Classificiation of musical scales

Motivated by a recent dicussion in the group I have tried to provide a
comprehensive and clear classification of musical scales.

I have collected the ideas in the attached short file. Writting all this in
the text of the email seemed a bit too much, besides I added a small figure
to clarify the ideas. The file is in PDF.

Comments would be most wellcome.

Thanks

Carlos

Carlos wrote:

>Motivated by a recent dicussion in the group I have tried to provide a >comprehensive and clear classification of musical scales. >
>I have collected the ideas in the attached short file. Writting all this in >the text of the email seemed a bit too much, besides I added a small figure >to clarify the ideas. The file is in PDF.
> >
It's easier to get at, and takes up less bandwidth, if you quote it in the e-mail. I also saw something about Yahoo removing attachments from the archives.

>Comments would be most wellcome.
> >
It may work as well as any other branching classification. I prefer to think of scales as having a number of orthogonal properties -- so the equivalence interval may be an octave, the scale may approximate some subset of just intonation, and so on.

You're counting the dimensions according to the number of odd primes in the JI being apporoximated, are you? Like Lindley and Turner-Smith.

Graham

On Thursday 21 August 2003 21:10, Graham Breed wrote:
> Carlos wrote:
> >Motivated by a recent dicussion in the group I have tried to provide a
> >comprehensive and clear classification of musical scales.
> >
> >I have collected the ideas in the attached short file. Writting all this
> > in the text of the email seemed a bit too much, besides I added a small
> > figure to clarify the ideas. The file is in PDF.
>
> It's easier to get at, and takes up less bandwidth, if you quote it in
> the e-mail. I also saw something about Yahoo removing attachments from
> the archives.

I will keep it in mind for future ocassions.

>
> >Comments would be most wellcome.
>
> It may work as well as any other branching classification. I prefer to
> think of scales as having a number of orthogonal properties -- so the
> equivalence interval may be an octave, the scale may approximate some
> subset of just intonation, and so on.
>

I understand the orthogonality in the sense that, just thinking about just
intervals as generators, each inteval is an indepent dimension and then a
particular interval in the interval space is a projection in each axis.

Is this what you are meaning?

To be more precise, I guess one would have to define a scalar product type of
operation. I guess if you define the vectors in the usual sense
(first_JI_interval, second_JI_interval, ...) an so on and use the regular
product among vectors, but of course we are taking about vectorial space in
which the the components of the vector are number mod_N. I am not sure if
this arrays will have all the properties of a proper vector space.

Is this the case? Is there a writting about this somewhere?

> You're counting the dimensions according to the number of odd primes in
> the JI being apporoximated, are you? Like Lindley and Turner-Smith.
>

Not necessarily I was thinking about each dimension as a indepedent interval
that you need to complete your octave (say).

For example in the 1/4 comma tunning, you have tempered fith such that 4
firfhts equates a pure third. That would be one generator and one dimension.
Then you need another generator to complete you octave set. That would be
wolf fifth.

In this case we of all the product that you could generate in the interval 2d
space, for the actual scale you would take only points in the temperd fifht
axis and then only one point outside that axis that would be needed to
complete the cycle.

Is this Ok with you?

Thanks,

Carlos
>
> Graham
>
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Carlos wrote:

>I understand the orthogonality in the sense that, just thinking about just >intervals as generators, each inteval is an indepent dimension and then a >particular interval in the interval space is a projection in each axis. >
>Is this what you are meaning? > >
I mean "orthogonal" in the sense of the parameters being independent. So the Bohlen Pierce is a nonoctave just intonation scale. But there's nothing about it being just intonation that tells you it isn't likely to have the octave as an equivalence. However, your classification first puts it in a "nonoctave" category that means it has no connection with other just intonation scales.

>To be more precise, I guess one would have to define a scalar product type of >operation. I guess if you define the vectors in the usual sense >(first_JI_interval, second_JI_interval, ...) an so on and use the regular >product among vectors, but of course we are taking about vectorial space in >which the the components of the vector are number mod_N. I am not sure if >this arrays will have all the properties of a proper vector space. >
>Is this the case? Is there a writting about this somewhere?
> >
We have been talking about scalar products recently, although that's a different question. What we would end up with is a lattice (in the algebraic sense of the word) but I don't know much about them.

>For example in the 1/4 comma tunning, you have tempered fith such that 4 >firfhts equates a pure third. That would be one generator and one dimension. >Then you need another generator to complete you octave set. That would be >wolf fifth. > >
Ah, well, in group theory terms an octave equivalent meantone only has one generator -- the fifth. The wolf is simply one of the intervals that gets generated (by 11 fifths). So meantone would be one dimensional (or two dimensional if you include the octave as a generator).

>In this case we of all the product that you could generate in the interval 2d >space, for the actual scale you would take only points in the temperd fifht >axis and then only one point outside that axis that would be needed to >complete the cycle.
>
>Is this Ok with you?
> >
You don't need the point outside the axis. Only the tempered fifth is a generator.

Graham

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> Motivated by a recent dicussion in the group I have tried to
provide a
> comprehensive and clear classification of musical scales.
>
> I have collected the ideas in the attached short file. Writting all
this in
> the text of the email seemed a bit too much, besides I added a
small figure
> to clarify the ideas. The file is in PDF.

Yahoo has quit supporting file attachments; you might put this up in
the files area so we all can see it.

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> Motivated by a recent dicussion in the group I have tried to
provide a
> comprehensive and clear classification of musical scales.
>
> I have collected the ideas in the attached short file. Writting all
this in
> the text of the email seemed a bit too much, besides I added a
small figure
> to clarify the ideas. The file is in PDF.
>
> Comments would be most wellcome.
>
> Thanks
>
> Carlos

it says attachment not stored. you need to put it in the files folder
or somewhere like that.

On Thursday 21 August 2003 22:13, Graham Breed wrote:
> Carlos wrote:
> >I understand the orthogonality in the sense that, just thinking about just
> >intervals as generators, each inteval is an indepent dimension and then a
> >particular interval in the interval space is a projection in each axis.
> >
> >Is this what you are meaning?
>
> I mean "orthogonal" in the sense of the parameters being independent.
> So the Bohlen Pierce is a nonoctave just intonation scale. But there's
> nothing about it being just intonation that tells you it isn't likely to
> have the octave as an equivalence. However, your classification first
> puts it in a "nonoctave" category that means it has no connection with
> other just intonation scales.
>

I understand. I separated right away the octave equivalence ones from others
because of their much higher practically implications, but you are right that
the same type of sub-classification that applies to the octave equivalence
group would apply to the non-octave and then the equivalence interval could
be considered like another dimension.

