Dictionary query

My Dictionary entries
http://www.ixpres.com/interval/dict/positive.htm
http://www.ixpres.com/interval/dict/negative.htm

define positive and negative tuning systems as those
which have "5ths" larger or smaller, respectively,
than the 700-cent 12-EDO "5th".

Isn't that wrong? Doesn't the 3:2 ratio define the
boundary between positive and negative? Is there
more than one accepted usage? Help!

-monz

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It's correct. The interval of 12 fifths minus 7 octaves
(the Pythagorean comma) defines whether a system is positive
or negative. So with a fifth of 3/2 it's positive because
P is greater than 1/1.

Manuel

In-Reply-To: <003f01c19866$03a02e80$af48620c@dsl.att.net>
monz wrote:

> define positive and negative tuning systems as those
> which have "5ths" larger or smaller, respectively,
> than the 700-cent 12-EDO "5th".

That's correct.

> Isn't that wrong? Doesn't the 3:2 ratio define the
> boundary between positive and negative? Is there
> more than one accepted usage? Help!

You remember when we were at the Huygens Fokker Institute in Amsterdam,
and I pulled a Bosanquet book off the shelves? That's where the
"positive"/"negative" terminology is defined, and it is relative to
12-equal, not Pythagorean. If you'd been paying attention, you could
have checked it. One of Erv Wilson's early Xenharmonikon articles
reiterates this, and another (I think "On Linear Notations ...") extends
it for ETs other than 12.

Graham

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

That's where the
> "positive"/"negative" terminology is defined, and it is relative to
> 12-equal, not Pythagorean. If you'd been paying attention, you could
> have checked it. One of Erv Wilson's early Xenharmonikon articles
> reiterates this, and another (I think "On Linear Notations ...") extends
> it for ETs other than 12.

Now all that's left is to get it to make a particle of sense.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> That's where the
> > "positive"/"negative" terminology is defined, and it is relative
to
> > 12-equal, not Pythagorean. If you'd been paying attention, you
could
> > have checked it. One of Erv Wilson's early Xenharmonikon
articles
> > reiterates this, and another (I think "On Linear Notations ...")
extends
> > it for ETs other than 12.
>
> Now all that's left is to get it to make a particle of sense.

This is a rich and deep literature we're referencing. Your arrogance
has won you few fans on the other lists, and if you are to correspond
with other mathematically-oriented tuning theorists, you may be well
advised to try to keep it in check.

So you've read all of Bosanquet and Wilson and come to the conclusion
that they were total ignorami? That's the impression you're giving.
Though you and I may have come up with different definitions
of "positive" and "negative" had we been the first in this field,
fortunately we're not, and the terms have been quite fruitful for
generations of theorists who would disprove your seeming attitude
that you possess 100% of the world's intelligence on these matters.

Perhaps a conversation with John Chalmers would be well-advised for
you at this point.

All that said, please let it be known that I am greatly humbled by
the depth of your vision and the quantity and quality of your
efforts, especially on the subjects I am particularly interested in.
Does becoming a leading contributor in this field mean that you have
to take a dismissive and patronizing attitude toward all others who
have touched it? If nothing else, sheer _politics_ would dictate that
a different tone might be more fruitful.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> So you've read all of Bosanquet and Wilson and come to the conclusion
> that they were total ignorami? That's the impression you're giving.

To you, perhaps. I would imagine other people might come to a more correct impression, namely that 12-et-centric thinking in my estimation makes no sense in this context, since it has nothing whatever to do with the 12-et. It ain't how a mathematician would define things, let's put it that way.

> Does becoming a leading contributor in this field mean that you have
> to take a dismissive and patronizing attitude toward all others who
> have touched it? If nothing else, sheer _politics_ would dictate that
> a different tone might be more fruitful.

My interest in that sort of politics is virtually nil.

