More N-out-of-M

19-out-of-26, which Dan Stearns suggested, only has 12 consonant 7-
limit tetrads.

The maximally even scales I advocated a while back were 19-out-of-31,
which has 18 7-limit tetrads (beating blackjack in meantone, with
only 16); and 22-out-of-41, which has 20 7-limit tetrads. Thus I was
surprised to see 31-out-of-72 with 36 7-limit tetrads.

By 7-limit tetrads I mean 1:3:5:7 and 1/(1:3:5:7).

--- In tuning@y..., paul@s... wrote:
> The maximally even scales I advocated a while back were
19-out-of-31,
> which has 18 7-limit tetrads (beating blackjack in meantone, with
> only 16); and 22-out-of-41, which has 20 7-limit tetrads. Thus I was
> surprised to see 31-out-of-72 with 36 7-limit tetrads.

Yes. But the real miracle is not the 31-out-of-72, but the generator
that it embodies.

By the way I calculated an optimum for this generator when we weight
the errors in the 11-limit intervals by their odd-limit and minimise
the maximum absolute weighted error. It is 116.71c, so there doesn't
seem much point in leaving 72-EDO (116.67c).

I agree that tuning these scales in 31-EDO or 41-EDO is to miss the
point, since they aren't 11-limit quasi-just in 31 or 41. And there is
a better 7-limit choice available in 31-EDO

I propose that a tuning is only "quasi-just" at the M-limit (M is odd)
if for every odd number N less than or equal to M, no approximation to
a ratio of N has an error greater than 35.2/N cents, and where N=3 the
error must be less than 5.4 cents. I believe this is in line with one
of Partch's observations.

Ratios of 3 5 7 9 11 13
Max error 5.4 7.0 5.0 3.9 3.2 2.7 (cents)

Of course this is an entirely self-serving definition, designed so
that 72-EDO just scrapes in at the 11-limit (the 4:9 being the worst)
and 1/4-comma meantone just scrapes in at the 7-limit (the 2:3 being
the worst).

-- Dave Keenan

I know this is the wrong thread, but Graham Breed are you
reading?
I think you will agree that 31-EDO and 41-EDO are "miracle" tunings in
exactly the way that 12-EDO and 26-EDO (respectively) are "meantones".

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> I propose that a tuning is only "quasi-just" at the M-limit (M is
odd)
> if for every odd number N less than or equal to M, no approximation
to
> a ratio of N has an error greater than 35.2/N cents, and where N=3
the
> error must be less than 5.4 cents. I believe this is in line with
one
> of Partch's observations.
>
> Ratios of 3 5 7 9 11 13
> Max error 5.4 7.0 5.0 3.9 3.2 2.7 (cents)
>
> Of course this is an entirely self-serving definition, designed so
> that 72-EDO just scrapes in at the 11-limit (the 4:9 being the
worst)
> and 1/4-comma meantone just scrapes in at the 7-limit (the 2:3
being
> the worst).
>
> -- Dave Keenan

Quasi-just at the n-limit should mean no more than 4 cents error in
any n-limit consonances.

--- In tuning@y..., paul@s... wrote:
> Quasi-just at the n-limit should mean no more than 4 cents error in
> any n-limit consonances.

But that would exclude 1/4-comma meantone at the 5-limit. And is too
generous for ratios of 11 and beyond. But I'd be willing to bring the
ratios of 5 down to 5.4c as well.

Dave Keenan wrote:

> I know this is the wrong thread, but Graham Breed are you
> reading?

Yes, I've got a big backlog of messages to reply to.

> I think you will agree that 31-EDO and 41-EDO are "miracle" tunings in
> exactly the way that 12-EDO and 26-EDO (respectively) are "meantones".

I thought "miracle" was the 31-note scale, so I'm calling the tuning
"Erlich-Keenan" now.

Yes, I agree entirely, the analogy to meantone is very good indeed.
Although I wouldn't class 26-equal as strictly meantone. Furthermore, I
can see every point in using decimal notation with 31-equal.

I now have Kyma set up to cross-fade between 31 and 41 and it sounds good!
I haven't actually worked out where 72 is yet. But some of the 11-limit
chords do sound better when you blur them towards 31, like 7:9:11. I can
also see great opportunities for polymicrotonality when three 11-limit ETs
can all be notated the same way.

And I'm planning to extend the range to get even more blurring. Borrowing
Blackwood's terminology, I think the whole range from 10 to 11 equal can
be termed recognizable Erlich-Keenan tunings, even though the 11-limit
approximations won't hold. Once you've got the notation and instruments
set up, you can play with it as much as you like, see what comes out.