> >To be more precise, I guess one would have to define a scalar product type
> > of operation. I guess if you define the vectors in the usual sense
> >(first_JI_interval, second_JI_interval, ...) an so on and use the regular
> >product among vectors, but of course we are taking about vectorial space
> > in which the the components of the vector are number mod_N. I am not sure
> > if this arrays will have all the properties of a proper vector space.
> >
> >Is this the case? Is there a writting about this somewhere?
>
> We have been talking about scalar products recently, although that's a
> different question. What we would end up with is a lattice (in the
> algebraic sense of the word) but I don't know much about them.
>

I might try to give it a though and come back on this.

> >For example in the 1/4 comma tunning, you have tempered fith such that 4
> >firfhts equates a pure third. That would be one generator and one
> > dimension. Then you need another generator to complete you octave set.
> > That would be wolf fifth.
>
> Ah, well, in group theory terms an octave equivalent meantone only has
> one generator -- the fifth. The wolf is simply one of the intervals
> that gets generated (by 11 fifths). So meantone would be one
> dimensional (or two dimensional if you include the octave as a generator).
>

OK, you are right. Then the example would have to be those mean tones that
have more than size of tempered fifths. I will correct this in the small
note.

Thanks for discussion.

Carlos

> >In this case we of all the product that you could generate in the interval
> > 2d space, for the actual scale you would take only points in the temperd
> > fifht axis and then only one point outside that axis that would be needed
> > to complete the cycle.
> >
> >Is this Ok with you?
>
> You don't need the point outside the axis. Only the tempered fifth is a
> generator.
>
>
> Graham
>
>
>
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

> Thanks for discussion.

You've still not put your pdf file up under files, so many of us are
locked out of the discussion. This is not the way to do things.

On Thursday 21 August 2003 22:49, Paul Erlich wrote:
> --- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> > Motivated by a recent dicussion in the group I have tried to
>
> provide a
>
> > comprehensive and clear classification of musical scales.
> >
> > I have collected the ideas in the attached short file. Writting all
>
> this in
>
> > the text of the email seemed a bit too much, besides I added a
>
> small figure
>
> > to clarify the ideas. The file is in PDF.
> >
> > Comments would be most wellcome.
> >
> > Thanks
> >
> > Carlos
>
> it says attachment not stored. you need to put it in the files folder
> or somewhere like that.
>

I'll do, as soon as I introduce some of the comments by Gene.

Thanks

>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

The file is up in the server. It happened that I had to re-subscribe to the
group to be able to uploaded, but there it is.

Thanks

Carlos

On Friday 22 August 2003 07:28, Gene Ward Smith wrote:

> > Thanks for discussion.
>
> You've still not put your pdf file up under files, so many of us are
> locked out of the discussion. This is not the way to do things.
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

> > --- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> > > Motivated by a recent dicussion in the group I have tried to
> > provide a
> > > comprehensive and clear classification of musical scales.
...

Carlos,

Your seem to be classifying methods of scale _construction_, rather
than how the scales sound or can be used. And that's fine, so long as
you recognise it for what it is, and realise that two scales may be in
different categories but be audibly (and even immeasurably)
indistingushable from each other, in part because these categories
rely on the purely mathematical distinction between ratios and their
neighbouring irrational numbers that may only be infinitesimally
different. I prefer to use the term "justly intoned" for an audible
distinction (with fuzzy boundaries), and "rationally constructed"
otherwise.

Even so, your scheme seems to have no place for tunings that have a
single generator but no particular interval of equivalence, such as
Gary Morrison's 88-cET. This is where successive pitches are simply
placed 88 cents apart for as far as you like in either direction from
a starting pitch.

From a purely mathematical point of view. 88-cET is a one-dimensional
system, while meantone and Pythagorean (3-limit rational) are two
dimensional. We call the latter "linear" temperaments because we are
taking the octave for granted as the other generator. ... Or are we?
We are certainly taking _something_ for granted as the other
generator. Let's look at another example.

Paul Erlich's pajara is a mathematically 2D system which is an
octave-equivalent linear temperament. One generator is a slightly wide
fifth, but the other generator is not the octave. It is the exact
half-octave. There are other octave-equivalent linear temperaments
that are mathematically 2D and that use 1/3 octave or 1/4 octave or
1/5 octave etc. Graham Breed even found an apparently useful one that
used 1/29 octave. When a generator is an exact submultiple of the
interval of equivalence like this, we call it a "period".

But it's unclear how important that distinction is (between a period
and an ordinary generator) for classification. Perhaps the so-called
interval-of-equivalence (IoE) should be considered as just another
just interval to be approximated. What's special about it? .. I think
that what's special about it is that when defining a scale, the number
of times the IoE is to be iterated is left undefined. It is considered
to depend on the physical properties of a particular instrument that
the scale might be realised on. How wide its compass.

Octave-equivalent 5-limit rational tunings are clearly mathematically
3D, although we call them "planar". And clearly planar temperaments

Now what about N equal divisions of the octave (N-tETs or N-EDOs)? Are
they (mathematically) closed 2D systems, or are they 1D? Do they have
a period of an octave and a generator that just happens to close on
the octave after a certain number of iterations, or do they have a
period of 1/N octave and no other generator.

The latter seems more meaningful to me, i.e. they are mathematically
1D. Since otherwise the generator can be arbitrarily chosen as any
interval of the ET. If one generator "closes" on another generator,
then it doesn't really belong to a different dimension, does it?

So I'd put -tETs and -cETs in the same mathematically 1D category
called "equal tunings". Then we have "linear tunings", "planar
tunings", etc. Then orthogonal to those categories we have the
question of what the IoE is if any, then the size of the period which,
for tunings that have an IoE, is expressed as how many periods there
are in the IoE, and for tunings that don't have an IoE, is simply
given in cents or whatever. Then we have the sizes of any subsequent
generators and how many times they are iterated.

All the best,
-- Dave Keenan

Dave,

Thanks for your comments.

I am providing a few remarks to some of them.

>
> Your seem to be classifying methods of scale _construction_, rather
> than how the scales sound or can be used.

That's actually right.

> you recognise it for what it is, and realise that two scales may be in
> different categories but be audibly (and even immeasurably)
> indistingushable from each other, in part because these categories
> rely on the purely mathematical distinction between ratios and their
> neighbouring irrational numbers that may only be infinitesimally
> different.

Are you referring to the "fuzzy neigbourghoods" of Lindley and Tuner-Smith. I
think they refer to it like a way to allow things like vibrato in violin
playing and the like. I personally found that disturbing for in the paper I
have by them, they do no end up using that concept in any practical way and
it complicates things from the point of view of the algebraic clasification
of the scales.

