Yahoo Tuning Groups Ultimate Backup tuning George Secor: a terminological nightmare ("middle path")?

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

> Hi George,
>
> For quite a few years on these lists, the term "middle path" has
been
> used by Margo Schulter, Dave Keenan, and myself, in discussions
with
> Jacky Ligon and others, to refer to a tuning concept or philosophy
> that is between those of JI and the various ETs.

Back in the 70's I found myself sorely in need of a term that would
encompass what I thought of as a "third world" of tuning: all tunings
other than strict JI and ET's (which I understood at the time to
include only equal divisions of the octave), which are generally
thought to be at opposite ends of the tuning philosophical spectrum.
Your term "middle path" seemed to have filled that void very nicely.

However, there are two reasons that I think non-octave ET's more
properly belong in a "third-world" or "middle-path" category:
1) This category includes tunings that are rather esoteric and, from
a practical standpoint, in a category rather distinct from EDO's;
2) Theoretically, non-octave ET's are open systems.

> Some posts mentioning the concept include:
>
> /tuning/topicId_29348.html#29348
> /tuning/topicId_28984.html#29212
>
> The idea is that, while ETs (not EDOs) take JI and simplify it
> by "regularly" (or uniformly) tempering out a sufficient number of
> independent commas, the "middle path" tunings regularly temper out
a
> smaller number of independent commas. The most famous example is
> meantone temperament, whose various varieties are solutions to the
> problem of best tempering out the syntonic comma (81:80) without
> regard to any other commas vanishing. Schismic (or schismatic)
> temperament is another classic example (q.v. Helmholtz, Groven, and
> Sabat-Garibaldi), while recently we have seen the discovery and
> rediscovery of quite a few other such schemes, such as Miracle,
> Hanson, and the "Pajara" system which forms the underpinning for my
> paper on the decatonic scales.

Since "middle path" doesn't sound particularly technical, and since
all of the times I have seen it used it seemed to apply to "third
world" tunings (not a good name, of course, because of its political
and economic connotations), I assumed that this is how you were using
it, and I didn't see anything in the above examples that would have
given me a clue that I was mistaken. My apologies.

> Like JI and ETs, middle path tunings are "regular" -- a given
> consonance is represented the same way no matter where it appears
in
> the tuning system. It seems to me that if a property such
> as "regularity" is possessed by both JI and the ETs, then it ought
to
> be possessed by any "middle path" between them.

Yes, that seems logical. However, I could also argue that, since
well-temperament has historical importance in the transition from
meantone temperament to 12-ET, it therefore occupies a middle-ground
between the two and could thereby be included in the broader middle
ground (or middle path) between 5-limit JI and 12-ET.

> My XH18 paper is a complete survey of 5- and 7-limit, two-
dimensional
> (like those mentioned above) "middle path" systems within a certain
> boundary on error and complexity. In fact, the title of the paper
is
> _The Middle Path. Part 1: Fifty Floragrams_.
>
> It seems that you are now putting forth a different definition
> of "middle path", similar to the "well-temperament" concept, which
> refers to closed systems (while mine refers to open systems) and
> irregular systems (while mine refers to closed [s/b regular]
systems).

About my well-temperaments you are absolutely correct. However, my
high-tolerance temperament sets are basically multiple chains of
uniformly tempered fifths, hence are not really irregular, at least
from a philosophical standpoint, and they are closed sets only as a
matter of convenience, for the same sense that Blackjack or Canasta
might be considered closed sets. You could conceive of the 29-tone
high-tolerance temperament as starting with chains of fifths
including the tones 1/1, 5/4, 7/4, 11/8, and 13/8, then tempering the
fifths so that 7/4, 11/8, and 13/8 are all in the same chain. Fifths
perceived to be "irregular" are actually intervals between tones in
different chains, and otonal ogdoads in 6 keys (B-flat through A)
require 28 tones. Yes, there is one exception: an additional tone (E-
flat) is required to fill a hole in the 29-tone mapping, but even
that could be interpreted as 19/16 of C in a one-tone chain (hey, cut
me a little slack here, huh?).