It seems to me you are adopting the same line as Jon Szanto recently did, which amounts to saying that thinking for yourself and expressing your opinion amounts to heresy when a revered name is in question. I don't believe in thought control, and prefer to come to my own conclusions without someone helpfully reminding me what is and is not an acceptable thing to think. Nor do I think it is arrogant to come to a different conclusion than Partch, Bosanquet, or anyone else.

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

> To you, perhaps. I would imagine other people might come to a more correct impression, namely that 12-et-centric thinking in my estimation makes no sense in this context, since it has nothing whatever to do with the 12-et. It ain't how a mathematician would define things, let's put it that way.

One can, of course, turn this around--if Bosanquet looked at things relative to the 12-et, then he was *not* looking at them purely as temperaments, but presumably had 12-note keyboards and the like in mind. At least, I hope when he was tuning a 31-note keyboard he was not thinking in 12-et terms.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > So you've read all of Bosanquet and Wilson and come to the
conclusion
> > that they were total ignorami? That's the impression you're
giving.
>
> To you, perhaps. I would imagine other people might come to a more
> correct impression, namely that 12-et-centric thinking in my >
estimation makes no sense in this context,

In what context??? We're talking about a _definition_ of "positive"
and "negative". The only relevant context is the context in which
these terms have been used in tuning theory!

>since it has nothing whatever to do with the 12-et.

_What_ has nothing whatever to do with the 12-tET?

> > Does becoming a leading contributor in this field mean that you
have
> > to take a dismissive and patronizing attitude toward all others
who
> > have touched it? If nothing else, sheer _politics_ would dictate
that
> > a different tone might be more fruitful.
>
> My interest in that sort of politics is virtually nil.
>
> It seems to me you are adopting the same line as Jon Szanto
>recently did, which amounts to saying that thinking for yourself and
>expressing your opinion amounts to heresy when a revered name is in
>question.

That's not what I'm saying.

>I don't believe in thought control, and prefer to come to my own
>conclusions without someone helpfully reminding me what is and is
>not an acceptable thing to think.

God bless you for that.

>Nor do I think it is arrogant to come to a different conclusion than
>Partch, Bosanquet, or anyone else.

No. But to say that something doesn't make a "particle of sense",
when it makes absolutely perfect sense in the context in which it's
been used, is arrogant and insulting.

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

> No. But to say that something doesn't make a "particle of sense",
> when it makes absolutely perfect sense in the context in which it's
> been used, is arrogant and insulting.

I wasn't referring to the context in which it's been used, in fact I don't know what that is; for my guesses see my previous posting. I was referring to the context in which we were using it now--namely, to refer to linear temperaments with some type of fifth as generator, without reference to the 12-et or any other et. In that context, it manifestly makes no sense to drag in the 12-et. How does it even enter the conversation, so to speak? Who invited it, and why? Why not base everything on 53 instead, for example?

It seems to me you are reading a highly personal dismissal of Bosanquet and Wilson into what I wrote, instead of looking at it on its merits.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> My Dictionary entries
> http://www.ixpres.com/interval/dict/positive.htm
> http://www.ixpres.com/interval/dict/negative.htm
>
> define positive and negative tuning systems as those
> which have "5ths" larger or smaller, respectively,
> than the 700-cent 12-EDO "5th".
>
>
> Isn't that wrong? Doesn't the 3:2 ratio define the
> boundary between positive and negative? Is there
> more than one accepted usage? Help!

Hi Monz.

The accepted usage, not in some imagined "context" that Gene may wish
to dream up as his ego inflates to the point where all existing
tuning literature vanishes from existence, but in the actual context
in which the term has been used, has the dividing line at 700 cents.

Today, on these lists, we tend to call negative systems "meantone"
and positive systems "schismic". The reason 700 cents was chosen as
the dividing line between "negative" and "positive" is that when the
fifth is below 700 cents, the "meantone" (+4 fifths) approximation to
the 5/4 is better than the "schismic" (-8 fifths) approximation to
the 5/4. When the fifth is above 700 cents, the "schismic"
approximation to the 5/4 is better than the "meantone" approximation
to the 5/4. I might differ, saying that there is a "gray area", and
also factoring the 6/5 into consideration . . . but the definitions
are well-established and there is no reason to favor ones which could
breed potential contradictions.

