Maqam tuning systems, EDO and non-EDO

> Jz0 is 16:15 and jz1 is 14:13, remember.

Hi, Graham.

This may raise a point that my crude table didn't address: Jz0
would include 16:15 or the very slightly larger interval of
2187/2084 (a Pythagorean apotome at 114 cents) or 5 commas,
minutely smaller in 53-EDO, as a typical size, but also the
superparticular ratio of 15:14 or 119 cents. The "boundary"
generally might be at around 125 cents or a bit lower, depending
on the context.

This is worth briefly pointing out. Thus 126 cents would
represent Jz1 under lots of conditions -- but in a 16th century
European setting in 19-EDO or 1/3-comma meantone, it's really a
very large 5-limit diatonic semitone. I'm writing mainly about
maqam contexts where the fifths are likely to be pure or very
close to pure (Pythagorean, 53-EDO, 41-EDO), or possibly a bit
larger (e.g. O3, average 703.871 cents).

Your comment led me to focus on the point that I didn't discuss
15:14, giving me an opportunity to clarify that I see it near the
upper limit of Jz0.

And whatever your methodological questions about the reported
tetrachord of 123-137-228 cents, I would indeed call that example
as reported an illustration of Jz1-Jz2-A10 -- the "A10" being
borrowed from Turkish theory to show an "augmented" interval
(somewhat greater than 9:8) of about 10 commas, and here almost
exactly so.

As for your alternative reading:

117 148 223
0 117 265 488

I'd call this Jz0-Jz2.5-A10, with the "Jz2.5" showing that 148
cents is almost exactly half of a 32/27. And we agree that 117
cents, between 16:15 and 15:14, is within Jz0.

Anyway, regardless of how one chooses to interpret this example,
Ozan's observation about the main issue of how Turkish theory
varies from the YAEU system is quite correct. Here's a quick
summary.

In the YAEU system, a Maqam like Ushshaq is supposed to be 8-5-9
commas, or 180-114-204 cents or in approximate JI terms,
10:9-16:15-9:8. However, as skilled Turkish performers and also
investigators like Karl Signell (who measured some of the tunings
of these performers) have long known, the real tuning might be
more like 6-7-9 commas (136-158-204 cents), with two very clear
neutral second steps. Signell found that the lower step tended to
range from around 133 to 148 cents.

Politically, the YAEU system was designed to accommodate the
Turkish ultranationalist ideology of the early Republican era
(starting in 1923) which held that neutral intervals were a
product of Greek, Byzantine, or Arab "decadence," as opposed to
the simple, cheerful, and progressive Turkish spirit turning
toward the dynamic West and its 12-note diatonic scale, refined
through Pythagorean tuning and 5-limit schismatic intervals. Thus
"quartertones" could be avoided, and the maqamat notated in a a
form which the properly initiated could read in the traditional
way.

For example, Ushshaq might be notated (here I'm using arbitrary
rather than Turkish symbols) D E-c F G -- with E-c at a comma
below 9:8. However, those in the know would read that "-c" as
closer to three commas below 9:8, and play what Signell observed
and measured.

So a value like 148 cents in Ushshaq contradicts YAEU, unless we
take it as a momentary "mistuning" or "artistic liberty": and
steps of 133-148 cents are clearly the norm, not the exception.
This specific example doesn't really affect that conclusion.

> Where did you get the Yarman 24 paper?

<http://www.ozanyarman.com/files/yarman24.pdf>

> Ozan's thesis mentions that both 72- and 84-EDO have been used
> for tuning qanuns. He doesn't give good arguments against
> them. Each has a better 9/8 than his 79 note MOS, for example.

This is true for Ozan's 9/8 step at either 13 or 14 yarmans
(around 2/159 octave or about 15.09 cents each), a rounded 196 or
211 cents. There are some virtually just 9:8 steps at 13L yarmans
(12 usual yarmans plus the single "large" yarman in the 79-MOS at
a full Holderian comma or 1/53 octave), but not at this location.

As I see it, the big advantage of the 79-MOS -- or, better yet, a
complete 159-EDO if that were technical possible -- is the high
resolution of about 7.55 cents or 1/3-comma.