I did actually start with the crossfade range way to high -- very strange!

Graham

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:
> > Quasi-just at the n-limit should mean no more than 4 cents error in
> > any n-limit consonances.
>
> But that would exclude 1/4-comma meantone at the 5-limit.

I don't think that should be considered quasi-just.

> And is too
> generous for ratios of 11 and beyond.

It is, or we are?

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., paul@s... wrote:
> > > Quasi-just at the n-limit should mean no more than 4 cents error
in
> > > any n-limit consonances.
> >
> > But that would exclude 1/4-comma meantone at the 5-limit.
>
> I don't think that should be considered quasi-just.

I understand that many people already refer to 1/4-comma meantone as
5-limit quasi-just. But that is probably not because of any particular
upper limit on the errors, but because of the just 4:5's.

I guess I could go along with excluding it because of the error in its
fifths. But I still want to taper off the allowable errors for ratios
of higher odds. It would be ridiculous to call something 31-limit
quasi-just if the errors in the ratios of 31 were up to 4 cents.

> > And is too
> > generous for ratios of 11 and beyond.
>
> It is, or we are?

Your "max error of 4c" criterion is too generous for ratios of 11 and
beyond.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> I understand that many people already refer to 1/4-comma meantone as
> 5-limit quasi-just.

Really? Like who?

> But that is probably not because of any particular
> upper limit on the errors, but because of the just 4:5's.

Oh yeah, that could be.
>
> I guess I could go along with excluding it because of the error in its
> fifths. But I still want to taper off the allowable errors for ratios
> of higher odds. It would be ridiculous to call something 31-limit
> quasi-just if the errors in the ratios of 31 were up to 4 cents.

First, create a chord that is 31-limit just according to your definition of just. Then, decide empirically
whether 4 cent differences disturb you or not. I'll remain agnostic until then.

> > > And is too
> > > generous for ratios of 11 and beyond.
> >
> > It is, or we are?
>
> Your "max error of 4c" criterion is too generous for ratios of 11 and
> beyond.

When you listen to ratios of 11 just, and ratios of 11 four cents off, how do their sounds
compare? Using a harp timbre, say?

So you're saying 72-tET isn't 11-limit quasi-just?

--- In tuning@y..., paul@s... wrote:
> First, create a chord that is 31-limit just according to your
definition of just. Then, decide empirically
> whether 4 cent differences disturb you or not. I'll remain agnostic
until then.

Ok. I don't have time to set this up. But I understand your point that
in order to perceive something as 31 limit just it will need a very
large otonal context. Maybe 7 or 8 notes. In which case the errors
won't be so noticeable.

But I'm basing this on Partch's observation and my experience with
listening to bare 8:11 and 8:13 dyads. I can hear them as just, but a
4c error in them is _waaay_ more objectionable than a 4c error in a
4:5. Don't you agree?

> So you're saying 72-tET isn't 11-limit quasi-just?

Not at all. 72-EDO has a max error of 2.6 cents in its ratios of 11.
So it comes in under my 35.2/N cents where N is the odd limit of the
ratio.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:
> > First, create a chord that is 31-limit just according to your
> definition of just. Then, decide empirically
> > whether 4 cent differences disturb you or not. I'll remain
agnostic
> until then.
>
> Ok. I don't have time to set this up. But I understand your point
that
> in order to perceive something as 31 limit just it will need a very
> large otonal context. Maybe 7 or 8 notes. In which case the errors
> won't be so noticeable.
>
> But I'm basing this on Partch's observation and my experience with
> listening to bare 8:11 and 8:13 dyads. I can hear them as just, but
a
> 4c error in them is _waaay_ more objectionable than a 4c error in a
> 4:5. Don't you agree?

You're speaking of sawtooth timbres, correct?
>
> > So you're saying 72-tET isn't 11-limit quasi-just?
>
> Not at all.

You're saying it is . . .
>
72-EDO has a max error of 2.6 cents in its ratios of 11.

But 9:8 is 4 cents off . . .

> So it comes in under my 35.2/N cents where N is the odd limit of
the
> ratio.

Aha.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > But I'm basing this on Partch's observation and my experience with
> > listening to bare 8:11 and 8:13 dyads. I can hear them as just,
but
> a
> > 4c error in them is _waaay_ more objectionable than a 4c error in
a
> > 4:5. Don't you agree?
>
> You're speaking of sawtooth timbres, correct?

Correct.

> > > So you're saying 72-tET isn't 11-limit quasi-just?
> >
> > Not at all.
>
> You're saying it is . . .

Yes. I consider 72-EDO to be 11-limit quasi-just. Or should that be
quasi-11-limit-just?