As for your definition pages, Monz, they definitely give the wrong
idea. Positive systems should be characterized by the fraction of a
_schisma_ that the fifths differ from just -- this is the relevant
measure of them. Knowing what fraction of a syntonic comma a positive
system's fifth might have been _increased_ by is irrelevant for
understanding the functioning of the system, and is potentially
misleading.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > No. But to say that something doesn't make a "particle of sense",
> > when it makes absolutely perfect sense in the context in which
it's
> > been used, is arrogant and insulting.
>
> I wasn't referring to the context in which it's been used,

But look at your posts again -- YOU were the one who brought
up "context". What other context is there?

>in fact I don't know what that is; for my guesses see my previous
>posting. I was referring to the context in which we were using it
>now--namely, to refer to linear temperaments with some type of fifth
>as generator, without reference to the 12-et or any other et.

What do you mean, the context in which we are using it now? Who among
us has used the terms "negative" and "positive" in this context? None
of us have. The only relevant context is that in which the terms
_have_ been used.

>In that context, it manifestly makes no sense to drag in the 12-et.
>How does it even enter the conversation, so to speak? Who invited
>it, and why? Why not base everything on 53 instead, for example?

See my last post.

>It seems to me you are reading a highly personal dismissal of
>Bosanquet and Wilson into what I wrote,

Not highly personal.

>instead of looking at it on its merits.

If you look at the thread of this conversation again, perhaps you'll
see that you clearly came off as saying "whoever thought of this
definition was a nitwit".

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> The accepted usage, not in some imagined "context" that Gene may wish
> to dream up as his ego inflates to the point where all existing
> tuning literature vanishes from existence, but in the actual context
> in which the term has been used, has the dividing line at 700 cents.

I'm not sure why you keep attacking me--in what way have I injured you?

So far as this "positive" and "negative" business goes, my suggestion would be to use "sharp" and "flat" instead, so that a "sharp system" would have a sharp fifth, and a "flat system" (which would include the 12 et) would have a flat fifth.

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

> So far as this "positive" and "negative" business goes, my
>suggestion would be to use "sharp" and "flat" instead, so that
>a "sharp system" would have a sharp fifth, and a "flat system"
(which would include the 12 et) would have a flat fifth.

This set of definitions is independent of the "positive"
and "negative" definitions. The distinctions they draw are different
and the terms they use are different. So why is this a matter
of "instead"?

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

> This set of definitions is independent of the "positive"
> and "negative" definitions. The distinctions they draw are different
> and the terms they use are different. So why is this a matter
> of "instead"?

Your proposal is that 700 cents is the border between meantone and schismic, if I understand correctly, since the third you get by going round each way on a 12-et will of course be the same. I could object that a 701 cent tuning is more likely to be used in a meantone manner anyway, but more to the point, why not simply say "schismic" or "meantone" if that is what you mean? After all, a 22 or 46 et isn't either one but can also be using a generator of a fifth, and even discounting that kind of possibility, there are in theory at least an infinite number of other possible 5-limit temperaments using a fifth+octave scheme.

What is the point of saying "schismic is a positive system" and "meantone is a negative system" on your view--or does one say that sort of thing?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > This set of definitions is independent of the "positive"
> > and "negative" definitions. The distinctions they draw are
different
> > and the terms they use are different. So why is this a matter
> > of "instead"?
>
> Your proposal is that 700 cents is the border between meantone and
>schismic, if I understand correctly, since the third you get by
>going round each way on a 12-et will of course be the same.