In the other HTT sets there are two more chains of fifths that
include the tones 65/64 and 91/64, giving 9-limit otonalities on
(approximated) ratios of 11 and 13. Ratios of 11 and 13 may be
approximated on those roots using tones from the 1/1 chain, which is
tantamount to substituting tempered 22:27:32 intervals for 9:11:13.

While the preceding 2 paragraphs may seem to be a bit of a
digression, I was hoping to make 2 points:

1) Not all of my tunings that you were excluding from the "middle
path" category are well-temperaments;
2) Like JI vs. temperament or rational vs. irrational, a practical
distinction between regular vs. irregular or open vs. closed may not
always be as clear as we expected.

> I can
> certainly see how a JI periodicity block might be created, then
> tempered first near the edges, and finally tempered uniformly
> everywhere to yield an ET, such that the middle step would be a
well-
> temperament. However, I feel this unfairly omits the whole universe
> of possibilities where fewer of the periodicity blocks' unison
> vectors are tempered out, and which are thus also in the "middle"
> between JI and ETs.

As I said above, my impression was that the term "middle path" would
not include any tuning not covered by the two opposite extremes, JI
and ET (or as I would prefer, EDO).

> While I certainly respect your thinking process, and the musical
> value of well-tempered systems is unquestionable, I wonder if we're
> heading toward a terminological nightmare here.

Well, "middle path" is your term, so I think I'm obligated to abide
by whatever definition you choose for it. My problem is that you
have used the term to exclude certain tunings that, given a similar
number of tones, offer better intonation than EDO's and more
modulation than JI, thereby offering a practical middle-ground
between the two.

Hmmm, maybe I should call this broader category the "middle-ground",
to include tunings both on and off the "middle path". (I always
fancied most of my tunings as being off the beaten path, anyway. ;-)

Question: would all regular middle-ground tunings then be middle-path?

--George

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> /makemicromusic/topicId_6820.html#6852
>
> > Hi George,
> >
> > For quite a few years on these lists, the term "middle path" has
> been
> > used by Margo Schulter, Dave Keenan, and myself, in discussions
> with
> > Jacky Ligon and others, to refer to a tuning concept or
philosophy
> > that is between those of JI and the various ETs.
>
> Back in the 70's I found myself sorely in need of a term that would
> encompass what I thought of as a "third world" of tuning: all
tunings
> other than strict JI and ET's (which I understood at the time to
> include only equal divisions of the octave), which are generally
> thought to be at opposite ends of the tuning philosophical
spectrum.
> Your term "middle path" seemed to have filled that void very nicely.
>
> However, there are two reasons that I think non-octave ET's more
> properly belong in a "third-world" or "middle-path" category:
> 1) This category includes tunings that are rather esoteric and,
from
> a practical standpoint, in a category rather distinct from EDO's;

Where do you draw the line? Pianos are tuned to a stretched 12-equal -
- strictly speaking, a non-octave ET.

> 2) Theoretically, non-octave ET's are open systems.

Huh? I don't get it. ETs, whether octave-based or not, have a
discrete set of pitches, each one separated from the others by a
given interval, which would seem to make them closed rather than open
systems.

> > Some posts mentioning the concept include:
> >
> > /tuning/topicId_29348.html#29348
> > /tuning/topicId_28984.html#29212
> >
> > The idea is that, while ETs (not EDOs) take JI and simplify it
> > by "regularly" (or uniformly) tempering out a sufficient number
of
> > independent commas, the "middle path" tunings regularly temper
out
> a
> > smaller number of independent commas. The most famous example is
> > meantone temperament, whose various varieties are solutions to
the
> > problem of best tempering out the syntonic comma (81:80) without
> > regard to any other commas vanishing. Schismic (or schismatic)
> > temperament is another classic example (q.v. Helmholtz, Groven,
and
> > Sabat-Garibaldi), while recently we have seen the discovery and
> > rediscovery of quite a few other such schemes, such as Miracle,
> > Hanson, and the "Pajara" system which forms the underpinning for
my
> > paper on the decatonic scales.
>
> Since "middle path" doesn't sound particularly technical, and since
> all of the times I have seen it used it seemed to apply to "third
> world" tunings
> (not a good name, of course, because of its political
> and economic connotations), I assumed that this is how you were
using
> it, and I didn't see anything in the above examples that would have
> given me a clue that I was mistaken.