> 72-EDO has a max error of 2.6 cents in its ratios of 11.
>
> But 9:8 is 4 cents off . . .
>
> > So it comes in under my 35.2/N cents where N is the odd limit of
> the
> > ratio.
>
> Aha.

Now Paul, I'd expect you to be the last person to confuse "ratios of
N" (which is what I wrote right from the start) with "N-limit ratios".
;-)

So whadya think of the proposal now?

-- Dave Keenan

> So whadya think of the proposal now?
>
Whattabout wafso-just?

--- In tuning@y..., paul@s... wrote:
>
> > So whadya think of the proposal now?
> >
> Whattabout wafso-just?

I originally intended wafso-just to be about half the errors of
quasi-just. I considered 1/4-comma meantone to be quasi-just at that
stage (and maybe still do).

I consider that 72-EDO has wafso-just 7-limit ratios but only
quasi-just ratios of 9 and 11.

It's only the 4:9 and 9:11 that keep it from being 11-limit
wafso-just (= microtempered).

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

>
> I consider that 72-EDO has wafso-just 7-limit ratios but only
> quasi-just ratios of 9 and 11.

So Blackjack is wafso-just with respect to the lattice I posted.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> >
> > I consider that 72-EDO has wafso-just 7-limit ratios but only
> > quasi-just ratios of 9 and 11.
>
> So Blackjack is wafso-just with respect to the lattice I posted.

Yes!

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22522.html#22617

> --- In tuning@y..., paul@s... wrote:
> >
> > > So whadya think of the proposal now?
> > >
> > Whattabout wafso-just?
>
> I originally intended wafso-just to be about half the errors of
> quasi-just. I considered 1/4-comma meantone to be quasi-just at
that
> stage (and maybe still do).
>
> I consider that 72-EDO has wafso-just 7-limit ratios but only
> quasi-just ratios of 9 and 11.
>
> It's only the 4:9 and 9:11 that keep it from being 11-limit
> wafso-just (= microtempered).
>
> -- Dave Keenan

I'm having trouble with the term "microtempered" since it doesn't
imply just intonation in the term... It seems to imply the opposite
(??)

I much prefer WAFSO-just, but I propose that we think of flys wearing
little shirts.... so it becomes "within a fly's SHIRT of just..."

Isn't that sweet??

________ ______ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_22522.html#22903

> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> /tuning/topicId_22522.html#22617
>
> > --- In tuning@y..., paul@s... wrote:
> > >
> > > > So whadya think of the proposal now?
> > > >
> > > Whattabout wafso-just?
> >
> > I originally intended wafso-just to be about half the
> > errors of quasi-just. I considered 1/4-comma meantone to
> > be quasi-just at that stage (and maybe still do).
> >
> > I consider that 72-EDO has wafso-just 7-limit ratios but
> > only quasi-just ratios of 9 and 11.
> >
> > It's only the 4:9 and 9:11 that keep it from being 11-limit
> > wafso-just (= microtempered).
> >
> > -- Dave Keenan
>
>
> I'm having trouble with the term "microtempered" since it
> doesn't imply just intonation in the term... It seems to
> imply the opposite
> (??)

Dave! (and Paul),

Microtemper is not in my Dictionary!

PLEASE post a definition!

> I much prefer WAFSO-just, but I propose that we think of
> flys wearing little shirts.... so it becomes "within a fly's
> SHIRT of just..."
>
> Isn't that sweet??

Really cute, Joe, really cute...

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

--- In tuning@y..., jpehrson@r... wrote:
> I much prefer WAFSO-just, but I propose that we think of flys
wearing
> little shirts.... so it becomes "within a fly's SHIRT of just..."
>
> Isn't that sweet??

Hee hee. That's fine with me.

-- Dave Keenan

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Microtemper is not in my Dictionary!
>
> PLEASE post a definition!

Microtempered is currently a synonym for wafso-just. The definition
is, I think still in flux. But the errors for wafso-just are about
half those for quasi-just whatever that means. I currently think of
them as roughly 2.7 cents versus 5.4 cents (or maybe 3.5c versus 7c).
But I want to make the allowable error for these categories taper off
as the odd-limit of the ratio increases. Paul disagrees.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22522.html#22923

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > Microtemper is not in my Dictionary!
> >
> > PLEASE post a definition!
>
> Microtempered is currently a synonym for wafso-just. The
> definition is, I think still in flux. But the errors for
> wafso-just are about half those for quasi-just whatever
> that means. I currently think of them as roughly 2.7 cents
> versus 5.4 cents (or maybe 3.5c versus 7c). But I want to
> make the allowable error for these categories taper off
> as the odd-limit of the ratio increases. Paul disagrees.
>
> -- Dave Keenan

I agree with you, Dave.