The disadvantage is that we get lots of locations where the only
available fifths are impure by 7.5 cents, a somewhat greater
impurity that in 19-EDO or 22-EDO.

Of course, the reply might be: "Nevertheless, we have pure or
virtually pure 3/2 fifths in the places they are most needed for
the standard positions of untransposed maqamat -- plus advantages
like a meantone path to a 17-comma flavor of Rast."

Here's how I might see some of this, coming from an angle a bit
different than Ozan's, and also striving to recognize his
concerns that might not play much if any role in the kinds of
smaller systems I tend to use and design. People who read this
should also read my important disclaimer which follows:

(1) While 41-EDO is very nice for Pythagorean and 5-limit
schismatic intervals as well as ratios of 2-3-7, neatly
yielding a small major third at 380.5 cents or so at
the "just a tad smaller than 5/4" zone so much desired,
I see real limitations with neutral or Zalzalian
intervals. In short, the only way of dividing the 292.7
cent minor third is geometrically: 146.34-146.34.

Likewise, the closest approximation to a Zalzalian or
neutral third flavor of Rast is 205-146-146, with a
third at 351 cents, or precisely half of the fifth.
For a Rast, especially a Turkish Rast, I'd want
something close to 16/13 or higher. We have the
near-5/4 flavor, of course, but not a "higher neutral
third" flavor. In that aspect, apart from the size of
the minor third very close to 32/27, this is to me much
like the 150-150 division of 24/48/72-EDO. It's a
usable choice, but we'd like others as well.

(2) A qanun in 53-EDO seems to me the safe, conservative,
and theoretically friendly choice, covering both YAEU
(a 20th-century Turkish theory basically focusing on
Pythagorean and 5-limit schismatic intervals and
playing down neutral or Zalzalian ones, largely for
ideological reasons explained in Ozan thesis, and very
readably so), and two flavors of neutral intervals, for
example neutral seconds at 6 commas (136 cents) and 7
commas (158 cents). The fifths and fourths are
uniformly near-pure, and we could likewise use a
53-note Pythagorean cycle (one odd fifth narrow by
about 3.615 cents, Mercator's comma) to get a touch of
yet more subtle variety. Since the 53-comma system is
standard, with the neutral intervals fully admitted
into truly modern Turkish theory, there's a very nice
fit between the tuning and the prevalent model.

(3) With 72-EDO, we have three neutral seconds at 133, 150,
and 167 cents, as well as quite congenial positions for
the third degree in Rast at 367 or 383 cents. And
either 313 or 333 cents might likewise be conducive for
a 5-limit minor or a supraminor flavor of Maqam
Nihavend or Maqam Segah.

What are the disadvantages as I might see them, or
Ozan? Here the considerations might sometimes differ:

(a) We might want a regular minor third, here
300 cents, which is a bit smaller (with 32/27
as one traditional ideal), or more
specifically supports some unequal neutral
second division (the same issue as in
41-EDO). Our only choice is 150-150 --
although we have the attractive unequal
divisions of 133-150 or 150-133, as well an
attractive equal division for 7:6 of 133-133.

(b) To be fair, there are some other great
divisions, like 150-267-83 for Hijaz.
OTOH, I'd love an idiomatically Turkish
Huseyni with a minor third a bit smaller than
300 cents and a division into two neutral
seconds. We have 167-133, but I'd love
something like 167-127, for example. We have
150-133, but here we really want the first
step at 160 cents or larger. In fairness, I
can't get 166-128-204 in the 79-MOS either,
although all three interval sizes are
present.

(c) One might prefer having the regular limma a
bit smaller than 100 cents, maybe Pythagorean
(256/243, 90 cents) or a bit smaller.

(d) As Ozan might point out, 72-EDO is not a
meantone and has no path to meantone. Since
he is seeking a version of Rast from a single
chain of fifths, this is an impediment under
his criteria.

(4) With 82-EDO, we have the fine Pythagorean/schismatic
framework of 41, but three neutral seconds at 132, 146,
and 161 cents. This gives a fine Rast at 205-161-132
cents, and a fine Huseyni at 161-132-205. What it
doesn't have is a meantone path to Rast, nor the
resolution of the 79-MOS, which may reflect the basic
advantage of an irregular tuning as versus a complete
EDO in getting more interval sizes for a given tuning
size.