If you start with the above examples and click on "up thread" enough
times, you'll see more than a clue.

> > My XH18 paper is a complete survey of 5- and 7-limit, two-
> dimensional
> > (like those mentioned above) "middle path" systems within a
certain
> > boundary on error and complexity. In fact, the title of the paper
> is
> > _The Middle Path. Part 1: Fifty Floragrams_.
> >
> > It seems that you are now putting forth a different definition
> > of "middle path", similar to the "well-temperament" concept,
which
> > refers to closed systems (while mine refers to open systems) and
> > irregular systems (while mine refers to closed [s/b regular]
> systems).

I'm pretty sure you've distorted this quote. I wouldn't have
said "mine refers to open systems", and then said "mine refers to
closed systems"! Or did I really goof that badly?

> 2) Like JI vs. temperament or rational vs. irrational, a practical
> distinction between regular vs. irregular or open vs. closed may
not
> always be as clear as we expected.

But the theoretical distinction is clear -- in open temperaments,
including more and more of the lattice means adding more and more
tones between the existing ones, with no limit to how far this
process may proceed. In closed temperaments, the entire infinite
lattice is already mapped to a finite set of tones.

> > I can
> > certainly see how a JI periodicity block might be created, then
> > tempered first near the edges, and finally tempered uniformly
> > everywhere to yield an ET, such that the middle step would be a
> well-
> > temperament. However, I feel this unfairly omits the whole
universe
> > of possibilities where fewer of the periodicity blocks' unison
> > vectors are tempered out, and which are thus also in the "middle"
> > between JI and ETs.
>
> As I said above, my impression was that the term "middle path"
would
> not include any tuning not covered by the two opposite extremes, JI
> and ET

Exactly. Is there anything in the above paragraph that would suggest
otherwise?

> (or as I would prefer, EDO).

I would prefer EDO to refer to tuning systems conceived as equal
divisions of the octave and not as temperaments. ET should refer to
temperament, a tempering of JI, in particular that of a sort which
leads to a finite number of equally-spaced tones. 24-equal, for
example, would be awkward to derive as an ET, unless you began with
the restriction that only primes 3, 11, and perhaps 13 would figure
into your initial JI system. 24-equal, at least in the West, is
usually best considered an EDO. Not so for 12-equal.

> > While I certainly respect your thinking process, and the musical
> > value of well-tempered systems is unquestionable, I wonder if
we're
> > heading toward a terminological nightmare here.
>
> Well, "middle path" is your term, so I think I'm obligated to abide
> by whatever definition you choose for it. My problem is that you
> have used the term to exclude certain tunings that, given a similar
> number of tones, offer better intonation than EDO's and more
> modulation than JI, thereby offering a practical middle-ground
> between the two.
>
> Hmmm, maybe I should call this broader category the "middle-
ground",
> to include tunings both on and off the "middle path". (I always
> fancied most of my tunings as being off the beaten path, anyway. ;-
)

Sounds good to me.

> Question: would all regular middle-ground tunings then be middle-
>path?

Shoot me an example to test.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> >
> > Back in the 70's I found myself sorely in need of a term that
would
> > encompass what I thought of as a "third world" of tuning: all
tunings
> > other than strict JI and ET's (which I understood at the time to
> > include only equal divisions of the octave), which are generally
> > thought to be at opposite ends of the tuning philosophical
spectrum.
> > Your term "middle path" seemed to have filled that void very
nicely.
> >
> > However, there are two reasons that I think non-octave ET's more
> > properly belong in a "third-world" or "middle-path" category:
> > 1) This category includes tunings that are rather esoteric and,
from
> > a practical standpoint, in a category rather distinct from EDO's;
>
> Where do you draw the line? Pianos are tuned to a stretched 12-
equal -
> - strictly speaking, a non-octave ET.

Strictly speaking that's not an ET, since the octaves are not
stretched equally over the entire range of the instrument. The
amount of stretch varies according to the specific dimensions of each
string.

And even if the amount of stretch were equal over all octaves, this
example would still not be a tuning *theoretically* distinct from 12-
ET, since the amount of stretch would vary with the implementation of
the tuning on one particular instrument vs. another. The intention
is to tune the instrument so that the octaves sound exact (or as
nearly exact as possible), in order to make the result (what the ear
hears in *practice*) most closely match the *theoretical* tuning.