My experience with the Hendrix Chord debate convinced me
that 19:16 is much more sensitive to mistuning that 7:6.

I know that's just an isolated example, but long-time
acceptance by many listeners of quite severely mistune
3:2s strengthens the argument.

I've even noted in the past that in jazz any "dominant 7th"-type
chord works just as well with a "flat 5th" (i.e., tritone) as
it does with a "perfect 5th", in fact it's probably usually
better (more interesting) with the "flat 5th". And this
gets more true the more "extension" notes are piled on top
of the chord. That's an "acceptable" mistuning of 100 cents!
As always, context is everything.

-monz

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22522.html#22922

> --- In tuning@y..., jpehrson@r... wrote:
> > I much prefer WAFSO-just, but I propose that we think of flys
> wearing
> > little shirts.... so it becomes "within a fly's SHIRT of just..."
> >
> > Isn't that sweet??
>
> Hee hee. That's fine with me.
>
> -- Dave Keenan

It's mos "proppa..."

More serious posts after I catch up! @#$#@!#$#$#!!!!

_____________ ______ _____
Joseph Pehrson

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

>
> Microtempered is currently a synonym for wafso-just.

Is it really? If you just took a JI scale and introduced small random
errors into it, would it be microtempered, or just wafso-just?

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> > But I want to
> > make the allowable error for these categories taper off
> > as the odd-limit of the ratio increases. Paul disagrees.
> >
> > -- Dave Keenan
>
>
> I agree with you, Dave.

Actually, I mostly agree with Dave too, as countless posts on this
list since the Mills days should testify. I just wanted Dave to do a
few listening experiments to understand how, in some cases, the
relationship may not be quite so clear-cut.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> >
> > Microtempered is currently a synonym for wafso-just.
>
> Is it really? If you just took a JI scale and introduced small
random
> errors into it, would it be microtempered, or just wafso-just?

Aha! Excellent point. A microtempered scale is necessarily wafso-just
but not all wafso-just scales are microtempered. Thanks.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22522.html#22979

> --- In tuning@y..., paul@s... wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > >
> > > Microtempered is currently a synonym for wafso-just.
> >
> > Is it really? If you just took a JI scale and introduced
> > small random errors into it, would it be microtempered,
> > or just wafso-just?
>
> Aha! Excellent point. A microtempered scale is necessarily
> wafso-just but not all wafso-just scales are microtempered.
> Thanks.

Please clue me in.

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

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_22522.html#22979

> --- In tuning@y..., paul@s... wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > >
> > > Microtempered is currently a synonym for wafso-just.
> >
> > Is it really? If you just took a JI scale and introduced small
> random
> > errors into it, would it be microtempered, or just wafso-just?
>
> Aha! Excellent point. A microtempered scale is necessarily wafso-
just
> but not all wafso-just scales are microtempered. Thanks.

So, Dave... then in your definition there is *always* an attempt to
push a scale toward just intonation if it is "microtempered..."

There would be no other possible use of "microtemperment..."???

__________ _______ ______
Joseph Pehrson

P.S. I did like the "little flys with shirts," but I
admit "microtemperament" is a bit more dignified!

--- In tuning@y..., jpehrson@r... wrote:
>
> So, Dave... then in your definition there is *always* an attempt to
> push a scale toward just intonation if it is "microtempered..."

I would say it the other way around . . . "tempered" means pushed away from JI . . . there is
always an attempt to produce more consonances when a scale is "microtempered". It's
synonymous with plain old "tempered", but the deviations from JI are very small.

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22522.html#23307

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > So, Dave... then in your definition there is *always* an attempt
to
> > push a scale toward just intonation if it is "microtempered..."
>
> I would say it the other way around . . . "tempered" means pushed
away from JI . . . there is
> always an attempt to produce more consonances when a scale
is "microtempered". It's
> synonymous with plain old "tempered", but the deviations from JI
are very small.

OH! Gotcha! Thanks, Paul!

So "microtemperament" is essentially just BETTER or "JUSTER" (more
refined toward pure consonances_ temperament... same process..
_______ ________ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> So, Dave... then in your definition there is *always* an attempt to
> push a scale toward just intonation if it is "microtempered..."

Err. It's usually the other way 'round. A scale starts out as strictly
just (SJ) and is pushed slightly away from that particular SJ so as to
be close to a whole range of different SJ scales. i.e. Increase the
available consonances by introducing very small errors (say < 3c).

> There would be no other possible use of "microtemperment..."???

Who knows. It's early daze.

-- Dave Keenan