This is not "my proposal", Gene, this is the _established
definition_. Monz is not producing a dictionary of Gene-and-Paul-
tuning-terms -- it's a Dictionary of tuning terms, which means it
accounts, as well as possible, for the intended meanings of words as
they have been used in the tuning literature. Take a look at the
Microtonal Bibliography on the Huygens-Fokker page to get an idea of
how vast this literature is.

>I could object that a 701 cent tuning is more likely to be used in a
>meantone manner anyway,

I would agree, and objection was in my mind when I wrote, 'I might
differ, saying that there is a "gray area", . . . '

>but more to the point, why not simply say "schismic" or "meantone"
>if that is what you mean? After all, a 22 or 46 et isn't either one

Actually, I simplified my explanation slightly. There are other
possible subcategories of "negative" besides "meantone", and
of "positive" besides "schismic". You might want to look at
Bosanquet's discussion of 22, for example. He was very well aware of
this fact, as was Wilson, and did not leave it unaccounted for. In
fact, Bosanquet presents some rigorous mathematical formulae, which
may not be very interesting to you and I currently, and we are
certainly in agreement about the 12-centricity of them (which Paul
Rapaport, for example, would disagree with us), but they are
nonetheless mathematically correct and form part of the literature.

>What is the point of saying "schismic is a positive system"
>and "meantone is a negative system" on your view--or does one say
>that sort of thing?

The terms, as far as I've seen them used, have mainly been applied to
ETs.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 10, 2002 1:56 PM
> Subject: [tuning-math] Re: Dictionary query
>
>
> Today, on these lists, we tend to call negative systems "meantone"
> and positive systems "schismic". The reason 700 cents was chosen as
> the dividing line between "negative" and "positive" is that when the
> fifth is below 700 cents, the "meantone" (+4 fifths) approximation to
> the 5/4 is better than the "schismic" (-8 fifths) approximation to
> the 5/4. When the fifth is above 700 cents, the "schismic"
> approximation to the 5/4 is better than the "meantone" approximation
> to the 5/4. I might differ, saying that there is a "gray area", and
> also factoring the 6/5 into consideration . . . but the definitions
> are well-established and there is no reason to favor ones which could
> breed potential contradictions.
>
> As for your definition pages, Monz, they definitely give the wrong
> idea. Positive systems should be characterized by the fraction of a
> _schisma_ that the fifths differ from just -- this is the relevant
> measure of them. Knowing what fraction of a syntonic comma a positive
> system's fifth might have been _increased_ by is irrelevant for
> understanding the functioning of the system, and is potentially
> misleading.

Thanks very much for that, Paul. So how does it look now?
http://www.ixpres.com/interval/dict/positive.htm

-monz

_________________________________________________________
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Well, now the tuning-math list has 58 members instead of 57. And I
might not have even signed up had it not been for reading, for the
past week or so, a couple of histories of mathematics and
mathematicians. The second category was represented by the book "Mem
of Mathematics - The Lives and Achievements of the Great
Mathematicians from Zeno to Poincare", by E. T. Bell. Astonishing
that it was written over 60 years ago, with a style that is
remarkably fresh.

And reminded me that every single one of us, interests, occupations,
and motivations aside, are human beings. In all the variety that the
species encompasses. So to set the record straight:

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> It seems to me you are adopting the same line as Jon Szanto
recently did, which amounts to saying that thinking for yourself and
expressing your opinion amounts to heresy when a revered name is in
question.

That is not quite right, Gene: you are more than welcome to have any
opinion you may so please, informed or not. Any reasonable person can
very well object to that opinion as well. The aforementioned in no
way indicates that in all situations everything is open to
interpretation. I called you on a couple of issues because where you
read an author, I did so and *knew* the author (you probably didn't
realize I helped Partch proof-read the galley proofs of the second
edition of "Genesis of a Music"). You might have heard his music;
I've heard it, played it on his instruments, and rehearsed it with
him.

It's not heresy, and Partch goes to great lengths to ask that his
work - most pointedly his theoretical work - be subjected to
scrutiny. I've seen this over a number of years, and find
illumination (at times). His stance doesn't make it easy on someone
like myself, when I see his works performed in sub-par, bastardized
arrangements, but I'm working on it, both in refining my discourse
and learning to accept some inevitabilities.

