. The
225:224-based approximations of the following 7-limit ratios occur as
follows:

9:7 (actually 16: 25) appears 4 times
6:7 (actually 32: 75) appears 3 times
4:7 (actually 128:225) appears 2 times

I'm sure you know that so this must be just some language thing. To me I
use a fixed Duodene tuning in various "keys". I thought that was normal
lingo.

> But if so, what would your remark
> above -- "with hopefully a choice of several chords achieving that in
> most keys" -- mean? Did you really just mean "on most possible
> roots/tonics"?

Yes, exactly. I'd settle for "most" if I couldn't get "all".

>> Ah, I see. And if no prime limit is relevant then you would just
> run at the
>> highest prime-limit that your computer and algorithm can handle,
> right?
>
> Hopefully there's be no limitation there -- instead I was thinking
> that if you wanted to construct a scale out of pure JI consonances,
> and had a list of them, then the highest prime found in the list
> would be the best choice for a prime limit in which to address the
> problem. Clearly you won't be generating any higher primes by
> combining intervals in your list, so you might as well cap the
> dimensionality of the problem using the number of primes present (and
> thus, the number that could possibly arise).

Right, I get it.

>>> So you're asking about chords?
>>
>> Yes, I'm not inclined to simply project dyad results into a chord
> scenario.
>> I believe something else is going on. Thoughts about neural
> responses
>> reinforce that intuition.
>
> Sure, but one can do a lot with the false assumption that the dyad
> results are everything. All of Sethares's work is founded on this
> assumption, and would never see the difference between a 4:5:6:7 and
> 1/(7:6:5:4) chord. Clearly, these assumptions are flawed, but they
> led to some nice music and a theoretical treatise that seems to have
> a large number of people convinced (whatever that's worth).

Sure. I'm planning on trying to make my own dynamic retuning capable of
utonal/otonal in-place reciprocal modulations so that I can really get some
experience of the utonal/otonal difference. I'm laying some of the
groundwork for this in my software by more fully implementing Carl's XMW
ideas, as they have fleshed themselves out in the context of my organ
softsynth. Then a direct u/o modulation should be possible as a new option,
based on (tenatively) keeping the highest and lowest playing notes fixed.
I'm looking forward to hearing this so I can really come to know the
difference!

>>> When comparing simple otonal JI ones,
>>> I find the size of the numbers works great.
>>
>> I can't follow what you mean here.
>
> Aside from dyad and triad considerations, which usually appear less
> important anyway, a:b:c:d will generally be more 'concordant' than
> e:f:g:h if a*b*c*d < e*f*g*h, and all eight of these are quite small
> integers.

Ah, good. Yes, I remember that now. Just couldn't get it from your earlier
words.

>>> But you rapidly reach the
>>> limit of usefulness with that formula, particularly where tempered
>>> chords are concerned.
>>>
>>> Triadic and tetradic harmonic entropy (which would address this
>>> directly) are things I am working on on the harmonic entropy list,
>>> and the former is well-studied in part . . . I guess my work
> there is
>>> halted for now since no one else wanted to get involved in the
>>> calculations . . .
>>
>> Yes, I just refreshed myself briefly on H.E. and realize this is
> probably
>> very in line with my own intuition, even moreso based on what I
> perused this
>> time than what I had seen before.
>>
>> My first inclination might be to use some specific timbres and do
> some
>> nonlinear things with them in order to hopefully discover something
>> predictive, something perhaps involving how beating signal
> movements cross
>> each other. My impression is you are starting with a simpler
> distillation
>> to see what it predicts well. But I still haven't read enough. Is
> the best
>> resource the H.E. list archives? Or the tuning dictionary excerpts?
>
> The former.

You mean *not* a simpler distillation? But are you assuming some actual
timbres?

>>>> As far as appropriateness of the above material for this list,
> I'm
>>> not sure
>>>> whether it is on the one hand "too newbie/nieve" to interest most
>>> people or
>>>> "too theoretical" and belonging on that basis more on tuning
> math.
>>>
>>> It's right in line with the harmonic entropy list too.
>>
>> How is the volume on that list?
>
> Very slow. But I feel moving this discussion there or to the tuning-
> math list would be most appropriate now, though any non-technical sub-
> discussions could remain here, of course.

Well I had to just join to see your pictures. So I guess I'll go turn on
the email subscription option. You can reply there next time if
appropriate, perhaps with a note here that you are making the transition.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> I was thinking possibly worse than that. I was just thinking with
my
> "neural intuition" and figuring that any expectation of continuity
might not
> be based in neural reality.

Hmm . . . I don't see how continuity could fail to hold, because
frequency resolution is always finite in practice . . .
Discontinuities would seem to require infinite precision, which would
seem to require infinite duration . . .