>I prefer to use the term "justly intoned" for an audible
> distinction (with fuzzy boundaries), and "rationally constructed"
> otherwise.
>

I am not sure I am understanding you here. Do you use the term "justly" in a
reference to the just ratios of prime numbers? and the term rationally, I
guess you do not mean it in the sense of using rational numbers but rather to
having a rational appraoch to the construction. But if this is the case, well
all case are rational, arent't they?

> Even so, your scheme seems to have no place for tunings that have a
> single generator but no particular interval of equivalence, such as
> Gary Morrison's 88-cET. This is where successive pitches are simply
> placed 88 cents apart for as far as you like in either direction from
> a starting pitch.
>

You are right. I made the assumption that there has to be allways a interval
of equivalence. By making explicit the interval, I get rid of one dimension.
And you are right of course, you could make explicit the interval of
equivanlence and add one dimension. I did no do that because in practical
terms most of us use octave equivalent scales and I wanted to focus on them.

I guess the clasification tree should be open one brach for non equivalent
scales as the Gary Morrison you mention or the Wendy Carlos scales.

> From a purely mathematical point of view. 88-cET is a one-dimensional
> system, while meantone and Pythagorean (3-limit rational) are two
> dimensional.

I agree in the sense discussed above.

>
> Now what about N equal divisions of the octave (N-tETs or N-EDOs)? Are
> they (mathematically) closed 2D systems, or are they 1D? Do they have
> a period of an octave and a generator that just happens to close on
> the octave after a certain number of iterations, or do they have a
> period of 1/N octave and no other generator.
>
Well they only really close on the octave if you are using a modulo 2 type of
arithmetic and hence if you enforce the octave equivalence, otherwise they do
no really close, as they series will extend to intervals above the 2:1.

Regards,

Carlos

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> > you recognise it for what it is, and realise that two scales may be in
> > different categories but be audibly (and even immeasurably)
> > indistingushable from each other, in part because these categories
> > rely on the purely mathematical distinction between ratios and their
> > neighbouring irrational numbers that may only be infinitesimally
> > different.
>
> Are you referring to the "fuzzy neigbourghoods" of Lindley and
Tuner-Smith.

I don't think so. But I haven't read about them.

> >I prefer to use the term "justly intoned" for an audible
> > distinction (with fuzzy boundaries), and "rationally constructed"
> > otherwise.
> >
>
> I am not sure I am understanding you here. Do you use the term
"justly" in a
> reference to the just ratios of prime numbers? and the term
rationally, I
> guess you do not mean it in the sense of using rational numbers but
rather to
> having a rational appraoch to the construction. But if this is the
case, well
> all case are rational, arent't they?

No. By "rationally constructed" I did mean "constructed using ratios"
and in future I will use the latter phrase to avoid confusion.

Many take "justly intoned" to be a synonym for "constructed using
ratios". This is not necessarily so. One can easily construct a scale
using ratios, that no one in their right mind would call justly
intoned if they _listened_ to it, in a typical harmonic timbre. I
understand that a tuning indistinguishable from 12-tET can be
constructed using only ratios of numbers less than 100 - the famous
Hammond organ.

>
> No. By "rationally constructed" I did mean "constructed using ratios"
> and in future I will use the latter phrase to avoid confusion.
>
> Many take "justly intoned" to be a synonym for "constructed using
> ratios". This is not necessarily so. One can easily construct a scale
> using ratios, that no one in their right mind would call justly
> intoned

But justly intoned by most usual definitions (see for example Randell
Dictionary of Music) is a scale that tries to replicate a portion of the
natural harmonics, hence you have ratios of the numbers 1,2,3,4, .. and so
forth, which are normally expressed in terms of their prime components.

On the other hand, not all scales based in ratios have to be JI.

Carlos

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> But justly intoned by most usual definitions (see for example Randell
> Dictionary of Music) is a scale that tries to replicate a portion of
the
> natural harmonics, hence you have ratios of the numbers 1,2,3,4, ..
and so
> forth, which are normally expressed in terms of their prime components.

There's no doubt that justly intoned scales include many intervals
whose frequency ratio are very close to simple ratios. But one can't
really _define_ just intonation that way, because such mathematical
models of the perceptual quality called "justness" are not (yet)
sufficiently accurate. It's currently too difficult to say, with any
hope of wide agreement among experts, what counts as a simple enough
ratio and what counts as close enough, in what musical contexts. And
note that a complex ratio may or may not be justly intoned. For
example, in most contexts, 20001:30001 is just, while 34:55 is not.

> On the other hand, not all scales based in ratios have to be JI.

I'm very glad you realise that. The equal tempered, but rational,
Hammond Organ is very useful for making this point.

The other side of the coin is that not all just intonation involves
ratios. The preeminent example here is Barbershop singing. There need
be no intention on the part of the singers to produce exact ratios of
frequencies, and no measurement could, even in principle, determine
whether they are producing ratios or merely irrational relationships
that happen to be very close to simple ratios. And yet no one denies
that they are producing justly intoned harmonies.

Fortunately human genetics and environment are sufficiently stable to
make it possible to define just intonation _injunctively_. That is, by
giving a series of instructions, which if correctly followed will lead
a person to experience that salient perceptual quality of some
harmonies that we call "justness", or "purity". We usually do so by
first defining (injunctively) another perceptual quality called
"beating", etc.

In my opinion the Oxford English Dictionary has a far better definion
of "just" than most musical dictionaries and other publications that
attempt to define it purely in terms of ratios.

"Just ... /Mus/. in /just interval/, etc.: Harmonically pure; sounding
perfectly in tune 1811."

I think that equating justness with numerical rationality (rather than
merely associating it) is a recent abberation and I hope it will soon
fade.

Of course, such simplifications will always be understandable in
casual discussions, but not in dictionaries or other texts which claim
to be authoritative.

>
> In my opinion the Oxford English Dictionary has a far better definion
> of "just" than most musical dictionaries and other publications that
> attempt to define it purely in terms of ratios.
>
> "Just ... /Mus/. in /just interval/, etc.: Harmonically pure; sounding
> perfectly in tune 1811."
>

Your views are interesting. The problem surges when you try to represent or
view scales as mathematical objects, which is what I wanted to do. Your
definition is more difficult to tackle because it involves a subjective
perception by the observer: "sounding perfectly in tune".

But a classification is just a means to organize your views, so as long as we
know we are talking about, there is no disagreement here.