(5) With 84-EDO, one possible disadvantage is the step at
precisely 1/7 octave, valuable for many world musics,
but not so often visited in lots of maqam music where
one tends to jump from 11:10 or so the the territory
near 10:9. In 72-EDO or 82-EDO, this step is in the
"sweet zone" (for maqam music) close to 11:10.

(6) With 96-EDO, with have four neutral second sizes, at
125, 137.5, 150, and 162.5 cents. Shaahin Mohaheri has
used this tuning or subsets of it, and the variety of
neutral steps and intervals is a big plus.

Thus I might paraphrase Ozan's points about 72-EDO somewhat as
follows:

(1) Like 24-EDO, it has a regular 300-cent minor third
permitting only an equal neutral second division at
150-150 cents.

(2) There is no meantone path to a near-5/4 step for Maqam
Rast -- not a concept of traditional maqam theory, but
an important contemporary one to Ozan.

(3) The overall organization is different enough from
53-EDO (or a 53-note Pythagorean cycle, I might add) to
cause some complication for musicians attuned to the
53-comma system.

(4) One might prefer regular diatonic intervals closer to
Pythagorean, so that divisions like those of the
medieval treatises (e.g. 32/27 into two unequal neutral
seconds) can more often and more closely apply.

If I were investigating possible alternatives under Ozan's
groundrules, I might look at supersets of a complete 53-EDO
circle. Whether this might require a satisfactory set
considerably larger than 79/80, or a set of not more than 79/80
notes with considerably less utility than the actual 79/80=MOS,
I'm not sure.

MAJOR DISCLAIMER: Since I've dealt with 24 notes per octave as a
norm on a regularized keyboard (not a generalized one!), looking
at EDO or MOS sets of 41-84 notes is really a theoretical rather
than practical exercise for me.

From experience, I do know that an EDO is the least effective way
to get a variety of step sizes in a given tuning size. Thus I
might be a lot less passionate about "X-EDO vs. Y-EDO" debates,
because I find them mostly academic.

And, needless to say, I am much less well-situated than a Turkish
musician, especially one as well-informed about both tradition
maqam styles and some modern innovations as Ozan, to judge what
the criteria should be for a contemporary maqam tuning.

However, I can say that six neutral second sizes, lots of them
near-epimoric, are a very attractive proposition.

And one thing I _do_ know: different agendas can lead to very
different tuning systems.

For example, with my O3 system, Ozan could well point out that it
is impossible within the given 24 steps to find even a basic
9-note set for a 26/21 Rast (the notes from the lower tetrachord
plus the conjunct and disjunct forms of an upper Rast tetrachord)
without encountering some quirk of the tuning, and either having
to substitute a septimal for a regular step, or else having a
neutral sixth at a more or less notably low position.

We would quickly agree that this is so because a regular solution
would require a chain of at least 12 fifths, 3 up for a regular
major sixth, e.g. B-G# (~22/13 in this system); and 9 down for a
diminished sixth, e.g. B-Ab (~33/20 in this system).

Since a single chain has only 11 fifths (Eb-G#), we have a choice
of B-G# for the major sixth or F#-Eb for the large neutral sixth,
but not of having both above the same step chosen as the final or
resting note of Maqam Rast.

There are various compromises and strategies for handling this:

(1) Choose F#* for the final, and use septimal steps at
~8/7 and ~12/7 in place of ~9/8 and ~22/13. This can
be a wonderful incentive to use beautiful medieval
tunings more often!

(2) Choose B* for the final, and accept a low neutral
sixth at around 854 cents, and a 485-cent fourth
between this lower neutral sixth and the usual high
neutral third at ~26/21 or 369 cents.

(3) Focus on the usual form of sixth best supported for a
given final -- either ~22/13 or ~33/20.