That is had to come to Paul Erlich, your strongest supporter and a
person who has grown greatly in these areas on these lists, is pretty
sad. But I'm glad that, as of this juncture, you might have opened
the door to the fact that you have the ability to either foster
collaborative and collegeal dialogue, or to simply utilize
phraseology that - as an intelligent and learned person - must surely
know will come off as filled with both indifference and audacity.

That, I would propose, is a sure-fire way to go back to toiling in
isolation. Everyone here would be impoverished if that were to
happen, and there is no reason for it to be so. I hope Paul's wise
counsel has even the smallest of effect on you.

Respectfully submitted as a member of the tuning community,
Jon Szanto

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Thursday, January 10, 2002 1:56 PM
> > Subject: [tuning-math] Re: Dictionary query
> >
> >
> > Today, on these lists, we tend to call negative
systems "meantone"
> > and positive systems "schismic". The reason 700 cents was chosen
as
> > the dividing line between "negative" and "positive" is that when
the
> > fifth is below 700 cents, the "meantone" (+4 fifths)
approximation to
> > the 5/4 is better than the "schismic" (-8 fifths) approximation
to
> > the 5/4. When the fifth is above 700 cents, the "schismic"
> > approximation to the 5/4 is better than the "meantone"
approximation
> > to the 5/4. I might differ, saying that there is a "gray area",
and
> > also factoring the 6/5 into consideration . . . but the
definitions
> > are well-established and there is no reason to favor ones which
could
> > breed potential contradictions.
> >
> > As for your definition pages, Monz, they definitely give the
wrong
> > idea. Positive systems should be characterized by the fraction of
a
> > _schisma_ that the fifths differ from just -- this is the
relevant
> > measure of them. Knowing what fraction of a syntonic comma a
positive
> > system's fifth might have been _increased_ by is irrelevant for
> > understanding the functioning of the system, and is potentially
> > misleading.
>
>
> Thanks very much for that, Paul. So how does it look now?
> http://www.ixpres.com/interval/dict/positive.htm
>
>
>
> -monz

Unfortunately, 22 is not a schismic temperament . . . this is my
fault, of course . . . I later alluded to the correct definition in
conversation with Gene, as you can see . . . I'm a bit too tired to
correct this now, but I'm sure Graham or John Chalmers can help you
if they're available before I can get back to you.

P.S. Monz, why do you like to keep incomplete/incorrect
definitions/descriptions at the top of your dictionary pages, or even
in there at all? Why not attempt for the more precise, univerally
agreed-on definitions/examples first, and then post
alternate/intermediary-stages-in-someone's-thinking stuff later,
preferably on entirely separate webpages?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 10, 2002 7:05 PM
> Subject: [tuning-math] Re: Dictionary query
>
>
> > Thanks very much for that, Paul. So how does it look now?
> > http://www.ixpres.com/interval/dict/positive.htm
> >
> >
> >
> > -monz
>
> Unfortunately, 22 is not a schismic temperament . . . this is my
> fault, of course . . . I later alluded to the correct definition in
> conversation with Gene, as you can see . . . I'm a bit too tired to
> correct this now, but I'm sure Graham or John Chalmers can help you
> if they're available before I can get back to you.

OK, but if they don't post anything here, please do give me
more info.

> P.S. Monz, why do you like to keep incomplete/incorrect
> definitions/descriptions at the top of your dictionary pages, or even
> in there at all? Why not attempt for the more precise, univerally
> agreed-on definitions/examples first, and then post
> alternate/intermediary-stages-in-someone's-thinking stuff later,
> preferably on entirely separate webpages?

Umm... because I'm a decent author but a lousy editor?

The Dictionary is always a work-in-progress, and I prefer to
simply amend definitions that are not complete.