Regards,

Carlos

Hi Carlos and Dave,

Dave wrote:

<< I think that equating justness with numerical rationality (rather
than merely associating it) is a recent abberation and I hope it
will soon fade.>>

I think it's fairly standard that 'just' means constructed by ratios of
(usually small) whole numbers. On the other hand, the term
'pure' is often used to describe perceptions of such intervals.
'Beatless' is strictly a matter of physics. Note also that many
European language music theories classify intervals as 'just'
that we in the US call 'perfect' - that is, 'fifths' and 'fourths'. These
terms are not 'perceptual.' They are a product of music theory. In
my opinion, the dictionary entry you cite which defines 'just' as
'sounding pure' is a poor one because it conflates the theoretical
term 'just' with the perceptual term 'pure'. These terms should be
kept in their seperate spheres.

It sounds like you want to say that intervals which are just do not
necessarily sound pure. I find this to be true in my experience,
but this is of course subjective, because statements about
perceptions always depend on the listener.

Aaron

--- In tuning-math@yahoogroups.com, Carlos <garciasuarez@y...> wrote:
> Your
> definition is more difficult to tackle because it involves a subjective
> perception by the observer: "sounding perfectly in tune".
>
> But a classification is just a means to organize your views, so as
long as we
> know we are talking about, there is no disagreement here.

Agreed.

--- In tuning-math@yahoogroups.com, "pitchcolor" <pitchcolor@a...> wrote:
> Dave wrote:
>
> << I think that equating justness with numerical rationality (rather
> than merely associating it) is a recent abberation and I hope it
> will soon fade.>>
>
> I think it's fairly standard that 'just' means constructed by ratios of
> (usually small) whole numbers.

Yes unfortunately, it has become fairly standard. "A recent
abberation" as I said. When it's pointed out that the "just" =
"numerically rational" definition would lead to the 12-tET Hammond
organ and top-octave-divider electronic organs being called "justly
intoned", while no one could ever say for sure whether Barbershop
singing is justly intoned or not, then people usually realise that
what they really mean by "just" is a perceptual quality, not a
mathematical one, and they have been oversimplifying.

> On the other hand, the term
> 'pure' is often used to describe perceptions of such intervals.

An interesting idea, but I'm afraid it's unsupportable. A quick search
on "just pure intonation tuning" (without the quotes) in Google showed
that the terms "pure" and "just" are still used as synonyms, even by
those who define "just" purely in terms of ratios.

As just one example, the very first sentence of Kyle Gann's "Just
Intonation Explained" contains the phrase "just-intonation (pure) tuning".

> 'Beatless' is strictly a matter of physics.

Really? What about when the beats get faster and we no longer perceive
them as beats, only as roughness, and then eventually we don't
perceive them at all? Perhaps there are two meaning of "beatless" that
we should be careful to distinguish.

> Note also that many
> European language music theories classify intervals as 'just'
> that we in the US call 'perfect' - that is, 'fifths' and 'fourths'.
These
> terms are not 'perceptual.' They are a product of music theory.

We are of course discussing the English meaning of "just", however,
even if you are right about this usage in other languages, it doesn't
help the case for "just" = "rational" any more than it does the
perceptual definition.

> In
> my opinion, the dictionary entry you cite which defines 'just' as
> 'sounding pure' is a poor one because it conflates the theoretical
> term 'just' with the perceptual term 'pure'. These terms should be
> kept in their seperate spheres.

Can you provide any evidence that they have ever been in separate spheres?

> It sounds like you want to say that intervals which are just do not
> necessarily sound pure.

No, I want to say that intervals which are rationally constructed do
not necessarily sound just or pure. And that intervals which sound
just or pure are not necessarily rationally constructed.

> I find this to be true in my experience,
> but this is of course subjective, because statements about
> perceptions always depend on the listener.

Of course. But we needn't run in terror from this subjectivity and try
to bury our heads in an objective but oversimplified mathematical
definition. The fact is that most people can recognise what is meant
by justly intoned when you demonstrate it to them by, for example,
playing two notes simultaneously, in a sustained timbre, rich in
harmonics, and slowly varying the pitch of one of them while asking
the listener to tell you when they hear something unusual.

I raised this some years ago on the main tuning list, and I found that
some people had a strong emotional attachment to ratios, whether they
resulted in any audible quality or not. And the term "JI" had become
something like a badge of cult status for them, so they felt very
threatened by what I was saying. So much so that some went off and
started new Yahoo groups. And others who could see the logic of what I
was saying, nevertheless decided not to support it, for the sake of peace.

Some folks agreed to use the term RI (for rational intonation) instead
of JI in those cases where the ratios did not imply (in your terms)
sounding pure. I never felt quite comfortable about that term (RI) but
I couldn't say why, and let it go (for the sake of peace). In this
present discussion, thankfully far less heated, I have realised why.

I think it is because the word "intonation" also refers to perception,
not mathematics. One cannot rationally _intone_ something, because one
cannot _hear_ rationality. "intonation" and "tuning" seem to be almost
synonyms, but not quite. Otherwise Kyle Gann would be guilty of
redundancy when he wrote ""just-intonation (pure) tuning".

So I think "RT" for "rational tuning" would be better, or possibly we
could get away with "RCI" for "rationally constructed intonation".

Dave wrote:

> > I think it's fairly standard that 'just' means constructed by
ratios of
> > (usually small) whole numbers.
>
> Yes unfortunately, it has become fairly standard. "A recent
> abberation" as I said.

But you've supplied nothing which substantiates this claim.

> > On the other hand, the term
> > 'pure' is often used to describe perceptions of such intervals.
>
> An interesting idea, but I'm afraid it's unsupportable. A quick
search
> on "just pure intonation tuning" (without the quotes) in Google
showed
> that the terms "pure" and "just" are still used as synonyms,
even by those who define "just" purely in terms of ratios.

I really don't think a Google search for some internet text makes
what I said 'unsupportable'. It's a small observation anyway - not
a great claim.

> > 'Beatless' is strictly a matter of physics.
>
> Really? What about when the beats get faster and we no
longer perceive
> them as beats, only as roughness, and then eventually we
don't
> perceive them at all? Perhaps there are two meaning of
"beatless" that
> we should be careful to distinguish.

There are physical frequency thresholds for wave cancellation
phenomenae which are called 'beats', even if they are based on
thresholds of human perception. If there needs to be a
distinction as you describe it then we simply need to say that it
either '_sounds beatless' or it '_is beatless'.

>
> > Note also that many
> > European language music theories classify intervals as 'just'
> > that we in the US call 'perfect' - that is, 'fifths' and 'fourths'.
> These
> > terms are not 'perceptual.' They are a product of music
theory.
>
> We are of course discussing the English meaning of "just",
however,
> even if you are right about this usage in other languages, it
doesn't
> help the case for "just" = "rational" any more than it does the
> perceptual definition.