(4) If regularity is at a premium, use a 357-359 cent
Rast more in the Arab manner, where it is possible
to get a congruent set of up to 17 notes in the
manner of Wilson's Rast-Bayyati matrix (mixing
notes from both chains). Just to show that it's
possible:

G* A* B C* D* E E* F* F# G*
0 209 359 497 704 854 912 993 1062 1200

Since a system of this kind has scope closer to a 17-tone Persian
tuning (noncirculating) than to any EDO within the range of sizes
we're discussing, I'm probably not the best person to judge the
whole EDO question, except to say that Ozan has reasonably chosen
a non-EDO to get more step sizes, as well as features like a
meantone path to certain intervals and tetrachords.

But if we really want to look for something "competitive" to the
79/80-MOS, as I might imagine myself doing had I been his thesis
advisor or the like, I'd be looking not at EDO systems, which
simply won't give the same resolution and diversity for neutral
or middle intervals, but at irregular tunings like supersets of
53-EDO (maybe bringing in additional steps from 159-EDO or
212-EDO) offering other solutions including a complete 53-EDO
circle,a nd thus the virtually pure 3/2 fifths at all 53 primary
locations.

And there is a big complication which doesn't arise for an EDO
(all intervals available from all locations), but very well could
for such competitive alternatives: unity and coordination of
pitches in a way that will be idiomatically Turkish. Here O3 is an
excellent "how not to" <grin>.

In O3, the final of Rast might be B or F#* for a Turkish flavor
(~26/21), but maybe G* if we want more of an Arab or "_low _
Turkish" flavor (357-359 cents). This is a bit like a European
12-note well-temperament, although without the circulation!

For a good Turkish gamut, we'd need to consider what steps or
nuances are available at a given "standard" location, and how the
system holds together using a credible set of standard locations
for the different maqamat. That's what this a lot more
complicated than looking at accuracy of prime approximations, or
even variety of neutral steps in itself (as with O3).

There are also contextual and historical factors at play here. I
can certainly understand how a Turkish musician might point to
qanun-s tuned in 72-EDO as a possible sign of the encroaching
influence of 12-EDO or 24-EDO. If, hypothetically, we were
talking about 288-EDO (also a subdivision of 12-EDO), things
might be a bit different:

(1) While the regular third remains 300 cents, we have
tempered fifths at 695.83, 700, 704.17, and 708.33
cents as ingredients for various regular or
irregular "subtemperaments" producing, for regular
chains of fifths, minor thirds at 312.5 cents, 300
cents (of course), 287.5 cents, or 275 cents. And a
chain alternating 700-704.17 would closely
approximate Pythagorean or 53-EDO.

(2) For 300 cents, we'd have unequal divisions like
154-146 (subtly unequal); 158-142 (a bit like 7-6
commas). etc. Not to mention various divisions at
thirds at 292 or 296 cents, closest to Pythagorean.

(3) More generally, from 125 to 167 cents, we'd have no
less than 11 shadings of neutral seconds.

(4) The limmas at 87.5 cents or 91 cents should
especially congenial to 41-EDO or 53-EDO outlook.

If we were debating 288-EDO, the whole discussion might be
different, because we get better support for a lot of features
that the 79-MOS has and 72-EDO doesn't at least to the same
degree. Building a physically and ergonomically practical
instrument in 288-EDO, of course, might be another question; but
I feei it's a good thought-experiment to clarify the "12n-EDO"
question.

And some creatively diverting thought-experiements might be
helpful for this group and everyone concerned.

> Graham

Best,

Margo

Margo Schulter <mschulter@...> wrote:
> > Jz0 is 16:15 and jz1 is 14:13, remember.
>
> Hi, Graham.
>
> This may raise a point that my crude table didn't
> address: Jz0 would include 16:15 or the very slightly
> larger interval of 2187/2084 (a Pythagorean apotome at
> 114 cents) or 5 commas, minutely smaller in 53-EDO, as a
> typical size, but also the superparticular ratio of 15:14
> or 119 cents. The "boundary" generally might be at around
> 125 cents or a bit lower, depending on the context.

I forgot about 15:14 in that other message.

> As for your alternative reading:
>
> 117 148 223
> 0 117 265 488
>
> I'd call this Jz0-Jz2.5-A10, with the "Jz2.5" showing
> that 148 cents is almost exactly half of a 32/27. And we
> agree that 117 cents, between 16:15 and 15:14, is within
> Jz0.