> Goal: to make the *least* consonant intervals in a scale more
consonant by
> sacrificing the consonance of some of the more consonant ones. In
short to
> reduce the variation in consonance among all the intervals in the
scale such
> that the worse ones are less bad and the best ones less good.
>
> I'm not sure whether I should say some other word
besides "consonant" here,
> but remember one of my original questions was:
> >What is the 12-tone scale that maximizes the clarity of the least
clear
> interval?
>
> Well after this little exchange:
>
> Kurt wrote:
> >> Only now does it hit me that this implies I am looking for a
> >> temperament,not a just solution.
> on 4/30/04 12:23 PM, wallyesterpaulrus <paul@s...> wrote:
> > Phew! It's a lot easier to operate when you're not constrained by
> > unrealistic assumptions.
>
> I thought about it some more, thought about how prime factors work,
with the
> example of stacking two 5:4s as representative of the typical
problem which
> is 2 stacked intervals inevitably producing a higher-odd-limit
interval,
> nothing you can do about that basically, unless you are tempering
something
> away from just.
>
> That's what I *was* thinking and your comment about "unrealistic"
sort of
> inspired that thinking and tentatively confirmed it.
>
> But now I'm thinking that if you chose carfully what intervals are
available
> to stack, maybe you can improve things. But on the other hand you
had
> apparently thought about it and considered Duodene to be typical,
neither
> particularly good nor particularly bad, and I took that to apply to
my
> criterion of optimizing the least clear interval in a scale.

Probably the best you could do in this regard would be a big 12-tone
harmonic series (or subharmonic series, if you're only looking
dyadically it's just as good).

> > Did you see this message:
> >
> > /tuning/topicId_53249.html#53280
> >
> > ?
> >
> > Did you understand it?
>
> I understand abs exponential and pointy. So you are saying the
points ones
> are so pointy that inevitably any change will drive down the value
of the
> consonance function.

Pretty much.

> But seems to me that depends on how you add them up
> over all the intervals being optimized, and doing an ad-hoc
>smoothing step
> might not really be to the point.

Don't follow.

>
> > This regards the use of "pointy" rather than "smooth" consonance
> > functions. You can see some pictures here:
> >
> > smooth: /harmonic_entropy/
>
> Hmm. You just pointed me to the H.E. page. No picture there.

Really? That's odd . . . no picture on the homepage? I see one.

> > In two keys? Perhaps you really just meant "positions"
> > or "transpositions", not "keys"?
>
> I meant the 128:225 appears twice in any fixed tuning of the
scale. The
> 225:224-based approximations of the following 7-limit ratios occur
as
> follows:
>
> 9:7 (actually 16: 25) appears 4 times
> 6:7 (actually 32: 75) appears 3 times
> 4:7 (actually 128:225) appears 2 times
>
> I'm sure you know that so this must be just some language thing.
To me I
> use a fixed Duodene tuning in various "keys". I thought that was
normal
> lingo.

"Keys" to me means C major, D minor, etc.
>
> > But if so, what would your remark
> > above -- "with hopefully a choice of several chords achieving
that in
> > most keys" -- mean? Did you really just mean "on most possible
> > roots/tonics"?
>
> Yes, exactly. I'd settle for "most" if I couldn't get "all".

Then the big harmonic series scale probably isn't for you . . .

>>> enough. Is
>>> the best
> >> resource the H.E. list archives? Or the tuning dictionary
excerpts?
> >
> > The former.
>
> You mean *not* a simpler distillation?

I mean the H.E. list archives. (sorry I'm in a rush)

> You can reply there next time if
> appropriate, perhaps with a note here that you are making the
>transition.

I'm making the transition with my next post.

on 5/3/04 9:18 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> I was thinking possibly worse than that. I was just thinking with
> my
>> "neural intuition" and figuring that any expectation of continuity
> might not
>> be based in neural reality.
>
> Hmm . . . I don't see how continuity could fail to hold, because
> frequency resolution is always finite in practice . . .
> Discontinuities would seem to require infinite precision, which would
> seem to require infinite duration . . .

Faulty neural intuition! ;)

>> Goal: to make the *least* consonant intervals in a scale more
> consonant by
>> sacrificing the consonance of some of the more consonant ones. In
> short to
>> reduce the variation in consonance among all the intervals in the
> scale such
>> that the worse ones are less bad and the best ones less good.
>>
>> I thought about it some more, thought about how prime factors work,
> with the
>> example of stacking two 5:4s as representative of the typical
> problem which
>> is 2 stacked intervals inevitably producing a higher-odd-limit
> interval,
>> nothing you can do about that basically, unless you are tempering
> something
>> away from just.
>>
>> That's what I *was* thinking and your comment about "unrealistic"
> sort of
>> inspired that thinking and tentatively confirmed it.
>>
>> But now I'm thinking that if you chose carfully what intervals are
> available
>> to stack, maybe you can improve things. But on the other hand you
> had
>> apparently thought about it and considered Duodene to be typical,
> neither
>> particularly good nor particularly bad, and I took that to apply to
> my
>> criterion of optimizing the least clear interval in a scale.
>
> Probably the best you could do in this regard would be a big 12-tone
> harmonic series (or subharmonic series, if you're only looking
> dyadically it's just as good).