The word 'just' (which translates as 'just' into English) is used as
a theoretical term in South America and in Poland, for example,
where 'fifths' are assumed to be 3/2 and 'fourths' are assumed to
be 4/3. These are theoretical constructs. These terms apply to
piano music, for example. 'Just' is clearly used there as a
_theoretical term. Again, if you need to make the distinction, then
just say either it '_sounds just' or it '_is just'.

> > In
> > my opinion, the dictionary entry you cite which defines 'just'
as
> > 'sounding pure' is a poor one because it conflates the
theoretical
> > term 'just' with the perceptual term 'pure'. These terms
should be
> > kept in their seperate spheres.
>
> Can you provide any evidence that they have ever been in
separate spheres?

No, I can only hold that they _belong in seperate spheres.
People make errors with language and say unclear things all the
time. We have two words there. It's worthwhile to exploit their
connotations. If you choose not to do that, OK, but I think it's
inefficient and unnecessary to make them synonymous.

> > It sounds like you want to say that intervals which are just do
not
> > necessarily sound pure.
>
> No, I want to say that intervals which are rationally constructed
do
> not necessarily sound just or pure. And that intervals which
sound
> just or pure are not necessarily rationally constructed.

Yes, I see I had your meaning correct, and the language I used
was quite clear. Add a vice versa to mine and we've said the
same thing. Your way just uses more words.

> > I find this to be true in my experience,
> > but this is of course subjective, because statements about
> > perceptions always depend on the listener.
>
> Of course. But we needn't run in terror from this subjectivity and
try
> to bury our heads in an objective but oversimplified
mathematical
> definition. The fact is that most people can recognise what is
meant
> by justly intoned when you demonstrate it to them by, for
example,
> playing two notes simultaneously, in a sustained timbre, rich in
> harmonics, and slowly varying the pitch of one of them while
asking
> the listener to tell you when they hear something unusual.
>
> I raised this some years ago on the main tuning list, and I
found that
> some people had a strong emotional attachment to ratios,
whether they
> resulted in any audible quality or not. And the term "JI" had
become
> something like a badge of cult status for them, so they felt very
> threatened by what I was saying. So much so that some went
off and
> started new Yahoo groups. And others who could see the logic
of what I
> was saying, nevertheless decided not to support it, for the sake
of peace.
>
> Some folks agreed to use the term RI (for rational intonation)
instead
> of JI in those cases where the ratios did not imply (in your
terms)
> sounding pure. I never felt quite comfortable about that term
(RI) but
> I couldn't say why, and let it go (for the sake of peace). In this
> present discussion, thankfully far less heated, I have realised
why.
>
> I think it is because the word "intonation" also refers to
perception,
> not mathematics. One cannot rationally _intone_ something,
because one
> cannot _hear_ rationality. "intonation" and "tuning" seem to be
almost
> synonyms, but not quite. Otherwise Kyle Gann would be guilty
of
> redundancy when he wrote ""just-intonation (pure) tuning".
>
> So I think "RT" for "rational tuning" would be better, or possibly
we
> could get away with "RCI" for "rationally constructed intonation".

What Kyle wrote there is not redundant. 'Just-intonation' has a
theoretical connotation and 'pure tuning' has a perceptual
connotation. Anyway, it sounds like you may have an axe to grind
here, and I'm not interested in sparring, so this is my reply and it
will stand as my opinion and nothing more.

Best Regards,
Aaron

--- In tuning-math@yahoogroups.com, "pitchcolor" <pitchcolor@a...> wrote:
> Dave wrote:
>
> > > I think it's fairly standard that 'just' means constructed by
> ratios of
> > > (usually small) whole numbers.
> >
> > Yes unfortunately, it has become fairly standard. "A recent
> > abberation" as I said.
>
>
> But you've supplied nothing which substantiates this claim.

Hi Aaron,

I should have explained. That number at the end of the OED entry,
"1811". It's the year of publication of the earliest document that the
editors of the OED can find which uses the word "just" with this
meaning, in this context, in the English language. When the meaning of
a word changes with time, the OED editors show them all, and indicate
when each new meaning first entered the language. See for example the
entry for "Jazz". Apparently "just" was a synonym for "pure" in this
context back in 1811 and nothing has yet convinced them that this has
changed.

By far the best survey of definitions of just intonation that I know
of is Joe Monzo's "dictionary" entry (more like an encyclopedia).
http://sonic-arts.org/dict/just.htm
You will find "pure" used as a synonym for "just" in several of the
quotes there.

> The word 'just' (which translates as 'just' into English) is used as
> a theoretical term in South America and in Poland, for example,
> where 'fifths' are assumed to be 3/2 and 'fourths' are assumed to
> be 4/3. These are theoretical constructs. These terms apply to
> piano music, for example. 'Just' is clearly used there as a
> _theoretical term. Again, if you need to make the distinction, then
> just say either it '_sounds just' or it '_is just'.

The word "sounds" makes it clear that we have perception in mind, but
how does the word "is" make it clear that we are talking math or music
theory? In fact I have written elsewhere that "Just is as just
sounds." So I'm afraid this doesn't work for me.

But also, the justness of the fifths and fourths on a piano tuned to
12-equal is not so theoretical. They sound close enough to just for
many people, particularly given the tuning "stretch" on a piano.

> > Can you provide any evidence that they have ever been in
> separate spheres?
>
> No, I can only hold that they _belong in seperate spheres.
> People make errors with language and say unclear things all the
> time. We have two words there. It's worthwhile to exploit their
> connotations. If you choose not to do that, OK, but I think it's
> inefficient and unnecessary to make them synonymous.

Sorry Aaron,

I didn't mean to come down hard on you. I understand you were trying
to give us all a way out of a possible impasse. I agree it's
inefficient. But it's also a bad idea to have a discontinuity of
meaning for a term in the literature, and there's no doubt in my mind
that "pure" and "just" have been used as synonyms in this context for
a very long time, both having perceptual "connotations".

And besides, we've already got a word meaning "relating to ratios",
namely "rational", so if we were to use "just" for this as well, that
would be inefficient too.

> What Kyle wrote there is not redundant.

I don't think so either.

> 'Just-intonation' has a
> theoretical connotation and 'pure tuning' has a perceptual
> connotation. Anyway, it sounds like you may have an axe to grind
> here, and I'm not interested in sparring, so this is my reply and it
> will stand as my opinion and nothing more.