The original histogram shows a wide distribution of pitches
around this point. So the perde must cover both your Jz0
and Jz1. Whatever distinction you're making with them
doesn't seem to be observed in practice.

Do you have this performance? It wouldn't help me if I
did, but does it sound like a Jz1 to you?

> Anyway, regardless of how one chooses to interpret this
> example, Ozan's observation about the main issue of how
> Turkish theory varies from the YAEU system is quite
> correct. Here's a quick summary.

It looks like middle eastern music differs from any fixed
system. YAEU is unfortunate in making specific predictions
that it can be tested against. And it's not the issue
here, although it happens to be in Ozan's thesis.

> So a value like 148 cents in Ushshaq contradicts YAEU,
> unless we take it as a momentary "mistuning" or "artistic
> liberty": and steps of 133-148 cents are clearly the
> norm, not the exception. This specific example doesn't
> really affect that conclusion.

Right, you can take anything as a mistuning if you want to.

> > Where did you get the Yarman 24 paper?
>
> <http://www.ozanyarman.com/files/yarman24.pdf>

Thank you! Another Turkish one, then. But it gives the
scale (Tablo 7, right?) that's used in an English paper.

> As I see it, the big advantage of the 79-MOS -- or,
> better yet, a complete 159-EDO if that were technical
> possible -- is the high resolution of about 7.55 cents or
> 1/3-comma.

159-EDO would indeed be great if it were technically
possible. The 79-MOS loses that resolution.

> The disadvantage is that we get lots of locations where
> the only available fifths are impure by 7.5 cents, a
> somewhat greater impurity that in 19-EDO or 22-EDO.
>
> Of course, the reply might be: "Nevertheless, we have
> pure or virtually pure 3/2 fifths in the places they are
> most needed for the standard positions of untransposed
> maqamat -- plus advantages like a meantone path to a
> 17-comma flavor of Rast."

Yes, but the 9:8 isn't in the right place, and it's
supposed to support transposition.

> Here's how I might see some of this, coming from an angle
> a bit different than Ozan's, and also striving to
> recognize his concerns that might not play much if any
> role in the kinds of smaller systems I tend to use and
> design. People who read this should also read my
> important disclaimer which follows:
>
> (1) While 41-EDO is very nice for Pythagorean and
> 5-limit schismatic intervals as well as ratios of 2-3-7,
> neatly yielding a small major third at 380.5 cents or so
> at the "just a tad smaller than 5/4" zone so much desired,
> I see real limitations with neutral or
> Zalzalian intervals. In short, the only way of dividing
> the 292.7 cent minor third is geometrically:
> 146.34-146.34.

This is something that's often suggested in theory. I
don't now how seriously to take it. Some musicians in the
middle east have accepted 24-EDO. Have blind trials been
done on any musicians to show that they reject equal
neutral thirds?

Also, for the true style, you need variability of pitch.
So any system would only be the default pitches you bend
relative to. Given that, musicians can have their unequal
divisions in an equal system.

> Likewise, the closest approximation to a
> Zalzalian or neutral third flavor of Rast is 205-146-146,
> with a third at 351 cents, or precisely half of the fifth.
> For a Rast, especially a Turkish Rast, I'd want
> something close to 16/13 or higher. We have the
> near-5/4 flavor, of course, but not a "higher
> neutral third" flavor. In that aspect, apart from the
> size of the minor third very close to 32/27, this is to
> me much like the 150-150 division of 24/48/72-EDO. It's a
> usable choice, but we'd like others as well.

The choice is bound to be limited with a small number of
notes.

> (2) A qanun in 53-EDO seems to me the safe,
> conservative, and theoretically friendly choice, covering
<snip>
> 53-note Pythagorean cycle (one odd fifth
> narrow by about 3.615 cents, Mercator's comma) to get a
> touch of yet more subtle variety. Since the 53-comma
> system is standard, with the neutral intervals fully
> admitted into truly modern Turkish theory, there's a very
> nice fit between the tuning and the prevalent model.

This is one that Ozan does address, and doesn't like. But
still, it looks pretty good. I don't know how it would be
received outside Turkey, though.