Similarly, are we to insist that any implementation of JI on acoustic
instruments is a temperament, since it's impossible to achieve exact
phase-locked ratios? Of course not; this is a *practical* problem
rather than another (theoretical) tuning.

So I would draw the line at the point where the *theoretical tuning*
is not mathematically an EDO, where the octaves are intentionally
tempered by a specified number of cents, usually for the purpose of
approximating other consonances more accurately.

> > 2) Theoretically, non-octave ET's are open systems.
>
> Huh? I don't get it. ETs, whether octave-based or not, have a
> discrete set of pitches, each one separated from the others by a
> given interval, which would seem to make them closed rather than
open
> systems.

But the set can be extended indefinitely up or down without any pitch
ever repeating any previous pitch at a different octave (taking the
term "octave" in the strict sense of the word). The principle of
octave equivalence exists whether or not an ET is octave-based.
(Please read further before replying to this.)

> > ... I didn't see anything in the above examples that would have
> > given me a clue that I was mistaken.
>
> If you start with the above examples and click on "up thread"
enough
> times, you'll see more than a clue.

I didn't look that far, but I'll take your word for it.

> > > My XH18 paper is a complete survey of 5- and 7-limit, two-
dimensional
> > > (like those mentioned above) "middle path" systems within a
certain
> > > boundary on error and complexity. In fact, the title of the
paper is
> > > _The Middle Path. Part 1: Fifty Floragrams_.
> > >
> > > It seems that you are now putting forth a different definition
> > > of "middle path", similar to the "well-temperament" concept,
which
> > > refers to closed systems (while mine refers to open systems)
and
> > > irregular systems (while mine refers to closed [s/b regular]
systems).
>
> I'm pretty sure you've distorted this quote. I wouldn't have
> said "mine refers to open systems", and then said "mine refers to
> closed systems"! Or did I really goof that badly?

See for yourself (in the next-to-last paragraph):
/makemicromusic/topicId_6820.html#6852

Hey, this can happen to the best of us, especially when we start
getting on in years. Once you've reached 22 or so, it's all downhill
from there. ;-)

> > 2) Like JI vs. temperament or rational vs. irrational, a
practical
> > distinction between regular vs. irregular or open vs. closed may
not
> > always be as clear as we expected.
>
> But the theoretical distinction is clear --

Well, I'm glad that we seem to agree that restricting distinctions to
what is theoretically intended can clarify the issues at hand.

> in open temperaments,
> including more and more of the lattice means adding more and more
> tones between the existing ones, with no limit to how far this
> process may proceed. In closed temperaments, the entire infinite
> lattice is already mapped to a finite set of tones.

Ah, now I see that you're counting the tones to determine how many
actual pitches occur in the range of a given octave, whereas I'm
would count them according to how many actual pitches there are when
all of them are reduced to a single octave in order to determine how
many distinct pitches there might be when they are evaluated
harmonically. I realize that there's a practical limit to how many
pitches fall within the range of audibility, but think for a moment:
couldn't you have an organ tuned in a non-octave temperament with
ranks of pipes pitched an octave or two apart, or two instruments
(such as flute and piccolo) in that same temperament pitched an
octave apart? Are there any rules for avoiding such things with non-
octave temperaments, and if so, who's going to enforce them?

> > > I can
> > > certainly see how a JI periodicity block might be created, then
> > > tempered first near the edges, and finally tempered uniformly
> > > everywhere to yield an ET, such that the middle step would be a
well-
> > > temperament. However, I feel this unfairly omits the whole
universe
> > > of possibilities where fewer of the periodicity blocks' unison
> > > vectors are tempered out, and which are thus also in
the "middle"
> > > between JI and ETs.
> >
> > As I said above, my impression was that the term "middle path"
would
> > not include any tuning not covered by the two opposite extremes,
JI
> > and ET
>
> Exactly. Is there anything in the above paragraph that would
suggest
> otherwise?

Oops, I think my brain flew out of gear at that point. I meant to
say, my impression was that the term "middle path" would not exclude
any tuning covered by the two opposite extremes, JI and ET. But in
light of your response (below), I would now substitute "middle
ground" for "middle path" in that statement (and I rather think that
I would prefer EDO to ET for one of those extremes).