Sorry Aaron. I think you've got me wrong. Although I must admit that
my fondness for logic and consistency sometimes causes me to ride
roughshod over others' feelings, I don't think I have any "axe to
grind" in this case. Just consciousness to raise. My description of
those responses to my earlier raising of this issue on the tuning
list, was intended in the nature of a tersely summarised
anthropological observation. No hard feelings whatsoever.

Regards,
-- Dave Keenan

Hi Dave,

No hard feelings.

I have immediate access to the OED from my office, so here is a
copy of the current entry. Just intonation falls under number 5 as
letter b.

***

1. That does what is morally right, righteous. just before (with)
God or, simply, just: Righteous in the sight of God; justified. Now
chiefly as a Biblical archaism.
 
  1382 WYCLIF Ezek. xxxiii. 12 The ritwijsnesse of a iust man
[Vulg. justitia justi; 1388 The ritfulnesse of a ritful man]. Luke i. 6
Sothli thei bothe weren iuste [so 1388: Vulg. justi] bifore God.
Rom. iii. 26 That he be iust [so 1388: Vulg. justus], and
iustifyinge him that is of the feith of Ihesu Crist. 1526 TINDALE
Matt. v. 45 He..sendeth his reyne on the iuste and on the iniuste
[Vulg. bonos et malos]. 1560 J. DAUS tr. Sleidane's Comm. 6
Scripture, declareth playnly, howe it is faith that maketh us iust
before God. 1561 T. NORTON Calvin's Inst. III. iv. §28. 211 The
iustest man passeth no one day wherein he falleth not many
times. 1659 SHIRLEY Ajax & Ulysses iii, Only the actions of the
just Smell sweet and blossom in the dust. 1719 WATTS Hymn,
`Not to the terrors' iii, Behold the spirits of the just, Whose faith is
turn'd to sight! 1824 R. HALL Wks. (1832) VI. 355 God can be at
once the just and the justifier.
 

    b. absol. in singular. Obs. or arch.
 
  1382 WYCLIF Acts vii. 52 The prophetis..that bifore teelden of
the comynge of the iust [1611 the Iust one]. 1526 TINDALE Acts
vii. 52 That iust whom ye haue betrayed. 1535 COVERDALE Ps.
xxxvi[i]. 12 The vngodly layeth wayte for the iust, & gnassheth
vpon him with his tethe [so 1611 and R.V.].
 

    2. Upright and impartial in one's dealings; rendering every one
his due; equitable.
 
  1382 WYCLIF 1 John i. 9 If we knowlechen oure synnes, he is
feithful and iust [Vulg. justus] that he foriue to us our synnes.
1484 CAXTON Fables of Æsop II. Proem, The good ond Iuste be
not subget to the lawe as we fynde and rede of alle the
Athenyens. 1503 DUNBAR Thistle & Rose 122 Scho..bawd him
be als just to awppis and owlis, As vnto pacokkis. 1553 T.
WILSON Rhet. (1580) 209, I mistrust not the Iudges, because
thei are iuste. 1605 SHAKES. Lear V. iii. 170 The Gods are iust,
and of our pleasant vices Make instruments to plagve vs. 1725
POPE Odyss. XIII. 249 Some juster prince perhaps had
entertained, And safe restored me to my native land. 1771
Junius Lett. lvi. 294 How much easier it is to be generous than
just. 1850 TENNYSON In Mem. Prol., Thou madest man, he
knows not why..And Thou hast made him: Thou art just. 1853
LYTTON My Novel V. iii, He was just, but as a matter of business.
He made no allowances. 1860 RUSKIN Mod. Paint. V. IX. i. §13.
204 Just! What is that?..dealing equitably or equally.
 

    b. Faithful or honourable in one's social relations. Const. of, to.
Obs.
 
  1601 SHAKES. Jul. C. III. ii. 90 He was my Friend, faithfull, and
iust to me. 1624 CAPT. SMITH Virginia I. 3 He was very iust of his
promise. 1727 POPE Epit. R. Digby, Just of thy word, in ev'ry
thought sincere. 1809 CAMPBELL Gert. Wyom. III. xxix, Friend to
more than human friendship just.
 

    3. a. Consonant with the principles of moral right or of equity;
righteous; equitable; fair. Of rewards, punishments, etc.:
Deserved, merited.
 
  c1400 Destr. Troy 214 More it Ioyes me, Iason, of i iust werkes.
c1430 Hymns Virg. 114 The hiest lessoun at man may lere Is to
lyue iust lijf. 1553 EDEN Treat. Newe Ind. (Arb.) 5 If honest
commendacions be a iust reward dew to noble enterprises.
1590 R. HITCHCOCK Quintess. Wit 5 That warre is iust, that is
necessarye. 1632 J. HAYWARD tr. Biondi's Eromena 33, I will
never rest, till I have executed just vengeance on him that
unjustly slew thee. 1766 GOLDSM. Vic. W. viii, You'll think it just
that I should give them an opportunity to retaliate. 1840
DICKENS Barn. Rudge vi, Is this fair, or reasonable, or just to
yourself?
 

 
  quasi-n. 1667 MILTON P.L. VI. 381 Strength from Truth divided
and from Just..naught merits but dispraise.
 

    b. Constituted by law or by equity, grounded on right, lawful,
rightful; that is such legally; legally valid (obs.).
 
  c1430 LYDG. Min. Poems (Percy Soc.) 17 The degre be just
successioune..Unto the kyng is now descended doune. 1542 in
Marsden Sel. Pl. Crt. Adm. (1894) I. 116 Being in his lyfetyme
juste owner and possessor of a certayne waterboote. 1642
Perkins' Prof. Bk. ix. §581. 253 Where a just grant or other thing
cannot take effect without a deed. 1667 MILTON P.L. II. 38 We
now return To claim our just inheritance of old. 1712-14 POPE
Rape Lock III. 60 The rebel Knave, who dares his prince engage,
Proves the just victim of his royal rage. 1726-31 TINDAL Rapin's
Hist. Eng. (1743) II. XVII. 100 Another Person has a juster title
than she to the Crown. 1849 MACAULAY Hist. Eng. iv. I. 443 He
[James II] would still go as far as any man in support of her [his
country's] just liberties.
 

    4. Having reasonable or adequate grounds; well-founded.
 
  c1374 CHAUCER Troylus III. 1178 (1227) Al quyt from euery
drede and teene As she at Iuste cause hadde hym to triste. 1553
T. WILSON Rhet. (1580) 217 Images we maie chaunge, as the
matter shall give iuste cause. 1633 P. FLETCHER Purple Isl. XI.
xii, A simple maid, With justest grief and wrong so ill apaid. 1792
Anecd. W. Pitt II. xxix. 130 The excuse is a valid one if it is a just
one. 1796 E. HAMILTON Lett. Hindoo Rajah I. 45 Alas! my fears
were just. The pure spirit had fled. 1858 GEN. P. THOMPSON
Audi Alt. II. lxxiv. 23 The justest object of jealousy to wise men in
all ages.
 