> (3) With 72-EDO, we have three neutral seconds at
> 133, 150, and 167 cents, as well as quite congenial
> positions for the third degree in Rast at 367 or 383
> cents. And either 313 or 333 cents might likewise be
> conducive for a 5-limit minor or a supraminor flavor of
> Maqam Nihavend or Maqam Segah.

It's a good tuning in the 5-limit and beyond, and it has
the right precision.

> What are the disadvantages as I might see
> them, or Ozan? Here the considerations might sometimes
> differ:
>
<snip>
> step at 160 cents or larger. In
> fairness, I can't get 166-128-204 in the 79-MOS either,
> although all three interval sizes are
> present.

That's it. You can go into details about what you do or
don't want, but no fixed system of 70-80 notes will give
you everything. 72-EDO is an obvious choice and nobody's
convinced me why it's inferior to 79-MOS.

> (c) One might prefer having the regular
> limma a bit smaller than 100 cents, maybe Pythagorean
> (256/243, 90 cents) or a bit smaller.

I don't think we need to worry about 10 cent errors. The
data we've been looking at show a 4:3 coming out 10 cents
flat.

> (d) As Ozan might point out, 72-EDO is
> not a meantone and has no path to meantone. Since
> he is seeking a version of Rast from
> a single chain of fifths, this is an impediment under
> his criteria.

Yes, if that's a criterion there are going to be problems,
because maqam-like meantones are difficult to find.

> (4) With 82-EDO, we have the fine
> Pythagorean/schismatic framework of 41, but three neutral
> seconds at 132, 146, and 161 cents. This gives a fine
> Rast at 205-161-132 cents, and a fine Huseyni at
> 161-132-205. What it doesn't have is a meantone path to
> Rast, nor the resolution of the 79-MOS, which may reflect
> the basic advantage of an irregular tuning as versus a
> complete EDO in getting more interval sizes for a given
> tuning size.

As a temperament, it's not interesting because it's a
contorted version of 41.

> (5) With 84-EDO, one possible disadvantage is the
> step at precisely 1/7 octave, valuable for many world
> musics, but not so often visited in lots of maqam music
> where one tends to jump from 11:10 or so the the territory
> near 10:9. In 72-EDO or 82-EDO, this step is
> in the "sweet zone" (for maqam music) close to 11:10.

84 is interesting because it's an Orwell temperament, and
Orwell has some properties that may suit it to middle
eastern music. But 84-EDO is reportedly less popular on
qanuns than 72-EDO.

> (6) With 96-EDO, with have four neutral second
> sizes, at 125, 137.5, 150, and 162.5 cents. Shaahin
> Mohaheri has used this tuning or subsets of it, and the
> variety of neutral steps and intervals is a big plus.

Others have been proposed. According to the Theory vs
Practice paper (which I know you have) one Nail Yavuzoglu
proposed both 40 and 48 divisions. 60 is interesting
because it's Magic, along with 41. It gives two distinct
neutral seconds or thirds along with the 12-equal and
5-limit choices. It's what you'd naively expect people to
use if they have tuners with cents readouts, in that each
division is 20 cents, a nice round number. But in practice
nobody seems to be using it.

> Thus I might paraphrase Ozan's points about 72-EDO
> somewhat as follows:
>
> (1) Like 24-EDO, it has a regular 300-cent minor
> third permitting only an equal neutral second division at
> 150-150 cents.

But it obviously permits other divisions.

> (2) There is no meantone path to a near-5/4 step
> for Maqam Rast -- not a concept of traditional maqam
> theory, but an important contemporary one to Ozan.

Yes, no meantone, but nobody else seems to want this.

> (3) The overall organization is different enough
> from 53-EDO (or a 53-note Pythagorean cycle, I might add)
> to cause some complication for musicians attuned to the
> 53-comma system.

Could be, but musicians are already mixing the systems.

> (4) One might prefer regular diatonic intervals
> closer to Pythagorean, so that divisions like those of the
> medieval treatises (e.g. 32/27 into two
> unequal neutral seconds) can more often and more closely
> apply.

One might, yes. But I don't know how far the Pythagorean
chain goes in practice.