Well that's an interesting persepctive. Indeed that's literally correct.
And it proves that that wasn't what I was really looking for. So my other
point about "otonal clarity" (which I think may still be vague to you) turns
out to be part of the requirement.

>>> Did you see this message:
>>>
>>> /tuning/topicId_53249.html#53280
>>
>> I understand abs exponential and pointy. So you are saying the
> points ones
>> are so pointy that inevitably any change will drive down the value
> of the
>> consonance function.
>
> Pretty much.
>
>> But seems to me that depends on how you add them up
>> over all the intervals being optimized, and doing an ad-hoc
>> smoothing step
>> might not really be to the point.
>
> Don't follow.

You apply the consonance function to independently to a whole set of target
intervals (the ones being optimized) don't you? If so then you have to
combine all those results into a single metric that you can use to compare
one scale to another, right?

>>> This regards the use of "pointy" rather than "smooth" consonance
>>> functions. You can see some pictures here:
>>>
>>> smooth: /harmonic_entropy/
>>
>> Hmm. You just pointed me to the H.E. page. No picture there.
>
> Really? That's odd . . . no picture on the homepage? I see one.

Hmm. I tried another browser and no different. There is a *missing*
picture to the left of the description. It fails to load and fails to
reload upon manual request.

>>>> and Duodene has a 4:5:6:7 approximation in two keys,
>>> In two keys? Perhaps you really just meant "positions"
>>> or "transpositions", not "keys"?
>> I meant the 128:225 appears twice in any fixed tuning of the
>> scale. The 225:224-based approximations of the following 7-limit ratios
>> occur asfollows:
>>
>> 9:7 (actually 16: 25) appears 4 times
>> 6:7 (actually 32: 75) appears 3 times
>> 4:7 (actually 128:225) appears 2 times
>>
>> I'm sure you know that so this must be just some language thing.
>> To me I use a fixed Duodene tuning in various "keys". I thought that was
>> normal lingo.
>
> "Keys" to me means C major, D minor, etc.

Yes, exactly. For a Duodene with F-C-G-D in the middle row, the 128:225
appears in two keys: C# and G#. Right?

>>> But if so, what would your remark
>>> above -- "with hopefully a choice of several chords achieving
> that in
>>> most keys" -- mean? Did you really just mean "on most possible
>>> roots/tonics"?
>>
>> Yes, exactly. I'd settle for "most" if I couldn't get "all".
>
> Then the big harmonic series scale probably isn't for you . . .

Right, not when the otonal clarity requirement as I perhaps too vaguely
stated it is introduced. I'll try to restate it here to incorporate the
additional idea of key clarity....

The otonal clarity of a *chord* is the degree to which a single or a very
small number of implied fundamentals are clearly present. A strong single
implied fundamental is the condition of good clarity. A weaker implied
fundamental or a blending of several without a clear "winner" is indicative
of less clarity.

Otonal clarity is key-appropriate when the strong implied fundamental is in
agreement with the root of the chord. (I personally tend to call this the
key of the chord and I'm not sure that is good music-theory lingo.)

The otonally-clear chord variety of a key in a scale is the degree to which
there is a variety of available otonally clear chords appropriate to that
key.

The key-relevant otonal versatility of a scale over a set of specified
important keys is the degree to which all the important keys have good
otonally-clear chord variety. There may be a secondary set of keys that
have at least one otonally-clear chord but no variety, and this would be
part of the overall measure of otonal versatility of a scale.

>> You can reply there next time if
>> appropriate, perhaps with a note here that you are making the
>> transition.
>
> I'm making the transition with my next post.

Ok, well this is my final post on this thread to the tunings list. I will
also post it to the H.E. list, so you and others can reply there.

-Kurt

>>>> smooth: /harmonic_entropy/
>>>
>>> Hmm. You just pointed me to the H.E. page. No picture there.
>>
>> Really? That's odd . . . no picture on the homepage? I see one.
>
>Hmm. I tried another browser and no different. There is a *missing*
>picture to the left of the description. It fails to load and fails to
>reload upon manual request.

It worked here. Maybe some of the yahoo mirrors are broken.

-Carl