    5. Conformable to the standard, or to what is fitting or
requisite; right in amount, proportion, æsthetic quality, etc.;
proper; correct.
 
  c1430 LYDG. Min. Poems (Percy Soc.) 60 Iuste weight halte
justly the balaunce. 1588 W. SMITH Brief Descr. Lond. (Harl. MS.
6363 lf. 13) If they ffynd [the weights] not Iust: they breake them.
1598 YONG Diana 491 A maruellous sweete concent keeping
iust time and measure. 1671 R. BOHUN Wind 67 So that a just
and moderate condensation is necessary to the constitution of
Winds. 1734 J. WARD Introd. Math. II. xi. (ed. 6) 139 The First
Root is 300 being less than Just. 1750 JOHNSON Rambler No.
23 9 Rules for the just opposition of colours, and the proper
dimensions of ruffles and pinners. 1821 J. Q. ADAMS in C.
Davies Metr. Syst. III. (1871) 74 The first of these
injunctions..commands that the standards should be just. 1877
E. R. CONDER Bas. Faith v. 203 The just balance between the
moral and intellectual sides of his nature is often destroyed.
 

    b. Mus. in just interval, intonation, etc.: Harmonically pure;
sounding perfectly in tune.
 
  1850 GEN. P. THOMPSON (title) Theory and Practice of Just
Intonation. 1878 W. H. STONE Sci. Basis Music v. §90 The
differences of the old [mean-tone] and equal systems [of
temperament], and their respective departures from just
intonation. 1881 BROADHOUSE Mus. Acoustics 353 Just
Intonation, where all the Fifths and Thirds are perfect, used only
by singers and theorists.
 

    6. Of speech, ideas, opinions, arguments, etc.: In accordance
with reason, truth, or fact; right; true; correct. Often with mixture of
sense 3.
 
  1490 CAXTON Eneydos xxi. 77 He refuseth to lene his eeres for
to vnderstande my wordes that ben soo iuste and resonable.
a1610 HEALEY Theophrastus (1636) 20 He maintaineth, that
strangers speake wiser and juster things than his own
fellow-citizens. 1725 POPE Odyss. III. 306 Much he knows, and
just conclusions draws From various precedents, and various
laws. 1774 GOLDSM. Nat. Hist. (1776) V. 136 A single glance of
a good plate or a picture imprints a juster idea that a volume
could convey. 1888 BRYCE Amer. Commw. II. lxxv. 618 To
present a just picture of American public opinion one must cut
deeper.
 

    b. Of a copy, description, calculation, etc.: Exact, accurate. [So
F. juste.] Said also of personal agents. Obs.
 
  1563 WINET Four Scoir Thre Quest. To Rdr., Wks. 1888 I. 60
We sett furth this iust copie without altering or eiking ony thing.
1657 R. LIGON Barbadoes (1673) 33 Having given you a just
account..of the bread and drink of this Island. 1691 SWIFT Athen.
Soc., Like a just map. 1704 J. PITTS Acc. Mahometans Pref.
(1738) 7, I have since procured a just Translation. 1727 SWIFT
What passed in London, I am apt to think his calculation just to a
minute. 1798 G. FORSTER Journ. Bengal to Eng. I. 80 The
Hindoos of this day are just imitators, and correct workmen; but
they possess merely the glimmerings of genius.
 

    7. Adapted to something else, or to an end or purpose;
appropriate; suitable. Obs.
 
  c1384 CHAUCER H. Fame II. 211 [It] stant eke in so Iuste a
place That euery sovne mot to hyt pace. 1664 EVELYN Kal. Hort.
Introd. (1729) 187 How many Things to be done in their just
Season. c1665 MRS. HUTCHINSON Mem. Col. Hutchinson
(1846) 32 He was very liberal to them, but ever chose just times
and occasions to exercise it. 1684 R. WALLER Nat. Exper. 10
Our Instrument remains still unalterably just to every place where
'tis made use of.
 

    8. Of clothing, armour, etc.: Well adjusted, fitting exactly.
Hence, Fitting too closely, tight. [So F. juste.] Obs.
 
  a1400 Sir Perc. 273 His hode was iuste to his chynne. c1400
Destr. Troy 9505 Mekull iust armur. a1450 Knt. de la Tour (1868)
38 Streite and welle sittinge and iuste, that sum tyme the fruite
that was in me suffered payne and was in perelle. 1649
LOVELACE Poems, Aramantha, It [a robe] sate close and free,
As the just bark unto the Tree.
 

    9. Of a calculated result, measure, amount, number, date, etc.:
Exact, as opposed to approximate. Also with defining word: That
is exactly what is designated; = `(the) exact..'. Obs.
 
  c1391 CHAUCER Astrol. II. §3 To haue take a Iust Ascendent by
their Astrilabie. 1551 RECORDE Pathw. Knowl. I. iv, Open your
compasse to the iust length of ye line. 1594 Acc.-Bk. W. Wray in
Antiquary XXXII. 118 [He] owes me..the just some of iijli. xixs. id.
1596 SHAKES. Merch. V. IV. i. 327 If thou tak'st more Or lesse
then a iust pound. 1608 WILLET Hexapla Exod. 875 The forepart
of the court was a iust square. 1655 FULLER Ch. Hist. IX. iv. §3
We cannot exactly tell the just time thereof. 1723-4 CHAMBERS
tr. Le Clerc's Treat. Archit. I. 105 It shou'd be rais'd to the just
height of the Windows. 1759 B. MARTIN Nat. Hist. Eng. I.
Cornwall 4 Its Height and just Balance.
 

    b. Of an instrument, natural action, etc.: Exact or uniform in
operation, regular, even. Obs.
 
  c1386 CHAUCER Sompn. T. 382 Thou shalt me fynde as Iust
as is a squyre. 1579 GOSSON Sch. Abuse (Arb.) 26 The
vnfallible motion of the Planets, the iuste course of the yeere.
1665-6 Phil Trans. I. 61 An instrument composed of two
Rulers..will be no longer just at all. 1721 BAILEY, Just Divisors
are such Numbers or Quantities which will divide a given
Number or Quantity, so as to leave no Remainder. 1769 SIR W.
JONES Pal. Fortune in Poems, etc. (1777) 23 Mark'd the just
progress of each rolling sphere.
 