> If I were investigating possible alternatives under Ozan's
> groundrules, I might look at supersets of a complete
> 53-EDO circle. Whether this might require a satisfactory
> set considerably larger than 79/80, or a set of not more
> than 79/80 notes with considerably less utility than the
> actual 79/80=MOS, I'm not sure.

That's exactly what he looked at, of course. It gets
complicated.

> And, needless to say, I am much less well-situated than a
> Turkish musician, especially one as well-informed about
> both tradition maqam styles and some modern innovations
> as Ozan, to judge what the criteria should be for a
> contemporary maqam tuning.

Only Ozan's in that position, and he's not communicating
well.

> For example, with my O3 system, Ozan could well point out
> that it is impossible within the given 24 steps to find
> even a basic 9-note set for a 26/21 Rast (the notes from
> the lower tetrachord plus the conjunct and disjunct forms
> of an upper Rast tetrachord) without encountering some
> quirk of the tuning, and either having to substitute a
> septimal for a regular step, or else having a neutral
> sixth at a more or less notably low position.

What was O3? I'm getting lost now.

> But if we really want to look for something "competitive"
> to the 79/80-MOS, as I might imagine myself doing had I
> been his thesis advisor or the like, I'd be looking not
> at EDO systems, which simply won't give the same
> resolution and diversity for neutral or middle intervals,
> but at irregular tunings like supersets of 53-EDO (maybe
> bringing in additional steps from 159-EDO or 212-EDO)
> offering other solutions including a complete 53-EDO
> circle,a nd thus the virtually pure 3/2 fifths at all 53
> primary locations.

EDOs give you transposition. That's one of Ozan's
requirements.

We were talking about rank 2 temperaments before he settled
on the 79/80-MOS. I'll mention some obvious candidates,
that I havent had a good critique of. I've got an eye on
harmony/polyphony, as does Ozan. And in reality, this is
all I care about, because I'm not a maqam musician.

Schismatic (53&41)

An obvious one as it's implied in some Arab/Persian
documents. The story, as I remember it from Farmer, is
that some pitches were written both as string divisions
implying a 5-limit interval, and positions on a Pythagorean
chain implying the Schismatic equivalent. But then Farmer
says they probably intended the 5:4 intervals to be neutral
thirds. So we're supposed to assume they were smart enough
to notice that 5:4 was very close to a Pythagorean
interval, and then dumb enough to think that the two
intervals were neutral thirds. If that interpretation's
valid, how can outsiders possibly make sense of the theory?

Anyway, Schismatic may work. It ties in with some old
theory, it gives good 5-limit intervals, and you eventually
get neutral intervals. They're distant on the chain of
fifths so the scales will be complex but by the same rule
you get a choice of tunings of the neutral seconds by small
tweaks of the fifths. Also supports tetrachordal
structures.

Miracle (31&41)

Obvious because it's so good in the 11-limit, and you get
neutral intervals with the 11-limit. It doesn't give a
choice of neutral thirds, but it does of neutral seconds.
41 is the near-Pythagorean tuning it works with.

Orwell (22&31)

Has a choice of neutral intervals. Has simple 11-limit
harmony. Includes a 53 note MOS. The fifths are complex.
9:8 is a remote interval and you won't get much
tetrachordality.

Magic (22&41)

Works with 60 and 41. Has unequal neutral thirds. Works
reasonably with the 11-limit. Neutral intervals are
remote, but that means you get a choice of tunings. There
are plenty of them with 41 notes.

Catakleismic (72&53)

Unites the two EDOs we were talking about. Good 15-limit
harmony. Complex.

Harry (72&58)

Good 15 and higher limit harmony. Complex.

Tritikleismic (72&87)

Covers both 72- and 153-EDOs. An option in the 15-limit.
Complex.

Mystery (29&58)

Something Ozan considered but rejected, probably
correctly. Good 15-limit harmony if you can afford 58
notes and don't want to stray outside the 58 note MOS.

This is a quick reply, anyway because my Internet
connection's well overdue to time out. A detailed
discussion of desired pitches, would take a long time, and
it's not one I can really appreciate anyway.

Graham