But if they really are *incorrect*, then please, by all means,
not only give me the correct information, but also tell me what
to get rid of! I have no problem deleting something that really
is wrong. (I may not get around to it as quickly as any of us
would like... but that's another story...)

If even one other person here agrees with you that the commatic
description of positive systems is absolutely useless, then
they're history. Let me know.

-monz

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

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

> OK, but if they don't post anything here, please do give me
> more info.

I have some of the original Bosanquet papers . . . I'll try to bring
them in.

> If even one other person here agrees with you that the commatic
> description of positive systems is absolutely useless, then
> they're history. Let me know.

Well, let me put it this way. For meantone systems, the meaning of
_which_ comma you're talking about is clear from the way meantone
works. If you're describing a non-meantone system as "x/y-comma"
whatever, then it's ambiguous. At least specify _which_ comma you're
talking about, that will at least make the specification
mathematically precise. But functionally, schismic temperaments are
best described by the fraction of the schimsa that's tempered
out . . . 1/8 schisma is Helmholtian, 1/9 schism is Sabat-Garibaldi's
Dinarra tuning . . . am I making any sense?

Paul, Gene, Joe,

You've missed or ignored my answer to Joe's question,
which was the most concise I could give.
The borderline is the point where the Pythagorean comma
vanishes: 700 cents. This choice is not 12-tET centric
in my view.

> Thanks very much for that, Paul. So how does it look now?
> http://www.ixpres.com/interval/dict/positive.htm

You could add that systems with p=1 (scale steps) are
called singly positive, with p=2 doubly positive,
p=-1 singly negative, etc.

Manuel

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 11, 2002 1:49 AM
> Subject: Re: [tuning-math] Re: Dictionary query
>
>
> Paul, Gene, Joe,
>
> You've missed or ignored my answer to Joe's question,
> which was the most concise I could give.
> The borderline is the point where the Pythagorean comma
> vanishes: 700 cents. This choice is not 12-tET centric
> in my view.

Ahh ... so then it's not 12-*tET* centric, but it *is*
12-*tone* centric, because the Pythagorean comma is
("8ve"-invariant) 3^12.

> > Thanks very much for that, Paul. So how does it look now?
> > http://www.ixpres.com/interval/dict/positive.htm
>
> You could add that systems with p=1 (scale steps) are
> called singly positive, with p=2 doubly positive,
> p=-1 singly negative, etc.

Thanks, Manuel, good idea ... but what's "p"?

-monz

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>Ahh ... so then it's not 12-*tET* centric, but it *is*
>12-*tone* centric, because the Pythagorean comma is
>("8ve"-invariant) 3^12.

Yup.

>Thanks, Manuel, good idea ... but what's "p"?

Symbol for the Pythagorean comma.
If v is the size of the fifth, and a the size of the
octave, then p = 12 v - 7 a.
For example in 31-tET, v=18 and a=31, so p=-1.

Manuel

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 11, 2002 2:08 AM
> Subject: Re: [tuning-math] Re: Dictionary query
>
>
> > Thanks, Manuel, good idea ... but what's "p"?
>
> Symbol for the Pythagorean comma.
> If v is the size of the fifth, and a the size of the
> octave, then p = 12 v - 7 a.
> For example in 31-tET, v=18 and a=31, so p=-1.

OK, got it! Thanks for providing the concrete 31-tET
example ... now I understand.

-monz

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--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Paul, Gene, Joe,
>
> You've missed or ignored my answer to Joe's question,
> which was the most concise I could give.
> The borderline is the point where the Pythagorean comma
> vanishes: 700 cents. This choice is not 12-tET centric
> in my view.
>
> > Thanks very much for that, Paul. So how does it look now?
> > http://www.ixpres.com/interval/dict/positive.htm
>
> You could add that systems with p=1 (scale steps) are
> called singly positive, with p=2 doubly positive,
> p=-1 singly negative, etc.
>
> Manuel

Where p is the Pythagorean comma.