> > (or as I would prefer, EDO).
>
> I would prefer EDO to refer to tuning systems conceived as equal
> divisions of the octave and not as temperaments. ET should refer to
> temperament, a tempering of JI, in particular that of a sort which
> leads to a finite number of equally-spaced tones. 24-equal, for
> example, would be awkward to derive as an ET, unless you began with
> the restriction that only primes 3, 11, and perhaps 13 would figure
> into your initial JI system. 24-equal, at least in the West, is
> usually best considered an EDO. Not so for 12-equal.

Then should we call a non-octave equal spacing of tones that does not
approximate JI particularly well an ET, and, if not, then what name
should we use and where do we put the boundary between the two? ;-)

I would much prefer to make the two philosophical extremes JI and
EDO's rather than JI and ET's, because the former is unambiguous.

> > > While I certainly respect your thinking process, and the
musical
> > > value of well-tempered systems is unquestionable, I wonder if
we're
> > > heading toward a terminological nightmare here.
> >
> > Well, "middle path" is your term, so I think I'm obligated to
abide
> > by whatever definition you choose for it. My problem is that you
> > have used the term to exclude certain tunings that, given a
similar
> > number of tones, offer better intonation than EDO's and more
> > modulation than JI, thereby offering a practical middle-ground
> > between the two.
> >
> > Hmmm, maybe I should call this broader category the "middle-
ground",
> > to include tunings both on and off the "middle path". (I always
> > fancied most of my tunings as being off the beaten path,
anyway. ;-)
>
> Sounds good to me.

Agreed, then -- nightmare averted!

> > Question: would all regular middle-ground tunings then be middle-
> >path?
>
> Shoot me an example to test.

I didn't have anything particular in mind. I just wondered if you
could think of an example for which that would not hold true.

What about a linear tuning that doesn't approximate JI well? Or if I
considered JI and EDO's to be the two extremes between which there
exists a middle ground (while you keep JI vs. ET's): then non-octave
ET's would be regular middle-ground tunings that are not middle-
path. (Hmmm, that could get confusing.)

--George

🔗George D. Secor <gdsecor@yahoo.com>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> And even if the amount of stretch were equal over all octaves, this
> example would still not be a tuning *theoretically* distinct from
12-
> ET, since the amount of stretch would vary with the implementation
of
> the tuning on one particular instrument vs. another.

What if it didn't?

> The intention
> is to tune the instrument so that the octaves sound exact (or as
> nearly exact as possible), in order to make the result (what the
ear
> hears in *practice*) most closely match the *theoretical* tuning.

What if the theoretical tuning *does* have tempered octaves? For
example, the 3-limit TOP tuning of 12-equal has stretched octaves,
while the 5-limit TOP tuning of 12-equal has compressed octaves.

>
> Similarly, are we to insist that any implementation of JI on
acoustic
> instruments is a temperament, since it's impossible to achieve
exact
> phase-locked ratios? Of course not; this is a *practical* problem
> rather than another (theoretical) tuning.
>
> So I would draw the line at the point where the *theoretical
tuning*
> is not mathematically an EDO, where the octaves are intentionally
> tempered by a specified number of cents, usually for the purpose of
> approximating other consonances more accurately.

I still see these as ETs. I'm not sure what you would see them as.

>
> > > 2) Theoretically, non-octave ET's are open systems.
> >
> > Huh? I don't get it. ETs, whether octave-based or not, have a
> > discrete set of pitches, each one separated from the others by a
> > given interval, which would seem to make them closed rather than
> open
> > systems.
>
> But the set can be extended indefinitely up or down without any
pitch
> ever repeating any previous pitch at a different octave (taking the
> term "octave" in the strict sense of the word). The principle of
> octave equivalence exists whether or not an ET is octave-based.
> (Please read further before replying to this.)

OK.

> > in open temperaments,
> > including more and more of the lattice means adding more and more
> > tones between the existing ones, with no limit to how far this
> > process may proceed. In closed temperaments, the entire infinite
> > lattice is already mapped to a finite set of tones.
>
> Ah, now I see that you're counting the tones to determine how many
> actual pitches occur in the range of a given octave, whereas I'm
> would count them according to how many actual pitches there are
when
> all of them are reduced to a single octave in order to determine
how
> many distinct pitches there might be when they are evaluated
> harmonically.