    10. Corresponding exactly in amount, duration, position, etc.;
equal; even, level. Obs.
 
  1551 ROBINSON tr. More's Utop. II. iv. (1895) 141 Dyuydynge
the daye and the nyghte into xxiiii iust houres. 1594 BLUNDEVIL
Exerc. III. I. xxxiii. (1636) 343 Untill the last degree of the said
Signe do appeare just with the upper edge of the Horizon. c1630
RISDON Surv. Devon §46 (1810) 52 That..well in Derbyshire,
which ebbeth and floweth by just tides. 1725 POPE Odyss. XIV.
483 The destin'd victim to dis-part In sev'n just portions.
 

    b. Characterized by or involving exact correspondence. Obs.
 
  1753 HOGARTH Anal. Beauty xi. 83 They meet in just similitude.
1802 PALEY Nat. Theol. xvi. (1819) 258 In consequence of the
just collocation, and by means of the joint action of longitudinal
and annular fibres.
 

    11. That is such properly, fully, or in all respects; complete in
amount or in character; full; proper, `regular'. just battle, in quot.
1603, a regular (pitched) battle [= OF. juste bataille]. just age
(years), full age or age of discretion. Obs.
 
  1588 H. G. tr. Cataneo (title) Briefe Tables to know redily how
manie ranckes of footemen..go to the making of a iust battaile.
1588 D. ROGERS in Ellis Orig. Lett. Ser. II. III. 148 They are not
minded to Crowne the yonge kinge, before he come to just
yeares. 1603 KNOLLES Hist. Turks (1621) 663 The skirmish
was like to have come to a just battell. a1618 SYLVESTER Judith
To Rdr., I am the first in Fraunce who in a just Poem hath treated
in our tongue of sacred things. 1622 BACON Hen. VII 42 This
warre was rather a suppression of Rebels, then a warre with a
iust Enemie. 1624 BEDELL Lett. x. 136 It would require a iust
volume to shew it. 1668 CULPEPPER & COLE Barthol. Anat. III. i.
128 When a man comes to a just age. 1732 BERKELEY Alciphr.
I. §12 Published..sometimes in just volumes, but often in
pamphlets and loose papers. 1778 R. LOWTH Transl. Isaiah ix.
7 note, A just poem, remarkable for the regularity of its
disposition, and the elegance of its plan.
 

    12. nonce-use. That just is or takes place: cf. JUST adv. 5.
 
  1884 BROWNING Ferishtah, Two Camels 117 A lip's mere
tremble, Look's half hesitation, cheek's just change of colour.
 

    13. Comb.    a. with a pple. (or another adj.), where just is
adverbial in sense, = justly: as just-borne, -conceived,
-consuming, -dooming, -judging, -kindled, -tempered, -thinking;
just gentle.    b. parasynthetic, as just-minded (whence
just-mindedness).
 
   1595 SHAKES. John II. i. 345 Before we will lay downe our
*iust-borne Armes.
------------------------------------------------------------------------
1633 FORD Love's Sacr. V. i, The boundless spleen Of
*just-consuming wrath.
------------------------------------------------------------------------
1598 SYLVESTER Du Bartas II. ii. I. Noah 94 The deeds of
Heav'ns *just-gentle king.
------------------------------------------------------------------------
Ibid. 350 In my *just-kindled ire.
------------------------------------------------------------------------
1848 BUCKLEY Iliad 110 *Just-minded, wise-reflecting
Bellerophon.
------------------------------------------------------------------------
1887 Pall Mall G. 20 Aug. 2/2 Confidence in the
*just-mindedness of their employers.
------------------------------------------------------------------------
1829 E. S. SWAINE in Bischoff Woollen Manuf. (1842) II. 238 At
the very name of a drawback or bounty..the *just-thinking
legislator must shrink with an instinctive distrust.
------------------------------------------------------------------------

***

At any rate it seems to me that there is an emphasis is on
correctness of proportion, with aesthetic quality listed
secondarily. In this context consider that 'harmonically pure' may
refer to numbers in correct relation rather than to an aural
perception. I'm not sure where 1811 came from in your citation.

Regards,
Aaron

--- In tuning-math@yahoogroups.com, "pitchcolor" <pitchcolor@a...> wrote:
> I have immediate access to the OED from my office, so here is a
> copy of the current entry. Just intonation falls under number 5 as
> letter b.

Yes. It's only definition 5b that is relevant here.

> 5. Conformable to the standard, or to what is fitting or
> requisite; right in amount, proportion, æsthetic quality, etc.;
> proper; correct.
...
> b. Mus. in just interval, intonation, etc.: Harmonically pure;
> sounding perfectly in tune.
>
> 1850 GEN. P. THOMPSON (title) Theory and Practice of Just
> Intonation. 1878 W. H. STONE Sci. Basis Music v. §90 The
> differences of the old [mean-tone] and equal systems [of
> temperament], and their respective departures from just
> intonation. 1881 BROADHOUSE Mus. Acoustics 353 Just
> Intonation, where all the Fifths and Thirds are perfect, used only
> by singers and theorists.
> ***
>
> At any rate it seems to me that there is an emphasis is on
> correctness of proportion, with aesthetic quality listed
> secondarily.

I don't think you can conclude anything about relative importance from
order of listing.

> In this context consider that 'harmonically pure' may
> refer to numbers in correct relation rather than to an aural
> perception.

I guess it's _possible_, but I thought you were previously of the
opinion that "pure" had a perceptual connotation.

If we now have "rational", "just" and "pure" all referring to the same
mathematical or theoretical property then we no longer have any word
left for the perceptual one!

Here's another possibile way out:
/tuning/topicId_29654.html#29654
where I suggest the adjective "JI-system" as opposed to simply "JI",
for intervals like 64:81 or 32:45, and I propose a more sophisticated
mathematical model of justness derived from Paul Erlich's Harmonic
Entropy. You can see a proposed curve of justness versus interval size
in cents at
/tuning-math/files/Erlich/keenan.jpg
You may find that you have to join the harmonic_entropy group, at
least temporarily, to get to see the above.

> I'm not sure where 1811 came from in your citation.

I was quoting from a 1959 edition of the Shorter Oxford, which is the
best that I have ready access to, although I did look up the complete
one a few years ago. So thanks for the above. Of course a change from
1811 to 1850 makes no difference to my arguments.

I expect everyone else in this list is sick of hearing my arguments
for the perceptual definition of "just", and would rather I just gave
up. My apologies. I just couldn't resist another tilt at the windmill. :-)

Regards,
-- Dave Keenan