Slightly tempered octaves are really no different than pure octaves
in terms of the harmonic "equivalence" they convey. If you start with
an EDO, and then slightly stretch or compress the tuning, the
subjective effect is not one of a sudden and great increase in the
number of pitches.

> I realize that there's a practical limit to how many
> pitches fall within the range of audibility, but think for a
moment:
> couldn't you have an organ tuned in a non-octave temperament with
> ranks of pipes pitched an octave or two apart, or two instruments
> (such as flute and piccolo) in that same temperament pitched an
> octave apart? Are there any rules for avoiding such things with
non-
> octave temperaments, and if so, who's going to enforce them?

By this reasoning, 12-equal on most instruments is an open tuning,
since the harmonics don't coincide with the fundamentals. I don't
think this is a very practical way of viewing the situation.

> Then should we call a non-octave equal spacing of tones that does
not
> approximate JI particularly well an ET,

If it's not theoretically approximating JI, it's merely an equal
tuning, not an equal temperament.

> I would much prefer to make the two philosophical extremes JI and
> EDO's rather than JI and ET's, because the former is unambiguous.

"JI" is anything but unambiguous.

> What about a linear tuning that doesn't approximate JI well?

Again, if there's no approximation to JI intended, I'd call it a 2D
tuning rather than a 2D temperament (I'm not too thrilled with the
term "linear" -- it's equal tunings are better regarded as one-
dimensional).

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> > ...
> > So I would draw the line at the point where the *theoretical
tuning*
> > is not mathematically an EDO, where the octaves are intentionally
> > tempered by a specified number of cents, usually for the purpose
of
> > approximating other consonances more accurately.
>
> I still see these as ETs. I'm not sure what you would see them as.

Okay, I see your point. I will concede that ET's with slightly
tempered octaves can be considered closed systems, just as if they
were EDO's.

> > > > 2) Theoretically, non-octave ET's are open systems.
> > >
> > > Huh? I don't get it. ETs, whether octave-based or not, have a
> > > discrete set of pitches, each one separated from the others by
a
> > > given interval, which would seem to make them closed rather
than open
> > > systems.
> >
> > But the set can be extended indefinitely up or down without any
pitch
> > ever repeating any previous pitch at a different octave (taking
the
> > term "octave" in the strict sense of the word). The principle of
> > octave equivalence exists whether or not an ET is octave-based.
> > (Please read further before replying to this.)
>
> OK.
>
> > > in open temperaments,
> > > including more and more of the lattice means adding more and
more
> > > tones between the existing ones, with no limit to how far this
> > > process may proceed. In closed temperaments, the entire
infinite
> > > lattice is already mapped to a finite set of tones.
> >
> > Ah, now I see that you're counting the tones to determine how
many
> > actual pitches occur in the range of a given octave, whereas I'm
> > would count them according to how many actual pitches there are
when
> > all of them are reduced to a single octave in order to determine
how
> > many distinct pitches there might be when they are evaluated
> > harmonically.
>
> Slightly tempered octaves are really no different than pure octaves
> in terms of the harmonic "equivalence" they convey. If you start
with
> an EDO, and then slightly stretch or compress the tuning, the
> subjective effect is not one of a sudden and great increase in the
> number of pitches.

Okay. When I wrote the above I was thinking principally of tunings
such as the Bohlen-Pierce scale, where intervals with new harmonic
significance are encountered in each successive octave and therefore
do not behave harmonically as closed systems. Since one may
construct a continuum of ET's with the closest interval to an octave
ranging all the way from a few cents difference to an entire diesis
(or more), I was thinking that there may be no clear boundary between
a harmonically open or closed system, so I was inclined to lump
together all non-EDO ET's as open systems and therefore middle-ground
tunings.

Now I'm inclined just to throw in the towel and put the ET's
(including all EDO's) and JI at opposite extremes and just let
everything else be middle-ground (of which the subset consisting of
regular temperaments is middle-path). Is this okay by you?

But I'm still not convinced that tunings such as Bohlen-Pierce should
be considered closed systems.

--George